THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Chemical and Metallurgical Engineering Progress Report 04650-16-P THE PHASE EQUILIBRIA OF SOME COMPOUND SEMICONDUCTORS BY DTA CALORIMETRY Bernard M. Kulwicki Project Supervisor., Professbr Donald R. Mason ORA PrQjec.t 04650 Under contract with: TEXAS INSTRUMENTS, INC. DALLAS 22, TEXAS Administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR July 1963

THE PHASE EQUILIBRIA OF SOME COMPOUND SEMICONDUCTORS BY DTA CALORIMETRY By Bernard Michael Kulwicki A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan 1963 Doctoral Committee: Professor Donald R. Mason, Chairman Professor Lee O. Case Professor Ernst Katz Professor Joseph J. Martin Professor Giuseppe Parravano

ACKNOWLEDGMENTS The author would like to express his sincere gratitude to Professor Donald R. Mason for his guidance and assistance which was given throughout the course of this work. The assistance in the form of fellowships from American Cyanamid Corporation and Union Carbide Corporation is likewise appreciated. Special thanks is due to Texas Instruments, Inc., whose financial support was particularly welcome. Acknowledgment is also due to the Department of Chemical and Metallurgical Engineering for their partial support of this research. The author also wishes to thank Dr. Raymond C. Sangster of Texas Instruments and Dr. Alan J. Strauss of Lincoln Laboratories for the samples which they provided, and the Computing Center of the University of Michigan for the computer time which was donated. The following students also deserve credit for their assistance: Messrs. R. G. Grammens and E. K. Parrott who helped with the DTA measurements, Messrs. N. Delgass and R. J. Bassett who performed the chores of glass-blowing and sample preparation, Mr. S. J. Lewis who carried out many of the fusion operations, and last but not least Mr. A. N. Currim whose expert assistance in solving the boundary value problems associated with heat conduction in the system is greatly appreciated. ii

THE PHASE EQUILIBRIA OF SOME COMPOUND SEMICONDUCTORS BY DTA CALORIMETRY Bernard Michael Kulwicki ABSTRACT This work may be divided into three parts: 1) a theoretical study of heat transfer in the differential thermal analysis (DTA) equipment with subsequent development of two methods for the measurement of latent heats of fusion and transition; 2) the determination of the heats of fusion and transition of twenty compound semiconductors and the study of the nature of these transformations; and 3) the study of solid-liquid equilibria in the systems cadmium-tellurium, zinc-tellurium and indiumselenium. The theoretical analysis showed that the thermal conductance of the system could be well approximated by its steady state value, and a theoretical expression for the time constant for exponential decay of the DTA curve was derived. The first method for the determination of the thermal conductance of the system is based on its theoretical calculation from the steady state formulae. Two variations of this method for indirectly measuring the thickness of the gas film between the silica sample tube and the nickel sample holder (a quantity which can be directly measured only imprecisely) are presented. The first is based on a comparison of the theoretically calculated time constant with its experimental counterpart. This method was tested with data on a number of standards, and good agreement was found between theory and experiment. The second variation, which could not be adequately evaluated, is based on the calculation of the film thickness from two measurements of the area under the DTA curve when gases of widely different thermal conductivity fill the system. The second method for finding the thermal conductance depends on its calculation from the experimentally measured time constant and the estimated thermal capacity of the system. This method requires calibration of the equipment. iii

These methods are employed to measure the heats of fusion and transition of the following compounds whose melting points range from 800 to 1560 K: AgZIn8Se13, Ag2Se, Ag2Te, BiZSe3 BiZTe3, CdSe, CdTe, GaAs, GaSb, InAs, InSb, In Se3, InTe, In Te3, PbSe, PbTe, Sb Se3, Sb Te3, SnTe and ZnTe. The entropies of fusion plus transition are found to vary from 2. 2 cal/g atom K for AgZIn8Sel3 to 9. 6 cal/g atom K for GaAs. The experimental precision is about ~ 15%o on the average. Finally, phase equilibria in the systems Cd-Te, Zn-Te and In-Se were studied. Thermodynamic calculations suggest that CdTe and ZnTe molecules are stable in the melt, and it was established that the systems Cd-CdTe and Zn-ZnTe exhibit positive deviations from Raoult's law whereas the systems CdTe-Te and ZnTe-Te exhibit negative deviations. A two parameter correlation of the liquidus curves of these two systems is postulated. The In-Se system contains five compounds two of which melt congruently, InSe (614 C) and In2Se (885 C) and three of which decom23 pose peritectically, In54Se46 (5530C), In47Se53 (660C) and In 2Se80 (?) (745 C). Two monotectic reactions, at 5200C (indium-rich) and at 760 C (selenium-rich), were observed, as well as polymorphic transformations in In2Se3 (2010C) and In20Se80 (?) (6500C). iv

TABLE OF CONTENTS Page ACKNOWLEDGMENTS............................ ii ABSTRACT.................................. i LIST OF TABLPEES................................i LIST OF FIGURES.............................. ix LIST OF APPENDICES............................ xi I. INTRODUCTION............................. A. State of the Art.................... 1 B. Statement of the Problem................... 6 1. Direct Calculation of the Thermal Conductance 6 2. The Calculation of the Thermal Conductance from the Time Constant.................... 8 II. HEAT TRANSFER IN THE DTA SYSTEM................ 10 A. Heat Conduction Model..................... 10 B. Heat Conduction Prior to Melting -- Regime A....,.. 13 C. Heat Conduction During Melting — Regime B....... 17 D. Heat Conduction Following Melting -- Regime C. 23 E. The Thermal Conductance.................... 28 F. Radiation in the DTA System............... 30 III APPLICATION OF THE THEORY................. 34 A. Calculation of the Thermal Conductance Using the Theoretical Time Constant to Find the Gas Film Resistance............................. 34 V

TABLE OF CONTENTS (Cont'd) Page B. Calculation of the Thermal Conductance Using Two Measurements of the Area Under the DTA Curve to Find the Gas Film Resistance............. 3 38 C. Calculation of the Thermal Conductance from the Time Constant and the Thermal Capacity............. 42 IV. HEATS OF FUSION AND TRANSFORMATION BY DTA CALORIMETRY........................... 45 A. Experimental Data.................... 45 B. Experimental Results....................... 48 C. AgzIn8Se13............................ 51 D. Ag2Se...................... 53 E. Ag2Te............................... 53 F. Bi2Se3, Bi Te3, Sb2Se3 and Sb2Te............. 54 G. GaAs, GaSb, InAs and InSb.................. 55 H. CdSe, CdTe and ZnTe..................... 55 I. InZSe............................... 56 J. InTe and In Te......................57 K. PbSe, PbTe and SnTe...................... 58 L. Experimental Precision................. 59 M. Discussion............................ 60 V. SOLID-LIQUID EQUILIBRIUM STUDIES.............. 63 A. The System Cadmium-Tellurium............... 63 B. The System Zinc-Tellurium,.................. 71 vi

TABLE OF CONTENTS (Cont'd) Page C. The Indium-Selenium Phase Diagram............ 75 D. Discussion....................... 79 VI. CONCLUSION.............................. 83 A. Summary............................... 83 B. Advantages and Disadvantages of the Methods Used to Measure Latent Heats...................... 84 C. Recommendations................... 85 APPENDICES.............................. 86 BIBLIOGRAPHY............................ 175 vii

LIST OF TABLES Table Page 3. 1 Results of Computer Program DTA-19 for Standards........... e o......... 37 3. 2 The Correlation Factor Z = C/C'........... 43 4. 1 Heats of Fusion of Compound Semiconductors..... 49 4. 2 Heats of Transition of Compound Semiconductors.. 50 4. 3 Comparison of the Sum of the Entropies of Transition and Fusion with the Theoretical Prediction........................... 52 4. 4 Entropies of Transition and Fusion of Compound Semiconductors............... 61 5. 1 Experimental Data for the Cd-Te Phase Diagram.. 64 5. 2 Experimental Data for the Zn-Te Phase Diagram.. 72 5. 3 Experimental Data for the In-Se Phase Diagram... 76 A-III-1 Elements Used in the Preparation of Compounds... 99 A-III-2 List of DTA Samples.................... 103 A-IV-1 Specific Heats of Elements and Compounds at the Melting Point........................ 113 A-IV-2 Specific Heats of Compounds at the Transition Point............................. 114 A-IV-3 Thermal Conductivities of the Solid Near the Melting Point.................... 116 A-IV-4 Emissivities of Elements and Compounds....... 118 viii

LIST OF FIGURES Figure Page 2.1 Model for the Investigation of Heat Conduction in DTA........................... 1 2. 2 Relationship between the Differential Temperature, 9, and the Thermal Conductance, G, during DTA. 29 3. 1 Analogous Electrical Circuit for Calculation of the Overall Thermal Conductance.3............. 39 4. 1 Some Typical DTA Curves Showing Differential Temperature versus Time and Selected Sample Temperatures......................... 46 4. 2 Some Typical Semilogarithmic Plots for Finding the Time Constant for Exponential Decay of the DTA Curve......................... 47 5. 1 The Phase Diagram for the Cadmium-Tellurium System............................ 65 5. 2 The Integrand of the Excess Free Energy Function versus Composition for the Cadmium-Tellurium System............................. 68 5.3 The Liquidus Curves for the Cadmium-Tellurium System Plotted as log 4 NCdNTe versus Reciprocal Reduced Temperature................. 70 5.4 The Phase Diagram for the Zinc-Tellurium System.. 73 ix

LIST OF FIGURES (Cont'd) Figure Page 5. 5 The Liquidus Curves for the Zinc-Tellurium System Plotted as log 4NznNT versus Zn Te Reciprocal Reduced Temperature............. 77 5.6 The Proposed Phase Diagram for the IndiumSelenium System............................... 78 5. 7 Semilogarithmic Plot of the Average Cluster Size, n, versus Reciprocal Reduced Temperature.... 82 A-I-1 The DTA Samples inside the Nickel Block for Sample Arrangement B................... 87 A-I-2 Block Diagram Showing Operation of the DTA Equipment........................... 89 A-II-1 Plot of Equation A-2. 9 showing Relationships between Furnace Temperature and Autotransformer Setting for Heating and Cooling at Rates of 2. 5 0C/min and Steady State Conditions.......... 96 A-V-1 Contour in the Complex Plane for Inversion of the Laplace Transforms, Equations A-5. 29 and A-5. 30. 125 x

LIST OF FIGURES (Cont'd) Figure Page 5. 5 The Liquidus Curves for the Zinc-Tellurium System Plotted as log 4NZnN T versus Reciprocal Reduced Temperature............. 77 5.6 The Proposed Phase Diagram for the IndiumSelenium System....................... 78 5. 7 Semilogarithmic Plot of the Average Cluster Size, n, versus Reciprocal Reduced Temperature... 82 A-I-I The DTA Samples inside the Nickel Block for Sample Arrangement B. * * - 87 A-I-2 Block Diagram Showing Operation of the DTA Equipment........................... 89 A-II-1 Plot of Equation A-2. 9 showing Relationships between Furnace Temperature and Autotransformer Setting for Heating and Cooling at Rates of 2.5 0C/min and Steady State Conditions.......... 96 A-V-1 Contour in the Complex Plane for Inversion of the Laplace Transforms, Equations A-5. 29 and A-5. 30. 125 x

LIST OF APPENDICES Page I. DTA EQUIPMENT AND PROCEDURE................ 86 II. DYNAMICS OF THE DTA FURNACE................ 93 A. Determination of the Time Constant............ 93 B. Operating Conditions and Procedures........... 94 C. Discussion............................ 95 III. SAMPLE PREPARATION...................... 98 A. Materials.............................. 98 B. Preparation of Alloys....................... 101 C. Preparation of DTA Samples................. 101 IV. PHYSICAL PROPERTIES AND THEIR ESTIMATION........ 110 A. Heats of Fusion and Transformation.............. 110 B. Densities........................... ill C. Heat Capacities........................ 111 D. Thermal Conductivity...................... 112 E. Emissivities........................ 115 V. CONDUCTION OF HEAT IN THE SYSTEM DURING MELTING. 119 VI. COMPUTER PROGRAMS AND EXPERIMENTAL DATA....... 133 A. Program Number DTA-19 for Direct Calculation of the Thermal Conductance...................... 133 1. List of Variables.................... 133 2. MAD'Listing of Program................ 135 3. Data Card Format,................... 140 4. Tabulation of Data Cards............... 141 xi

LIST OF APPENDICES (Cont'd) Page B. Program Number DTA-16 for Direct Calculation of the Thermal Conductance...................... 145 1. List of Variables................... 145 2. MAD Listing of the Program............. 147 C. Program Number DTA-17 for Analysis of DTA Data.,. 152 1. List of Variables................... 152 2. MAD Listing of Program............... 153 3. Data Card Format................. 1 55 4. Tabulation of Data Cards.............. 156 VI NOMENCLATURE.......................... 170 xii

CHAPTER I INTRODUCTION Differential thermal analysis (DTA) is one method of recording time-temperature information for the purpose of studying thermal transformations, and consists in measuring the temperature difference between a sample and a reference as a function of time and/or temperature. Other methods have been described as well (3), but DTA remains as one of the oldest (50, 51) and most useful. DTA has met with widespread acceptance in many areas wherein a change in the rate of heat transfer to a sample generates a temperature difference. Many examples may be cited. The principle objective here is to determine the latent heats of fusion and transition of some twenty semiconducting compounds whose melting points range from 80.0 K to 1560 K by means of DTA calorimetry. The compounds are: Ag2In8Sel3, Ag2Se, Ag2Te, Bi2Se3, BiZTe3, CdSe, CdTe, GaAs, GaSb, InAs, InSb, In Se3, InTe, In2Te3, PbSe, PbTe, Sb Se3, Sb Te3 SnTe, and ZnTe. A second objective is the investigation of solidliquid equilibria in the systems Cd-Te, Zn-Te, and In-Se. State of the Art DTA has long been recognized as being a powerful tool in the investigation of phase equilibria. Transition temperatures may be accurately pinpointed. Furthermore, the technique is amenable to the use of very small samples, making possible the study of rare materials. A typical example of this type of study is the work of O'Kane (66) who investigated solid-liquid equilibria in pseudobinary systems such as CdSe-In Se, ZnTe-In Te, etc. Solid-vapor equilibria may also be readily investigated as has been done by Markowitz and Boryta (56) who studied the behavior of NH4C1 in a controlled pressure DTA unit. The purpose here is to cite illustrative references, rather than to compile a bibliography, which has been done by Smothers and Chiang (78). Over 1500 references are given. 1

2 DTA has also been used to investigate reactions in the solid state (26, 38, 78) thereby making possible the accurate definition of reaction temperatures and relative speed under a variety of experimental conditions. The study of thermal transitions in clays by DTA was first undertaken by Norton (64), and similar studies have been carried out by numerous workers. Locke and Rase (53) have indicated the utility of a controlled pressure unit in the screening of catalysts so as to provide rapid, qualitative information on the chemical and physical changes which affect catalytic activity. Even such small heat effects as those due to specific heat changes accompanying the glass transition temperature of polymers may be readily investigated by DTA (41) and Steiner and Johnston (80) have discussed the application of DTA to the quantitative determination of specific heats. In principle, DTA is also a calorimetric technique in which the heat absorbed or evolved during a thermal transition or chemical reaction may be determined by comparing the area under the DTA curve with that produced by a standard whose latent heat or heat of reaction is well known. In practice, however, its usefulness has been limited because of difficulties associated with accurately predicting the appropriate thermal conductance or apparatus constant of the system, particularly where a wide range of temperature is of interest. Among the first to recognize the possibility of obtaining a quantitative measure of changes in heat content from cooling curves was Plato (69) who estimated the heats of fusion of a number of inorganic salts from the time of freezing on the cooling curves. His formulae as well as those of other investigators up to about 1925 were strictly empirical in nature and are thus of little interest here. Among the first to derive the relationship of proportionality between the heat of reaction and the area under the DTA curve were Steiner and Johnston (80) whose thorough article provides an excellent review of the art through 1928 as well as a comprehensive discussion of the merits and limitations of the technique as a Cfo reference (78) for bibliography,

3 quantative tool. This relationship has been derived by many other authors including Berg and Anosov (6) who measured the heats of dissociation of dolomite and related substances; Speil (79) who studied the thermal transitions in clays and related minerals; Vold (87) who measured the heats of fusion of stearic and benzoic acids, and Borchardt and Daniels (12) who were interested in determining the kinetic parameters of homogeneous, liquid phase chemical reactions. At this point, the quantitative theory of DTA will be reviewed briefly. The equations presented first are essentially the same as those given by Vold (87), but using different symbols. The model for the DTA system comprises a sample and a reference material having substantially identical properties and physical geometries. They are placed inside a furnace and heated together under essentially identical conditions. Assuming that the sample temperature is uniform and that heat losses are negligible, a differential energy balance may be written for the sample as follows: CdT = dE + K(T - T)dt 1.1 s where C is the total heat capacity (cal/ K). E is the energy evolved during thermal transition (cal), and K is the thermal conductance (cal/ K-sec.). Similarly for the reference C dT K (T -T )d t 1.2 r r s r where no energy is evolved and hence the dE term does not appear;. In the above equations, the subscript s referrs to the surroundings and the subscript r to the reference. Defining 9 as the temperature difference between sample and reference (T - T), and y as the constant heating rate (d T = -y d t), the following relation is obtainedCd9 = dE + Cy(g - l)dt - KG dt 1.3 where g = KC /K C and is dimensionless. Choosing the initial condition r r

4 9 = 0 at t = 0, Equation 1. 3 may be integrated for the condition dE = 0 to yield: 9 = ( - 1)(C/K)(1 -KtC) 1.4 whence the steady state value of G is obtained: 9 = y( - 1) C/K 1.5 ss Thus Equation 1.3 becomes: dE K(98- ) dt + Cd9 1.6 ss or t AE = K ( - s)dt 1.7 ss Equation 1.7 is strictly valid only if 9 does not vary during the transiss tion, i. e., the baseline remains constant. Even if the baseline does shift due to a change in the specific heat of the sample as a result of phase transformation, however, Steiner and Johnston (80) have shown that Equation 1. 7 may still be used provided that the area is measured with respect to the extension of the final baseline. Boersma (7) as well as Kronig and Snoodijk (46) have obtained this relation in a somewhat more elegant manner, and have shown that the assumption of uniform sample temperature is unnecessary provided that 9 is the temperature difference between the center of the sample and the center of a matched reference. In addition, both authors have derived expressions for the thermal conductance in terms of thermal conductivity of the sample, that of the holder, and the geometry. Boersma has also derived correction terms which take into account the transfer of heat along the thermocouple wires.

5 In addition to the possibility of computing K from the geometry and thermal properties of the system, several other techniques are available. The first, due to Berg and Anosov (6) and also applied by Kornilov and Matveeva (45), consists in the admixture of a standard, whose transition heat is known, with the sample. The second possibility, proposed by Vold (87) consists in calculating K from the exponential decay of 9 after the transition has been completed. A third method, recently proposed by Sturm (82) consists in computing the effective conductance from data taken by controlling the heating rates so as to obtain a constant thermal gradient. A fourth possibility consists in measuring the area under the curve produced by standards whose transition heats are known. This method is perhaps the least satisfactory particularly where a wide range of temperature is of interest, since at high temperatures radiation is an important contributor to the heat transfer mechanism. This method is quite useful, however, over short temperature intervals particularly where a large thermal resistance intervenes between the heat source and the: sample. This fact has been pointed out by Boersma (7) as well as by Smith (76). Further ingenious variations have been proposed by Eyraud (23), Wittig (91) and Lueck, Beste and Hall (55). The first two authors proposed methods in which the need to determine a numerical value for K is eliminated. Eyraud proposed using a temperature controller to regulate the temperature difference between the heat source and the sample while measuring the power input to the furnace as a function of time, with the area under the power-time curve being equal to the heat of transformation. Wittig suggested using an auxiliary heater within the sample. When varying quantities of electrical energy were injected via this heater, differing values of the area under the DTA curve were obtained, extrapolation to zero area yielding the heat of transformation. Lueck et al., who were interested in studying the kinetics of liquid phase chemical reactions, have developed a technique whereby nearly isothermal conditions may be maintained in the reaction cell, sensitive thermistors being used to measure the differential temperature (0. 1 K at most). The thermal conductance was

6 determined from the exponential decay of the DTA curve to baseline after a drop of warm water was added to the reaction solution. Statement of the Problem The particular properties of the compounds studied in this work required modifications and refinements to the existing techniques. The volatility and reactivity of our samples required their being sealed under vacuum in silica glass tubes. Because of the fact that the dimensions of the silica tubing, and therefore the thermal conductance, varied from sample to sample, a simple calibration of the apparatus would not suffice. The method of Berg and Anosov could not be used because of the unavailability of suitable standards. In Sturm's method, as he himself admits, the heating rates resulting in a constant thermal gradient may be found only with difficulty. And the methods of Eyraud and Wittig not only do not appear to be any more direct or reliable than the two remaining methods which were used in this work; viz. direct calculation of the thermal conductance and measurement of the time constant, but also would require extensive modification of our equipment. The application of these two methods to our particular system is now considered. Direct Calculation of the Thermal Conductance Instead of using the aforementioned definition of the thermal conductance, viz. heat flux across system x area temperature difference cal/sec K 1.8 a slightly different definition will be used _ K _ heat flux across system x area 1 _9 L temperature difference x sample height Cf. Appendix III for description of the samples and their preparation.

8 and 3) an experimental measurement of the area under the DTA curve. The DTA system consists in samples which are sealed under vacuum in silica sample tubes, each of which contains a concentric thermocouple well in the bottom. The samples are placed in a nickel block in which holes have been drilled in order to accommodate them. A gas film thus separates the silica tube from the wall of the nickel block. The primary contributions to the overall conductance come from the gas film and the sample tube. Secondary contributions are due to the sample itself and, at high temperatures, thermal radiation. It is clear then, that a precise measurement of the physical dimensions of the gas film is required. This measurement is difficult to achieve directly because the gas film is very thin, so that its thickness is a small difference of large numbers. This complication may be resolved, however, if an independent method of evaluating the thickness of the gas film is available. Two such methods are proposed in the following. In the first method, two measurements of the area under the DTA curve are obtained using different gaseous atmospheres having widely different thermal conductances, such as nitrogen and helium. The thickness of the gas film can then be computed from the two areas and the known thermal conductivities of nitrogen and helium respectively. In the second method, the thickness of the gas film can be found by comparing the experimental time constant for exponential decay (see Equation 1.4). d In I 9- 9 1 Te bd~n ~G I 1.14 exp d t of the DTA curve after completion of melting with the theoretical value found from a consideration of the differential equations governing heat conduction in the system. The precise formulation of these methods will be given in Chapters II and III. The Calculation of the Thermal Conductance from the Time Constant Inspection of Equations 1. 4 and 1.o 14 leads to the following expression for the thermal conductance:

9 K = C/T 1.15 exp where C is the thermal capacity of the system. A number of authors have used this formula including Vold (87) and Lueck, et al. (55). It is evident that the thermal conductance can be estimated provided that the thermal capacity of the system can be estimated. In this regard several questions arise: Exactly what constitutes the thermal capacity of the system? Should C include the heat capacity of the sample plus that of the container, or that of the sample alone? If the container is included, should the entire container, or just that portion of it which is in thermal contact with the sample be included? Viold, who studied fusions or organic materials in a nickel sample container, found that her data were best explained by including the heat capacities of the sample and the entire container in the estimation of C. In the particular equipment used in this work, the heat flux is, for all practical purposes, radial, and the container is constructed from fused silica, a material of relatively low thermal conductivity. It is therefore unlikely that changes in the temperature of the container a few centimeters away from the sample would appreciably affect the heat flux to the sample. This would seem to imply that only a portion of the heat capacity of the container would, in this case, contribute to the thermal capacity of the system. In succeeding chapters the theory of heat conduction and radiation in this system will be examined, the methods sketched here will be more fully developed and subsequently applied to experimental data, and the results will be critically examined in the light of present knowledge and theory.

CHAPTER II HEAT TRANSFER IN THE DTA SYSTEM In this section, the differential equations governing the flow of heat in the DTA system are considered together with appropriate boundary conditions which approximate the experimental system. Three regimes may be discerned: A) heat conduction in the system immediately prior to melting (or transformation); B) during melting; and C) immediately following melting of the sample. The equations describing the first regime are amenrable to direct analytical solution in cylindrical coordinates. Those describing the second regime cannot be readily solved in cylindrical coordinates but the analogous problem in cartesian coordinates may be solved. The long time approximation in the cylindrical case may then be deduced by analogy. In the third regime, owing to the tediousness of the mathematical manipulations which are required, an expression for the temperature profile as a function of space and time cannot be easily obtained, but a great deal of useful information including a theoretical expression for the time constant of exponential decay can nevertheless be extracted. With these solutions, procedures for measuring latent heat effects can be formulated. Heat Conduction Model The model used in attacking this problem is depicted in Figure 2. 1. The figure shows the DTA tube inside the nickel block as a series of concentric annular volumes. The central core is reserved for the thermocouple and hence is vacant free space. The first concentric annular volume (region 5) is the inner wall of the thermocouple well, the next (region 4) is the sample material, the next (region 3) is the outer wall of the sample tube, the next (region; 2) is the gas film between the sample tube and the nickel block, (region 1). During the operation of the DTA equipment, three different sets of boundary conditions may exist, establishing three distinct regimes of operation. 10

11 0 0 0 o 0o H 0 ~r 0 0 LLJU 0 o 0 o 0 o Rs R4 3 R2RI RADIUS, r -~ Figure 2. 1. Model for the Investigation of Heat Conduction in DTA (Not to Scale).

12 Regime A defines the system under constant heating conditions wherein the temperatures at all; points in the system are increasing linearly with time. Regime B defines the system during the time that the sample is melting, and Regime C defines the system as it is returning to the constant heating condition (regime A) after the sample has melted. In all three regimes of operation it is assumed that, owing to its massiveness, the nickel block acts as a source for heat, and owing its high thermal conductivity, its temperature is uniform, and also increases linearly with time, i. e., T2(R1 t) = (t+t 2. 1 where y = heating rate, K/unit time. The differential equation governing the flow of heat in each contiguous region may be written in cylindrical coordinates as follows: 65T 1 6T P_ p T2 2 r 6r k 6t 6r where T = temperature r = radius p = density c = specific heat p k = thermal conductivity t = time Internal sources and sinks are assumed to be absent. In terms of the normalized variables and parameters for region 4, Equation 2. 2 becomes 2 6 v 1 6v 6v + = T- 2. 3 where v(, t) = T4(rt) - T 4 ~O

13 = -r/R4 T = R4 (PC)4/k4 T = reference temperature 0 For region (3) 2 8 +y 1 6y 62y 1,2 65' 6t where. y(',t)= T3(r,t)-T C, = r/R3 T = R3. (pC )3/k For region (2) 6z 1 6z 6z + - - = T" 2. 5.112,, 6" t where z( ",t) = T2(rvt)-T'" == r/R2 = R2 (P p) 2/k2 After determining the boundary conditions which satisfactorily approximate the physical system, a solution of the simultaneous differential equations, Equations 2. 3 through 2. 5 may be sought. This problem will be considered next, for regimes A, B, and C, each in turn. Heat Conduction Prior to Melting - Regime A Assuming that the heating rate is constant and equal to y K/unit time, Equations 2. 3 through 2. 5 become:

14 d v 1 dv + -' - T 2 d y 1 dy 12 r5' df' = 2.6 d z 1 I dz + = 0 112,, " d~"d It has been assumed that the rate of accumulation of heat in the gas film is negligible compared to the rate at which heat is transferred across the film, or, in other words that the thermal diffusivity of the gas film is very large. For the particular system being considered here this approximation seems to be excellent although in general it may not be a valid assumption. Since the thermocouple well acts as a sink for heat: 2 2 1 2 2 (pc) r (R -R ) k t4 R (P 5 4 5 t 4 4 kr Ir R or = y t5 = 5ff't =j i, =1. 7 where t5= (pcp)(R - R )/2k Continuity of the temperature profile and heat flux at the inner boundaries lead to the relations: v(I,t) = y(l,t) 2.8 y(l,t) = z(l,t) 2.9 6v, &

15 k6y' i, 1 2. 11 where = R3/R4 = k4/k3' =R2/R' = k3/k2 2 3 3 2,, = R/R Finally, at the outer boundary, the temperature rises linearly with time z(r ",t) = y(t + t) 2.12 0 The reference temperature, yto has been taken to be the temperature of the thermocouple well at zero time, T5(0). These equations can be integrated directly to yield: v = y~T2/4 + Aln + B 2 y = yT'S /4 + C n' + D 2.13 z = P n l" + Q where A, B, C, D, P and Q are independent of r and may be evaluated from Equations 2. 7 through 2. 12 to obtain v(F,t) = (t + to) (- (rl2) - ( - 1) 0-4 4 + (2t-T)n -)n [ 2P t + T(11 - 1) - T'] In 2 5 1 2 5 -[2 t + T(2 -1) + T'(1' -1)]' In 11 2. 14

16 y(', t) = (t + to) - (l )' o 4 + z[ t5 + 0 T 2-1)-T'] In'In " [ 2. t5 2 5 Ti' 2 5 + T(12 1) + T'( 2 _ 1)] 2. 15 2 2 5 Ti' z (, t) = y(t + to) +i [ 22 t5 + T( - 1) + T'( -1)] 2 16 Lastly, the overall temperature lag across the system may be computed. yt = z(",t) - v(l,t) -= (TI 1) + U(2- 1) + 2 (2t - T) In + + t+ T( -1) - T] ln T + t [ 2 t + T( 1 T( - 1) + T'n (l -2 1] n 2 5 For a typical set of parameters: Tr = 1.56 T = 1.83 sec = 3. 13' = 1.30 T' = 26.6 sec' = 35.7 T" = 1.0384 t = 1.06 sec y = 0.05 K/sec 5 a value of 1.30 K was found for y t, of which 1. 08 K was contributed by a gas film (N), 0. 17 K by the silica tube, and 0.05 K by the sample (InSb). The thermal conductance, defined in Equation 1. 9 as the heat flowing across the system per unit height per unit temperature difference, may also be computed. Using the results in Equations 2. 14 through 2. 16:

17 2-r k v 4 6.-, = 1 G = = 2Tk t /t 2.18 z (l "t) - v(l,t) 4 o 5. For the above set of parameters, G is found to be 0. 0161 cal/sec-cm-~C. Heat Conduction During Melting - Regime B In this case it is assumed that the sample is melting as the temperature of the nickel block increases linearly with time. The heat conduction in the sample is ignored as its temperature is assumed to remain uniformly at the melting point Tf. Taking T = Tf the differential equations which must be solved are: 2 1 + = T 2.19.2 5' 65' 6t 2 z z 6 -+ -- = 0 2.20 11 2 C" A At the inner boundary, y(1,t) = 0 2.21 Initially, let us take the condition: y(I',0) = 0 2. 22(a) ( ", 0) = 0 2. 22(b) The remaining three boundary conditions are given by Equations 2. 9, 2.11, and 2. 12, respectively with yto in Equation 2. 12 being replaced by Tf. The solution to Equation 2. 20 is: z(,",t) = A(t) In," + B(t) 2.23

18 Taking the Laplace transform of Equation 2. 23 with respect to time: z(, s) = A(s) In Y" + B(s) 2.24 Equation 2. 19 may be likewise transformed: 1 - 2 y" + ~ y = T'sy 2.25 Equation 2. 25 is the modified Bessel equation, whose solution is: y(',s) = C(s) (it' VT-) + Ds )Y (it' ls) 2.26 The nomenclature used for the Bessel functions is that used by Mickley, Sherwood, and Reed (59. The transformed boundary conditions are: y(l,s) = 0 y(rn',s) = z(l,s)'n''y(l',s) = z'(1, s) z(r",s) = y/s2 2.27 Combining Equations 2. 27, 2. 24 and 2. 26, four equations in A, B, C and D may be obtained: CJ (s) + DY (P) = 0 CJ ( r'P) +DY o('i) = B 0 0 C P J (n'p) + Dp Y (I'1) = A/, n' A In " + B + y/s 2.28 where i /f7

19 Solving Equations 2. 28 for A, B, C and D, the transformed variables y and z become: y (I,w) = ^_____________ __o(P3)Yo(6') - Y (p)J0(_ ('p) 1P (Jo(p)Yo(' ) Y()J (ip) -r -'' In qr"[ po(Y)Yl(qI ) - PYo(p)J (r )] 2.29 z ( Ip) = y,2 Jo(p )YO('p)- Y (P)Jo( - -'1' In p" [ PJo(p)Yl(' p) - Yo(p)Jl(l' p)] P4 (P)Y ( - Y (P)J (') -' IInn 1" [ pJ(p)Y1(I'p) - PY (P)J (l' ) 2.30 Equations 2. 29 and 2. 30 cannot be easily inverted because they contain terms of the form Y (p) and Y1( n'). The series expansion of each of these Bessel functions contains a term in In i 1V, and therefore y(,', s) and z(~, ", s) have a branch cut along the negative real axis. In order to perform the inversion.M+ioo y(',t) = 2ir Y ('s) exp (st) ds M-i O A contour in in the s-plane must be chosen which excludes the negative real axis. This is a very difficult problem, and efforts to achieve a solution were finally abandoned. Instead the analogous problem of linear heat flow between infinite parallel planes was considered. The differential equations simplify to:

20 62 T (p c) 6T T3 3 3 6x2 k 3 6t 2 6 T2 = 0 2.32 2 6x Or, in terms of the parameters defined previously: 2 ='T 2.33 6,2 5 6t 2 = 0 2.34 where C' = x/R, " = x/R etc. Although the parameters'',,T', etc. are defined unambiguously in the cylindrical case, since the region is fixed by the axis of the cylinder, the same is not true in the case of infinite parallel planes. It is convenient, however, to retain this notation in order to preserve the analogy, realizing that the choice of the origin is not arbitrary but fixed. That is, the origin is chosen such that r = R3 (cylindrical) corresponds exactly to x = R3 (parallel planes) The boundary conditions are then unchanged and are given by Equations 2.9, 2.11, 2. 12, 2. 21 and 2. Z22, with yt in 2.12 being replaced by Tf. Equations 2. 33 and 2. 34 may be written in operational form as before and solved to give

21 y(', s) = C(s) sinh' i-'s + D(s) cosh T'- S 2. 35 z( ",s) = A(s) " + B(s) 2. 36 The coefficients A, B, C and D are found by using the boundary conditions, with the final expressions being given by: Y (', s) = __ sinh (1 - t') /s' 2. 37 s sinh (1 - I') -vT/s + ]'' (1 - 1") -T cosh (1 - r]')-''s z(, s) = y_ sinh (1 - n') 7s + I''(1 - ") J s cosh (1 -') - s.38 2................. 38 s sinh (1 - Ti') -T'S + n''(1 - n") T'S cosh (1 - I') rVT'S Equations 2. 37 and 2. 38 are well behaved; that is, they do not have any branch cuts. The inversion integral may thus be evaluated by the method of residues (18 ). Complete inversion yields: y('t) = a (') + a 1(')t + b sin ( ) n exp - n 1 0 n n' n=l (-1) T2.39 and z( ",t) = c( ") + c( ")t + o b sin + T''a cos a exp- n

22 a cot ac + = 0 2.41 and where al(') =' - 1+'r'("-) a l,,) = ~l (t'-1) + l''( "-1)! -l' ^~- 1 + tn''( "-1) T, ~f (C-1)^('-1) -1 =..('-l) -31 + n' ( — 1)2 o^^ 6 Ti' -1+ ti'~'(Ti"-l ) c (y )' = -'''( "- l)') (i'-1) and 2 -3 2yT'(Tl'-l) a b = n~~~~~~n b - a sin a -cos a - Cos a T'1- n n n T'-1 n The general solution of Equations 2. 31 and 2. 32 with the time derivative included in Equation 2. 32 is discussed in Appendix V, and it is concluded that the error introduced by neglecting the time derivative is small. For large values of the time Equations 2. 39 and 2. 40 become: r' - 1 + r' ("- ) Yt 2. 42 r'-i_ +D ~' (C"-l)] t2.43

23 The corresponding approximations for the cylindrical case may be found by replacing' - 1 by In,', etc. Moreover, because of the curvature of the bounding surfaces, the factors r-'' become simply'. This analogy yields: _ t ln r' y = n~' + r' + n" t large 2. 44 In n' +' Inn 1" - yt (In 11' +;' In ") z = (In~ tlni~ + Y. ") t large 2.45 In n' +: in 1: Equations 2. 44 and 2. 45 are in agreement with all of the boundary conditions. Furthermore, the conductance for heat transfer is defined by T 3 -6y Z k3 r 6 r = R3 2r k3' =1 2.46 T (R ) - T (R ) yt Using 2. 44, it is evident that: 23 k C In n + 01' In " t large 12 k + 2ni k3 2. 47 3 k4 That is, as time becomes large, the conductance for heat transfer during melting approaches its steady state value, as would be expected. For the set of parameters used in the preceding section a value of 0.0225 cal/ sec-cm- K is found for the steady state value of G during melting. Heat Conduction Following Melting - Regime (C) In this regime, it is assumed that melting is completed, and the sample temperature is rising more rapidly than the nickel block temperature towards its constantly rising value. The differential equations whose solutions are examined in this section are:

24 2 5 v 1 5 v'Sv + =' 2. 3 +. +, 6 6 =T 2. 4 22 5' 6' at 62z z + 60 2. 20 11.2 ^ 5^~-^ The initial temperature profile may be found from the large time approximation in regime (B). v(0,0) = 0 y(f - [0)ln'+ 0 In " where tf is the time interval over which melting takes place. As in Regime (A), the thermocouple well is assumed to behave as a sink for heat: to where v = T/t5. The remaining boundary conditions are given by Equations 2. 8 through 2. 12, with tf replacing to in Equation 2. 12. Equation 2. 3 becomes, after taking Laplace transforms with respect to time: 1 - _ V + VI = T SV 2. 49 which has the solution:

25 v:(",s) = A(s) J (it F-s) + B(s) Y (i'-Fs) 2.50 Similarly, Equation 2.4 may be transformed to give y"+'= T' y + T tf ln +' 2 2 51 If fln )'+' In 1" whence y( s',s) = C(s) J (i' f s) + D(s) Y (i' 1/ s) O 0 tf in 2'52 2. 52 s In t'+ 0' In 1 " Finally, z(",t) = P(t) ln " + Q(t) 2.53 or z( ",s) = P(s) In +" + Q(s) 2.54 The transformed boundary conditions are: V 56v v ( s)= T = T i v(,s) = y(,s v On, S) - 1,:S 0 v(, s) = y'(l,s),.. ) y(T] s) = z(l,s) ~'''('/,s) = z' iis) z(q1", S) = Y + 2.55 The boundary conditions, Equation 2. 55 together with Equations 2. 50, 2. 52 and 2. 54 yield six equations in the coefficients A, B, C, D,.P, and Q, viz.:

26 AJ (p)+BY ( = -A J (p) BP Y (1 A J (rp) + BY (n) = CJ (E3) + DY (EP) 0 yo tf t f ln' n ICJo(E"'p) + DY (E') - tf IK n' In Q ApJ (TIP) + BpY ('p) = 1 1 cE pC (e ):+ D E Y1 ( ) + s (n rl' + In l ) - 1 - ytf 2 f _tf C E PJ1(Er'P) + DEPY1(EN'P) + sl'(ln v' + q' In r") -l o P In T" + Q = ~ + 2.56 L^ s where p - i Ts and E = T'/T The simultaneous solution of six linear equations, Equations 2. 56, for the coefficients A, B, C, D, P and Q, subsequent determination of the transformed variables v, y, and z, and inversion are not practically feasible. Furthermore, even if the above operations were carried out, the result would be of questionable value since it would be very complex indeed. On the other hand, past experience gives a clue as to the general form of the expected solution. It would be expected that the solution would contain a term dependent on the geometry and thermal properties of the system alone, a term linear in the time, and a term of the form Z C e-Pr t/ 2. 57 n n where the C are complex functions of the geometry and thermal properties n of the system, and the summation is taken over all the eigenvalues of the system.

27 Furthermore, the 3 are found from the roots of the principal determinant, D, formed from the Equations 2.56; that is: D =F() E F (13)- " F (PI) f 111'II' 2 F3(1 3 + F4 (I) n l11 F5 (11 )- 1 F6 2. 58 where F1 () = al ()Yo (P1) - A2 (13)1 (') F2(P) = 1 (EP)Yo (E Y') - Y1(E) o (E ) 1) F3 () = I(E )Y1(Erl'3) - Y1(E3)J1(E'p) F4 () = A1()Y1(r) - AZ (P) 1 (iP) F5(p) = ( (E1)Yi(E'P) - Y (E ) 1(E1('3) F6 () = I (EY) Yo(E1'1) - Y ((E3) (E1T'1) 6 o o o o B1 ( =) 1 3o (P) - v Jl(P) A2 (P) = Yo (P) - vY1 (1) 0 (p c))4 R 4 5 p 5 R4 _ R5 Now it is experimentally known that after melting, the DTA curve decays back to the baseline in an exponential fashion, which means that only the leading term of the sum, Equation 2.57, need be considered. In other words the theoretical expression for the time constant of exponential decay becomes: _ 2 exp T/ 2.59 exp'

28 where 1 is the smallest root of the principal determinant, Equation 2. 58. Equation 2. 59 does in fact predict the correct order or magnitude of the time constant of exponential decay. For example, using the following typical set of parameters: -n = 1.56 T = 1.83 sec = 3.13 T]' = 1.30 T' = 26.6 sec 0' = 35.7 n" = 1.0384 t = 1.06 sec i was, fourn'dlt be;.- 70W o whi.ch lea s.tova uer-':, i-1LLL83./(0. 270) 1 * exp = 25. 1 sec, which is the correct order of magnitude. The Thermal Conductance It is now possible to examine the behavior of the thermal conductance during the course of a DTA experiment. Consider Figure 2. 2 which depicts the variation of the thermal conductance side by side with the DTA curve. Just prior to melting and a few minutes after melting the value of the thermal conductance will be well below its steady state value. Upon the initiation of melting the conductance will immediately rise to its steady state value with a time constant of about one second. It will remain at this value throughout melting which may take about ten minutes. Immediately following the completion of melting the conductance will revert to the value it had before melting with a decay constant of about 25 seconds. Based on these considerations it is evident that the steady state approximation will be quite valid because dt oGdt 0 o G Kats ss o/ dt Cf. Equations 1.10 through 1.13 as well as the discussion following Equations 2.17 and 2.45. Cf. Appendix V.

29 LzJ a_ bIJ L-J < Lr LU LL LL r sec.,, r- /25sec. LuJ:) z 0 0 tf - t2 10min TIME, t Figure 2.2. Relationship between the Differential Temperature, 9, and the Thermal Conductance, G, during DTA (Not to Scale).

30 as can be seen from Figure 2. 2. In the foregoing, it was tacitly assumed that thermal conduction was the only mechanism by which heat could be transferred across the DTA system. At high temperatures, however, it would appear that thermal radiation would also contribute even though the temperature gradient is small (A T across thegas film is only about 1 K before melting and may increase to about 30 K during melting). These matters will be examined next. Radiation in the DTA System In this section the equations governing the transfer of heat by means of thermal radiation are considered and expressions are derived for this contribution to the thermal conductance. Consider Figure 2. 1 where radiation is emanating from the "gray" oxidized nickel surface and reflects diffusely from the surfaces of regions (4) and (1). It is assumed that radiation which is absorbed is absorbed at the surface of each region, that the reflectivity of the silica tube is zero (p = 0), that the transmissivity of the sample is zero (T4 = 0), and that only wavelengths less than 3. 7 microns pass through the silica tube, so that multiple reflections of wavelengths greater than. 3. 7 microns need not be considered; that is to say none of the radiation which reflects from region (4) will be absorbed by region (3). With these assumptions, together with Planck's distribution function for black body radiaion, it is possible to estimate an effective emissivity for fused silica as a function of temperature. 4 Keeping this in mind, denoting cA, T by Xi, and noting that f43 = f3 = 1, where fjk is the fraction of the radiation from j which is "seen" by k, the following relations are readily found~ Radiation emitted by 1, 3, and 4 respectively: X1E1 X3E3 x33 4 4

31 Radiation reflected by 1: X3E3 (1-E1) + X4E4 (-E3)(1-E1) + (1-E1)(l-E )(1 — ) f14 (X1E + X3E3 (1-E3) + X4E4 (1-El)) + 1 (lc^(-^)(-c^3 4 14 1 3 3 3 44 El (-E1)(1-E 3)(1- E 4) f14 (X1E1 + X3E3 (1-E 3) + X4E4 (1-E1)) + *... e ~ * ~. Radiation absorbed by 1: X3E3E1 + X4E4 (1-E3) E1 + El l-E3)(I-E 4 )f14 (X 1 + X3 (1-El) E3 + X4 (1 -E1) E4) + (1-E)(1- )( E) f 2 (XE +X (-E)E +X (1-E) E) 1 3 4 1 4 1 3 1 3 4 1 4 Radiation transmitted by 3: X (l-E 3) f13 + X3E3 (1-E )(1-E3) f13 + X4E4 (-E3) Radiation absorbed by 3: X1El 3f13 + X3E3 (1-E ) f13 + X E4e3 1 313 3 ^ 3"3 1 13 4 4-3 Radiation reflected by 4: (1-E4)(I-E3) f14 (X1 E + X3 (1-E) E3) + 4 ( -E4) E4 (-E3)(1-E ) f14 + (l-E 1)(-E 3)(1-E ) f14 (X E + X3 (1-E1) E + X4 (1-E) E4 +3 4 14 1.1...0. 1 4

32 Radiation absorbed by 4: E4 (-E3) f14 (X1E1 + X3E3 (1-E1)) + XE42 (l -E3)(1-El) f14 4 3) 14 1 1 3 3 4 (1 3)(1-) f 4 E (-E )(1-E )(1 -E )14 f (X E + X (1-E ) E + X (l -E ) E ) + E (-E ) (1-E 3)(1- 4) f 43 (Xl + X (- ) E + X3 (1-E 4^1 3 4 14 1 1 3 1 3 4 1 4 The net radiation transferred to 3 is: Q, = Radiation absorbed by 3 - Radiation emitted by 3 ~5 = XE1 E13 + X3E3 (1-E ) f13 + X443 - X E 1 1313 3 3 1 13 4 443 3-3 But Q5 = 0 when T T3 = T so that: a- bE4 f;Z. 60 f3 E1 + aE3 (1-E) where a =A3/A = 27r R2L/27rr RL = R/R1 = D2/D1 and b = A4/A1 = R3/R = D3/D1 2.61 Substituting the value of f13 back into the expression for Q' and assuming that T3 T it is readily found that: E 1E3 (a-bE 4 4 4 Q5; A (T T 5 = A1 E+ (+aE (1-E) - 3 4 Q = rA1F5 (T - T ) where 1E3(a - bE 4) F5 = 2.62 E1 + E3 ( 1-E1

33 The thermal conductance for radiation to 3 may be defined as: G5 = Q/L(T1 - T3) and if T I T 1 3 G = 8nCR1 T F 2.63 The analogous procedure when applied to the net radiation transferred to region 4 is somewhat more complex because an infinite series results. The situation is rapidly solved, however, since the series obtained is of the form: 2 3 1 + z + z + z +.. where z = (1-E)(l-E4) f14 < 1, so that: + z +z.. - 1-z 1 - z Rather than going through the complete derivation, however, as it is lengthy, only the results will be stated. They are: f b 2. 64 14 ( l-E 3)(E +a ( -E l)3 + b(l-E)E4) + b(l-E1)(l-4) 4 4 Q = CA F (T - T4 bEIE _ "^16-1 4 F =2.65 F6 - E+ a(l-E1)3 + b(l-E )E4 G = 6 - a 8nrRT F 2. 6 6 L(T1 -) - T 24. All these factors are comb in the following chapter to formulate procedures for measuring latent heats of transformation and fuslon.

CHAPTER III APPLICATION OF THE THEORY In this chapter the equations which were derived in Chapters I and II will be applied to the determination of latent heats in the DTA calorimeter. It may be recalled that two methods of obtaining the thermal conductance were proposed: 1) theoretical calculation from the properties and geometry of the system; and Z) calculation from the experimental time constant of exponential decay of the DTA curve plus the thermal capacity of the system. Furthermore, in the first method, it was noted that direct measurements of the dimensions of the gas film thickness were imprecise, necessitating the use of an indirect method. Two such methods were proposed: 1) calculation of the film thickness by comparison of the experimental time constant of exponential decay of the DTA curve with its theoretical counterpart; and 2) calculation from two different measurements of the area underneath the DTA curve using two different gaseous atmospheres of widely different thermal conductivity, such as nitrogen and helium, in the system. These methods will now be developed in more detail. Calculation of the Thermal Conductance using the Theoretical Expression for the Time Constant to find the Gas Film Resistance In this method the gas film thickness is found by a trial and error calculation wherein " = R /R is varied until the time constant of exponential decay as computed from Equation 2. 59 agrees with the experimentally measured value given by Equation.lo 14 to within one percent. With this information it is then possible to compute the thermal conductance as follow So It may be recalled that the individual conductance, Gi, is defined by the relation: See Figure 2. lo n " - ng 4 34

35 G = = K./L (3.1) i L(T - Til) 1 where Qil is the net rate of heat flow across the conductance, L is the height of the sample which is a measure of the effective heat transfer area, Ti0 is the temperature at the outside radius and Til is the temperature at the inside radius. For thermal conduction across the gas film and the silica tube the value of the individual conductance may be obtained from the steady state formula: G. = 2 k./ln n. (3.2) 1 1 1 where.i represents the radius ratio, Ri -/R.. For the sample one obtains: R 3 G (r) dr -1 R4 G (3.3) 4 R3 - R4 After integration and inversion the result is 2 k 4 -7 (3.4) 0T - 1~ where the average is taken because the solid-liquid interface traverses the sample, starting at the outside diameter and ending at the inside diameter. The overall conductance is obtained by combining the individual conductances according to the steady state formula: - q4; - 3; ~ n3' 2

36 = + - + 1 (3. 5) G G2 G3 G4 and the computation of the heat of fusion according to Equation 1. 7 becomes a simple matter. In order to perform the above computations, a computer program was devised for use with the IBM-7090 Data Processing System. The program (Number DTA-19) was written in the MAD language and is reproduced in Appendix VI together with all of the data used in the computations. The estimation of the physical properties for use in these calculations and others to be soon described is discussed in Appendix IV. In order to test this method of computation, the DTA data for a number of standards whose latent heats of fusion are well known were processed. The results of this calculation are presented in Table 3. 1. The values of r "' computed here agree quite well with the directly measured values (1. 04 to 1. 07), and the calculated heats of fusion agree with the values reported in the literature. It may be noted that the contribution of thermal radiation was neglected in this analysis. Its effect, however, does not appear to influence the computation of the thermal conductance. Furthermore, the calculation of the thermal conductance by this method does not require precise values for the thermal properties of the various components of the system. Reasonable estimates are usually sufficient. The uncertainties which result from the neglect of thermal radiation in the model and from the use of approximate values for thermal properties show up in the calculation of a radius ratio for the gas film which is somewhat different from its true value without greatly affecting the computation of the thermal conductance or the heat of fusion. In this respect this method is self-compensating and relatively insensitive to experimental errors and to estimations of the parameters which appear in the calculation. Michigan Algorithm Decoder, 1963 Version, University of Michigan Computing Center, Ann Arbor, Michigan.

37 Table 3.1 Results of Computer Program Number DTA-19 for Standards T a Run L Lf lit Material f Number Cycle " cal/ cal/g Ag 1234 C-4 1C-N2 1.066 19.4 24.9 2H-N2 1. 055 22.8 C-5 1C-N2 1.065 21.4 2H-N2 1.060 22.3 C-13 2H-N2 1.037 21.8 In 430 C-12 2C-N2 1.049 6.6 6.8 3H-N2 1.047 6.8 Pb 600 C-3 1C-N2 1.080 4.8 5.9 2H-N2 1. 071 5.8 4H-N2 1.072 5.3 Sb 904 C-2 2C-N2 1.059 33.3 39.0 3H-N2 1.057 34.4 Te 723 C-1 2H-N2 1.076 33.6 32.7 2C-N2 1.084 33.9 Kubaschewski and Evans (47 ).

38 Calculation of the Thermal Conductance using Two Measurements of the Area under the DTA Curve to find the Gas Film Resistance In this method the gas film thickness is found by means of a double trial and error calculation wherein the two unknowns, r" and G, are evaluated from two pieces of experimental data: 1) the area under the DTA curve when a gas of known thermal conductivity (e. g. nitrogen) fills the system; and 2) the change in the area under the DTA curve when a second gas of different thermal conductivity (e. g. helium) replaces the first. A double trial and error calculation is necessary because an estimate of G is required in order to calculate ". Figure 2. 1 schematically depicts the physical system under consideration and also illustrates the nomenclature to be used in the following discussion. It is assumed that the nickel block due to its high thermal conductivity has a uniform temperature and due to its high total heat capacity (about 240 cal/ C) acts as a source for heat while the melting interface acts as a sink. It is assumed that two parallel mechanisms contribute to the overall thermal conductance of this system: conduction of heat from the nickel block through a series of intervening media to the solid-liquid interface, and exchange of radiation between the oxidized nickel surface and the silica tube and sample. Note that subscripts 2, 3, and 4 refer to the conductances of the gas film, the silica tube, and the sample, respectively, whereas the subscripts 5 and 6 refer to equivalent conductances for radiation. The individual conductances for thermal conduction of heat across the system are given by Equations 3o 2 and 3. 4. It may be recalled that the parallel conductances for thermal radiation are given by: G. = 8i R F.I-T3 i = 5,6 (2.63; 2.66) E1E3 (R- - R3 E ) F (1. 6z) 5 R1E1 + RZ E3(1-E)

39 G6 G2 G3 G4 ~ G ~G Figure 3.1. Analogous Electrical Circuit for Calculation of the Overall Thermal Conductance.

40 R3 EE4 F6 R1l R R2E3 (1 - E R E () + R65)'4(1 E1) 1 1 R 13 1 3 4 1 where cr is the Stefan-Boltzmann constant and F5 and F6 represent the fractions of the total radiation emitted from the oxidized nickel surface which are absorbed by the silica tube and sample, respectively. In order to combine the individual conductances into a working formula for the overall conductance, it is useful to consider the analogous electrical circuit depicted in Figure 3. 1. Combining the individual conductances in the usual fashion, the final equation which represents the overall thermal conductance is readily obtained, viz: G + (G2+ G5) (1 + G6/G3) G - (3.6) 1+ (G +G5)( + G6 ) 2 G3 G4 G3G4 The radius ratio of the gas film may be found from two different measurements of the area under the DTA curve as follows: -RR In r " = in R1 -+ ) -Z 2~k~k/ (GI' G ") In -2 kk2 ( (3.7) k2 -k2 (G-') (G -a) where a and p include the effect of radiation, and are defined by the relationships: Gp + ~GG GG (3. 8) G3 G 4 G3G4 a = (G + G6 + GG/G3) (3.9)

41 For G5 = G6 = 0, then = and = 0. In the above equations the starred and unstarred values refer to helium and nitrogen gases, respectively, and G = K/L is obtained from Equation 1.7. It may be noted that at sufficiently low temperatures Equation 3.7 reduces, as it should, to: lim Inra" = (/) (3. 10) G5, G6 - 0 Thus the increase in complexity introduced by the radiation terms is evident. In order to solve Equations 1.7, 2. 63, 3. 2, 3.4, 3. 6, and 3. 7 simultaneously a computer program (Number DTA-16) in the MAD language was devised. The program listing is presented in Appendix VI.. Unfortunately, it was not possible to check this method as thoroughly as the previous method because much of the data was unsuitable. Nearly all of the:T ddta Wa:s taken by alternating the flows of nitrogen and helium into the system. It was found, however, that in replacing the nitrogen with helium, not all of the nitrogen could be removed in this manner. Thus the data with helium as the gas atmosphere really represent data with a mixture of helium and nitrogen of uncertain composition. Furthermore, the calculation which has been described requires that the exact thermal conductivity of the gas be known, since otherwise quite erroneous values for the film thickness were obtained. It was decided, therefore, to reconstruct the DTA furnace so that it could be evacuated and backfilled with the desired gas. The system has not as yet been fully perfected, but some data has been obtained (DTA run number C-32 with sample number 152, indium) which substantiates the above conclusions. In this DTA run it was estimated that about 90% of the nitrogen was removed and replaced by helium. Brokaw's rule (7T07.) was used to estimate the thermal conductivity of the mixture. It was found that at 430 K, the melting point of indium, the thermal conductivity of helium was reduced by 20% if 10% nitrogen was present as an impurity. Using this correction, the calculation of ar" and the heat of fusion by this method

42 (Program DTA-16) agreed quite well with the values obtained by the previously described method (Program DTA- 19). Calculation of the Thermal Conductance from the Time Constant and the Thermal Capacity It was pointed out in Chapter I that the application of Equations 1. 4 and 1. 14 to the calculation of the thermal conductance depends on the estimation of the thermal capacity, C, and that some questions arise in regard to the constitution of C. These questions may now be examined. In considering the DTA data on standards whose latent heats of fusion are known, a value of the thermal capacity may be computed as follows: r AE C - -exp (3.11) e.;dt e d t and a value may also be estimated by summing the heat capacities of the sample and the silica in contact with the sample: C' = m c + m c (3.12) s p, s Q p, Q Let Z denote the ratio C/C' which may be computed for each standard. If the theory and experiment were completely matched then Z would be equal to unity. The results of such a calculation are given in Table 3. 2. It is evident that, within the precision of the data, Z may be considered to be a constant dependent only on the geometry of the system, and then used to compute the heats of fusion of other materials. For the Series B runs Z = 1.30 ~ 0. 274, and for the Series C runs Z =1.115 ~ 0.187. B C Of course, it may be argued that this approach is naive, and so it is. In the first place, the system is one containing distributed parameters, that is distributed thermal resistance and distributed thermal capacitance, so that the model through which Equation 1.4 was derived is in fact oversimplified. In the second place, it is known that after melting the See Appendix VT; MAD program Number DTA-17.

43 Table 3, 2 The Correlation Factor Z = C/C' Series B DTA Runs Standard ~ Number Deviation of Cycles Mean Value Devaton Material N Z Z Ag 6 1.310.0.. -132 Cu 14 1.345 0. 202 In 20 1.240 0. 256 Pb 8 1.425 0.306 Sb 6 1.285 0.140 Te 8 1.235 0. 244 Overall 62 1.300 0. 274 Series C DTA Runs N Z Z Ag 10 1.085 0. 071 In 4 1.090 0.145 Pb 5 1.195 0.191 Sb 4 1.232 0. 080 Te 4 1.005 0.113 Overall 27 1.115 0.187

44 conductance is not constant and is not identical to the conductance during melting. Notwithstanding these objections, however, within the precision of the data, this method of finding the thermal conductance is satisfactory. The pragmatic viewpoint is taken that the use of this method does provide useful and realistic estimates of the latent heats and is therefore justified. These two methods for measuring the latent heats of fusion are used in the following chapter to analyze the data on about 20 semiconducting compounds and to determine their latent heats of fusion and latent heats of transformation. See Figure 2. 2.

CHAPTER IV HEATS OF FUSION AND TRANSFORMATION BY DTA CALORIMETRY In this chapter the methods described in the preceding chapter are applied to the determination of the latent heats of fusion and transition of twenty compound semiconductors: Ag In8Sel3, AgzSe, Ag2Te, Bi Se3, Bi Te3, CdSe, CdTe, GaAs, GaSb, InAs, InSb, In Se3, InTe, In2Te3, PbSe, PbTe, Sb2Se3, Sb2Te3, SnTe and ZnTe. Experimental Data The experimental data consists in the tracings from a recording potentiometer which portray the difference in temperature between the DTA sample and reference as the amplified millivoltage output of a differential thermocouple versus time. A few typical DTA curves are shown in Figure 4. 1 with the shaded region representing the area under the DTA curve. The melting point was chosen as the temperature of the sample over the time interval that the DTA curve was linear. Three basic types of DTA curves have been observed: 1) those for very pure materials, such as silver, which have a very sharp melting point, and the decay of the differential temperature after melting is truly exponential for both heating and cooling curves; 2) those for impure materials or compounds which contain an excess of one of the constituents, such as CdTe, wherein melting occurs gradually at first and then more rapidly; although the decay of the heating curve may yield the true time constant, the cooling curve generally will not because of the continued evolution of latent heat; and 3) those for materials which undergo a large volume decrease on melting, such as the III-V compounds, Ge, and Bi. The latter curves often contain "wiggles", and may or may not be useful for finding the time constant for exponential decay. The time constant may be found from a plot of log I 9 - 9 I versus time. A few such typical plots are shown in Figure 4. 2. The experimental time constant,'T, may be found from the slope of such graphs via Equation 1.14. 45

9570 10910 10910 944~ 9370, 92?g70 10090 112, 10940 1010 1076 10 G=Ts-TR 960 1092~ Tm9600 0.1 my Time 9 (PT vs PT + 10~o Rh Thermocouples) I X;^T 586_ //\ _ 5s9 ^ ^ ^ ^^57 381- y/- 711 525~ ~ 5 min. 5~ Figure 4. 1. Some Typical DTA Curves Showing Differential Temperature vs Time, and Selected Sample Temperatures: A- No. 115, Ag, Run C-13, Cycle IC-N; B- No. 93, Ag, Run C-4, Cycle IIH-N2; C- No. 104, CdTe, Run C-10, Cycle I -N; D- No. 104, CdTe, Run C-10, Cycle IIH-N E- No. 110, Bi Te Run C-9, Cycle IIC-N F- No. 126, InSb, Run C-18, Cycle I H-N; G- No. 92, Te, Run C-1, Cycle IIC- N; H- No 102, GaSb, Run C-6, Cycle IIH-N.

47 7 \ \ \:\ B 0 0" \0\\.CUD 0 04 0.8 1.2 1,6 20 24 t, min Figure 4. 2. Some Typical Semilogarithmic Plots for Finding the Time C 1.0 —' A Constant for Exponential Decay of the DTA Curve. The Legends Correspond to the DTA Curves of Figure 4.1. Legends Correspond to the DTA Curves of Figure 4. 1.

48 The remainder of the experimental data consists in the measurement of the dimensions of the silica sample tubes by means of a micrometer and calipers during their fabrication. The estimation of the physical properties of the components of the system is discussed in Appendix IV, and all the numerical values used in the calculations are listed side by side with the computer programs in Appendix VI. Experimental Results The heats of fusion calculated from the DTA data via computer programs DTA-17 (using Z values of 1.300 and 1.115 for the series B and C DTA runs, respectively) and DTA-19 appear in Table 4. 1 where they are compared with the values reported in the literature and with the values computed from the heats of fusion of the elements according to the method of Kubaschewski and Evans (47): N. L Lf = TfA Sf = Tf[ + T 1 4.1 f, - T^ - T^. Z -^ ] 4.1 i f,i r= -R Z N, In N, 4.2 1 1 i where T = melting point of compound, K T = melting point of element, i, K f, i L = heat of fusion of compound, cal/mean gram atom L = heat of fusion of element i, cal/g-atom f, i A S = entropy of fusion of compound, cal/m. g. a. - K N. = atom fraction of element i in the compound R = gas constant = 1.987 cal/g-atom- K The heats of transition are summarized in Table 4. 2. The confidence intervals given in columns IV and V of Table 4. 1 and in columns III and IV of Table 4. 2 are simply the standard deviations from the mean

49 Table 4. 1 Heats of Fusion of Compound Semiconductors Material Latent Heat of Fusion - Calories/gram I II III IV V VI AgZIn8Se3 - - 49.8 20.2~3.3 19.1~ 2.8 20~3 2 8 13 Ag2Se - - 44.4 9.3~1.1 7.9 0.8 8~2 Ag2Te - - 50.1 8.9~3.0 7.0~1.3 8~2 Bi Se - - 37.9 39.2 3.0 32.2 3.1 36~ 6 Bi Te 36.2 (8) 35.8 31.6 ~4.4 31.4 ~2.7 33 ~ 4 CdSe - - 71.5 56.5~15.0 53.5~ 14.2 55 ~15 CdTe - - 63.3 (62.2) 46.3 ~ 5.8 50 8 GaAs -- 192. 199 ~ 23 200~25 GaSb 62.6 (73) 63.7 90.8~ 3.5 75.8~15.3 85~12 InAs - - - 98.0~ 15.6 95.5~ 9.7 96 12 InSb 47.3;51.5 (62, 73) 33.4 46.3~1.0 51.7~6.0 50~5 In2Se3 - - 48.2 30.0~1.8 35.0~7.1 30~4 InTe - - 41.2 (36.7) 24.7 ~ 2.6 28 ~ 8 In Te - - 42.5 19.7~5.1 18.4~4.3 19~5 PbSe - - 36.7 64.6~ 23.0 32.2~5.8 38 ~12 PbTe - - 37.3 37.4~12.6 25.0 ~3.2 28~9 Sb Se - - 48.5 36.6~ 3.5 31.8 ~4.8 35 5 2S3 SbTe - - 42.1 45.1~8.5 33.4~4.1 37~6 2T3 SnTe - 40.2 31.3 ~0.1 33.2 ~1.5 32 3 ZnTe - - 89.7 - 80.7~11.0 81~11 Code: l.-Heat of Fusion from Literature II - Literature reference III - Value calculated from Heats of Fusion of Elements; after Kubaschewski and Evans (47). IV - Program DTA-19; L ~ c Lf V - Program DTA-17; Lf ~ L f "Lf VI - Recommended Value.

Table 4, 2 Heats of Transition of Compound Semiconductors Heats of Transition - Calories/gram T ~K Material t I II III IV V AgzIn Se3 1022 - - 6..0 ~ 0. 9 4.1 ~ 1.0 5~ 2 Ag2Se 406 5.43 (47) 10.1 ~1.7 6.5 ~ 0.4 8 3 Ag2Te 420 - - 5.9 ~ 3.8 6.9 ~ 0.5 7 1 Ag Te 1072 - -.1 0.2 0,75 ~ 0.12 1.0 ~ 0.3 In2Se3 474 0.73 (94) 2.8 2.3 ~ 0.2 2.5 ~ 0.5 In Te - 893 - - - (0.8) (0.8) Code: I- Literature Value II- Literature Reference III- Program DTA-19; L ~ oL t t t IV- Program DTA-17; L ~ ot V- Recommended Value.

51 for the number of heating and cooling cycles used in computing the mean, and represent the scatter of the data. The confidence intervals quoted in column VI of Table 4. 1 and column V of Table 4. 2 represent, in this writer's judgment, the probable error associated with the recommended values cited. An examination of Table 4.1 reveals that in most instances the method of Kubaschewski and Evans yields a high estimate of the heat of fusion, though in the case of the III-V compounds the estimate is low. Of course, some of the materials undergo crystalline transformation before they melt in which case the sum of the entropies of transition and melting must be compared with the value predicted from Equation 4. 1. Such a comparison is provided in Table 4. 3 wherein it is evident that the prediction of Equation 4. 1 is still high. It is of interest to examine each transformation in more detail in order to learn as much as possible from the phenomena which occur. Ag2In8Se13 O'Kane (66) has investigated the thermal transformations in this compound and reported the melting point to be 8150C with a thermal transformation occurring at 7530C on heating and 7450C on cooling. In this work, DTA measurements were obtained on samples of high resistivity (> 10 ohm-cm), zone refined AgzIns Se3. Initial melting was somewhat gradual, but the constant melting point, 812 C, was quickly attained about one to two minutes after the first deflection from the baseline. The thermal transition was moderately sharp and occurred at 748 C on heating but supercooled to 743 C on cooling. These figures are in general agreement with the findings of O'Kane. The heats of fusion calculated by both of the methods described agreed with one another quite well, and the recommended value is 43. 5 ~ 6. 5 kcal/g mol. The heats of transition by the two methods disagreed somewhat, and the average value of 11 ~ 4 kcal/g mol is recommended.

52 Table 4.3 Comparison of the Sum of the Entropies of Transition and Fusion with the Theoretical Prediction A S Melting or A AoS.r S Transition ASf or t j Theoretical Temperature cal cal cal Material OK q - g - O - K Ag In Se 1022 0.. 0049 1088 0.0184 0. 023 0.0409 Ag2Se 406 0.0197 1170 0.0068 0.027 0.0380 Ag2Te 420 0.0167 1072 0.0009 1232 0.0065 0.024 0.0406 In2Se3 474 0.0053 1158 0.0259 0.031 0.0417 In2Te (893) (0.0009) 940 0.0203 0.021 0.0453 j refers to the number of transformations, including melting. After Kubashewski and Evans (47).

53 Ag Se Walsh, Art and White (89) who have measured its heat capacity from 16-300 K have reported the true composition of this compound to be Agl 99Se. These authors report no anomalies in the specific heat in this temperature range. The crystalline transformation at 133 C and the melting point of 89 C have been well established (31, 39, 47, 58) and the heat of transition has been reported to be 1. 6 ~ 0. 4 kcal/mole (47). These figures have been confirmed by our measurements which indicate that the transition occurs sharply at 133-136 C on heating at 2.5 C/min and supercools to 1220C on cooling at a rate of 0.5 0C/min, and that the melting point, also quite sharp, occurs at 894 C. The heat of transition measured here, however, is somewhat higher than that reported in the literature, being 2.36 ~ 0. 59 kcal/g mol where the average of the two measurements has been chosen. The heats of fusion found by the two methods agree within experimental precision, and the recommended value is 2. 3 ~ 0. 5 kcal/g mol. It is probably fortuitous that the heat of transition is substantially equal to the latent heat of fusion. Ag2Te The investigations of Miyatani (60) indicate that two compounds of nearly identical composition, Ag2 00Te and Agl 93Te exist in the silver-tellurium system at room temperature. Walsh, Art, and White (89) report that the composition of the latter phase is actually Agl 88Te and have measured its heat capacity from 16-3000K finding no anomalies. The existence of a thermal transition in Ag2Te at 147 C has been established (31) and its melting point has been reported to be 959 C (31, 39). In this work it was found that the low temperature transition occurs very sharply at 147-149 C on heating at 2. 5 C/min, but supercools to 143 C on cooling at 0.7 C/min. A second, previously unreported, thermal transition has also been observed which occurs very sharply

54 on both heating and cooling at 799 C. Melting was somewhat gradual, beginning at about 951 C with a constant melting point of 960 C being established within a few minutes. The heats of fusion computed by the two methods described agree within experimental precision and a value of 2. 74 ~ 0. 69 kcal/g mol is recommended. The heats of transition at 147 C found by the two methods disagree somewhat and a larger value of 2.40 ~ 0. 35 kcal/g mol is preferred because the experimental scatter is much lower. The heat of transition at 799 C is equal to 0. 34 kcal/g mol. Bi2Se3, Bi2Te, SbSe3 and Sb2Te These four compounds melt congruently and no crystalline transformations have been reported for them. Hansen (31) gives their melting points as 706 C, 585 C, 6220C and 617 C, respectively. Offergeld and Cakenberghe (65) have investigated the stoichiometry of three of the compounds and list the true compositions as Bi40 065Te59 935, B40.02Se59. 98 and Sb40.40Te59.60 Except for the sample of Bi2Te3 which was zone refined, the materials measured here exhibited somewhat gradual initial melting. Our melting points for Bi2Se, Bi Te, Sb Se and Sb Te are 700 C, 23 2 3' 2 3 2 3 586 C, 613 C and 617 C, respectively. Bolling (8) has measured the heat of fusion of Bi Te3 and found a value of 29. 0 ~ 3. 2 kcal/g mol which compares favorably with the value of 26.4 ~ 3. 2 kcal/g mol found in this work. The values found by the two methods for the remaining three materials differ somewhat. Where the standard deviations were equal the recommended value was found by simply averaging the two measurements, but if the standard deviations were unequal, the preferred value was found by adjusting to a point where the standard deviations overlap. In this manner the heats of fusion for BizSe, SbzSe3 and Sb Te3 are found to be 23.6 ~ 3.9, 16.8 ~ 2.4, and 23.5 ~ 3.8 kcal/g mol, respectively.

55 GaAs, GaSb, InAs and InSb Hansen (31) gives the melting points of these compounds, each of which melts congruently and undergoes no crystalline transformations, as 1238 C, 706 C, 943 C and 525 C, respectively. Nachtrieb and Clement (62) have measured the heat of fusion of InSb which they report as 11. 2 ~ 0. 4 kcal/g mol; and Schottky and Bever (73) report the heats of fusion of InSb and GaSb to be 12. 2 ~ 0. 7 and 12. 0 ~ 0.7 kcal/g mol, respectively. In this work, the melting points of GaAs, GaSb, InAs, and InSb' were found to be 1236 C, 712 C, 942 C and 524 C with all occurring sharply except GaSb which was somewhat gradual. Our heat of fusion for InSb, 11. 7 ~ 1. 2 kcal/g mol, is in good agreement with the previous measurements, but our value for GaSb, 16.3 ~ 2. 3 kcal/g mol is substantially higher. Furthermore, the two measurements for GaSb disagree substantially, the higher value being preferred because of less scatter. The two measurements of the heat of fusion of InAs and GaAs are in good agreement with one another with the preferred values being 18. 2 ~ 2. 3 and 28. 9 ~ 3. 6 kcal/g mol, respectively. CdSe, CdTe and ZnTe These compounds melt congruently and do not undergo crystalline transformation on heating. The melting points of CdSe and ZnTe have been reported by Mason and O'Kane (58) to be 12580 and 1300 C, respectively, whereas that of CdTe has been variously reported as 1090~C (63), 1092 C (54), 10980C (57,58) and 1106 C (49). In this work all three materials were observed to melt gradually at first with a constant melting point being achieved a few minutes after melting was initiated. The corresponding melting points and heats of fusion of CdSe, CdTe, and ZnTe were found to be 1250 C and Cf. Figure 4. 1.

56 10.5 ~ 2.9 kcal/g mol; 109100 and 12.0 ~ 1.9 kcal/g mol; and 12900C and 15. 6 ~ 2. 1 kcal/g mol. The heats of fusion of CdTe and ZnTe represent primarily the results of the calculation from the time constant for exponential decay plus an estimate of the thermal capacity via Program DTA-17, as only one data point, that being for CdTe, was available for processing by the second method. The two methods yielded values in good agreement with one another for CdSe, although the standard deviation was high. In Se3 Miyazawa and Sugaike (61) have investigated the crystal structure of In Se at room temperature and the thermal properties by 2 3 DTA, and found that a - InZSe3, the stable phase at room temperature, is hexagonal and undergoes sharp heat absorption at 200 C on heating but below 100C on cooling. Semiletov (74) has extended the crystallographic investigation to higher temperatures and reports four crystalline modifications of In Se3 1) an a, graphite-like, hexagonal phase which is stable below 200 C; 2) a P, hexagonal phase which is stable above 200 C; 3) a y, cubic modification which exists above 500-600 C; and 4) a possible 6, monoclinic phase whose region of stability is uncertain. He further reports that the a -~ p transformation is sluggish, requiring a seven to eight hour anneal at 350 C for completing the transformation of a thin film 400-600 A thick. O'Kane (66) has observed a thermal transformation 2100 C and also reports a small transition at 7400C. The melting point of In2Se3 has been well established (58, 66) and has been reported to be 888 C. Yoshioka (94) has studied the thermal properties of In Se in the range 20-300 C and has estimated the heat of 2 3 the a - t transformation from heat capacity measurements and found it to be 0. 34 kcal/g mol. In this work, the low temperature phase change was observed to occur very abruptly on heating at 201-202 C, and supercooled to below 1000 Con cooling. Melting began gradually about 50 C below the melting

57 point which was 885~C, and as soon as the melting point was achieved, proceeded at constant temperature. Our value for the latent heat of transition at 202 C is 1.17 ~ 0.23 kcal/g mol which is larger than that reported by Yoshioka by a factor of three. There is good agreement between the two methods of calculating the heat of transition from our data. The alleged phase transformation at 740 C, although it was observed, is thought not to belong to the compound, which is believed to be non-stoichiometric and deficient in selenium. The basis for this judgment is the phase diagram study of the InSe system which is reported in Chapter V, and which exhibits a peritectic reaction at 7450C which extends from nearly pure selenium to InzSe3. This explanation would also account for the anamalous 5-phase postulated by Semiletov. Finally, the a - p reaction did not seem to be as sluggish as Semiletov has suggested. It is condeded, however, that the value of the heat of transition reported here would be too low if all of the latent heat were not absorbed during the five to ten minutes required to traverse the transition peak in our DTA experiments. This probability, however, appears slim. The two methods of calculation yield values for the latent heat of fusion which are in good agreement with one another. The preferred value is 14. 0 ~ 1 9 kcal/g mol. InTe and In Te 2 3 Hansen (31) gives the melting points of these compounds as 6960C and 667 C, respectively. Zaslavskii and Sergeyeva (95) have investigated polymorphism in InzTe3 and report that two phases exist: an a-phase which is stable at low temperatures and decomposes between 500 and 600 C into a P-phase. The In-Te phase diagram has been recently investigated by Grochowski and Mason (29) who give the exact compositions of these compounds as In Te and In Te. Unlike In Se3, the a a- transformation in In Te3 is an orderdisorder transition, not accompanied by a large change in the crystal

58 structure (95) and has been almost too elusive to detect in our equipment. Minute deflections in the neighborhood of 5800C have been observed, however, and they may or may not be associated with this transformation. A more substantial heat absorption of about 0.5 kcal/mole has, however, been detected at about 620 C on heating. This is presently believed to be associated with the peritectic decomposition of the compound In3Te5 which decomposes at 625 C. Both InTe and In Te have been observed 2 3 to melt gradually with their respective melting points being 693 and 667~C. The two methods for finding the heat of fusion yield results for In2Te which are in good agreement with one another, but only one data point, that for InTe, was available for processing by the second method (Program DTA-19). The recommended heats of fusion are 6. 9 ~ 1.9 kcal/g mol InTe and 11. 6 ~ 3. 1 kcal/g mol In2Te3. In the case of InTe, it was impossible to obtain time constants from the cooling curves because of continued evolution of heat as freezing continued. This was due to the deviation of the composition from the In-deficient compound. Similar troublesome behavior was encountered with SnTe. PbSe, PbTe and SnTe These three compounds all melt congruently and do not undergo any other phase transformations above room temperature. Hansen (31) has listed their melting points as 10880, 917, and 790~C. In this work, PbSe has been observed to melt sharply at 1083 C while PbTe and SnTe melted gradually at 923 and 804 C, respectively. Umeda, Jeong and Okada (85) have found the true composition of SnTe to be SnTe. 38 A large number of tin vacancies have also been inferred from the anomalous thermal conductivity data of Damon (19). The heat of fusion of SnTe has been determined from heating data to be 7. 9 ~ 0. 8 kcal/g ool. The cooling curves could not be used because the discrepancy of the composition of our material from the tin-deficient compound resulted in continu ed evolution of latent heat during the

59 the decay portion of the DTA curve, and thus the true time constant could not be extracted. The two measurements for PbSe and PbTe did not agree with one another very well. The lower values were favored because the scatter was less, so that the recommended values are 10.9 ~ 3.4 kcal/g mol PbSe and 9.4 ~ 3.0 kcal/g mol PbTe. Experimental Precision The melting points and transition temperatures reported here are considered to be accurate, for the materials measured, to within ~ 2 C. The largest sources of variability are changes in thermocouple calibration and the melting of samples over a range of temperature which is occasioned by a deviation from the congruent composition. The precision in the measurement of quantities of heat is not particularly high, perhaps ~ 15%0 on the average. Numerous sources for this variability may be cited: error in the measurements of the dimensions of the system or of the area under the DTA curve, error in the values of the physical property estimates, error in the measurement of the time constant, improper choice of a physical model on which to base the calculations, etc., but it is not apparent that one of these choices should be preferred over the others. It is presumably a combination of all of them which leads to the observed variability. The question arises as to which of the two methods for finding latent heats is to be preferred. In this writer's judgment, it is the first, that is direct calculation of the theirmal conductance from a knowledge of the geometry and physical properties of the system with calculation of the dimensions of the gas firlm by matching the time constant for exponential decay of the DTA curve with its theoretical counterpart. This statement nevertheless must be qualified because, although the method is moderately insensitive to the estimate of the thermal conductivity of the sample, the latter may not be chosen indiscriminately, particularly when it is low. This fact is borne out by the results on PbSe and PbTe where large discrepancies betrween the two methods appear. The method

60 of direct calculation is also preferred because it is constructed on a much firmer theoretical foundation and does not depend on any calibration of the equipment. Discussion A comparison of the entropies of fusion plus transition of these twenty compounds in Table 4.4 reveals that they can be grouped as follows: AS = 2-3 Ag2In8Se13, Ag2Se, Ag2Te, In2Se3, In2Te3 A S = 3-5 CdSe, CdTe, ZnTe, InTe, PbSe, PbTe, SnTe AS = 4-7 Bi2Se3, Bi2Te3, Sb2Se3, Sb2Te AS = 7-10 GaAs. GaSb, InAs, InSb. These entropy changes represent changes in the state of order of the compound in transforming from the crystalline state at room temperature to the liquid state. The compounds having low values of A S undergo less pronounced configurational changes than those having high values of A S. It is known, for example, that the coordination number of the III-V compounds in the crystalline state is 4 whereas in the liquid the coordination number is about 6 (25). Furthermore, the change in density for these materials on melting is large, being about l. 6%o for InSb and 7. 3% for InAs (37). Thus the change in the state of order on melting is large for these materials, and the entropy of fusion is correspondingly large. Finally, it may be pointed out that the variations exhibited in Table 4.4 are great -- from 2. 2 for Ag2In8Se13 to 9. 6 for GaAs. It is clear that no simple correlation can be found for predicting heats of fusion from entropies of fusion which sould be analogous to Trouton's rule for heats of vaporization. This conclusion has been reached on numerous occasions before and is confirmed by our data. E.g. see Kubaschewski and Evans (47).

61 Table 4.4 Entropies of Transition and Fusion of Semiconducting Compounds T Sf AS. t A St Tf it Sf 1 Compound OK cal/g-atK — K Ag2In8Se13 1020 0.47 1085 1.74 2.2 Ag2Se 408 1.93 1169 0.68 2.6 Ag2Te 420 1.91 1233 0.74 2.7 1072 0.1 Bi Se 973 4.85 4.9 Bi Te3 859 6.15 6.2 Sb Se3 886 3.80 3 8 Sb2Te3 890 5.28 5.3 GaAs 1508 9.60 9.6 GaSb 985 8.25 8.3 InAs 1216 7.50 7.5 InSb 797 7.34 7.3 CdTe 1364 4 40 4.4 CdSe 1523 3.45 3.5 ZnTe 1563 4.99 5.0 In2Se3 474 0.50 1158 2.42 2. 9 InTe3 940 2.47 2. 5 InTe 966 4.1 4.1 PbSe 1356 4.02 4.0 PbTe 1196 3.93 3.9 SnTe 1077 3.67 3.7

62 Finally it is of interest to compare the entropies of fusion of these compounds with values for other materials. The entropies of fusion of metals are about 2. 2 cal/g at- K whereas those of ordered and disordered alloy phases are about 3. 5 and 2. 2 cal/g at- K, respectively (47). Dworkin and Bredig (21) have measured the heats of fusion of the alkali halides and have found that the corresponding entropy changes average 2. 9 cal/mean gram atom- K. The entropies of fusion of Ge and Se are 6. 7 (28, 71) and 7. 2 (67) cal/g at- K, respectively, which are about the same order of magnitude as the III-V compounds.

CHAPTER V SOLID-LIQUID EQUILIBRIUM STUDIES In this chapter the results of studies on the systems cadmiumtellurium, zinc-tellurium, and indium-selenium are presented and discus sed. The System Cadmium-Tellurium This system was first studied by Kobayashi (42) who reported that the system contained one compound, CdTe, which was congruently melting at 31050 C and the two eutectics between the pure elements and CdTe which occur very close to the Cd and Te ends of the diagram at 322 and 437 C, respectively. He was unable to measure the liquidus curve for compositions wherein the mole percent of cadmium was greater than 50%o because of the high vapor pressure and evaporation of his samples. The discrepancies in the melting point of CdTe and the unavailability of liquidus data and vapor pressure data especially for high percentages of cadmium in solution prompted several reinvestigations of this phase diagram (54, 57, 58, 63). Our work (57, 58) has not been formally published in detail as yet because it was desired to determine the heat of fusion of CdTe so as to be able to make thermodynamic calculations from the liquidus data. Estimates of the heat of fusion by two different methods, calculation from the liquidus data and calculation from the entropies of fusion of the elements (47) did not agree with one another and necessitated an experimental determination. Our experimental data for the cadmium-tellurium phase diagram are presented in Table 5. 1 and are plotted in Figure 5. 1 together with Cf. Chapter IV. See Appendix I for discussion of the interpretation of the DTA curves. 63

64 Table 5.1 Experimental Data for the Cd-Te Phase Diagram Maximum Sample Composition Fusion Eutectic Liquidus Number Atom %o Te Temperature Temperature Temperature ~C ~C ~C — 0 ---- — 321 376 1.0 980 (333) 730 380 3.0 970 (340) 808 379 10. 0 975 322 895 384 25.0 1050 323 963 385 40. 0 1100 325 1035 402 45.0 1150 325 1067 104-F 50.0 1200 --- 1091 520 50.0 1200 -- 1091 389 54.0 1100 450 1075 383 62.5 1100 450 1000 375 75. 0 1000 450 885 382 85.0 900 449 755 381 95.0 750 450 602 388 98.2 850 448 480 396 98.7 800 450 480 92-F 100 ---- --- 449 151-F 100 ---- --- 449

1200 1091~ 20 U1000 - 800 r.I 600 I 449 ~ 20 4 r001- ~324 ~20 3210 200 L I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ATOM FRACTION TELLURIUM Figure 5.1. The Phase Diagram for the Cadmium-Tellurium System. O Data of Kobayashi; A de Nobel; o Lorenz; O This Work.

66 the data of other workers. In the regions of high cadmium and high tellurium concentrations, the agreement is quite good, but in the neighborhood of 40-50% Te our liquidus temperatures are substantially higher than those reported by de Nobel, (63) and in the neighborhood of 60-70%o' Te they are substantially higher than the results of Lorenz (54). The cadmium-rich eutectic temperature was 324 ~ 2 C and the eutectic composition was practically pure cadmium. The tellurium-rich eutectic was found to be 449 ~ 2 C and 98. 7 atom percent tellurium. Knowing the liquidus curves and the thermodynamic properties of the constituents it is possible to make some thermodynamic calculations in order to gain some insight into the nature of the system. For example, Wagner (88) has derived equations for computing the excess free energy of a liquid phase from the liquidus of a compound. If a term associated with the excess molar entropy of the liquid is neglected, his Equation 61 for the case of an equimolar compound becomes NB 1 F 2LA 1 B B B 1 B B /o NB 5. 1 where N = atom fraction of element A in solution N = 1-N = atom fraction of B B A E F = excess molor free energy of the solution, kcal/mean gram atom and the integrand I1 (NB) is defined by ASf[ Tf - (N) T] I1(N) = 5.2 1 B ( 12 (NB where A Sf is the entropy of fusion of the compound in cal/mean gram atom - K, Tf the melting point of the compound in K, T the liquidus temperature and 9 (NB) is defined as: B

67 1 R 1 (N) = 1 + In 5. 3 ASf 4 NAN B where R is the gas constant. Equation 5. 3 also represents the liquidus curve of a hypothetical ideal liquid, that is T d(N) = T/ (N) 5.4 where the standard state is chosen as the pure, completely dissociated, equimolar solution. The function I1(NB) is plotted in Figure 5. 2 wherein it is apparent that the excess free energy exhibits a steep minimum at the compound CdTe. This would infer that CdTe molecules are very stable in solution, which is remarkable in view of the fact that the compound dissociates completely in the gas phase (44, 63) as do the rest of the II-VI compounds (44, 92, 93). Further support to this argument is given by the fact that the two liquidus lines intersect at 50% Te rather than joining in a single smooth curve. This behavior is also characteristic of a compou nd which exists as such in solutiono Integration of Equation 5. 1 for N gives the excess free B g energy of liquid CdTe at the melting point to be -16. 0 kcal/g mol which, when used with Wagner's Equation 34, results in a value for the standard free energy of formation of the compound from the pure solid elements at the melting point of CdTe of - 24,4 kcal/g mole. Using the thermodynamic properties listed in the Handbook of Chemistry and Physics (30) a corresponding figure of - 26. 5 kcal/g mole is obtained. The agreement is good in view of the high probable error associated with drawing the curve I (NB) in the neighborhood of the compound. It should be noted that a value of 11 2 kcal/g mol for the heat of fusion of CdTe has been used in these calculations. Although this figure is slightly lower than the value recommended in the preceding chapter (12. 0 ~ 1.9 kcal/g mol), it is within the probable error cited, See Lewis and Randall (52) p. 220; for a more quantitative discussion of the effect of interactions in solutin on on the curvature of the liquidus, see also Bonnier and Desre (10)o

60 I I I I I 50 I\ I I ^ <J~~~ I 40 ok 0 0 30 ~~~H~~~~~~~0 ~~~~~~20-~~0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ATOM FRACTION TELLURIUM Figure 5.2. The Integrand of the Excess Free Energy Function versus Composition for the CadmiumTellurium System. O Data of Kobayashi; O de Nobel; A Lorenz; o This Work.

69 and the use of the slightly higher value would not substantively alter the conclusions. Bonnier and Desre (11) have derived an expression for computing the heat of mixing at infinite dilution from liquidus data in binary systems containing a compound and a eutectic close to the pure element. Their expression for the case where the compound is equimolar is: ___ dln NANB 0A (B) AHB LfB Lf A R (l/T) where A HAB) = partial molar enthalpy of mixing of 1 mole of A and 0 an extremely large quantity of B. A H = standard enthalpy of formation of compound AB AB L = latent heat of fusion of A f,A L = latent heat of fusion of B f, B and the last term represents the gas constant times the slope of the liquidus curve in the region of dilute A when plotted as ln N N AB versus reciprocal temperature. Such a plot for the cadmium-tellurium system appears in Figure 5. 3 where the straight, dashed line represents the liquidus of a hypothetical ideal solution based on Equation 5. 4. A brief calculation using A H - 24. 52 kcal/g mol, L CdTe f, Cd 1.53 kcal/g atom and L = 4. 18 kcal/g atom (47), yields f, Te A HdTe) = -19.0 kcal/g atom and A HTe Cd = +4.4 kcal/g atom Ca(Te) a~Te(Cd) These figures indicate that dissolution of Te in Cd requires the expenditure of energy whereas the dissolution of Cd in Te releases energy, or in other words the liquidus for the system CdTe-Te exhibits negative deviations from Raoult's law while the system Cd-CdTe exhibits positive deviations. Finally it might be noted that correlation of the liquidus data by means of the sub-regular solution theory of Thurmond and Kowalchik

70 I I I I I i 1.0-' U 0.3 11 1.2 1.3 1.4 1.5 1.6 Tf/T Figure 5. 3. The Liquidus Curves tor the Cadmium-Tellurium System Plotted as log 4 NCdNTe versus Reciprocal Reduced Temperature. O Data of Kobayashi; O de Nobel; A Lorenz; [ This Work. For Open Symbols NT < 1/Z and for Closed Symbols NTe> 1/2. The Dashed Line Represents E quatizoiin 5.. LLL12.4 0.03 1 1.1 1.2 1.3 14 1.5 1.6 TfI1 Figure 5. 3. The Liquidus Curves tor the Cadmium-Tellurium System Plotted as log 4 No NTe versus Reciprocal Reduced Temperature. O Data of Kobayashi; Q de Nobel; A Lorenz; D This Work. For Open Symbols N < 1/2 and for Closed Symbols N > 1/2. The Dashed Line Represents Equation 5. 4.

71 (83) was tried. These authors found that the solubility of many elements in silicon and germanium could be explained by such a correlation. This method requires that a plot of the quantity RT In y/ (1 - x) be a linear function of T, where x is the mole fraction of the compound in solution and Lf 1 1 n y = -ln x + f (1 R T T These equations are based on a standard state of pure undissociated, supercooled liquid compound, so that the activity coefficient y represents the deviation of the liquidus from that given by Van't Hoff's equation Lf 1 1 In x = R (T - ) R T T f id rather than from Equation 5.4. When these plots were prepared for the systems Cd-CdTe and CdTe-Te, however, they were highly non-linear. The System Zinc-Tellurium The only previous study of this system is due to Kobayashi (43) who found that one congruently melting compound, ZnTe, exists as do two eutectics whose compositions are very close to the pure elements. His measurements were performed in an open system, however, so that evaporation of the alloys at high temperatures was a problem, and therefore he was unable to measure liquidus temperatures of mixtures containing less than 50 atom percent tellurium. In this work eighteen DTA runs were performed on alloys containing from 3.0 to 100 atom percent tellurium, and the results are presented in Table 5. 2 and Figure 5. 4. On the high-tellurium side of the phase diagram, the liquidus temperatures were found to occur from 30 -70 C higher than the values found by Kobayashi. They are higher partly because our measurements were taken at constant volume while his were taken at constant pressure, but more probably because our elements

72 Table 5.2 Experimental Data for the Zn-Te Phase Diagram Maximum Sample Composition Fusion Eutectic Liquidus Number Atom % Te Temperature Temperature Temperature 0C 0C ~C 0 ---- -- 419.5 407 3.0 1175-1200 424 1190 ~ 10 447 10.0 1130 423 1208 ~ 5 461 20.0 1310 420 1215~ 5 475 30.0 1280 420 1223 ~ 5 466 40.0 1300 421 1250 ~ 5 476 45.0 1310 420 1270 ~ 5 125 50.0 ---- --- 1290 ~ 3 432 50.0 1350 --- 1290 ~ 3 531 50.0 1350 --- 1290 ~ 2 477 55.0 1310 449 1260 5 436 60.0 1250 445 1205 ~ 5 457 70.0 1250 448 1100 ~ 15 478 78.0 1100 449 996 5 404 85.0 1100 446 915 ~ 5 484 92.5 980 449 790 ~ 15 401 95.0 1025 448 740 ~20 92-F 100 ---- --- 449~ 2 151-F 100 - --- 449 ~ 2

1290 ~ 2 1200 I o I 6 600 - h - Da4222oi0 i 447 2 44 Lt\ 200 10 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ATOM FRACTION TELLURIUM Figure 5. 4. The Phase Diagram for the Zinc-Tellurium System. * Data of Kobayashi; 0 This Work. O Data of Kobayashi; O This Work.

74 were much purer and less likely to become contaminated by oxidation. The tellurium-rich eutectic was observed to occur at 447 ~ 2 C and greater than 99 atom percent Te. The decomposition of the liquid phase at the solidus was observed to occur by peritectic reaction as the arrest temperatures were 422 ~ 2C - higher than the melting point of pure zinc (419. 5 C) - with the composition at the peritectic point being indistinguishable from pure zinc. The liquidus curve for alloys containing less than 50 atom percent Te was very flat, but it is felt that the possibility of the presence of a monotectic is remote, since no range of constant arrest temperatures was observed. The liquidus temperature also increases continuously from the pure Zn to the ZnTe compound. It should be pointed out that the exact composition of the samples containing 30-45%) Te was somewhat different from the nominal composition reported here, because of the difficulty in recovering a substantial portion of the sample which stuck to the quartz tube on fusion. This error is tolerable, however, because of the relative flatness of the liquidus curve in this region. One additional point should be mentioned. It may be noted that the liquidus curve shows a point of inflection. That this is the case is supported by the fact that the DTA curves for the 40 and 45%o Te samples show double peaks in the neighborhood of the liquidus. The lever rule demands that, at equilibrium, in a two phase region, the two phases are present in a given mass ratio which is determined in part by the position of the liquidus curve. The shape of the DTA curve will depend on how this mass ratio varies with time. When the liquidus curve is very flat and contains a point of inflection, as is the case here, the ratio of solid/liquid will change as follows. On heating a sample whose composition lies to the right of the inflection point but to the left of ZnTe, the above ratio will be approximately constant from 4230 to about 12000; as the bend of the liqudus curve is passed, this ratio will decrease very rapidly; as the inflection point is passed the decrease will become less rapid; finally, on approaching the liquidus the decrease will become

75 more and more rapid until no more solid is left at the liquidus point. Granted that this analysis is merely a qualitative one as other factors also influence the shape of the DTA curve (e. g., the rate of energy input to the sample, and the variation of the heat of dissolution with temperature and composition), it is nonetheless evident that such a double peak should not be entirely unexpected. In this case the above explanation appears to be the most plausible one. As regards the thermodynamics of Zn-Te solutions, it may be stated that, as with Cd-Te solutions, the excess free energy of the solution exhibits a sharp minimum at the 50% composition which fact indicates that ZnTe molecules are stable in the liquid phase. The heats of mixing at infinite dilution may also be computed as described in the previous section (See Figure 5. 5). We find AH =- 21. 7 oo Zn(Te) kcal/g atom and a HTe( = + 300. kcal/g atom. It is evident that oo Te(Zn) the positive deviation from ideality in the Zn-ZnTe system is exceedingly strong indeed, which fact is also indicated by the extreme flatness of the liquidus in this region. Finally, it may be noted that as with the previous systems, the liquidus temperature data for the systems Zn-ZnTe and ZnTe-Te could not be explained by sub-regular solution theory based on the non-dissociative solubility of ZnTe in atomic zinc and atomic or diatomic tellurium. The Indium-Selenium Phase Diagram The only data reported on this system as yet have been a few selected studies on isolated compositions (31, 61, 74, 94) within the system, and no liquidus temperatures, other than the melting points of InSe, 660 ~ 100 and In2Se3, 890 ~ 100C (31) have been reported. In this work sixteen thermograms have been obtained on compositions from 10. 0 to 90. 0 atom percent selenium. The DTA results are summarized in Table 5. 3, and the proposed phase diagram is depicted in Figure 5. 6. Five compounds are believed to exist -- two which melt congruently, and three which decompose pertectically. The congruently melting compounds,

76 Table 5.3 Experimental Data for the In-Se Phase Diagram Maximum Sample Atom % Fusion Liquidus Number Se Temperature Transition Temperature Temperature C ~C ~C 0 ---- ---- 157 493 10.0 950 158, 521 521 494 20.0 950 157, 518 518 495 30.0 950 157, 520 520 496 40.0 950 159, 520, 554 (560) 844 45.0 950 521, 553 598 497 50.0 950 Melts gradually, 613 beginning at 605 509 54.0 1000 663 686 510 56.0 950 659 765 846 58.0 1000 (195), 660 850 676 60.0 950 201, 745 885 294 62.0 1000 201, 220, 640, 744 865 533 64.0 1000 220, 742 275 66.0 975 201, 270, 650, 744, 795 (760) 499 70.0 900 214-220, 745, 759 759 500 80.0 625 201, 220, 650, 745, 759 759 501 90.0 625 220, 658, 743 822 100 ----- 217 All samples were water quenched.

77 1.0- ~ 0.3 -n H AH 3.98 z 0.1 AH RTf:107 D f k.83 0.031.1 1.2 1.3 1.4 1.5 1.6 Te Temperature. O Data of Kobayashi; U This Work. For Open Symbols N < 1/Z and for Closed Symbols N > 1/2. The Dashed Line Represents Equation 5.4. Te

78 900 1 1 8850 800 760 \ 7450 700 - 600~~-/ \61 6500+1 / / 500- "i I cr-I 400 - 3001I _^220_ 2001 2010 1580 0 0.2 0.4 0.6 0.8 1.0 ATOM FRACTION SELENIUM Figure 5.6. The Proposed Phase Diagram for the Indium-Selenium System.

79 InSe (M. P. = 6140C) and InzSe3 (8850C), both appear to be deficient in selenium. The first peritectic compound contains about 46 atom percent selenium and decomposes at 553 C. The second contains about 53% Se and decomposes at 660 C, the previously reported melting point of InSe. The location of the third compound which decomposes at 745 C is uncertain, but it probably contains about 80% Se. There appears to be crystalline transformations associated with In2Se, at 201 C, and InSe4 (? ), at 650 C. The indium-rich eutectic (?) occurs at 158 C and is thought to have a composition of nearly pure indium. The selenium-rich liquid decomposes by a peritectic reaction at 220 C (vs 2170 for the melting point of pure Se) and contains nearly pure selenium. The anomalous variations in the low temperature transition for alloys containing 60-1 00o Se is ascribed to non-equilibrium, local variations in the composition of the ingots, since they were prepared by quenching in water from the two-liquid region, and were not annealed prior to DTA. Two monotectic (S + L - L + L2 on heating) transformations were observed. In the first, two liquid phases of nominally 5%o and 35% selenium form at 520 C, and in the second, two liquid phases of nominally 68% and 95% Se form at 7600C. Clearly, more work is indicated in order to completely characterize the nature of this system as there are many uncertainties which still must be resolved. Yet, with relatively few thermograms it has been possible to formulate a fairly complete, qualitative picture of the phase diagram. Discussion In attempting to correlate the liquidus data for the Cd-Te and Zn-Te systems it has been observed that sub-regular solution theory based on the non-dissociative dissolution of the compound in either See Table 5. 3.

80 element does not suffice. Furthermore, regular solution theory based on the complete dissociation of the compound is also insufficient, for -.t it requires that the integrand I1 be constant, and this is obviously untrue. The phase diagrams are, however, of a simple type which would infer that their theoretical explanation would also be reasonably uncomplicated. In this regard the following scheme was tried. Consider the following model: the compound AB dissolves as the molecular species in element B which exists in the form of a cluster of n atoms on the average, B, thereby forming an ideal solution of the Van't Hoff type. A Sf Tf In x = R - 5.6 where A Sf is the entropy of fusion of the compound, Tf is its melting point, R is the gas constant, and x the mole fraction of AB in a solution containing AB and B. Equation 5.6 may also be written as follows: a Sf Tf AS _ T x =exp R 1 T 5. 7 Now x may also be found by material balance as follows. Let NB denote the atom fraction of B in solution, whence n - 1 - (n-2)N 5. B Solving Equation 5. 8 for n, Equation 5. 9 is obtained: Z N - 1 x B n = 1 5. 9 x - I 1 -N Thus for each liquidus point, x may be found from Equation 5.7 and n from Equation 5. 9.,.. See Wagner (88); also Equation 5.1 ff. and Figure 5.2.

81 Applying this calculation to the liquidus curves of the systems Cd-Te and Zn-Te using A Sf = 8. 8 and 10.0 cal/g mol K for CdTe and ZnTe respectively, the results which appear in Figure 5. 7 are readily obtained. The indicated correlations may also be stated as follows: System CdTe-Te n = constant = 1.53 ZnTe-Te n = constant = 1.35 Cd-CdTe n = 1.62 exp 8.38[ (Tf/T) - 1] Zn-ZnTe n = 0.30 exp 75.[ (Tf/T) - 1] 8 f Hence this model does provide a simple, two parameter, correlation for the liquidus curves of these two systems. In fact, for the systems CdTe-Te and ZnTe-Te only one parameter is needed, for n is constant. Admittedly the theoretical basis for these results is shaky, but on the other hand the idea of atoms clustering is comprehensible. Perhaps the theory presented is a simplification of a more general hypothesis. It cannot be said at this time. The results are only presented here as being interesting.

82 Zn-ZnTe 50.0-'] +,/ Cd-CdTe 10.050- n CdTe -Te g v v v 1.0 -ZnTe-Te 0.510 1.1 1.2 1.3 1.4 1.5 1.6 Tf/T Figure 5. 7. Semilogarithmic Plot of the Average Cluster Size, n, versus Reciprocal Reduced Temperature.

CHAPTER VI CONCLUSION In this final chapter, the results of the work are summarized and evaluated, and recommendations are made for possible future work. Summary In Chapter I several methods for determining the thermal conductance for the transfer of heat to the DTA sample were discussed. Of these, two — direct calculation from the geometry and physical properties of the system and calculation from the time constant plus the thermal capacity — appeared to be well suited to our system, but the first method was imperiled because direct observation of the dimensions of the gas film yielded imprecise values for its thickness. Subsequently, two indirect methods for calculating the thickness were proposed, viz. comparison of the theoretical time constant for the exponential decay of the DTA curve and comparison of the areas under the DTA Curve when two gases of widely different thermal conductivity filled the system. Next, in Chapter II the problems of heat conduction in the system prior to melting, during melting, and immediately following the completion of melting were considered. From these studies two significant facts emerged. The first important result was that the steady state formulae for the conductances could indeed be used to predict the thermal conductance of the system. The second result of significance was the extraction of expressions which would allow a theoretical computation of the time constant for exponential decay of the DTA curve, and thereby afford a method for evaluating the thickness of the gas film. In Chapter III the methods proposed were further developed and applied to data on standards whose latent heats of fusion were well known. The agreement between the calculated heats of fusion and those taken from the literature was good, but it was learned that a calibration of the apparatus was necessary in order to apply the second method -- 83

84 calculation of the thermal conductance from the observed time constant plus the thermal capacity. The two methods via computer programs DTA-19 and DTA-17 were applied to the measurement of latent heats of fusion and transition of twenty compounds in Chapter IV. The entropies of fusion plus transition were found to vary from 2. 2 cal/g atom K for Ag2In8Se3 to 9.6 cal/g atom K for GaAs. It was further found that the compounds could be grouped. into broad categories on the basis of their entropies of fusion and that similar compounds could be placed in the same category. In Chapter V the phase diagrams for the systems Cd-Te, Zn-Te, and In-Se were presented and discussed, and the measured values of the heats of fusion of cadmium telluride and zinc telluride were used in performing thermodynamic calculations with the liquidus data for these systems. It was concluded that cadmium telluride and zinc telluride exist as the molecular species in solution although some dissociation is probable. A speculative model which correlates the liquidus data of these two systems was also proposed. This model was based on the formation of an ideal solution of the Van't Hoff type between undissociated compound AB and clusters of atoms of type B, B, where n represents the average number of B atoms in a cluster and is a function of temperature. Advantages and Disadvantages of the Methods Used to Measure Latent Heats It is clear that a principal advantage of both methods for measuring latent heats is that they permit the measurement over a very wide temperature range, from 300 to 15600K. Secondly, the large thermal resistances of the gas film and the silica sample tube minimize uncertainties due to the properties of the sample which may not be well known. As to the two methods proposed, the first, direct calculation of the thermal conductance from the geometry and thermal properties of the system, is clearly preferred because no calibration of the apparatus

85 is required. It does require an estimate of the thermal conductivity of the sample, however, and the estimate must be reasonably precise if the thermal conductivity of the sample is low. With regard to experimental precision, both methods leave a great deal to be desired since the probable error in the measured heats if 15%o or more. Recommendations Three avenues for further work along these lines are open. In the first place, efforts would have to be made to determine the major sources of variability which contribute to the probable error associated with the latent heats measured. This would have to be accomplished by a process of elimination and would entail considerable effort. Secondly, the method for evaluating the gas film dimensions by alternating the gas atmospheres in the system should be examined more closely. At the same time the contribution of thermal radiation to the heat transfer mechanism could be studied. In order to carry out these studies the DTA furnace would have to be modified so that it could be evacuated and backfilled with the desired gas. This appears to be the only way in which one can be sure that one gas is completely replaced by another. Finally, the methods could be adapted to the study of other types of heat effects such as heats of reaction and heats of solution. These areas provide a fertile field for future research activity. It is felt that DTA will perform a useful role in the field of calorimetry in the future because it is rapid, the measurements are performed with ease, and the equipment required is simple, easily constructed, and relatively inexpensive.

APPENDIX I DTA EQUIPMENT AND PROCEDURE Two different sample arrangements were used: A) Crushed samples were sealed under vacuum below 0. I micron in specially cleaned, clear fused silica tubes which were 11 mm in inside diameter by about 10 cm long. The sample tubes contained concentric thermocouple wells 6 mm in outside diameter by about 2. 5 cm deep. The tubes were placed in holes drilled in the nickel holder, with a concentric graphite spacer being inserted between the sample tube and the nickel block. The weight of each sample was adjusted in accordance with its density so as to maintain about the same sample volume from one run to the next, a volume of about three cubic centimeters being chosen as the norm. Thi s volume corresponds to a sample weight of from 15 to 35 grams. Chromel-P versus alumel thermocouples were used. B) The second sample arrangement utilized is depicted in Figure A-I-1. In this arrangement 10 mm (rather than 11 mm) tubes were used in which the thermocouple well was 6 cm (rather than 2.5 cm) deep. The graphite spacer was eliminated; and the nickel block was oxidized by heating in air at 11000C. Platinumplatinum + 10%o rhodium thermocouples were used. Sample arrangement A corresponds to the series B DTA runs, whereas sample arrangement B corresponds to the series C DTA runs. The DTA furnace was designed around a 3" diameter, 2. 5 kva Kanthal furnace winding which was installed in a vertical position, surrounded with bubble alumina insulation, and contained inside a 12" diameter stainless steel shell. Copper cooling coils were soft-soldered to the shell in order to control the cooling rate at lower temperatures. With this arrangement it is possible to maintain a heating rate of 2. 5 C/min. from room temperature to 1100 C and a cooling rate of 2. 5 C/min. from 1300-5000C. Above 11000C and below 500 C the maximum heating and cooling rates, respectively are limited by the time constant of the system, the latter being about 200 min. at room temperature and decreasing to about 120 min. at 1300 C. 86

87 [ NICKEL I SAMPLE I oSAMPLE 2 FUSED SILICA AMPOULE P -ALUMINA PROTECTION Pt + TUBE 10% Rh TAT2 — Pt T -T2 Figure A-I-1. DTA Samples inside the Nickel Block for Sample Arrangement B.

88 A 2-1/2" diameter by 21" long alumina (McDanel AV-30) tube which is sealed at the top is mounted inside the winding. The sample holder consists of a nickel block, 2" in diameter by 6" long, which contained two or three symmetrically located sample wells. The nickel block is supported inside the furnace by an alumina tube (McDanel AV-30) which is 2" in diameter by 12" long and flanged at the bottom to accommodate an arrangement for holding the assembly in the furnace. This tube also serves as a holder for the thermocouples and contains a port for introducing the helium or nitrogen purge gases. It too is filled with bubble alumina for insulation. A block diagram showing the salient features of the temperature control and measurement circuit is shown in Figure A-I-2. All the operations in the system are actuated by a pulse generator. A short 110 volt, 60 cycle, AC pulse, about 1/4 to 1/3 second in duration is generated each 20 seconds by the microswitch which rides against a cam on a clock motor, which makes one revolution every 20 seconds. The pulses are used to control a motor-driven autotransformer which feeds power to the Kanthal furnace so as to generate a substantially linear heating or cooling rate. The pulse length is adjusted so that each pulse can drive the autotransformer through about 1/200 of its full range. (It requires 64 seconds for the motor to drive the autotransformer motor, the heating rate might be too rapid, and hence it was necessary to devise a Pulse Selector Circuit. With such a circuit it is possible to vary the time between the successive pulses which drive the autotransformer and thus to establish a heating rate of from 1 C/min to 6 C/min. The pulse selector circuit uses a Guardian stepping relay, Model MER-115, which comprises a 21-point stepping circuit and a reset circuit. As the stepping coil is triggered by the pulse generator the contact arm advances along the various contacts from number 1 to number 21. Whenever the reset coil is triggered, the contact arm is returned by a detent spring to contact number 1. The short AC pulses are used not only to trigger the stepping coil but are

220 v A.C. 0~ P-2 TRANSFORMER P-2 r~.~~~____________________^__,>~ -~^i^~0~~1 ICu WIRE, F r o + Co TURN. FLOW_ I~1I I s^ g 19 COOL Hl 6000 n Lo RESET 0 I ~~2 0 — IN~ o I IN 3 0 —-2 2 3 P NMOTOR DRIVEN 0 /_ 5 4 AUTOTRANSFORMER 0 6 I0 el CONTINUOUS 6 AUTOTRANS FORMER 0 20 i () DIVE. COOL H20 STEP 19 OUT 2o~^0 OFF CONTAOS p s TIME DELAY RELAY NICKEL L~~(NORMALLY CLOSED) BBSAMPLE INERT GAS PULSE SELECTOR CIRCUIT HOLD PURGE _____________________________^ SYSTEM RIV~___________________________ SAMPLE REFERENCE o MOTO 110I?~~~~~~~~ o 1MVI 0M|'-ICE FiueAI22lc1iga 10%!rat Pt (NORMALYCLOSED SP NET_____________ ~ CRAM S-I T PULSE GENERA TOR6 -y - + oSPEEDOMAX D.C. AMPLIFIER STRIPCHART 66 o RECORDER 5MV 0-10 MV vs TIME D C. or 10 MV vs 10 MV PU Figure A-I-2. Block Diagram Showing Operation of the DTA Equipment.

90 also fed through the contacts of the stepping switch. Each of the contacts on the stepping switch from contact number 2 to contact number 21 is connected on one terminal (A) on each of the 20 outlets on the panel. Hence, by connecting the autotransformer motor to the appropriate outlet the motor is activated only when the stepping switch has advanced sufficiently to feed the pulse to the outlet, and the time interval between each selected pulse can be varied in 20-second steps from 60 seconds to 420 seconds. A control switch is mounted in parallel with a pulsing switch so that the initial or final position of the autotransformer can be set as desired. The second terminal (B) on each outlet is connected to the subsequent contact on the stepping switch, and leads to the reset coil. The next pulse then triggers the reset coil and returns the contact arm to contact number 1 for another cycle. For example, in order to select a pulse once every 60 seconds, the plug would be inserted into the second outlet. As the contact arm advances to contact number 3, the pulse is fed through terminal (A) of outlet number 2 and activates the autotransformer motor. The next pulse advances the contact arm to contact number 4 which feeds the pulse to terminal (B) of the second outlet. At this point the reset coil is triggered and the contact arm is returned to contact number 1 to start the cycle over again. An ordinary power control relay was required in the circuit as shown in order to open the stepping circuit for the duration of the resetting pulse, in order to allow the stepping contact to return all the way to contact number 1. The measuring and recording circuits use a Leeds and Northrup Model G Speedomax strip chart recorder to read the differential emf. The differential thermocouple is connected through a Leeds and Northrup DC amplifier to the x-axis of the recorder and the differential emf was shifted to read zero at mid-scale, by providing a 0-10 mv auxiliary variable emf in series with the output from the DC amplifier. The thermocouple used to measure the sample temperature (with

91 respect to the reference junction temperature) was monitored by means of a portable precision potentiometer. By using a 10 mv x 10 mv x-y recorder (not shown) a plot of differential emrnf versus sample temperature can be obtained directly, by introducing the sample thermocouple directly to the y-axis of the x-y recorder. The portable potentiometer is ret ained in the circuit. At low temperatures corresponding to less than 10 my output from the sample thermocouple, the potentiometer output is set at zero. Above 10 mv output, the x-y recorder was reset onto the next 10" of chart, and the first 10 mv of thermocouple emf is balanced out with the auxiliary precision potentiometer. In this way the record of sample thermocouple emf is plotted on a scale of 1. 0 mv per inch, which permits good separation of the transitions and careful indication of the transitions points, using either chromel-alumel or platinumplatinum rhodium thermocouples. In studying phase equilibria the x-y recorder was found to be considerably more convenient and records the data in a form which permits ready interpretation. The following procedure was carried out for each heat of fusion run. First, both samples were melted so that each would present substantially the same effective heat transfer area. Then each sample in turn was frozen and re-melted with a nitrogen purge atmosphere in the DTA furnace and the procedure was repeated with a helium purge atmosphere in place of nitrogen. In each case the melting point and the heating rate were noted. The differential emf versus time curves were then graphically integrated so as to obtain the area under the curve. Upon removal of the samples from the DTA furnace the height of the sample material in each tube was noted. During each run, the furnace chamber was purged with an inert gas, the flow rate being maintained at about 2 cc/sec (at STP).. The liquidus temperatures of the binary alloys were estimated from both the heating and cooling curves. In the former case the temperature at which the last deflection returned to baseline was noted, and in the latter, the first deflection on cooling was used.

9Z In most cases these temperatures agreed with one another to within less than 5 C. Difficulty in establishing the liquidus temperature was encountered in two instances: 1) for very high concentrations of tellurium, nucleation of the solid phase on cooling was sometimes suppressed to below the eutectic temperature; and 2) in regions where the liquidus curve was very steep, nucleation of the solid phase at the liquidus produced only a small deflection which sometimes could not be detected precisely. The latter behavior is a consequence of the lever rule, since in such a region the amount of solid which nucleates at the liquidus will be very small and thus the heat evolved will also be very small.

APPENDIX II DYNAMICS OF THE DTA FURNACE Determination of the Time Constant The dynamics of the furnace can be determined by placing a thermocouple into the thermocouple well of a sample which is contained in the nickel block and establishing a constant temperature in the furnace. When the power input to the furnace is changed abruptly to a new constant value, it is assumed that a new source temperature is established instantaneously and the sample thermocouple temperature is measured as a function of time. The differential equation describing a single time constant system which is absorbing heat by both conduction and radiation is 4 4 VC C dT/dt = UA(T - T) + r A(T -T) (A-2. 1) where 3 V = volume of the system, cm = density, gm/cm C = specific heat, cal/gm C Zo U = heat transfer coefficient cal/cm K min 2 A = area of heat transfer, cm E = emissivity of receiver 0( = Stefan-Boltzmann constant T = heater (source) temperature T = thermocouple temperature. Over small temperature ranges the radiation term can be factored and an average temperature defined such that T - T = T (T -T) (A-2.2) 00 m o I3(T + 2T T+T T2 -3l1/3 l where T =... (T 3+ T 2 T + T T2+ T3), -(T +T). m l.59 o oo oo Z coo Substitution and rearrangement of the differential equation gives 93

94 dT K dt + T = To (A-2. 3) where K = a/(T + b) = effective time constant of the system. This m has the solution T T T = Z= 1 - exp t/K (A-2.4) To -T Therefore plots of log (1 - Z) vs. t should give a straight line of slope -1/K. If there is an initial lag, then this must be subtracted from the time. Alternately the point where 63. 2% of the change is complete also can be used as a time constant. Actually all three methods were used in this work. By carrying out several step function experiments over the temperature range, K can be evaluated as a function of T. In the furnace described, different results were m obtained from heating and cooling experiments. For heating K = 667/[(T1000)3 + 3.60] (A-2. 5) and for cooling K = 667/[ (T/1000)3 + 2. 94 ] (A-2. 6) When steady state operating conditions were established for each setting of the autotransformer, then the furnace temperature was found as a function of autotransformer setting. This relationship is T 20 + 0.8 Y/ (OC) (A-2.7) co where Y = autotransformer position, %o of full scale. These results now can be used to determine the operating conditions which are required for the DTA runs. Operating Conditions and Procedures In DTA measurements it is usually desirable to maintain a uniform heating or cooling rate, m. That is

95 T -T p -i T h] Tb~( dt) (t-tb) = Tb t (A-2. 8) I b dt b b K b where dT/dt = m = constant, C/min tb = time at beginning of DTA run, min T = temperature at beginning of DTA run, C. b The upper sign (+) is chosen for heating and the lower value (-) for cooling. When the expressions for T and K are combined, then the final expression is Thi = Tb m(t t (A-2. 9) -C- r 20+0.8Y m 5~3~667 3.60 /566 + T +0.8Y5/3 F] L2.94 2000 This equation is unique for the particular furnace and sample arrangement, and must be solved by trial and error. For heating and cooling rates of 2.5 C/min, the solutions are plotted in Figure A-II-l along with the correlation for T as a function of Y. co It is apparent then that for heating, the original setting of the autotransformer must be almost at midscale, but the rate of motion of the autotransformer (dY/dt) is less than that required to maintain Tc. The slope of the Y vs. t curve then defines the rate of motion for the autotransformer, which must be established by the Pulse selector circuit. Discussion The time constant and equilibrium temperature correlations are functions of the furnace and sample designs, and must be determined experimentally for any particular system. Although similar sample geometries should be used for any consistent series of runs, there is

LLJ 100 0 90-^0: 80 LL 0 70 0' 60 fop~~~~~~~o z750 0 I, >^ ^/^. 40 Cf) o CL 30 > 20 0 0 0 CL 0 100 200300 400 500 600 700 800 900 1000 1100 1200 o^~ ~SAMPLE TEMPERATURET, (0C) < 0 32 72 112 152 192 232 272 312 352 392 432 472 TIME DURING HEATING CYCLE,t, (MIN.) FOR m=2.50/MIN. Figure A-II-i. Plot of Equation A-2. 9 Showing Relationships between Furnace Temperature and and Autotransformer Setting for Heating and Cooling at Rates of 2. 5 C/min and Steady State Conditions.

97 no need to retain a particular sample holder if it is found to be unsatisfactory for any reason. A new calibration can be obtained, and in general will be found not to vary significantly for a particular furnace. Although the apparatus requires careful attention at each change point on the y-scale, it represents a good compromise between reliability, accuracy, cost and ease of operation. No temperature controller or precision program circuit is required, and the only expensive items in the system are the recorder, the DC amplifier, the motor-driven autotransformer, and the precision potentiometer. The rest of the system, being build from small, standard components, is relatively inexpensive.

98 APPENDIX III SAMPLE PREPARATION Materials The materials used in this work originated from three sources. Samples of the III-V compounds, GaAs, GaSb, InAs, and InSb were donated by Texas Instruments, Inc. These samples were either portions of Czochralski grown single crystals, or portions of zone refined ingots. One sample of GaSb was received in the form of a "button" formed by direct fusion from the elements. Specimens of a number of compounds, including Ag2Se, Ag2Te, PbTe, PbSe, and SnTe, were donated by Dr. Alan J. Strauss of Lincoln Laboratories. Exact sample histories for the preparation of the samples mentioned above are largely unavailable. A qualitative measure of their purity, however, may be ascertained from their respective melting points and their thermal behavior (sharp versus gradual) during melting. The remainder of the samples were prepared in our laboratory by direct fusion from the elements. Table A-III-1 contains a summary of all of the elements, whether used as standards or used in the preparation of compounds, together with their respective sources and purities. And Table A-III-2 contains a list of all the DTA samples used for the measurement of heats of transformation. Specimens of Bi2Te3, Bi2Se3, CsSe, CdTe, InZSe3, InTe, In2Te3, PbSe, PbTe, SnTe, ZnTe, and Ag2In8Se13 were prepared from the elements in our laboratory. First, the elements were wherever possible etched with nitric acid or aqua regia in order to remove any surface oxide, and then weighed out to ~0. 25 mg (total weight of sample = 25 to 70 g) by means of an analytical balance using standardized weights. In each case the stoichiometric composition was assumed. Thus in the following, Ag2Te Te gZIn Te In 0Te3001 etc. That is, the compounds "' Hodgkinson (32) has pointed out, however, that the maximum melting point does not in general correspond to the stoichiometric composition. This behavior is borne out experimentally. Thus, InTe, In Te3, Ag2Se, BiTe, Bi Se, Sb Te and SnTe are really In Te In Te (2) 60_ e2 2 (89), Bi4 Te 4,2 50,8' 40. 0b 7' e 3g4 - 40. 06.935' 40.02 e59.98' 4040 59: 60 6a 49. 06 50. 94 -

TABLE'A- III-1 ELEMENTS USJD AS STANDARDS IN THE PREPARATION OF COMPOUNDS Element Source Purity Ag Cominco Products, Inc. 5N, deoxidized Bi American Smelting and Refining Co. 5N Cd American Smelting and Refining Co. 4N Cu American Smelting and Refining Co. 5N Ge Zone-levelled single crystal 2-4 ohm-cm. In Indium Corp. of America 5N Pb Cominco Products, Inc. High Purity Sb Johnson, Matthey and Co., Ltd. High Purity Se American Smelting and Refining Co. 5N Sb Johnson, Matthey and Co., Ltd. High Purity Sn Johnson, Matthey and Co., Ltd. High Purity Te American Smelting and Refining Co. 5N Zn American Smelting and Refining Co. 4N 5N = 99. 999%o 4N = 99. 990 99

100 prepared here did not necessarily correspond precisely to the composition * at the maximum melting point. An exception was AgZIn8Se13, which was purified by zone refining. The elements were then placed in specially cleaned, clear, fused silica tubes, subsequently evacuated to below 0.1 micron and sealed. The encapsuled sample was next reacted by controlled heating to well above the melting point or liquidus temperature. For most samples, the heating and cooling cycles were accomplished in a digitally programmed resistance furnace. The latter equipment has been described by Hozak, Cook, and Mason (33). For cadmium and zinc telluride, however, it was found more convenient to use a much more rapid reaction cycle than could be achieved with the resistance furnace. For these compounds, the encapsuled elements were placed in a graphite susceptor and heated by radio frequency induction to a temperature about 50-100 C above the melting point of the compound. The entire heating cycle could be consummated within 3-5 minutes, thus suppressing any reaction with or wetting of the silica sample tube. The compound Ag2In8Sel3, the existence of which has been reported by O'Kane ( 63), was prepared as follows. 70 grams of Ag, In, and Se, in the proper stoichiometric ratios, were placed in a 10 mm I. D. silica tube. The Ag was placed in the bottom of the tube and the In and Se were cut into small pieces, mixed together and placed on top of the Ag. After seal-off the overall sample length was about 30 cm. The sample was reacted by carefully heating it in a rotating, rocking, resistance furnace. Extreme caution was taken as the temperature approached 245 C, since the In and Se react violently with one another at this point. Following the exothermic reaction at 245 C, heating was continued until the sample was completely molten (MP. = 814 C). The molten sample was then transferred to a preheated zone * In this case the composition Ag2n8Se is approximate, since the exact influence of the zone refining process on the composit ion is as yet undetermined. This fact was confirmed by DTA experiments on a sample of 21n + 3Se. Secondary reaction peaks were observed at 610 and 835 C.

101 refiner and subjected to three or more passes, using a zone length of about 3.. 8 cm and a zone travel rate of about 1. 9 cm/hr. Upon completion of zone refining, portions of high resistivity (> 10 ohm-cm) material were removed for further processing. Preparation of Alloys Alloys of Cd-Te and Zn-Te for the phase diagram determinations were prepared in much the same manner as the intermetallic compounds. In the case of Cd-Te alloys, the elements in the desired ratio were weighed into fused silica tubes and sealed under vacuum as described above. They were reacted by heating to temperatures well above the liquidus in a rotating, rocking resistance furnace, and were rapidly cooled or air-quenched to room temperature to obtain a homogeneous sample. In many instances the samples wet the fused silica tube and induced cracks and fractures in the containers during cooling. Only samples which remained bright, shiny and unoxidized after fusion were processed further. In the case of the Zn-Te alloys, the R-F induction method described above was found to be much more satisfactory. Reaction with and wetting of the silica tube were completely suppressed. Although water-quenching of some samples proved satisfactory, a few materials, notably 30-50 a/o Te in Zn, were found to explode when rapidly plunged into cold water. As a result, most of the samples were rapidly cooled in air by turning off the power to the R-F unit. Preparation of DTA Samples Samples for the heat of transformation determinations were processed as described in Appendix I. For the phase diagram studies, the alloys were completely crushed and 15 grams of material were sealed under vacuum below 0.1 micron, in clear, specially cleaned, fused silica tubes which were 10 mm in I. D. and each of which contained a concentric thermocouple well about 2. 5 cm deep. Exceptions were samples containing

102 more than 80 a/o Cd or Zn, in which case crushing was impossible. These materials were recovered in the form of a solid ingot which was rebottled in its entirety and rested on top of the thermocouple well until melting was initiated.

TABLE A-III-2 LIST OF SAMPLES USED IN THE HEAT OF TRANSFORMATION STUDIES DTA Mass DTA Run Melting Characteristic and Sample No. Composition Grams Number(s) Source/Purity Melting Point Observed 6 In 21.93 B-l; B-3 In Corp; 5N sharp; 1570C 9 Cu 26.88 B-1 AS and R; 5N sharp; 10820C 12 Ge 15.96 B-2 2-4 ohm-cm fairly sharp; 9380C 13 Te 18.72 B-3 AS and R; 5N sharp; 450 C 14 CdTe 18,60 B-4 SL-47,9 gradual; 10920C 16 InSb 17.37 B-5 TI; "P" ends sharp; 5260C 17 InAs 17. 10 B-5 TI; "N"; 0.02 sharp; 9430C ohm-cm 18 GaAs 15.93 B-6 TI; zone-refined fairly sharp; 12350C 19 CdSe 17.43 B-7 SL-705+SL-717 gradual initial melting; 1251 C 20 ZnTe 16.62 B-6 SL-463 gradual; 12890C 21 GaSb 16.86 B-7 TI; "button" somewhat gradual; 712~C 22 Pb 34.02 B-10 Baker; Anal. Reag. sharp; 3270C 23 Sb 19.04 B-8 Matthey sharp; 6310C 24 Ge 15.96 B-8 2-4 ohm-cm fairly sharp; 938~C 25 Te 18.72 B-ll AS and R; 5N sharp; 450 C 26 Cu 26.88 B-14 AS and R; 5N sharp; 10820C

TABLE A-III-2 (Cont.) DTA Mass DTA Run Melting Characteristic and Sample No. Composition Grams Number(s) Source/Purity Melting Point Observed 27 CdTe 18.60 B-9 SL-479+SL-482 gradual; 10920C 29 Sb 19.04 B-ll Matthey sharp; 631 C 30 In 21.93 B-9; B-10 InCorp; 5N sharp; 1570C 32 Ge 15.96 B-12 2-4 ohm-cm fairly sharp; 9370C 33 CdTe 18.60 B-12 SL-479; SL-482 gradual; 10900C 35 ZnTe 16.62 B-20 SL-1099 gradual; 12890C 37 Sb 19.04 B-15 Matthey sharp; 6310C 38 InAs 17.10 B-16 TI sharp; 9430C 39 GaAs 15.93 B-17 TI fairly sharp; 12350C 40 Ag 31.53 B-14 Cominco; deoxi- very sharp; 958 C dized 41 In 21.93 B-19 In.Corp; 5N sharp; 1580C 42 InSb 17.37 B-15 TI; "P" ends sharp; 5240C 43 Bi 29.40 B-16 AS and R; 5N sharp; 2750C 45 Cu 26.88 B-18 AS and R; 5N sharp; 10820C 46 Ag 31.50 B-18 Cominco; deoxi- very sharp; 9600C dized 47 CdTe 18.60 B-22 SL gradual; 10920C 48 Te 18.72 B-19 AS and R; 5N sharp; 4490C 49 CdSe 17.43 B-24 SL gradual initial melting; 12490C

TABLE A-III-2 (Cont.) DTA Mass DTA Run Melting Characteristic and Sample No. Composition Grams Number(s) Source/Purity Melting Point Observed 52 GaSb 16.86 B-26 TI; "button" somewhat gradual; 715 C 53 Pb 34.02 B-23 Cominco; 5N sharp; 3280C 54 Bi Te 22. 10 B-27 SL; zone refined fairly sharp; 587 C 2 3 55 Bi Se 20. 46 B-27 SL-489 somewhat gradual; 2 3 7020C 56 Sb Te 19.80 B-28 SL-491 somewhat gradual; o 3 617O0C 57 Sb Se 19.20 B-29 SL-490 somewhat gradual; 2 3 6120C 58 In Te3 17.37 B-25 SL-488 gradual; 668 C 59 In Se 17.70 B-25 SL-492 slightly gradual; 885 C 61 Cu 26.88 B-24 AS and R; 5N sharp; 10820C 62 Ag2Se 24.00 B-29 A.J.S. sharp; 894 C 63 Ag2Te 25.50 B-28 A.J.S. sharp; 9590C 64 PbSe 24.30 B-30 A.J.S. fairly sharp; 10850C 65 PbTe 24.48 B-30 A.J.S. slightly gradual; 9260C 68 InSb 17.37 B-33 TI; "P" ends sharp; 527~C 69 InAs 17.. 10 B-33 TI; "N"; 0.02 sharp; 9440C ohm-cm

TABLE A-III-2 (Cont.) DTA Mass DTA Run Melting Characteristic and Sample No. Composition Grams Number(s) Source/Purity Melting Point Observed 73 ZnTe 16.62 B-35 SL-463;SL-1099 gradual; 12860C 90 Bi 29.70 C-l AS and R; 5N sharp; 271 C 91 Sb 19.50 C-2 Matthey sharp; 9310C 92 Te 18.60 C-I AS and R; 5N sharp; 4490C 93 Ag 28.80 C-4 Cominco; deoxi- very sharp; 9600C dized 94 Pb 32.40 C-3 Cominco; 5N sharp; 3260C 96 Ge 16. 20 C-2 Intrinsic fairly sharp; 9370C 97 Ag 28.80 C-5 Cominco; deoxi- very sharp; 9610C dized 99 In 21.30 C-12 InCorp; 5N sharp; 1570C 100 InSb 18.30 C-6 TI sharp; 5240C 101 InAs 17.70 C-7 TI fairly sharp; 9420C 102 GaSb 17. 40 C-6 TI; single crystal slightly gradual; 0 712 C 103 SnTe 19.50 C-7 SL-512 gradual; 8040C 104 CdTe 18.60 C-10 SL gradual; 10910C 105 PbTe 24. 60 C-8 A. J. S. very gradual; 922 C 106 PbSe 24.30 C-8 A.J.S. fairly sharp; 10830C 107 GaAs 16.50 C-12 TI sharp; 12360C

TABLE A-III-2 (Cont.) Mass DTA Run Melting Characteristic and Sample No. Composition Grams Number(s) Source/Purity Melting Point Observed 110 BiTe3 23. 10 C-9 SL-zone refined fairly sharp; 586 C 2 3 111 Bi Se, 20.40 C-9 SL-489 somewhat gradual; 700~C 112 Ag2Te 25.50 C-10 A.J.S. sharp; 9580C 113 Ag2Se 24.00 C-11 A.J.S. sharp; 8940C 115 Ag 28.80 C-13 Cominco, deoxi- gradual; 957 C; dized oxide present 116 Sb Te 19.80 C-14 SL-515 somewhat gradual; 2 3 6150C 117 Sb2Se 19.20 C-15 SL-516 somewhat gradual; 613 C 118 In2Te3 17.40 C-21 SL-522 gradual; 667 C 119 In Se 17. 10 C-14 SL-492+SL-504 slightly gradual; 884 C 120 InTe 18.90 C-20 SL-529 gradual; 6930C 121 CdTe 18.60 C-15 SL-520 gradual; 1091 C 123 CdSe 17. 50 C-16 SL-514 gradual initial melting; 1251 C 124 Ag 28.80 C-29 Cominco; deoxi- sniarp; 9600C dized 126 InSb 18.30 C-18 TI sharp; 5230C

TABLE A-III-2 (Cont.) DTA Mass DTA Run Melting Characteristic and Sample No. Composition Grams Number(s) Source/Purity Melting Point Observed 127 InAs 17.70 C-18 TI fairly sharp; 9410C 128 GaSb 17.40 C-19 Tl slightly gradual; 711 C 129 GaAs 16. 50 C-19 TI fairly sharp; 12360C 130 Bi Te 23. 10 C-22 SL-528 fairly sharp; 5860C 2 3 131 Bi Se 20.35 C-22 SL-523 somewhat gradual; 700 C 2 3 - 132 PbTe 24.60 C-23 SL-526 gradual; 9230C 133 PbSe 24.30 C-23 SL-527 fairly sharp; 1082 C 135 Ag2Te 25.50 C-28 A.J.S. sharp; 9580C 2 136 Ag2Se 24.00 C-24 A.j.S. sharp; 8930C 137 Sb Te 19.80 C-17 SL-515 somewhat gradual; 2 3 6l70C 138 Sb Se 19.20 C-24 SL-521 somewhat gradual; 613 0 2 3 139 In Te 17.40 C-26 SL-522 gradual; 6670C 2 3 141 SnTe 19.50 C-16 A.J.S. gradual; 8040C 142 InTe 18.90 C-17 SL-529 gradual; 693 C 145 CdSe 17. 50 C-25 SL-498 gradual initial melting; 1248 C 146 ZnTe 16.60 C-28 SL-531 gradual; 1290 C

TABLE A-III-2 (Cont.) DTA Mass DTA Run Melting Characteristic and Sample No. Composition Grams Number(s) Source/Purity Melting Point Observed 148 Ag In Se 18.50 C-27 SL-281; zone slightly gradual; 8110C refined; midc4e section;'10 ohm-cm 149 Ag In Se 18.50 C-29 SL-741; zone slightly gradual; 813C refined; midci-e section; >10 ohm-cm o 150 In 21.30 C-30 InCorp; 5N sharp; 1570C 151 Te 18.60 C-29 AS and R; 5N sharp; 4490C Code 5N = 99. 999% 4N- 99. 99+4% SL = Samples prepared by the Semiconductor Materials Laboratory, University of Michigan A..S. = Samples donated by Dr. Alan J. Strauss, Lincoln Laboratories TI = Samples donated by Texas Instruments, Inc. AS and R = Materials purchased from American Smelting and Refining Co. Matthey = Materials purchased from Johnson, Matthey and Company, Ltd. Cominco = Materials purchased from Cominco Products, Inc. In Corp = Materials purchased from the Indium Corporation of America

APPENDIX IV PHYSICAL PROPERTIES AND THEIR ESTIMATION In order to perform heat transfer calculations, values are needed for the heat capacities, thermal conductivities and so forth of the various components of the system. Furthermore, values are needed for the latent heats of fusion of the "standards" so as to have a basis for comparison of the results. For many of the components these properties are well known, but for most of the intermetallic compounds they have to be estimated. In this section, the values used will be summarized and the references given. Methods of estimation will also be discussed Heats of Fusion and Transformation The heats of fusion for all of the standards, except Ge, were taken from Kubaschewski and Evans (47). More recent values for Ge have been reported by Greiner and Breidt (28) as well as by de Roche (71). Among the intermetallic compounds, values are available only for Bi2Te3 (8), InSb (62, 73) and GaSb (73). Heats of transition have been reported for In2Se3 (94) and Ag Se (47). Two methods are available for the estimation of heats of fusion. The first is based on a correlation between the entropy of fusion and the melting temperature by Turkdogan and Pearson (84). The data scatter considerably so that the probable error is high, perhaps ~ 25%o or more, but the correlation does point up a trend of increasing entropy of fusion with increasing melting temperature. Certain classes of compounds, such as those of zincblende structure, however, do not fit into this correlation. The second method, which is due to Kubaschewski and Evans (47), consists in additively combining the entropies of fusion of the elements to obtain the entropy of fusion of the compound. For disordered alloys, this method provides an excellent estimate. For ordered alloys and compounds it has been found that a better estimate is obtained if a factor 110

111 C = -R[ N In N + N2 In N2 is added to the sum of entropies of fusion of the elements. Here N1 and N are the respective atom fractions of each element in a binary compound, R is the gas constant (1.987 cal/g atom K), and the units of c are cal/mean gram atom OK. If transitions occur below the melting point of the compound the above calculation would give an estimate of the sum of the entropies of transition and the entropy of fusion. Densities The densities are perhaps the best known of all the physical properties needed. The densities of nearly all of the materials of interest here are given in the standard handbook references (27, 30, 35, 48, 77). The density of a and P - In Te at room temperature was measured by Zaslavskii and Sergeyeva (95), and those of the III-V compounds at high temperature were reported by Joffe and Regel (37). Although the values at high temperature are desired, corresponding measurements at room temperature usually provide a sufficiently precise estimate. If the density is unknown, it can be estimated from the crystal structure. The density of silica is practically independent of temperature from 3000K to 1600 K and has a value of 2.20 grams per cubic centimeter. For the purge gases, the perfect gas law provides a good estimate: p(g/cm3) = 0.0122 M/T (OK) where M is the molecular weight of the gas and the pressure is taken to be one atmosphere. Heat Capacities The specific heat of silica glass (77) in cal/g K is given as follows:

112 C SO T T P, Sio = 2 0.223 + 0.0613 (1 0. 00575 (10) T = OK A summary of the specific heats of a number of elements and compounds at their respective melting points appears in Table A-IV-1. Kubaschewski and Evans (47) have pointed out that the specific heat of many compounds is around 7.25 cal/mean gram atom K at the melting point. The data reproduced here is in general agreement with this figure, so it was used to predict specific heat when they were unknown. The specific heats of some compounds at the transition temperature are reproduced in Table A-IV-2. These figures were used as a guide in estimating heat capacities of similar materials at the transition temperature. Thermal Conductivity The thermal conductivity of fused silica is a nearly linear function of temperature and may be represented by the following equation (27): k'i (cal/min cmK) = 0. 155 + 0. 190 (- ) T > 300 K SiO 1000 The thermal conductivities of nitrogen (22) and helium (22, 90) are well approximated by the relations: k (cal/min cm K) = 0.00060 + 0. 01085 (7 ) - 0. 00180 (100) N 1000 1000 2 T > 3000K oT0 T 2 k (cal/min cm K) = 0.01163 + 0. 03625 (0) - 0. 00278 ( 0)0 T > 300~K

113 Table A-IV-1 Specific Heats of Elements and Compounds at the Melting Point s 1 T c c f DP P Material 0C cal/m. g.a. - C Reference Ag 961 7.63 7.3 (77) Bi 271 7.43 7.29 (77) Cu 1083 7.31 7.5 (77) Ge 937 7.27 (47) In 157 6.88 7.10 (77) Pb 327 7.03 7.55 (77) Sb 631 7.14 7.5 (77) Te 450 8.40 9.0 (77) AgBr 430 9.35 7.45 (47) AgCl 455 7.50 8.00 (47) Bi S3 777 7.05 (47) 2 3 Bi Te3 586 7.58 (9) CdS( (1750) (7.35) (47) InSb 525 6.55 7.4 (62, 72) Sb2S3 547 6.98 (47) SnSI 881 7.05 (77) ZnS (1650) 7.26 (47)

114 Table A-IV-2 Specific Heats of Compounds at the Transition Point T c Cc T c cY -mp P p P p P Material 0C cal/m. g.aC C cal/m.g.a. C Reference Ag2S 179 7.36 7.21 (47) Ag2Se 133 7.23 6.80 (47) Ag2Te 147 7.70 7.36 799 (47) Cu2S 103 6.50 7.75 350 7.75 6.77 (47) Cu2Se 110 7.07 6.73 (47) In Se3 200' 9.35 9.35 725 (94) 23

115 The thermal conductivities of the standards near their melting points are well known and are summarized in Table A-IV-3o Those of the intermetallic compounds, on the other hand, are in general unknown. Moreover, due to the large number of processes which influence the thermal conductivity at high temperatures, their prediction is extremely difficult without detailed information about the electrical properties of each sample. The latter information was not available for our samples. As a result, a reasonable estimate of the thermal conductivity at the melting point could be made only in the case of the III-V compounds for which experimental data were available. For the remainder of the compounds, values of the thermal conductivity were selected in a somewhat arbitrary fashion and represent orders of magnitude only. Some of the estimates used in our calculations are given in Table A-IV-3. Emissivities Even though no calculations in which the equivalent conductances for radiation appear were performed in this work, the values of the emissivities and other methods available for their estimation are of interest because knowledge of them will be useful in future work where it may be desired to perform such calculations. The emissivity of oxidized nickel (30) may be expressed as follows NiO = 0. 143 + 0.48 ( 00 An effective emissivity for fused silica was found by assuming that all wavelengths less than 3. 7 microns are totally transmitted through the quartz while all wavelengths greater than 3. 7 microns are completely absorbed, and that this distribution is independent of temperature. In truth the absorption spectrum is a complicated function of both wavelength and temperature. See, for example, ref. (27). It is a fact, however, that some radiation will penetrate the tube and impinge on the sample. As a zeroth approximation this factor may thus be taken into consideration.

116 Table A-IV-3 Thermal Conductivities of the Solid Near the Melting Point Material or Value Used Temperature Method of Property cal/sec-cm- K 0C Estimation References In 0. 053 156 A (77) Pb 0.070 327 E, A (77) Bi 0.018 270 E, A (77) Sb 0.052 630 E (77) Ag 0.65 960 E (27) Ge 0. 042 937 E (1, 5, 36, 75) Te 0.009 450 E (2, 20) InAs 0. 020 942 E (13, 14) InSb 0. 018 525 E (13, 15, 40, 81) GaSb 0.013 706 E (86) GaAs 0.018 1237 Code: E -- Extrapolated from experimental data. A - From the temperature coefficient of electrical resistance, a. kT/k00 = T/300 (1 + a(T- 300)). L — From the extrapolation of the lattice component of thermal conductivity to the melting point and the Wiedemann-Franz law: k= k + Lr. ph

117 Secondly, it is assumed that the reflectivity of the quartz is zero. Kirchoff's law then takes the form E - =1 - T The transmissivity as a function of temperature may then be found from the first assumption and the black-body radiation distribution function. The following empirical expression was obtained for the emissivity SiO exp 0 100 39 T > 420 K Emissivities of the various intermetallic compounds are largely unknown. The reflectivity of GaAs has been measured (4) but data on the other compounds is nonexistent. Where sample emissivity data were not available, they were estimated, from the index of refraction according to the formula: 2 E 4 4n/(n+ 1) Use of this formula implies that the transmittance is effectively zero and that the extinction coefficient is small compared to n. It may be noted that this equation appears to work quite well for Ge and GaAs at room temperature, so that it might at least be expected to provide a reasonable estimate. Some values of the emissivity for standards as well as compounds are summarized in Table A-IV-4. In reality the reflectivity of fused silica varies from 17%o at room temperature to 3%o at 800~C (68).

118 Table A-IV-4 Emissivities of Elements and Compounds Material or Method of Property Value Used Estimation Reference In 0.10 A Pb 0. 10 E (77) Ag 0.03 E (77) Bi 0.08 E (77) Te 0.45 E (77) Sb 0.35 E (77) Ge 0.52 E (27) GaSb 0.66 N (16, 24, 34) InSb 0.64 N (16, 24, 34) GaAs 0.70 E, N (4,16, 24,34) InAs 0.70 N (24,34) Code: E - Experimental; N - From index of refraction A - Arbitrarily assumed.

119 APPENDIX V CONDUCTION OF HEAT IN THE SYSTEM DURING MELTING In this Appendix, the complete temperature problem encountered during melting of a DTA sample [ Regime (B)] is considered and the solution established for the case of semi-infinite planar bounding surfaces. A continuous solution of the following boundary value problem is to be found. The differential equation governing the temperature in regions (2) and (3) is: 6._ T 6 T,56 k i; = p- (- \A-5. 1 6T 3x Pxp 6t X3 < x< X1 where k, p, and c, are functions of x only and are defined in the region X3 < x < X such that k3 for X3 < x< X2 k (x) = k2 for X2 < x< X1 P3 for X3 < x< X P 3x) = 2 p(x) = P2 for X < x< X1 cp 3 for X3 < x< X2 c (x) = p c for X < x < X p,2 2 1 where k3, kz, p3, p2, etc. are constants, so that k, p, and c have a discontinuity of the first kind at x = X2. The author is deeply indebted to Mr. A. N. Currim for his assistance in solving this problem. Cf. Figure 2. 1, where x is written in place of r and X. in place of R, since in this problem the bounding surfaces are planar.

120 The initial condition and the boundary conditions are: As t - 0+, T(x,t) 0; X < x< X1 A-5. 2 As x -> X+, T(x,t) - 0; t > 0 A-5. 3 As x - X1-, T(x,t) - yt; t > 0 A-5.4 The temperature is continuous at x = X 2 lim T(x,t) = lim T(x,t); t> 0 (A-5. 51 x - X- x X2+ 2 2 The flux is continuous across x = X2 6 T 6T lim kim k A-5. 6 3 5x 2 6x x X2- x X2+ The differential equation A-5. 1 takes the following form in region (3) and region (2), respectively: 56T 1 6T 2 2 6t 3 2 A-5.7 X3 < x< X2 A-5.7 6 x h3 ~2 62T 1 6T 6 —-- 6 T X < x K X1 A-5. 8 2 2 6t 2 6x h2 where k, h = i= 2,3 A-5.9 1 Pi Pic is the thermal diffusivity.

1 21 x - X Let q = X3 A-5. 10a X-X and 3 X2 X3 d = A-5.1 Ob Substituting Equations A-5. 10 and A-5. 11 into Equations A-5. 7, A-5. 8, and A-5. 2 through A-5. 6 transforms the inhomogeneous boundary problem for T (x, t) into the following homogeneous boundary value problem for v (q, t): l5 v H2 vq 0< A-5. 12 H 6 q 0 < q< d Aq2 3 6t H 3 q v 2 6v 2 = 2 + y HH q; d< q< 1 A-5.13 2 2 6t 2 5q As t 0+, v (q, t) - 0; for 0 < q< 1 A-5. 14 As q - 0+, v(q,t) - 0; for t>0 A:-5. 15 As q - 1-, v(q,t) -+ 0; for t > 0 A-5.16 lim v(q,t) = lim v(q,t) A-5. 17 q - d- q -* d+ lim k 6v + (3-k2) yt = li A-5. 18 q -- d- q -3 d+ where 2 2 (x1 - x3) H =, i = 2,3 A-5. 19 h. 1

122 The solution of the boundary value problem given by Equations A-5. 12 and A-5. 18 is obtained by using the Laplace transform. Let Lt v(q, t) e st(q,t)dt = V(q,s) A-5.20 0 Then Equations A-5. 12 to A-5. 18 become: 2 yH 2 6V 2 3 q 6V = H V + 0< q< d A-5. 21 2 3 s q 52V 2 _2 = H s sV+ 2 d < q< 1 A-5.22 q2 2 s 6q As q - 0+, V(q,s) - 0 A-5.23 As q - 1-, V(q,s) - 0 A-5. 24 lim V (q, s) = lim V(q, s) A-5.25 q - d- q - d+ 6 u (k3- V / 2v lim k + li A-5.26 3 56q )2 2 6q q ~ d- q - d+ Equations A-5. 21 and A-5. 22 are elementary and have the respective general solutions: V(q, s) = - -_ + A(s) sinh (H q /s) + B(s) cosh(H3q is), S 0 < q< d A-5.27 V(q, s) = - 2 + C(s) sinh(H2q fs) + D(s) cosh(H2q'Ts) s A-5. 28 Using Equations A-5. 23 to A-5. 26 in A-5. 27 and A-5. 28 there results a set of four simultaneous equations for A(s), B(s), C(s) and

123 D(s). Solving these for A(s), B(s), C(s) and D(s), substituting these results in to Equations A-5. 27 and A-5. 28 and simplifying, the following expressions for the Laplace transform, V(q, s), of v(q, t) result: V(q,s) = - 2 2 sinh (H q rs) s [ cosh (H3d is) sinh (H2 (l-d) is) + c sinh (H3d i-s) cosh (H2(1-d) Js)] valid in 0 < q< d A-5.29 V(q,s) = - -q s [ cosh (H3d fs) sinh (H2(q-d) s) + c sinh (H3d Fs) cosh (Hz(q-d) 7Fsi] s [ cosh (H d-s) sinh (H (l-d) ) + c sinh (H3d ) cosh (H2(1-d) )s)] 3 2 3 2 valid in d < q < 1 A-5. 30 where k H k2 H2 c = A-5.31 k"H k3 3 These transform functions may be inverted and will give the solution for the boundary value problem. The inverse transform is given by the inversion integral: M+i3 v (q, t) = lim V(q,s) e ds A-5.32 M-ip Cf. Churchill (18) p. 176 ff. In particular all the conditions of theorem 5, p. 178, are satisfied.

124 which satisfies the condition v(q,t) = 0 for t < 0. Here M is a real number so large that all the singularities of the complex function V(q, s) of the complex variable s lie in the left half plane Re(s) < M. By expanding the hyperbolic functions, appearing in V(q, s), in Maclaurin series in the complex s plane, it is easy to see that V(q, s) has no branch cuts in the s-plane. Hence we may evaluate the integral A-5.32 by closing the contour in the left half plane Re(s) < M, as in Figure A-V-1. The curve BB'CA'A is an arc of a parabola with focus at the origin and so chosen that it passes through no singularities of V(q, s). By Cauchy's theorem, it then follows that 1 st 21 i P V(q, s) et ds = sum of the residues of V(q, s) inside,/7^~~~~~ ~~~A-5. 33 where 6 is the contour ABCB'A'A of Figure A-V-1. By arguments similar to those in Churchill (18) p. 204 ff, it is easily seen that the integral st of V(q, s) e over the arc of the parabola BB'CA'A vanishes as 3-oo. Hence it follows that the inverse transform v(q, t) of V(q, s) is given by: v (q, t) = sum of the residues of V(q, s)et in the left half plane Re(s) < M A-5. 34 The function V(q, s) defined by Equations A-5. 27 and A-5. 28 has a double pole at s = 0 and it also has singularities at the roots of the transcendental equation: cosh H3d -s sinh H2(1-d) s + c sinh H3d)s cosh H2(1 -d) i = 0 A-5. 35 which may be written as sin Hz(1-d)p cos H3d3 + c cos Hz(1-d)p sin H3dp = 0 A-5. 36 2 3) 20^

125 I m(s) M+i/ I ARe(s M-i R Figure A-V-1. Contour in the Complex Plane for Inversion of the Laplace Transforms, Equations A-5. 29 and A-5. 30.

126 with p = i A-5.37 It has been shown that all the roots, n of Equation A-5. 36 are real and simple, so that it follows that V(q, s)e has only simple poles at the non-zero roots of A-5. 35. The residue po(t) at the double pole s = 0 is: p (t) = A + A2t A-5 38 where Hq3 q H3 q A1 = yc 6HZ2(1 -d) + c3d73 JH2(1-d) cH3d2 22 3 3 H d 2HZ(1-d) Hz(1-d) 3 3 c3HZ d(l-d)Z'2 + 2+ f H3 d 3+ 3 2 6 6 3 2 valid for 0 < q< d A-5.39 H2 H2 d (q d)3 + 3 +L 2 (qd) 2 H~cH(1-d) + cH3d H (q-d) -c H d H2 H d (l-d) H2 (-d) H {H2(1-d) + c H3d 6 cH H d(l-d) + ~I~ valid in d < q< 1 A-5. 41 Cf. Carslaw and Jaeger (17), p. 324 ff.

127 H2(q-d) - c H d A2 Y H (-d)+cH d - q valid in d < q < 1 ~~^2 3 A-5. 42 The non-vanishing roots of Equation A-5. 35 are designated, n =, 2, 3,... and order so that 1 < 32 <.... <... Putting 1P = i Vs- A-5.43a n n one sees that the non-vanishing roots of Equation A-5. 35 in the complex s-plane are all negative: s = - A-5. 43b n n so that we need only take pn to be the positive roots of A-5. 36. The st n residue p (t) of V(q, s) e at P is: n * n B (q) exp(-p t) P (t) = -2y 3 A-5.44 nP n where B (q) = csinH3P q 0<q<d n 3 n and Bn(q) = cos (H3dp )sin(Hz(q-d)n ) + c sin(H3dp )cos(H2(q-d)3 ) for d < q < 1 A-5.45 and C = H2(i-d) + cH3d cos (H3d3 )cos(H2(1-d)p) n 2 3 3 n n + -{cH2(1-d) + H3dj sin(H3d0 ) sin(H(1-d)P ) 0< q< 1 A-5.46

128 So it follows that o B (q) 2 v(q,t) = A +Azt-2y 3~ exp (-n t) n=l C n n 0< q< 1 A-5.47 where p are the positive roots of Equation A-5. 36, and A, A, B (q) n 1' n and C are defined above. It is now easily verified that Equation n A-5. 47 satisfies all the boundary conditions A-5. 15 to A-5. 18 and the differential equations A-5. 12 and A-5. 13 of the boundary value problem for v(q,t). It has already been shown in Equation A-5. 32 that v(q, t) satisfies the initial condition A-5. 14. Hence Equation A-5. 47 is completely established as the solution of the boundary value problem defined by the expressions A-5. 12 to A-5. 18. It then follows, via Equation A-5. 11, that the solution of the boundary value problem defined by the expressions A-5. 2 to A-5. 8 is given by: T(x,t) = v(q,t) + yqt A-5.48 where q, d are defined by Equation A-5. 10. It is of interest to know the behavior of Equation A-5. 48 as H - 0. This represents the situation of the gas film corresponding to 2 a negligibly small heat sink. If one takes the limit of Equation A-5. 47 as H2 0, one finds after a slightly tedious (but straightforward) calculation:

129 2 3 yqt yH q lim T (q, t) - k + + H2 O k (1 —d) 6d + - (-d) 22d + L1 — yH d q yk3 H3 d (l-d) k k - 2 + 6 d + k3 (-d) 3k [d + (1-d 2 k n n 1 t n t2 2d2 sinLd exp - H3d t -2yH3d Fo d n=l 3 3 (-d) k3 (-d) an { - cos a + cos a + + a sin a n 2 d n n k2d n valid in 0 < q < d A-5.49 where a = H3dP. n 3 n and also imT (qt) = k3 H d (q-I) t d + (q-d lim T (q, t).. 3 [3 2 H2 k 2 k O 2 k k 3k d +(l-d) + k- (1d) k 2. 0 1 Isin a + 3 a cos v exp -n t 2 2 y d2 L H. d 3 3 k k n=l aH n3 (I-d) 3 (1-d) n ~k d cos a + cos a a sin a k d n n k2 d n n valid in d < q< 1 A-5.50

130 where a are the positive roots of n 2k d a ot a + (l-d) A-5.51 The parameters and variables C', ", r', ",', T' of Chapter II are related to the variables above as follows: q(X - X) i( 1_) = 1 3 A-5. 52 X3 r' = X /X A-5. 53 23 q(X - X) - (X2 X1) (" - 1) = 3X A-5.54 x2 -q" = X/X2 A-5. 55, = k /k A-5, 56 3 4 T' = X /h A-5. 57 3 3 The region 0 < q < d corresponds to the region 1 < K' < 6 and the region d < q < 1 corresponds to the region 1 < " < n ". If one designates lim T(q, t) as y(,',t) in 0 < q < d and lim T(q, t) H -O H — O 2 2 as z( C",t) in d < q < 1, then Equations A-5. 49 and A-5. 50 become identically Equations 2. 39 and 2. 40. Thus Equations 2. 39 and 2. 40 result if the approximation can be made that 2 l - d 2 l- d sin H2 (l-d) = sin Ha d c H3 d A-5. 58

131 In our problem typical values of the parameters are X - X = 0. 175 cm, / X2 = 2~ X2- = 0.150 cm, H = 0. 127 sec and H 1.81 sec so that H2(1-d)/H3d = 0.0117. Therefore the approximation A-5. 58 is valid for a< 0. = 10 radians 0.01 Now the roots of Equation A-5. 51 are tabulated by Carslaw and Jaeger (17). In particular, for k3/k = 35.7 the roots of a cot a + 0.168 = 0 are a1 = 1.65 2 = 4.75 a3 = 7.87 a4 = 11.0 so that it may be concluded that the approximate solution given in Chapter II is sufficiently precise. Finally it is of interest to investigate the value for the time constant for damping out of the transient portion of the solution. It is clear from Equations A-5. 49 and A-5. 50 that this time constant may be written as follows: 2 2 2 T = H3 d /a1 A-5.59 whence 2 2 1.81 x 0.857 T = 857 0.89 sec. 1 1.652 For the cylindrical problem it is expected that this time constant would be given by T1 = T'/31 A-5.60 Cf. Chapter II, Equations 2. 30.

132 where 1 is the first positive root of J o (P) Y ('13) Y (P) J (') -'' In " Jo(p) Y1 (') - Yo (P)(N'3) = 0 A-5. 61 For the typical values of the parameters given on page 16, which correspond to the values used above, it may be found that the values of 3 computed from Equation A-5. 61 are: p13 = 5.34 6 = 15.74 13= 26.20 etc., whence 2 2 T1 = T'/p = 26.6/(5.34) = 0.94 sec. Thus the agreement between Equations A-5. 59 and A-5. 60 is excellent. It is thus concluded that as melting of the DTA sample is initiated the thermal conductance will increase from its value prior to melting to the steady state value within one or two seconds as shown schematically in Figure 2. 2.

APPENDIX VI COMPUTER PROGRAMS AND EXPERIMENTAL DATA In this appendix the computer programs used in evaluating the experimental data are described, and the numerical values of the necessary parameters are reported. Program Number DTA-19 for Direct Calculation of the Thermal Conductance This calculation consists in finding the gas film thickness by a trial and error comparison of the theoretical and experimental time constants for exponential decay of the DTA curve until agreement within a specified error (one percent) is achieved. The conductance and latent heat are then found in a straightforward manner via Equations 3. 2 through 3. 5 and Equation 1. 7 respectively. The variables used are defined below, and the MAD* listing of the program together with the data processed follow. MAD Variable Variable in Text Explanation AREA Io - Ie dt Area under DTA curve, Cmin ss P A Lower value of p in interval to be scanned B Upper value of 3 in interval to be scanned CYCLE Cycle of DTA run D3 2R Outside diameter of silica tube, cm D4 2R4 Outside diameter of sample, cm D5 2R Outside diameter of thermocouple well, cm * Michigan Algorithm Decoder, University of Michigan Computing Center, Ann Arbor, Michigan. 133

134 Variable Listing for Program DTA-1 9 (Cont'd) MAD Variable Variable in Text Explanation DELTA Increment function for changing l," DELX Increment for changing P in scanning the interval, A to B DP D Cf. Equation 2. 58 P EPS E Cf. Equation 2. 56 ERROR Percent disagreement between TAUX and TAUC ETAPP1 Initial estimate of ETAPP 7" R /R I R2 ETAP R/R ETA R3/R4 G3 G Thermal Conductance of the silica tube, cal/cm min OK G4 G Thermal Conductance of sample 4 GS2 G2 Thermal Conductance of gas film GS G* Overall Thermal Conductance I Subscript of DELTA JOX, etc. Jo(P), etc. Bessel function of first kind K3 k3 Thermal conductivity of silica tube, cal/cm min OK K4 k4 Thermal conductivity of sample KS2 k2* Thermal conductivity of gas film 2* LF Lf L Latent heat, cal/g L L Sample height, cm M m Sample mass, g NUMBER DTA sample number NU a) Cf. Equation 2. 48 PHIP' k/k PHI k4/k3

135 Variable Listing for Program DTA-19 (Cont'd) MAD Variable Variable in Text Explanation R1 Maximum value of R RHOCP3 (P )3 Specific heat of silica, cal/cc OK RHOCP4 (Pc )4 Specific heat of sample, cal/cc OK R Running index RUN DTA run number SAMPLE DTA sample composition T1 t5 Cf. Equation 2.7 TAUC Calculated value of ", min exp TAUX T exp TAUP' Cf. Equation 2.4 TAU T Cf. Equation 2. 3 TF T Absolute temperature, K x 10 U V E3 W Et X P Cf. Equation 2. 28 YOX, etc. Y (P), etc. Bessel function of the second kind 0

MAD LISTING FOR COMPUTER PROGRAM NUMBER DTA-19' $COMPILE MAD, EXECUTE, DUMP, PRINT OBJECT, PUNCH OBJECT DTA19 INTEGER N, J, I, K DIMENSION L(20), DELTA(10) READ DATA START READ FORMAT INPUT, NUMBER, SAMPLE, RUN, CYCLE, D3, D4, 1 D5, L, K4. RHOCP4? M, TF, TAUX, AREA PRINT COMMENT $1$ PRINT FORMAT LABEL, NUMBER, SAMPLE, RUN, CYCLE KS2 = 0.00060 + 0O01085*TF - 0.00180*TF*TF K3 = 0.155 + 0.19*TF RHOCP3 = 0.58 + 0.13*TF PRINT COMMENT $ODATA$ PRINT RESULTS D3, D4, D5, L, KS2, K3, K4, RHOCP3, 1 RHOCP4, M, TF ETA = D4/D5 ETAP = D3/D4 PHI = K4/K3 PHIP = K3/KS2 TAU = D5*D5*RHOCP4/(4.*K4) TAUP = D4*D4*RHOCP3/(4.*K3) EPS' (TAUP/TAU).PO.65 T1 = RHOCP3*(D5*D5 - 0.16)/(8.*K4) NU TAU/T1 PRINT RESULTS ETA, ETAP, PHI, PHIP, TAU, TAUP, T1, NU Cl = EPS/(ETA*ETAP*PHI*PHIP) C4 = 1./(ETAP*PHIP) I 1 R: 14 ERROR = 2,*EPS3

PROGRAM DTA-19 (CONT.) THROUGH ENDFOR ETAPP=ETAPP1,DELTA(I),.ABS.ERROR.LE.EPS3 WHENEVER ETAPP.GE. 1*200 PRINT COMMENT $OETAPP HAS BECOME GREATER THAN 1.200$ TRANSFER TO START END OF CONDITIONAL EXECUTE UITR2.(ADELX,BEPS1iEPS2,N X) OPEN U = X*ETA V = X*EPS W = X*EPS*ETAP C2 = (EPS*EPS*ELOG.(ETAPP))/(ETA*PHI) C3 = EPS*ELOG.(ETAPP) L(1)=BSL1. (X,1,0 OJOXK) L(2)=BSL1.(X1,1,J1XK) - DELl = X*JOX - NU*J1X L(3)=BSL1. (X,4,0,YOXK) L(4)=BSL1. ( X,4 1YIX ~K ) DEL2 = X*YOX - NU*Y1X L(5)=BSL1. ( U,0 JOU K) L(6) BSL 1 ( Us 1 JlUU K) L(7)=BSL1. (U,4,0YOUK) L(8) =BSL* (U,4,Y1 Y1U UK) L(9)=BSL1. (V,10.OJOVK) L( 10 )=BSL1 (V, 191JlVK) L(11)=BSL1 (V,4,0,YOV9K) L( 12 )= BSL1. (V,4,,YlVVK) L(13)=BSLl. (WW1O 0,JOWK) L(14)=BSL1.(W, l 1,J1WK) L(15)=BSLl (W,40,tYOWWK) L( 16 )BSL1. (W,4, 1,YIWKi

PROGRAM DTA-19 (CONT.) SUM = 06 THROUGH ALPHA, FOR = 1, 1 3 *G. 16 ALPHA SUM = SUM + L(J) WHENEVER SUM *G. 16.5 PRINT COMMENT $OLSUM IS GREATER THAN 16$ PRINT RESULTS L(1)*..L(16) TRANSFER TO START END OF CONDITIONAL DPI = DEL1*YOU - DEL2*JOU DP2 = DEL1*YIU - DEL2*J1U DP3 = J1V*YOW - YIVJOW DP4 = JIV*Y1W - YIV*JIW DP5 = JOV*Y1W - YOV*J1W DP6 = JOV*YOW - YOV*JOW DP = DP1*(C1*DP3 - X*C2*DP4) + DP2*(X*C3*DP5 - C4*DP6) II = UITR2A*(DP) WHENEVER II *L. 1.5t TRANSFER TO OPEN WHENEVER II *L. 2.5 PRINT COMMENT SOTHERE ARE NO ROOTS IN THIS INTERVAL$ PRINT RESULTS A, Bo II TRANSFER TO END END OF CONDITIONAL TAUC =TAU/(X4X) ERROR = (TAUX - TAUC)/TAUX WHENEVER R/5~.G. Rl PRINT COMMENT $OINTERMEDIATE VALUES$ PRINT FORMAT OUTPUT9 ETAPP, DELTA(I), DP, Xt TAUC, ERROR R 0. END OF CONDITIONAL

PROGRAM DTA-19 (CONT-. R = R + I. WHENEVER ERROR.GE. iO.00EPS4 I = 1 TRANSFER TO END END OF CONDITIONAL WHENEVER ERROR.GE* 10.*EPS4 I = 2 TRANSFER TO END END OF CONDITIONAL WHENEVER ERROR.GE. EPS4 I = 3 TRANSFER TO END END OF CONDITIONAL I = 4 WHENEVER ERROR.L. 0G. I = 5 END CONTINUE GS2 = 6.2832*KS2/ELOG.(ETAPP) G3 = 6.2832*K3/ELOG (E 1AP) G4 = 6.2832*K4/(1-(ELOG.(ETA)/(ETAP -.))) GS = 1./((1./GS2) + (1./G3) + (1./G4)) LF - GS*L*AREA/M PRINT COMMENT $ORESUL i-S$ PRINT RESULTS X, 11 PRINT RESULTS ETAPP DELFA(I) PRINT RESULTS GS, LF TRANSFER TO START VECTOR VALUES INPUT = $13,C6tC4,C6,10F6.3*$ VECTOR VALUES LABEL = $S5,I3,S5,C6,S5,C4,S5,C6*$

PROGRAM DTA-i9 (CONT.) VECTOR VALUES OUTPUT = $S10, 9H ETAPP F7.5,SiO0 1 9H DELTA = F7.5/S10, 6H DP = E10*5, S10, 5H X = F11.5/ 2 S10i 8H TAUC F8.5 510, 9H ERROR F F7*4*$ END OF PROGRAM $DA A K=-6 N=100 A=0.01, B=100 EPSr11.E-59 EPS2 1iE —6 EPS3=0*01, EPS4=0.029 DELX = 0*10, ETAPP1 - 1.010, Ri = 0.99 DELTA( I) = 0.04, 0*02, 0.004, 0*002, -0.OOl DATA CARD FORMAT FOR PROGRAM DTA-19 CODE VARIABLE COLUMNS ALLOTTED I NUMBER 1-3 II SAMPLE 4-9 III RUN 10-13 IV CYCLE 14-19 V D3 20-25 VI D4 26-31 VlI D5 32-37 VIlI L 38-43 IX K4 44-49 X RHOCP4 50-55 XI M 56-61 XII TF 62-67 XIii TAUX 68-73 XIV AREA 74-80

DAT A CARDS FOR MAD PROGRAM NO. DTA-19 STANDARDS AT THE MELTING POINT I II III IV V VI VII VIII IX X XI XII XIII XIV 93 AG C-4 1C-N2 1.244 0.968 0.593 6.0 39. 0.58 28.80 1.226 0.690 93.9 93 AG C-4 2H —-N2 1.244 0.968 0.593 6.0 39. 0.63 28.80 1.247 0.620 93.0 97 AG C-5 IC —N2 1.221 0.965 0.606 6.3 39. 0.58 28.80 1.229 0.679 96.6 97 AG C-S 2H —-N2 1.227 0.965 0.606 6.3 39. 0.63 28.80 1.2 3 7 0.631 93.5 115 AG C-132H-N2 1.303 0.992 0.605 5.9 39. 0*63 28.80 1.244 0.491 65.9 99 IN C-123H-N2 1.310 0.992 0.602 6.1 6.0 0.46 21.30 0.439 1.045 38.7 99 IN C-122C-N2 1.310 0.992 0.602 6.1 3.18 0.44 21.30 0.429 1.140 39.3 152 IN C-32 IH-N2 1.290 0.980 0.590 6.6 6.0 0.46 21.3 0.432 1.40 58.848 152 IN C-32 IC —N2 1.290 0.980 0.590 6.6 3.2 0.44 21.3 0.428 1.58 57.024 94 PB C-3 1C-N2 1.252 0.978 0.585 6.15 4. 2 0.36 32.40 0.592 1.170 51.8 94 PB C-3 2H-N2 1.252 0.978 0.585 6.15 2.4 0.39 32.40 0.632 1.025 52.6 94 PB C-3 4H-N2 1.252 0.978 0.585 6.15 2. 4 0.39 32.40 0.607 1.075 49.8 91 SB C-2 2C-N2 1.250 0.988 0.585 6.0 3.12 0.38 19.50 0.860 0.693 122. 91 SB C-2 3H-N2 1.250 0.988 0.585 6.0 3.12 0.38 19.50 0.919 0.645 116. 92 TE C-1 2H-N2 1.250 0.980 0.593 6.7 2.0 0.31 18.60 0.740 0.890 148.3 92 TE C-l 2C —-N2 1.250 0.980 0.593 6.7 0.54 0.31 18.60 0.672 1.050 160.7 COMPOUNDS AT THE MELTING POINT 148AGINSEC-273H-N2 1.314 1.000 0.592 6.5 1.0 0.42 18.50 1.095 0.673 61.6 148AGINSEC-273C-N2 1.314 1.000 0.592 6.5 1.0 0.42 18.50 1.075 0.893 66.6 149AGINSEC-291C-N2 1.286 0.984 0.572 6.9 1.0 0.42 18.50 1.072 1.00 69 7 149AGINSEC-292C-N2 1.286 0.984 0.572 6.9 1.0 0.42 18.50 1.072 0.714 55.2 149AGINSEC-293H-N2 1.286 0.984 0.572 6.9 1.0 0.42 18.50 1.095 0*791 63.2 i49AGINSEC-293C-N2 1.286 0.984 0.572 6.9 1.0 0.42 18.50 1.072 1.14 66.9

COMPOUNDS DTA-19 (CONT.) I II III IV V VI VII VIIIII IX X XI XII XII XIV i.49AGINSEC-294C-N2 1.286 0.984 0.572 6.9 1.0 0.42 18.50 1.072 1.008 61.3 113AG2SE C-111C-N2 1.318 1.039 0.597 6.1 1.0 0.55 24.00 1.160 0.483 18.6 136AG2SE C-243H-N2 1.306 1.020 0.610 6.4 1.0 0.55 24.00 1.173 0.473 16.7 136AG2SE C-243C-N2 1.306 1.020 0.610 6.4 1.0 0.55 24.00 1.159 0.522 17.1 112AG2TE C-102C-N2 1.290 0.998 0.578 6.5 1.0 0.46 25.50 1.225 0.302 13.2 135AG2TE C-283C-N2 1.310 1.000 0.570 6.3 1.0 0.46 25.50 1.223 0.614 19.63 1liBI2SE3C-9 3C-N2 1.310 1.016 0.586 6.4 0.5 0.32 20.40 0.968 0.635 98.0 131BI2SE3C-221C-N2 1.300 1.019 0.602 5.9 0.5 0.32 20.35 0.958 0.710 125. 131B12SE3C-222H-N2 1.300 1.019 0.602 5.9 0.5 0.32 20.35 0.991 0.740 109. 110BI2TE3C-9 2C-N2 1.310 1.016 0.610 6.6 0.5 0.29 23.10 0.855 0.792 112. 130BI2TE3C-222H-N2 1.303 0.993 0.616 6.1 0.5 0.29 23.10 0.868 0.580 106.6 130BI2TE3C-222C-N2 1.303 0.993 0.616 6.1 0.5 0.29 23.10 0.852 0.730 110. 145CDSE C-251C-N2 1.280 0.994 0.600 7.1 1.0 0.37 17.50 1.511 0.421 60.7 145CDSE C-252H-N2 1.280 0.994 0.600 7.1 1.0 0.37 17.50 1.535 0.315 58.8 104CDTE C-102H-N2 1.315 1.010 0.585 6.7 1.0 0.32 18.60 1.378 0.318 62.1 129 GAAS C=191C-N2 1.294 1.010 0.598 6.1 1.08 0.46 16.50 1.503 0.295 164. 102 GASB C-6 2H-N2 1.282 1.013 0.606 6.4 0.78 0.41 17.40 1.001 0.625 194.5 102 GASB C-6 2C-N2 1.282 1.013 0.606 6.4 0.78 0.41 17.40 0.977 0.576 186. 128 GASB C-192C-N2 1.309 0.990 0.597 6.2 0.78 0.41 17.40 0.976 0.602 187.5 101 tNAS C-7 2H-N2 1.302 1.033 0.606 6.4 1.20 0.37 17.70 1.225 0.396 145.6 127 INAS C-181C-N2 1.280 0.983 0.597 7.0 1.20 0.37 17.70 1.198 0.435 124.0 127 INAS C-182H-N2 1.280 0.983 0.597 7.0 1.20 0.37 17.70 1.223 0*455 151.0 100 INSB C-6 2C-N2 1.255 0.985 0.606 6.4 1.08 0.30 18.30 0.786 0*852 184. 126 iNSB C-182C-N2 1.314 0.989 0.595 6.2 1.08 0.30 18.30 0.786 0.717 149.2 1191N2SE3C-143H-N2 1.318 0.990 00593 6.8 0.5 0.39 17.10 1.167 0.639 57.6 119IN2SE3C-143C-N2 1.318 0.990 0.593 6.8 0.5 0.39 17.10 1.148 0.694 61.1

COMPOUNDS DTA- 19 (CON.T ) I 11 III IV V VI VII VIII IX X XI XII XIII XIV 1421N]E C-172H —N2 1i275 1.022 0.615 6.7 0.50 0.32 18.90 0.977 0.600 69.0 118IN2TE3C —212C-N2 1.303 0.991 0.615 6.5 0.50 0.29 17.40 0.930 1.220 7 2 2. 118IN2TE3C-213H-N2 1.303 0.991 0.615 6.5 0.50 0.29 17.40 0.955 0.762 70.7 139IN2TE3C-261C-N2 1.289 1.013 0.607 6.6 0.50 0.29 17.40 0.930 1.175 80.6 106PBSE C-8 IC-N2 1.285 1.013 0.606 6.6 0.40 0.33 24.30 1.353 0.352 54.7 IO6PBSE C-8 2H-N2 1.285 1.013 0.606 6.6 0.40 0.33 24.30 1.363 0.338 54.1 133PBSE C-232C-N2 1.275 0.997 0.597 6.8 0.40 0.33 24.30 1.347 0.447 56.5 105PBTE C-8 3H-N2 1.278 1.005 0.606 6.5 0.40 0.39 24.60 1.213 0.440 65.6 132PBTE C-231C-N2 1.312 1.010 0.612 6.8 0.40 0.39 24.60 1.189 0.575 59.4 132PBTE C-232H-N2 1.312 1.010 0.612 6.8 0.40 0.39 24.60 1.204 0.409 58.5 117SB2SE3C-153H-N2 1.289 1.021 0.602 6.5 0.5 0.41 19.20 0.897 0.823 106.3 117SB2SE3C-153CC-N2 1.289 1.021 0.602 6.5 0. 5 0.41 19.20 0.861 0.714 98.5 138SB2SE3C-241C-N2 1.291 1.026 0.600 7.1 0.5 0.41 19.20 0.833 0.851 88.6 138SB2SE3C-242H-N2 1.291 1.026 0.600 7.1 0. 5 0.41 19.20 0.903 0.936 97.7 116SB2TE3C-141C-N2 1.300 1.016 0.600 6.6 0. 5 0.32 19.80 0.884 0.82 115.2 137SB2TE3C-171C-N2 1.276 1.044 0.602 6.6 0.5 0.32 19.80 0.881 0.741 126.0 137SB2TE3C-172H-N2 1.276 1.044 0. 602 6.6 0.5 0.32 19.80 0.903 0.740 114.5 103SNTE C-7 3H-N2 1.292 1.025 0.607 6.0 1.0 0.29 19.50 1.100 0.527 76.0 141SNTE C-162H-N2 1.276 0.983 0.591 6.7 1.0 0.29 19.50 1.083 0.514 72.0 COMPOUNDS AT THE TRANSITION POINT 148AGtNSEC-27T3H-N21.314 1.000 0.592 665 0*40 0.42 18.50 1.025 06624 10.2 148AGINSEC-27T3C-N21.314 1.000 0.592 6.5 0.40 0.42 18.50 1.010 1.42 9*08 149AGINSEC-29T1C-N21.286 0*984 0*572 6*9 0*40 0.42 18.50 1.010 2.85 11.85 149AGINSEC-29T2C-N21.286 0.984 0*572 6.9 0.40 0.42 18.50 1*010 1.474 7.77

TRANSITIONS DTA-19 (CONT.) I II I IV V VI VII VIII IX X XI XII XIII XIV 149AGINSEC-29T3H-N21.286 0.984 0.572 6.9 0.40 0.42 18.50 1.025 0.736 14.06 149AGINSEC-29T3C-N21.286 0.984 0.572 6.9 0.40 0.42 18.50 1.010 1.404 9.77 149AGINSEC-29T4C-N21.286 0.984 0.572 6.9 0.40 0.42 18.50 1.010 1.663 6.57 113AG2SE C-IllTC-N21.318 1.039 0.597 6.1 0*20 0.55 24.00 0*396 1.85 72.6 113AG2SE C-IIT2H-N21.318 1.039 0.597 6.1 0.20 0.55 24.00 0.427 1.85 72.3 136AG2SE C-24T2H-N21.306 1.020 0.610 6.4 0.20 0.55 24.00 0.436 1.825 56.3 112AG2TE C-11O2C-N21.290 0.998 0.578 6.5 0.20 0.46 25.50 0.410 3.00 50.0 135AG2TE C-28T4H-N21.310 1.000 0.570 6.3 0.20 0.46 25.50 0.447 1.43 60.6 135AG2TE C-28T4C-N21.310 1.000 0.570 6.3 0.20 0.46 25*50 0.407 2.80 59.6 112AG2TE C-10OTC-N21.290 0.998 0.578 6.5 0.40 0.46 25.50 1.605 0.485 1.66 112AG2TE C-10T2H-N21.290 0.998 0.578 6.5 0.40 0.46 25.50 1.075 0.543 1.58 135AG2TE C-28T2H-N21.310 1.000 0.570 6.3 0.40 0.46 25.50 1.081 0.547 2.14 135AG2TE C-28T2C-N21.310 1.000 0*570 6.3 0.40 0.46 25.50 1.065 0.529 2.12 119IN2SE3C-14T3H-N21.318 0.990 0.593 6.8 0.20 0.57 17.10 0.483 1.42 10.1

145 Program Number DTA-16 for Direct Calculation of the Thermal Conductance In this method a double trial and error calculation is required because a value of G is needed in order to compute the gas film thickness from two measurements of the area under the DTA curve obtained when two gases of widely different thermal conductivity alternately fill the system. The definitions of the variables which are different from those used for Program DTA-19 are tabulated below, and the MAD listing of the computer program follows. MAD Variable Variable in Text Explanation A a Cf. Equation 2. 61 ALPHA a Cf. Equation 3. 9 B b Cf. Equation 2. 61 BETA P Cf. Equation 3.8 CHECK Function for choosing C to insure convergence C Function for incrementing LF D1 ZR1 Units are cm DELTAD 2(R -R ) Units are cm El e1 Emissivity of nickel oxide E3 3 Equivalent emissivity of silica E4 E4 Emissivity of sample EPSLN Desired limiting value of ERROR ERROR Percent difference between calculated and assumed values of LF F5, F6 F5, F6 Shape factors, Equations 2. 62, 2. 63 F, FS Multiplicative factors for correcting kZ and k~ G, GS G, G Overall conductances with helium and nitrogen, cal/cm min OK

146 Variable Listing for Program DTA-1 6 (Cont'd) MAD Variable Variable in Text Explanation I, IS Area under DTA curve with helium and nitrogen, ~C min JMAX Maximum value of J J Variable Subscript of F and FS K Variable subscript of C K2, KS2 kz, k2 Thermal conductivity of helium and nitrogen, cal/cm min OK LFO Initial estimate of LF NMAX Maximum value of N N Variable subscript of XI SMAX Maximum value of S S Running index, subscript of LF XI Arbitrary parameter, normally unity ZETA2 l / n A"j

MAD LISTING FOR COMPUTER PROGRAM NUMBER DTA-16 $COMPILE MAD, EXECUTE, DUMP, PRINT OBJECT, PUNCH OBJECT DTA-16 INTEGER S, J, JMAX, N, NMAX, NUMBER, SAMPLE, RUN, CYCLE INTEGER SMAX, K DIMENSION XI(20), F(10), FS(10), LF(750), C(10), ERROR(750) READ DATA START READ DATA PRINT COMMENT $1$ PRINT FORMAT LABEL, NUMBER, SAMPLE, RUN, CYCLE PRINT COMMENT $0$ K2 = 0.01163 + 0.03625*TF -0.00278*TF*TF KS2 = 0.00060 + 0.01085*TF - 0.00180*TF*TF K3 = 0.155 + O.19*TF El = 0.143 + 0.48*TF WHENEVER TF.LE. 0.39/0.93 E3 = 1. OTHERWISE E3 = EXP.(-(TF.P.O.5)x((O.93*TF) 0.39)) END OF CONDITIONAL PRINT COMMENT $ODATA$ PRINT RESULTS D3,D4,D5,K2,KS2,K3,K4,E1,E3,E4,TFI,ISM,LFOL G3 = 6.2832*K3/ELOG.(D3/D4) G4 = 6.2832*K4/(1.-(ELOG.(D4/D5)/((D4/D5) - 1.))) A-D3/D1 B=D4/D1 F5 = E3*(A-B*E4)/(1.0+A*E3*((1.0/E1)-i.O)) F6 = B*E4/(1.0+(A*E3+B*E4)*((1.o/E1)-1*0)) G5 = 1.021*D1*(TF.Pi3)*F5 G6 1.021*D1*(TF.P,3)*F6

PROGRAM DTA-16 (CONT.) ALPHA=(G5+G6+G5*G6/G3)/ (1.+(G5/G3)+((G5+G6)/G4)+(G5*G6/(G3 1 *G4))) BETA=1.O/((1.O+(G5/G3)+((G5+G6)/G4)+(G *G6/(G3G4) ) ) ) THROUGH ENDJ, FOR J= 1, 1, J *G. JMAX PRINT COMMENT $0$ PRINT COMMENT $0$ PRINT COMMENT $0 THE VALUES OF F ARE$ PRINT RESUL TS F(J), FS(J) PRINT COMMENT $0$ THROUGH ENDN, FOR N 1, 1, N.G. NMAX PRINT COMMENT $0$ PRINT COMMENT $0 THE VALUE OF XI IS$ PRINT RESULTS XI(N) PRINT COMMENT $0 RESULTS$ K = R = 1. S = 1 LF(S) = LFO IN G = M*LF(S)/(I*L) GS = M*LF(S)/(IS*L) G = G/XI(N) GS' GS/XI(N) ZETA2=(1/( 6.2832*(BETA.P.2 ) ) ) 1 ((1./(FS(J)*KS2)) - (1./(F(J)*K2)))* 2( (G-ALPHA) (GS-ALPHA) / (G-GS)) WHENEVER ZETA2.LE. 1.442 PRINT COMMENT $0 ETA2 IS LARGER THAN 2.*00$ PRINT RESULTS ZETA2, LF(S)

PROGRAM DTA-16 (CONT.) TRANSFER TO ENDN END OF CONDITIONAL WHENEVER ZETA2 *GE. 200. PRINT COMMENT $0 ElA? IS SMALLER THAN 1.005$ PRINT RESULTS ZETA2t LF(S) TRANSFER TO ENDN END OF CONDITIONAL ETA2 - FXP#(3./ZFTA2) DELTAD = D3*(ETA2 - 1) D2=D3+DELTAD G2 = F(J)*6*2832*K2/ELOG6(D2/D3) GS2 = FS(J)*6.2832*KS K2/ELOG.(D2/u3) G (G6+(G2+-G5)*(10+G6/G3) ) /(10i+(G2+G5 )( ( 10/G4)i+-iO/G3)+ I (G6/G3*G4) ))) GS=(G6+(GS2+G5)*1(0O+G6/G3))!/(O0+~(GS2+G5)*((iO/G4)+ I(1./G3)+(G6/(G3*G4)))) G G*XI(N) GS GS*XI(N) ERROR(S) = 1.0 - G*L*I/(M*LF(S)) WHENEVER S *GE. 10 CHECK - *ABS. ERROR(10)-.ABSo ERROR(S) OTHERWISE CHECK = - 0.5 END OF CONDITIONAL WHENEVER R *G* Rl PRINT COMMENT $0 INTERMEDIATE VALUES$ PRINT RESULTS LF(S)t C(K)9 DELTADi ERROR(S) R = 0. END OF CONDITIONAL

PROGRAM bTA- i6 (CONT.) R R + i* WHENEVER AABS. ERROR(S).L. EPSLN/1OO, PRINT COMMENT $0 FINAL VALUEb$ PRINT RESULTS F 5 F6 PRINT RESULTS G, GS PRINT RESULTS DELTADo LF(S) TRANSFER TO ENDN OTHERW ISE WHENEVER S *.G SMAX - 1 PRINT COMMENT $0 SMAX HAS BEEN EXCEEDED$ PRINT RESULTS LF(S), C(K), DELTADs ERROR(S) TRANSFER TO ENDN END OF CONDITIONAL WHENEVER ERROR(S),L. O. TRANSFER TO OPEN WHENEVER ERROR(S) *GE, EPSLN K 1 OTHERWISE WHENEVER ERROR(S) *GE. EPSLN/10, K = 2 OTHERWISE WHENEVER ERROR(S).GE. EPSLN/100. K=3 END OF CONDITIONAL END OF CONDITIONAL WHENEVER CHECK.G. O. TRANSFER TO ENTR2 ENTR1 S - + 1 LF(S) (i - C(K))*LF(S-1) TRANSFER TO IN OPEN WHENEVER ERROR(S).LE. -EPSLN

PROGRAM DTA-16 (CONT.) K = I OTHERWISE WHENEVER ERROR(S) *LE* -EPSLN/1O. K = 2 K=2 OTHERWISE WHENEVER ERROR(S) *LE. -EPSLN/100. K=3 END OF CONDITIONAL END OF CONDITIONAL WHENEVER CHECK *G. O. TRANSFER TO ENTRI ENTR2 S = S + 1 LF(S) = (lI + C(K))*LF(S-1) TRANSFER TO IN END OF CONDITIONAL ENDN CONTINUE ENDJ CONTINUE TRANSFER TO START VECTOR VALUES LABEL = $S5,13,S5,C6S55,C4,S5,C6*$ END OF PROGRAM $DATA Dl = 1.37t JMAX = 1, NMAX =, F(l) * 8, FS(1) = 1., C(l)=o089#02tO005 XI(l) 1., SMAX = 750, Rl =19ot EPSLN 1. l D3 = 1.290, D4 = 0.980, D5 = 0.590, K4 = 6.0, E4 = 0i10, IF = 0.432, I 181048) IS = 58*848s M = 21.3~ LFO = 6*8o L 6,66 RUN - $C-32$, NUMBER = 152* CYCLE = $HEATNG$, SAMPLE - $IN$

152 Program Number DTA-17 for Analysis of DTA Data This short program was devised in order to facilitate the processing of the experimental data according to the method whereby the conductance is estimated from the experimental time constant plus the estimated thermal capacity. It was designed so that either the data on standards could be used for computing the correlation factor Z, or the latent heat could be computed from the data on compounds using a specified value of Z. The definitions of the MAD variables are listed below, and the computer programs plus the experimental data follow. MAD Variable Variable in Text Explanation AREA j - I dt Area under DTA curve, C min ss C C "Correct" thermal capacity, cal/ C CEST C' Estimated thermal capacity, cal/ C CP c Specific heat of sample, cal/g 0C CPQ c Specific heat of silica, cal/g C INDEX Variable subscript; INDEX = 1 refers to data on compounds for series B; 2compounds, series C; 4- standards, series B; 5- standards, series C MQCONT Mass of silica in contact with the sample, g TAU Units are min exp TF T Temperature, K x 10 VQCONT Volume of silica in contact with the sample, cc Z Correlation factor, C/C' The remaining symbols used in this program have already been defined.

MAD LISTING FOR COMPUTER PROGRAM NUMBER DTA- I SCOMPILE MAD, EXECUTE9 DUMPq PRINT OBJECT, PUNCH OBJECT DTA17 INTEGER INDEX, NUMBER DIMENSION Z( 1O) L(10) PRINT COMMENT $1$ READ DATA START READ FORMAT INPUT, INDEX, NUMiBER, SAMPLE?, PUN, CYCLE, I D3, D4, D5, Lg Mg CP, LF- TF# TAU, AREA PRINT COMMENT$4$ PRINT FORMAT LABEL NUMBER, SAMPLE, kRUN CYCLE CPQ = 0.223 + 0.0613*TF - 0O OO75/(TF*TF) PRINT COMMENT $ODATA$ PRINT RESUl.TS 03, D4, D5, L, M, CP, CPQ, TF PRINT RESULTS TAU, AREA VQCONT 0.a785k(t*(D3*D3 - D4*D4) + L(INDEX)*(D5L*Db -_ 0.10)) 1 + 0.03 MQCONT = 2*20*VQCONT CEST = MQCONT*CPQ + M*CP WHENEVER INDEX *L. 39 TRANSFER TO OVER C = M*LF*TAU/AREA Z(INDEX) = C/CEST PRINT COMMENT $ORESULTS$ PRINT RESULTS LF, MQCONT, Z(INDEX) TRANSFER TO START

PROGRAM D'A —17 (CONT.),OVER C S = C ES Z (. ND EX LF' C*AREA/ (M'TAU) PRINT COMMENT $ORES ULTS$ PRI NI RESULTS Z( INDEX), iMiCON" Li-'TRANSFEI TO START VECTOR VALUES NPUT $i i S'L?C C4- tC ^' -5 F6 6 4S VECO TOR VALL US LAB EL $S5,i 3 S -5C6: s C S5 C 6 $:NDO OF PRO R.. A.- 2 1. *".1.300w 1 O i) 2... 2 r4 4 - 6*.0 n

DATA CARD FORMAT FOR PROGRAM )'-i,A,-1 CODE VARIABLE COLUMNS ALLOTTED I INDEX. TI NUMBER 2-4 III SAMPLE 5-10 TV RUN 1 ]-1 4 V CYCLE 15-20 VI )3 21-25 VII 0D4 26-30 VI I D5 31-35 IX i 36-40 X M 41-46 - XI C(P 47-52 X!I F 53-58 X iII T F 59-64 XIV TAU 65-70 XV AREA 71-76

DAIA CARDS FOR PROGRAM DFA-1i STANDARDS AT THE MEL TING POINT' I Il III IV V Vi Vii VIII IX X Xi XII X'lil XV XV 4 40 AG B-141C-HE 135 1.18 0.59 3.6 31.53 0.0707 25.0 1.229 0.269 51.7 4 40 AG B -1 42H-HE 1 35 1 3 18 0.59 3.6 31 53 0 0677 25.0 1.236 0.260 50*6 40t AG B 42 CN2 1 35 11 8 0 59 3 6 31 5 3 0 070 7 250 227 02 5-44 96 1 4 46 AG B-18 J.C-HE 1i3561.1930*59 3.'7 31*50 0.0707 25.0 1.226 0.345 61.0 4 46 AG B-.82H-..HE- 1.3551.1930.59 3.7 31.50 0.06 7 2.50 1.234 0.344 62*7 4 46 AG B —182C-N2 1.3561.1930.59 3.7 31.50 0.0707 25.0 1.216 0.725 122.0 5 93 AG C —4 iC-t 2'. 2440.9680.593 6.0 28.80 0 70 -7 25.0 1.226 0.690 93.9 5 93 AG C( —4 2H-N2 1.2440*9680.593 6.0 28.80 0.06? 2 25.0 1.247 0.620 93.0 5 93 AG C —4 2C-HE 1.2440.9680.593 6.0 28.80 0. 707t 25.0 1.225 0.363 51.4 5 93 AG C —4 3H-HE 12440.968i0.593 6.0 28.80 0.o 0 7 25.0 1.2m3 0.364 49,7 5 97 AG C I.. C-N2 1.2270.9650.606 6.3 28.80 0.8 * 07 25.U 1.229 0679 96.6 5 97 AG — 5 2H-N2 -!. 22'70.9650.606 6.3 28.80 0.07 7 25.0 1.2) / 0.631 93.5 5 97 AG C - 3C-HE 1. 22 0 9650. 606 6 3 28.80 0. 07 25 0 1.229 0. 3 72 5u 9 5 97 AG C —5 3H —HE 1,22-70.9650.606 6.3 28.80 0.067i 25.0 1.237 0.366 53.5 5115 AG C -3 2H-N 2 1 3030.9920.605 5 9 28.8u 0.067-7 25.,0 1.244 u.491 65.9 5115 AG C-133H-HE I13030.9920.605 5.9 28.80 0.0677 25.0 1.238 0.310 42.7 4 9 C U B-i C — H 1.34 1.18 0.59 3.9 26.88 0115 48 7 1.351 0.322 7 6 5 4 9 CiJ B.-. 2H —HE 1!.34 1.18 0.59 3.9 26*88 0.118 48.7 1.359 0.309 84.0 4 9 CU B- 2CH —-F 134 1.18 0.59 3.9 26*88 0*115 48.7 1.350 0.349 80*0 4 26 CU B -14C- N2 1. 135 1.18 0.59 4.0 26*88 0.115 48.7 1.346 0.640 138. 4 2F- CU B. — -142H —IHF 1.-3 5 1.18 0.59 4.0 26.88 0.118 48.7 1362 0345 81.5 4 26 CU B -142C —HF 1.35 1.18 0.59 4.0 26.88 0.11.5 48.7 1l348 0.336 75*0 4 45 CU B-181C'.-N2 1.,3527 1.730.59 3*8 26.88 0.115 48.7 1.346 0.691 134. 4 45 CU B- 82H- HF 1.3521.1730.59 3.8 26.88 0.118 48.7 1.359 0.382 86.6

STANDARDS DTA-17 (CONT.) I II III IV V VI VII VIII IX X XI XII XIII XIV XV 4 45 CU B-182C-N2 1.3521.1730.59 3.8 26.88 0.115 48.7 1.348 0.368 82.9 4 61 CU B-241C-N2 1.35 1.1750.59 3.75 26.88 0.115 48.7 1.345 0.752 160. 4 61 CU B-242H-N2 1.35 1.1750.59 3.75 26.88 0.118 48.7 1.363 0.736 163. 4 61 CU B-242C-HE 1.35 1.1750.59 3.75 26.88 0.115 48.7 1.333 0.411 88.7 4 61 CU B-243H-HE 1.35 1.1750.59 3.75 26.88 0.118 48.7 1.365 0.400 90.5 4 61 Cu B-243C-HE 1.35 1.1750.59 3.75 26.88 0.115 48.7 1.332 0.415 89.6 4 6 IN B-1 IH-HE 1.34 1.17 0.59 3.5 21.93 0.0677 6.8 0.437 0.28 16.6 4 6 IN B-1 IC-HE 1.34 1.17 0.59 3.5 21.93 0.0599 6.8 0.428 0.265 15.3 4 6 IN B-1 2H-HE 1.34 1.17 0.59 3.5 21.93 0.0677 6.8 0.434 0.28 16.1 4 30 IN B-9 2H-HE 1.32 1.14 0.59 3.7 21.93 0.0677 6.8 0.436 0.345 20.6 4 30 IN B-9 2C-HE 1.32 1.14 0.59 3.7 21.93 0.0399 6.8 0.429 0.355 19.9 4 30 IN B-9 3H-HE 1.32 1.14 0.59 3.7 21.93 0.0677 6.8 0.436 0.345 19.6 4 30 IN B-9 4H-HE 1.32 1.14 0.59 3.7 21.93 0.0677 6.8 0.446 0.321 18.0 4 30 IN B-9 4C-HE 1.32 1.14 0.59 3.7 21.93 0.0599 6.8 0.429 0.341 18.0 4 30 IN B-9 5H-HE 1.32 1.14 0.59 3.7 21.93 0.0677 6.8 0.433 0.311 18.1 4 30 IN B-9 6H-A 1.32 1.14 0.59 3.7 21.93 0.0677 6.8 0.441 0.942 48.9 4 30 IN B-9 6C-A 1.32 1.14 0.59 3.7 21.93 0.0599 6.8 0.428 0.992 49.7 4 30 IN B-IOIH-A 1.32 1.14 0.59 3.7 21.93 0.0677 6.8 0.437 0.635 31.0 4 30 IN B-101IC-A 1.32 1.14 0.59 3.7 21.93 0.0599 6.8 0.428 0.700 31.8 4 30 IN B-102H-HE 1.32 1.14 0.59 3.7 21.93 0.0677 6.8 0.433 0.265 13.5 4 30 IN 8-103H-HE 1.32 1.14 0.59 3.7 21.93 0.0677 6.8 0.433 0.246 12.9 4 30 IN B-9 IC-HE 1.32 1.14 0.59 3.7 21.93 0.0599 6.8 0.429 0.362 19.5 4 41 IN B-191C —N2 1.36 1.18 0.59 3.7 21.93 0.0599 6.8 0.429 0.831 37.9 4 41 IN B-192H-N2 1.36 1.18 0.59 3.7 21.93 0.0677 6.8 0.434 0.819 36.9 4 41_ IN B —192C-N2 1.36 1.18 0.59 3.7 21.93 0.0599 6.8 0.430 0.287 14.6 4 41 IN B-193H-N2 1..36 1.18 0.59 3. 7 21.93 0.0677 6.8 0.439 0.255 13.7

SrANDARDS DTA -7 (CONT.) I II iI IV V VI VI VIII IX X XI XII XIII XIV XV 5 99 IN C-121C-HE 1.3100.9920.6026.1 21.30 0.0599 6.8 0.427 0.560 20.4 5 99 IN C-122H-HE 1,3100.9920.6026.1 21.30 0.0677 6.8 0.434 0.517 19.6 5 99 IN C-122C-N2 1.3100.9920.6026.1 21.30 0.0599 6.8 0,429 1.140 39.3 5 99 IN — 123H-N2 1.3100.9920.6026.1 21.30 0.0677 6.8 0.439 1.045 38.7 4 22 PB B-O01C-HE 1*3181.14 0.59 3.7 34*02 0.03405.89 0,596 0.219 14.7 4 22 PB B-102H-HE 1.3181.14 0*59 3.7 34.02 0*0365 5*89 0*602 0.208 14*2 4 22 PB B-102C-HE 1.3181.14 0.59 3.7 34.02 0.0340 5.89 0.595 0.225 14.4 4 53 PB B-231C-N2 1.3261.1750.59 3.7 34*02 0.0340 5.89 0.597 0.845 49.0 4 53 PB B-232H-N2 1.3261.1750.59 3.7 34.02 0.0365 5.89 0.607 0.836 49,2 4 53 PB B-233H-HE 1.3261.1750.59 3.7 34.02 0.0365 5.89 0.605 0.290 21.2 4 53 PB B-233C-HE 1.3261.1750.59 3.7 34.02 0.0340 5.89 0.597 0.300 21.9 2 4 53 PB B-234H-HE 1.3261.1750.59 3.7 34.02 0.0365 5.89 0.608 0.288 22.5 5 94 PB C-3 IC-N2 1.2520.9780.5856.15 32.40 0.0340 5.89 0*592 1.17 51.8 5 94 PB C-3 2H-N2 1.2520.9780.5856,15 32*40 0.0365 5*89 0.632 1.025 52.6 5 94 PB C —3 2C-HE 1.2520.9780.5856.15 32*40 0.0340 5.89 0.595 0.46 21,0 5 94 PB C-3 3HH-HE 1.2520.9780o5856.15 32.40 0.0365 5.89 0.608 0.423 21.4 5 94 PB C-3 4H-N2 1.2520.9780.5856.15 32.40 0.0365 5.89 0.607 1.075 49.8 4 23 SB B-8 2H-f-E 1.35 1.18 0.59 3.5 19.04-0.0616 39.0 0.908 0.230 61.5 4 23 SB B-8 3H-HE 1.35 1.18 0*59 3.5 19.04 0*0616 39*0 0.908 0.232 62.5 4 29 SB B-112H-N2 1.3451.17 0.59 3.5 19*04 0.0616 39.0 0.911 0.404 111. 4 37 SB B-151C-.N2 1.3501.16 0.59 3.5 19.04 0.0586 39.0 0*861 0.594 140. 4 37 SB B-152H-I-IE 1.,3501.16 0*59 3.5 19*04 0.0616 39.0 0.910 0.244 68.1 4 37 SB B-152C-HE 1.3501.16 0.59 3.5 19.04 0.0586 39.0 0.864 0.262 65.5 5 91 SB C-2 IC —HE i*2500.9880.5856.0 19*50 0*0586 39.0 0.854 0.382 69.4 5 91 SB C-2 2H-HE 1.2500.9880*5856.0 19.50 0*0616 39.0 0.909 0.395 66h. 5 91 SB (-2 2C-N2 1.2500.9880.5856.0 19.50 0.0586 39.0 0.860 0.693 122.

STANDARDS DTA-17 (CONT.) II III IV V VI VII VIII IX X XI XII XIII XIV XV 5 91 SB C-2 3H-N2 1.2500.9880.5856.0 19*50 0.0616 39.0 0.919 0.645 116. 4 13 TE B-3 2H-HE 1.33 1.117 0.59 3.7 18.72 0.0705 32.7 0.727 0.252 62.2 4 13 TE B-3 3H —HE 1.33 1.17 0.59 3.7 18.72 0.0705 32.7 0.727 0.251 61.7 4 25 TE B-112H-N2 1.32 1.14 0.59 3.9 18.72 0.0705 32.7 0.732 0.574 103. 4 25 TE B-113H-HEF 1.32 1.14 0.59 3.9 18.172 0.0705 32.7 0.726 0.245 57.1 4 25 TE B-114H-HE 1.32 1.14 0.59 3.9 18.72 0.0705 32.7 0.731 0.254 56.0 4 148 iE B-191C —N2 1.3441.1740.59 4.1 18.72 0.06 58 32.7 0.659 0.897 185.5 4 48 TE B-,-192H —N2 1.3441.1740.59 4.1 18.72 0.0705 32.7 0.736 0.850 167. 4 48 TE B-192C-HE 1.3441.1740.59 4.1 18.72 0.0658 32.7 0.664 0.459 92.2 4 48 TE B-193H-HE 1.3441.1740.59 4.1 18.72 0.0705 32.7 0.731 0.356 79.5 5 92 TE C-l 2H-N2 1. 2500.9800.593 6.7 18.60 0.0705 32. 7 0.740 0.890 148.3 5 92 TE C-l 2C-N2 1.2500.9800.593 6.7 18.60 0.0658 32.7 0.672 1.050 160.7 5 92 TE C-1 3H-HE 1.2500.9800.593 6.7 18.60 0.0705 32.7 0.731 0.464 77.7 5 92 TE C-l 3C-HE 1.2500.9800.593 6.7 18.60 0.0658 32.7 0.752 0.503 82.0 COMPOUNDS AT THE MELTING POINT 2148AGINSEC-271C-HE 1.3141.0000.592 6.5 18.50 0.0596 1. 1.069 0.695 55.7 2148AGiNSEC-272H-HE 1.314J.0000.592 6.5 18.50 0.0596 1. 1.095 0.603 49.8 2148AGINSEC- 272C-HE 1.3141.0000.592 6.5 18.50 0.0596 1. 1.071 0.660 55.5 2148AGINSEC-273H-N2 1.3141.0000.592 6.5 18.50 0.0596 1. 1.095 0.673 61.6 2148AGINSEC-273C-N2 1.3141.0000.592 6.5 18.50 0.0596 1. 1.075 0.893 66.6 2149AGINSEC-291C-N2 1.*2860.9840.572 6.9 18.50 0.0596 1. 1.072 1.00 69.7 2149AGINSEC-292H-HE 1.2860.9840.572 6.9 18.50 0.0596 1. 1.095 0.714 64.8 2149AGJNSEC-292C-N2 1.2860.9840.572 6.9 18.50 0.0596 1. 1.072 0.714 55.2

COMPOUNDS DTA-17 (CON'I ) I II III IV V VI VII VIII IX X XI XII XIII XIV XV 2149AGINSEC-293H-N2 1,2860.9840.572 6.9 18*50 0.0596 1. 1.095 0.791 63.2 2149AGINSEC-293C-N2 1.2860.9840.572 6.9 18.50 0.0596 1. 1.072 1.14 66.9 2149AGINSEC —294H-HE 1.2860.9840.572 6.9 18.50 0.0596 1. 1.095 0.781. 51.1 2149AGINSEC-294C-N2 1.2860.9840.572 6.9 18.50 0.0596 1. 1.072 1.008 61*3 1 62 AG2SEB-291C-N2 1.3471.1720.59 3.6 24*00 0.0738 1. 1.154 0.632 30.2 1 62 AG2SEB-292H-N2 1.3471.1720.59 3.6 24.00 0.0738 1. 1.173 0.621 29.2 1 62 AG2SEB-292C-H E 1.3471.1720.59 3.6 24.00 0.0738 1. 1.153 0.443 19.0 1 62 AG2SEB-293H-HE 1.3471*1720*59 3.6 24.00 0.0738 1. 1.176 0.431 18.7 2113AG2SE C-111C-N2 1.3181.0390.597 6.1 24.00 0.0738 1. 1.160 0.483 18.6 2113AG2SE C —-112H-HE 1.3181.0390.597 6.1 24.00 0,0738 1. 1.173 0.331 13.6 2113AG2SF C-112C-HE 1.3181*0390.597 6.1 24*00 0.0738 1. 1.158 0.291 13.3 o 2136AG2SE C — 2 —41C-HE 1.3061.0200.610 6.4 24.00 0.0738 1. 1.157 0.345 13.0 2136AG2SE C-242H-HE 1.3061.0200,610 6*4 24.00 0.0738 1. 1.170 0.351 12.8 2136AG2SE C-242C-HE 1.3061.0200.610 6.4 24.00 0.0738 1. 1.158 0.336 12.6 2136AG2SEF C-243H.-N2 1.3061.0200*610 6*4 24.00 0.0738 1. 1.1o73 0.473 16.7 2136AG2SE C —243C-N2!.3061.0200.610 6.4 24.00 0.0738 1. 1.159 0.522 17.1 I 63AG2TE B-281C-N2 1.3451.1700.59 4.0 25.50 0.0633 1. 1.225 0.650 26.0 I 63AG2TE B-282H-N2 1.3451.1700*59 4.0 25.50 0.0633 1. 1.246 0.579 26.7 I 63AG21E B —282C-HE 1.3451.1700.59 4.0 25.50 0.0633 1. 1.223 0.548 19.8 I 63AG2TE B -283H-IIF 1.3451.1700.59 4.0 25.50 0.0633 1 1243 0.401 19.1 1 63AG2TE B-283C-HE ].3451.1700*59 4.0 25.50 0.0633 1. 1.224 0.532 20.0 2112AG2TE C-IO1C-HF 1.2900.9980*578 6.5 25.50 0.0633 1. 1.224 0.658 13.3 2112AG2TE C-102H-HE 1.2900.9980,578 6.5 25.50 0.0633 1. 1.240 0.895 16.4 2112AG2TE C —102C —N2 1.2900,9980.578 6.5 25.50 0.0633 1 1 1225 0.302 13.2 2135AG2TE C-282H-HE 1.3101.0000.570 6.3 25.50 0.0633 1. 1.241 0.406 15.1 2135AG2TI C-282C.I-E 1.3101.0000.570 6.3 25.50 0.0633 1. 1.222 0.421 15.62

COMPOUNDS DTA-17 (CONT.) I II III IV V VI V VII VIII IX X X XII XIII XIV XV 2135AG2TE C-283H-HE 1.3101.0000.570 6.3 25.50 0.0633 1. 1.240 0.307 14.5 2135AG2TE C-283C-N2 1.3101.00000570 6.3 25.50 0.0633 1. 1.223 0.614 19.63 I 55BI2SE3B-272C-N2 1.33 1.17 0*59 3.5 20*46 0.0553 1. 0.963 0.750 193. 1 55BI2SE3B-273H-N2 1.33 1.17 0.59 3.5 20.46 0.0553 1. 0.983 0800 160. 1 55BI2SE3B-273C-HE 1.33 1.17 0.59 3.5 20.46 0.0553 1. 0.963 0.454 109. 1 55BI2SE3B-274H-HE 1.33 1.17 0.59 3.5 20.46 0.0553 1. 0.983 0.344 93. 2111BI2SE3C-9 2C-HE 1.3101.0160*586 6.4 20*40 0.0553 1. 0.966 0.407 59.1 2111B12SE3C-9 3H-HE 1.3101.0160.586 6.4 20.40 0.0553 1. 0.981 0.426 56.0 2111BI2SE3C-9 3C-N2 1.3101.0160.586 6.4 20.40 0.0553 1. 0.968 0.635 98.0 2131BI2SE3C-221C-N2 1.3001.0190.602 5.9 20.35 0.0553 1. 0958 0*710 125. 2131BI2SE3C- 222H-N2 1*3001.0190.602 5.9 20.35 0.0553 1. 0.991 0.740 109. 2131B12SE3C-222C-HE 1.3001.0190.602 5.9 20.35 0.0553 1. 0O964 0.425 71.9 2131BI2SE3C-223H-HE 1.3001.0190.602 5.9 20.35 0.0553 1. 0.983 0*473 70.4 I 54BI2TE3B-272H-N2 1.33 1.18 0.59 3.65 22.10 0.0473 1. 0*877 0.728 213. I 54BI2TE3B-272C-N2 1.33 1.18 0.59 3.65 22.10 0.0473 1* 0*856 0.751 195. 1 54B 2TE3B -273H-N2 1.33 1.18 0.59 3.65 22.10 0.0473 1. 0*874 0.728 192* 1 54B12TE3B-273C-HE 1.33 1.18 0.59 3.65 22.10 0.0473 1. 0.859 0.352 105* 1 54BI2TE3B-274H —HE 1.33 1.18 0.59 3.65 22.10 0*0473 1. 0.868 0.385 97*3 2110Bi2TE3C-9 1C-HE 1.3101.0160.610 6.6 23.10 0.0473 1. 0.854 0.376 59i0 2110BI2TE3C-9 2H-HE 1*3101.0160.610 6.6 23.10 0.04-73 1. 0.863 0.378 58*5 2110B12TE3C-9 2C-N2 1*3101.0160*610 6.6 23.10 0.0473 1. 0855 0*792 1126 2130B12TE3C-222H-N2 1*3030*9930.616 6.1 23*10 0.0473 1. 0.868 0.580 106.6 2130B12TE3C-222C-N2 1.*30309930.616 6*1 23.10 0*0473 1. 0.852 0.730 110 2130BI2TE3C-223H-HE 1.030309930.616 6.1 23.10 0.0473 1. 0865 0.344 60*8 2130B12TE3C-223C —HE 1.3030.9930.616 6.1 23.10 0*0473 1. 0.853 0356 64.0 1 19 CDSE B —7 2H-HE 1.34 1.17 0.59 3.7 17.43 0O0757 1* 1*535 0.382 107.

COMPOUNDS DTA-17 (CONT.? i1ll III IV V VI VII VIII IX X XI XI XXIII XIV XV 1 49 CDSE B-242H-N2 13171.15 0.59 3.85 17*43 0.0157 1. 1.536 0.326 145. 1 49 CDSE B-243H-HE 1.3171.15 0.59 3.85 17.43 0.0157 1. 1.524 0.294 91.0 2123 CDSE C-162H-HE 1.3170.9900.594 6.9 17.50 0.0757 1. 1.535 0.238 54. 2145 CDSE C-251C-N2 1.2800.9940.600 7.1 17.50 0.0757 1. 1.511 0.421 60*7 2145 CDSE C-252H-N2 1.2800.9940.600 7.1 17e50 0.0757 1. 1.535 0.315 58.8 2145 CDSE C-252(-'HE 1.2800.9940.600 7.1 17.50 0.0757 1. 1.510 0.351 47.3 2145 CDSE C-'253H-HE 1.2800.9940.600 7.1 17.50 0.0(57 1. 1.528 0.244 42.6 2145 CDSE C-253C-HE 1.2800.9940.600 7.1 17.50 0.0757 1. 1.508 0.328 41.9 1 14 CDEE B-4 2H-HE 1.34 1.16 0.59 3.9 18.60 0.0604 1. 1.383 0.398 104. 1 27 CDTE B-9 2H-HE 1.34 1.17 0.59 3.8 18.60 0.0604 1. 1.375 0.330 94.7 1 33 CDTE B-122H- HE 1.35 1.17 0.59 3.8 18.60 0.0604 1. 1.368 0.387 94.5 1 47 CDIFT B-222H-HE 1.32 1.1730.59 4.? 18.60 0.0604 I. 1.373 0.338 86.1 2104 CDTE C-102H-N2 1.3151.0100.585 6.7 18.60 0.0604 1. 1.378 0.318 62.1 2121 CDTE C-152H-HE 1.2741.0090.575 6.5 18.60 0.0604 1. 1.368 0.255 58.9 1 18 GAAS B-6 2H-HE 1.33 1.16 0.59 3.2 15.93 0*100 1. 1.513 0.182 165.0 1 39 GAAS B-172H -HE 1.38 1.18 0.59 3.3 15.93 0.100 1. 1.517 0.207 160.0 2107 GAAS C-122H-HHE 1.3211.0210.602 6.9 16.50 0.100 1. 1.514 0.243 147.0 2129 GAAS C C191(-HE 1.?941*0i.00.598 6.1 16.50 0.100 1. 1.500 0.198 145.0 2129 GAAS C-191C-N2 1.2941.0100.598 6.1 16.50 0.100 1. 1.503 0.295 164t0 I 21 GASB B-7 IH-HE 1.34 1.17 0.59 3.6 16.86 0.0757 1. 0.991 0.263 131* 1 52 GASB B-262H-A 1.3411.1750.59 3.7 16.86 0*0757 1. 1.005 0.545 168. 1 52 GASB B-263H-A 1.3411.1750.59 3.7 16.86 0.0757 1. 1.010 0*535 168. 1 52 GASB B-264H-HE 1.3411.1750.59 3.7 16.86 0.0757 1. 1.002 0.252 82.1 2102 GASB C-6 2H-N2 1.2821.0130.606 6.4 17.40 0*0757 1. 1.001 0.625 194.5 2102 GASB C-6 2C-N2 1.2821.0130.606 6.4 17.40 0*0757 1. 0*977 0.576 186. 2102 GASB C-6 3H-HE 1.2821.0130.606 6.4 17.40 0*0757 1. 0.993 0.304 96*7

COMPOUNDS D'A- 17 (CONT.) I 1I IlI IV V VI VII ViiI IX X XI XII XIII XIV XV 2102 GASB C-6 3C-HE 1.2821.0130.606 6.4 17.40 0.0757 1. 0.980 0.424 102. 2128 GASB C-191C-HE 1.3090.9900*597 6.2 17.40 0.0757 1. 0,971 0.297 96.5 2128 GASB C-192H-HE 1.3090.9900.597 6*2 17.40 0.0757 1. 0.994 0.29 100.7 2128 GASB C-192C-N2 1.3090.9900.597 6.2 17.40 0.0757 1. 0.976 0.602 187.5 1 17 INAS B-5 2H-HE 1.32 1.14 0.59 3.8 17.10 0.0764 1. 1.223 0.282 134.7 1 38 INAS B-162H-HE 1.36 1.18 0.59 3.6 17.10 0.0764 1. 1.225 0.236 125.7 1 69 INAS B-332H-N2 1.3471.1710.59 3.8 17.10 0.0764 1 1*227 0.515 231. 1 69 INAS B —333H-HE 1.3471.1710.59 3.8 17.10 0.0764 1. 1.222 0.284 141.3 2101 INAS C-7 2H-N2 1.3021.0330.606 6.4 17.70 0.0764 1. 1.225 0.396 145.6 2101 INAS C-7 3H-HE 1.3021.0330.606 6.4 17.70 0.0764 1. 1.219 0.231 99.2 2127 INAS C-181C-N2 1.2800.9830.597 7.0 17.70 0.0764 1. 1.198 0.435 124*0 2127 INAS C-182H-N2 1.2800.9830.597 7. 1 0 770 0*0764 1. 1.223 0.455 151.0 2127 INAS C-182C-HE 1*2800.9830.597 7.0 17.70 0.0764 1. 1.200 0.27 94.0 1 16 INSB B-5 2H-HE 1.33 1.17 0.59 3.5 17.37 0.0626 1. 0.805 0.241 79.1 1 42 INSB B-152H-HE 1.3451.18 0.59 3.6 17.37 0.0626 1. 0.805 0.239 96.8 1 68 INSB B-331C-N2 1.3601.1830.59 3.9 17.37 0.0555 1. 0.764 0.770 239.0 1 68 INSB B-332H-N2 1.3601.1830*59 3*9 17.37 0.05626 1. 0*814 0.715 231. 1 68 INS B — 333H —-E 1.3601.1830.59 3.9 17.37 0*0626 1. 0.806 0.307 108.3 2100 INSB C-6 1C-HF 1.2550.9850.606 6.4 18.30 0.0555 1. 0.778 0.470 98*8 2100 INSB C-6 2C-N2 1.2550.9850.606 6.4 18.30 0*0555 1. 0.786 0.852 184. 2126 INSB C-181C-HE 1.3140.9890.595 6.2 18.30 0.0555 1. 0*776 0.337 79.8 2126 INSB C-182C-N2 1.3140.9890.595 6*2 18.30 0.0555 1. 0.786 0*717 149.2 I 591N2SF3B-251C —N2 1.3601.1750*59 40 170. 012 1 1. 1.145 1.02 138* I 591N2SE3B6-252H-N2 1.3601.1750.59 4.0 17.70 0.12 1. 1.175 0.817 151. I 591N2 F3B-252C-HF 1.3601.1750.59 4.0 17.70 0.12 1 1. 144 0.939 110* 1 5 9N2S 3B -253H —I E 1.3601.1750.59 4.0 17.70 0.12 1. 1.173 0*703 133.

COMPOUNDS DTA-17 (CONT ) I II III IV V VI VII VIIi IX X XI XII XIII XIV XV I 59IN2SE3B-253C-N2 1.3601.1750.59 4.0 17.70 0.12 1. 1.148 1.18 129. 2119IN2SE3C-142H-HE.3180.9900.593 6.8 17.10 0.12 1. 1.168 0.556 56.6 2119IN2SE3C-142C-HE 1.3180.9900.593 6.8 17.10 0.12 1. 1.148 0.518 47.2 21191N2SE3C-143H-N2 1.3180.9900.593 6.8 17.10 0.12 1. 1.167 0.639 57.6 2 11.9 N2SE3C —143C-N 2 1 3180.99 00593 68 1710 0.2 1. 1.148 0.694 61.1 1 591N2SE3B 251C-N2 1.3601 1750.59 4.0 17.70 0.0775 1. 1.145 1.02 138. 1 59IN2SE3B-252H-N2 1.3601.1750.59 4.0 17.70 0.0 775 1 1.175 0.817 151. 1 591N2SE'3B-252C-HE 1.3601.1750.59 4.0 17.70 0.0775. 1.144 0.939 110. I 59IN2SE3B=253H-HE 1.3601.1750.59 4.0 1 7.70 0.0775 1 1.173 0.703 133. I 59IN2SE3B-253C-N2 1.3601.1750.59 4.0 17.70 0.0775 1 1. 148 1.18 129. 21191N2SE3C —142H-HE 1.3180.9900.593 6.8 17.10 0.0775 1. 1.168 0.556 56.6 2119 N2SF3C 142C-HF 1. 3180.9900.593 6. 8 17.10 0077 1. 1 1. 1.48 0.518 47.2 2119IN2S E3C-143H —N2 1.3180.9900.593 6.8 17 10 0.0775 1. 1 167 0.639 57 6 21191N2SE3C-143C-N2 1.3180.9900.593 6.8 17.10 0.0775 1. 1.148 0.694 61.1 2142 INTLE C- - 7-1C-N 2 1.2751.0220.615 6.7 18.90 0059 1. 0.952 1.05 77.0 2142 INTE C-. 72H-N2 1.2751.0220.615 6.7 18.90 0.0599 1. 0.977 0.600 69.0 2142 INTE C-172C-HE 1.2751.0220.615 6.7 18.90 0.0599 1. 0.941 0.512 56.0 2120 INTE C-201C —N2 1.3401.0350.597 6.3 18.90 0.0599 1. 0.943 0.930 82.9 2120 INTE C-202C HE 1.3401.0350.597 6*3 18.90 0.0599 1. 0.940 0.426 51.6 2120 INTE C —203H —HE 1.3401.0350.597 6.3 18*90 0.0599 1. 0.972 0.376 37.1 2120 INTE C -203C-=HE 1.3401.0350.597 6.3 18.90 0.0599 1. 0.933 0.534 54*0 1 58IN2TE3B-251C -N2 1.3461.1750.59 3.6 ]'7.37 0.0592 1. 0928 1.170 136. 1 581N2TF3B-252H —N2 1*3461*1750*59 3*6 17.37 0*0592 1L 0.955 1.145 120* I 58IN2TE3B-253H-HE 1.*3461.1750.59 3.6 17.37 0.0592 1. 0.957 0.862 85.3 2118TN2TE3C-211C —HE 1.3030.9910.615 6*5 17.40 0.0592 1. 0*930 0.883 52.3 21181N2TE3C-212C -N2 13030*9910.615 6.5 17.40 0.0592 1. 0*930 1O220 72*2

COMPOUNDS DTA-I (CONT ) II II I IV V VI VII VIII IX X XI X I XIII XIV XV 21181N2TE3C-212H-HE 1.3030.9910.615 6.5 17.40 0*0592 1. 0.955 0.491 49.6 21181N2TE3C-21FI3H -N2 1.3030.9910.615 6.5 1740 0.0592 1. 0.955 0.762 70*7 2139IN2TE3C-261C-N2 1.2891.0130.607 6.6 17.40 0.0592 1. 0.930 1.175 80.6 2139IN2TE3C-262H-N2 1.2891.0130.607 6*6 17.40 0.0592 1. 0.955 0.302 74.2 2139IN2TE3C- 262C-HE 1.2891.0130.607 6.6 17.40 0.0592 1. 0.930 0.917 56.0 2139IN2TE3C-263H-HE 1.2891.0130.607 6.6 17.40 0.0592 1. 0.955 0.493 49.8 2139IN2Tf-3C-263 C-HE 1.2891.0130.607 6.6 17.40 0.0592 1. 0.930 0.956 61 8 1 64 PBSF B-301C-N2 1.3471.1720.59-3.8 24.30 0.0506 1. 1*349 0*426 111 1 1 64 PBSE B-302H-N2 1.3471.1720.59 3.8 24.30 0.0506 1. 1.364 0.479 109.0 1 64 PBSE B-302C-HE 1.3471.1720.59 3.8 24.30 0.0506 1. 1.345 0.218 76.0 - 1 64 PBSE B-303H-HE 1.3471.1720.59 3.8 24.30 0.0506 1. 1.360 0.309 71.3 1 64 PBSE B-303C-HE. 1.3471.1720.59 3.8 24.30 0.0506 1. 1.347 0*242 77.9 2106 PBSE C-8 1C-N2 1.2851.0130.606 6.6 24.30 0.0506 1. 1.353 0.352 54.7 2106 PBSE C-8 2i-N2 1.2851.0130.606 6.6 24.30 0.0506 1. 1.363 0.338 54.1 2106 PBSE C-8 2C-HE 1.2851.0130.606 6.6 2430 0630 0 0 1. 1.350 0.251 41,7 2106 PBSE C-8 3H-HE 1.2851*0130.606 6.6 24.30 0.0506 1. 1*366 0.240 41.0 2106 PBSE C-8 3C-HE 1.2851.0130.606 6.6 24.30 0.0506 1. 1.350 0.287 42.7 2133 PBSE C-231 C-HE 1.2750.9970*597 6*8 24.30 0.05061 1* *344 0.330 43*9 2133 PBSE C-232H-HE 1.2750.9970*597 6*8 24.30 0.0506 1. 1.363 0.240 40*9 2133 PBSE C-232C-N2 1.2750.9970*597 6.8 24.30 0.0506 1. 1.347 0.447 56.5 2133 PBSE C-233H-HE 1.2750.9970.597 6*8 24.30 0.0506 1 11.358 0.398 53.8 1 65 PBTE B-301C-N2 1.3441.1740.59 3.5 24.48 0.0433 1. 1.179 0.532 120.8 1 65 PBTE B-302H-N2 1.3441.1740*59 3.5 24.48 0.0433 1. 1.220 0.520 116.2 1 65 PBTE B-302C-HE 1.3441.1740.59 3.5 24.48 0*0433 1. 1.186 0.313 77*3 1 65 PBTE B-.303H-HE 1.3441.1740.59 3.5 24*48 0*0433 1 1.211 0*309 69*9 2105 PBTE C-8 1C-HE 1.2781.0050*606 6.5 24.60 0*0433 1. 1.205 0*300 47*6

COMPOUNDS DTA-17 (CONT.) I 11 III IV V VI VII VIII IX X XI XII XiI XIV XV 2105 PBTE C-8 3H-N2 1.2781.0050.606 6.5 24.60 0.0Q33 1. 1.213 0.440 65.6 2132 PBTE C-231C-N2 1.3121.0100.612 6.8 24.60 0.0433 1. 1.189 0.575 59.4 2132 PBTE C-232H-N2 1.3121.0100.612 6.8 24.60 0.0433 1 o 1.204 0.409 58.5 2132 PBTPE C-232C-HE 1.3121.0100.612 6.8 24.60 0.0433 1. 1.188 0.368 36.0 2132 PBTE C-233H-HE 1.3121.0100.612 6.8 24.60 0.0433 1. 1.201 0.300 37.8 2132 PBTE C-233C-HE 1.3121.0100.612 6.8 24.60 0.0433 1. 1.189 0.392 43.6 1 57SD2SE3B-292H-N2 1.3471.1720.59 4.3 19.20 0.0755 1. 0.898 0.976 209.5 1 57SB2SE 3B-292C-N2 1.3471.1720.59 4.3 19.20 0.0755 1. 0.849 0.822 158.0 1 57SB2SE3B-293H-HE 1.3471.1720.59 4.3 19.20 0.0755 1. 0.898 0.806 148.1 1 57SB2SE3B-293C-HE 1.3471.1720.59 4.3 19.20 0.0755 1. 0.858 0.495 101.0 2117S32SE3C-152H-HE 1.2891.0210.602 6.5 19.20 0.0755 1. 0.899 0.633 82.7 2117SB2StE3C-152C-HE 1.2891.02*10.602 6. 5 19.20 0. 0755 1. 0.865 0.396 65.8 2117S32SE3C-153H-N2 1.2891.0210.602 6.5 19.20 0.0755 1. 0.897 0.823 106.3 2117SB2SE3C-153C-N2 1.2891.0210.602 6.5 19.20 0.0755 1. 0.861 0.714 98.5 2138SS2SE3C-241C-N2 1.2911.0260.600 7.1 19.20 0.0755 1. 0.833 0.851 88.6 2138SB2SE3C-242H-N242-2 1.2911.0260.600 7.1 19.20 0.0755 1. 0.903 0.936 97.7 2138SB2SE3C-242C-HE 1.2911.0260.600 7.1 19.20 0.0755 1. 0.841 0.510 61.0 2138SB2SE3C-243H-HE 1.2911.0260.600 7.1 19.20 0.0755 1. 0.895 0.688 72.1 1 56S%2TE3B-281C-N2 1.3451.1700.59 4.0 19.80 0.0579 1. 0.876 0.671 147.3 1 56S32TE3B-282H-N2 1.3451.1700.59 4.0 19.80 0.0579 1. 0.913 0.715 143.2 1 56S32TE3B-282C-HE 1.3451.1700*59 4.0 19.80 0*0579 1. 0.878 0*425 83.0 I 56SB2TE83B-283H-H'E 1.3451.1700 o.59 4.0 19.80 0*0579 1. 0.900 0.278 72*9 2116SB2TE3C-141C-N2 1.3001*.0160*600 6.6 19.80 0.0579 1. 0.884 0.82 115.2 2116-S2Te L3C —142H-HE 1.3001.0160.600 6.6 19.80 0.0579 1. 0.900 0.348 65.0 2116R5F2TE3C- 142C-HE 1.3001.0160.600 6.6 19.80 0.*0579 1. 0.884 0*445 70* 5 2137SE2TF3C-171C-N27 1.2761.0440.602 6.6 19.80 0.0579 1. 0.881 0.741 126.0

COMP OUNrS ) - A-t (CONT ) I II III IV V v/ Vii VIII IX X XI XII XIII XIV XV 2137SB2TE3C-172H-N2 1.2761.30440.602 6.6 19.80 0.0579 1. 0.903 0.740 114*5 2137SB2T E3C-172C-HE 1*2761.0440.602 6*6 19.80 0.0579 1. 0.876 0o470 71o7 2103 SNTE C-7 IC-HE 1.2921.0250.607 6.0 19 50 0.0590 1. 1*063 0.653 52.5 2103 SNTE C-7 2H-HE,2 92`i*0250.607 6.0 1950 19 0.0590 1 1.081 0.311 48.3 2103 SNTE C-7 2C-N2 1L2921.0250.607 6.0 19*50 0.0590 1. 1.067 1.03 81*4 2103 SNTE C-7 3H-N2 1*2921.0250 607 6.0 19.50 0.0590 1. 1*100 0.527 76.0 2141 SNTE C-161C-N2 1*2760.9830.591 6.7 19.50 0.0590 1 1*065 0*982 85.2 2141 SNTE C-162H-N2 1*2760*9830.591 6*7 19.50 0.0590 1* 1.083 0*514 72*0 2141 SNTE C-162C-HF 1*2760.983.*591 6.7 19.50 0.0590 1. 1, 065 0*635 57*5 2141 SNTE C-163H —-E 1*2760.983C0.591 6.7 19*50 0.0590 1. 1.083 0.353 5 3 -3 1 20 ZNTE B-6 2H-HE 3 1 * 16 0 C.59 3.8 16.62 0 0752 1* 1*573 0*277 132*. 1 35 ZNTF- B-202H-HE 1.3521.155.C59 3*8 16*62 0*0752 1* 1*569 0*295 131* 1 73 Z7NTE B-352H-N2 1.3451.13 0.59 3.8 16*62 0.0752 1. 1.564 0*234 96.5 ] 73 ZNT'E B-353H-HE 1.34 51 *13 0*59 3.8 16*62 0*0752 1* 1.568 0.322 96.0 2 1.46 ZNT-E C-281C-HE 1.2390*9830.597 6*6 16*60 0*0752 1* 1*555 0*292 74*4 2146 ZNTE C-282H-HE 12890.9830.597 6.6 16.60 0.0752 1* 1.571 0*260 66.1 2 146 ZN T C -2 8 2 C-HE 1.28 0 9830.597 6 6 16*60 0.0752 1. 1.552 0 332 76 3 COMCiPOUitNDS AT THE TRANSITION TEMPERATURE 2148AGINSE-:C27TIC-HE1 314i.0000*592 6.5 18.50 0*0596 1. 1.010 0*675 5o85 2148AGINS EC- 27T2C-HEI*3 1- 0000*592 6*5 18*50 0*0596 1 1.010 1*245 10.2 2148AGINSEC-27T3H-N2].*3?i.*0000.592 6*5 18.50 0.0596 1. 1*025 0*624 10.2 2148AGINSVrC-27T3C-N2].S13 lI0000.592 6*5 18*50 0.0596 1* 1*010 1*42 9*08 2149AGINSECL- 29Ti:lC-N21.2860.9840.572 6*9 18,50 0*0596 1. 1,010 2.85 11*85

TRANSITIONS DTA-17 (CONT.) I I IV i I VI III IX X XI XI I XI II XIV XV 2149AGINSEC-29T2H-HEl.2860.9840.572 6 9 18.50 0.0596 1. 1 025 0.582 6.52 2149AGINSEC-29T2C-N21.2860.9840.572 6.9 18.50 0.0596 1. 1.010 1.474 7.77 2149AGINSEC-29T3H-N21.2860.9840.572 6.9 18.50 0.0596 1. 1.025 0.736 14.06 2149AGINSEC2C-29T3C-N21.28609840.572 6.9 1850 0.0596 1. 1.010 1*404 9.77 2149AGINSEC-29T4H-HE1.2860.9840.572 6.9 18.50 0.0596 1. 1.025 0.466 9.18 2149AGINSEC-29T4C-N21.2860*9840.572 6.9 18*50 0.0596 1. 1*010 1*663 6.57 1 62AG2SE B-29T4C-HE1.3471.1720.59 3.6 24.00 0.0736 1. 0.396 1*000 46.1 1 62AG2SE B-29T5H-HEI13471.1720*59 3.6 24.00 0.0691 1. 0.420 0.597 48.8 1 62AG2SE B-29T5C-N21*3471.1720*59 3.6 24.00 0.0736 1. 0.396 1.67 80.1 1 62AG2SE B-29T6H-N21.3471.1720.59 3.6 24.00 0.0691 1. 0.420 1.44 72.6 2113AG2SE C-11TIC-N21.3181.0390.597 6.1 24.00 0.0736 1. 0.396 1.85 72.6 o 2113AG2SE C-1lT2H-N21.3181.0390.597 6.1 24.00 0.0691 1. 0.427 1.85 72.3 2113AG2SE C-11T2C-HE1l3181.0390.597 6.1 24.00 0.0736 1. 0.397 0.935 34.7 2113AG2SE C-llT3H-HE1.3181.0390.597 6.1 24.00 0.0691 1. 0.416 0.833 31.8 2136AG2SE C-24T2H-N21.3061.0200.610 6.4 24.00 0.0691 1. 0.436 1.825 56.3 2136AG2SE C-24T2C-HE1.3061.0200.610 6.4 24 00 0.0736 1. 0.392 0.873 28.6 2136AG2SE C-24T3H-HE1.3061.0200.610 6.4 24.00 0*0691 1. 0.418 0.763 28.4 1 63AG2TE B-28T4H-HE1.3451.1700.59 4.0 25.50 0.0645 1. 0.440 0.600 29.8 1 63AG2TE B-28T4C-HEI.3451.1700.59 4.0 25.50 0.0674 1. 0.410 1.33 30.4 1 63AG2TF B-28T5H-N23.3451.1700.59 4.0 25.50 0.0645 1. 0*440 1.39 76.0 1 63AG2TE B-28T5C-N21.3451.1700*59 4.0 25.50 0*0674 1. 0.410 2.63 84.3 2112AG2TE C-1OT1C-HEI.2900.9980*578 6.5 25.50 0.0674 1. 0.410 1.88 27.2 2112AG2TE C-1OT2H-HEI.2900.9980.578 6.5 25.50 0.0645 1. 0.440 0.675 24.9 2112AG2TE C-1OT2C-N21.2900.9980.578 6.5 25*50 0.0674 1. 0*410 3.00 50.0 2135AG2TE C-28T3H-HE1.3101.0000.570 6.3 25.50 0.0645 1. 0.436 0.645 28.4 2135AG2TE C-28T3C-HE1.3101.0000.570 6.3 25.50 0.0674 1. 0.413 1.66 26.7

TRANSITIONS DITA- 17 (CONT.) II III IV V VI VII VIII IX X XI XII XIII XIV XV 2135AG2VE C-28T4H-N21i310IOO00.57/0 6.3 25.50 0.0645 1. 0.447 1.43 60.6 2135AG2TE C-28T4C-N21i3101.0000.570 6.3 25.50 0.0674 1. 0.407 2.80 59.6 2112AG2TE C-IOTIVC-N21.2900.9980.578 6.5 25.50 C,0.0633 1. 1.065 0.485 1.66 2112AG2TE C-I01-2H-N21.2900.9980.578 6.5 25.50 0u.0633 1. 1.075 0.543 1.58 2112AG2TE C-10 T 2 C-HE I.2900. 9 9 80.57 8 6.5 25.50 0.0633 1. 1.065 0.31 1.23 2135AG2TE C-28T2H-N2j.3101.0000.570 6.3 25.50 0.0633 1. 1.083 0.547 2.14 2135AG2TE CQ-2812C-N21.3101.0000o 7.570 6.3 25.50 o 0.0633 1. 1.065 0.529 2.*12 2135AG2TEL C-28T 3H-HEI.3101.-00000570 6.3 25.50 0.0633 1. 1.075 0.357 1.54 2135AG2TE C-28T3C-HE1.3101.0000.570 6.3 25.50 0.0633 1. 1.064 0.325 1.53 I 59IN2SE3B-25T4C-N21.3601.1750.59 4.0 17.70 0.10 1. 0.374 1.63 11. 1 1 59IN2SE3B-25T5H-N21.3601o.1750.59 4.0 17.70 0.11 1. 0.483 1.34 14.8 I 59IN2SE3B-25T6H-HEI.360o1.1750. 59 4.0 17.70 0.11 1. 0.483 0.762 8.63 1 591 N2.SE3B-25T6Cr-HEI.3601.1750.59 4.0 17.70 *I0.10 1. 0.374 1.33 7.28 2 1 1 9 1iN2.SE.3 C - 14i12H-H E 1.3180.9900.593 6. 8 17.10 0 o.. 1. 0.483 0.662 5.03 211 9 1N2SE3C-14T3H-N21.380.990.593 6.8 17.10 0.11 1. 0.483 1.42 10.1 1 58iN2TE:73B-25T4H-N21.3461.i750.59 3.6 17.37 0.0592 1. 0.900 0.545 3.68 1 581N2TE3B-25T5H-HE1.3461.1750.59 3.6 17.37 0.0592 1. 0.900 0.620 3.04 21181N2'LE3c-&2iT2C-N21 3030.9910.615 6.5 17.40 0.0592 1* 0.800 1.95 1.935

APPENDIX VII NOMENCLATURE 2 A = Area, cm A = Coefficient in solution of differential equation B = Coefficient in solution of differential equation B = Function defined by Equation A-5. 45 n C = Heat capacity, cal/ K C = Coefficient in solution of differential equation C = Function defined by Equation A-5. 46 n D = Determinant D = Coefficient in solution of differential equation D = Diameter, cm E = Energy, cal F = Free energy F = Shape factor (Cf. Eqs. 2. 62 and 2. 65) F = Various functions of p (Cf. Eq. 2.58) G = Thermal conductance per unit height, cal/cm sec K H = Square root of time constant (Cf. Eq. A-5. 19) H = Enthalpy I = Integrand of excess free energy function (Cf. Eq. 5.1 ff.) J = Bessel function of the first kind of order n n K = Thermal conductance, cal/sec K K = Time constant (Appendix II only) 170

171 L = Sample height, cm L = Lorentz number Lf or L = Heat of fusion or transition, cal/g M = A positive, real number (Cf. Eq. A-5. 32) M = Molecular Weight N = Atom fraction N = Number of items in computing an average P = Coefficient in the solution of a differential equation Q = Rate of heat flow, cal/sec Q = Coefficient in the solution of a differential equation R = A specific value of r (or x), cm R = Gas constant, 1.987 cal/g atom K S = Entropy T = Temperature, K T or T = Melting point or transition temperature t U = Overall heat transfer coefficient, cal/cm sec K V = Volume V = Laplace transform of v (Appendix V) X = A specific value of x, cm 4 X = The group of variables -AT (Chapter II) Y = Autotransformer setting, % of scale (Appendix II) Y = Bessel function of the second kind of order n n Z = Correlation factor defined in Chapter III a = Radius ratio R /R1 (Cf. Eqo 2.60) a = Function defined in Chapter II (Cf. Eq. 2. 39 ff.)

172 b = Radius ratio R3/R1 (Cf. Eq. 2.60) b = Function of a (Cf. Eqs. 2. 39 and 2.40 ff.) n n c = Function defined in Chapter II (Cf. Eq. 2. 39 ff.) c = Constant defined by Equation A-5. 31 0 c = Specific heat, cal/g K p d = Constant defined by Equation A-5. 10b e = 2.71828... f = Fraction of total radiation emitted by one surface which is seen by other h = Square root of thermal diffusivity (Cf Eq. A-5. 9) i =k = Thermal conductivity, cal/cm sec K m = Mass, g m = Heating rate, 0K/min (Appendix II only) n = Average number of atoms in cluster n = Index of refraction q = Dimensionless variable defined by Equation A-5. 1 Oa. r = Radius, cm s = Laplace transform variable t = Time, sec or min; with subscript, a time constant v = Temperature variable (Cf. Eq. 2. 3) v = Temperature variable (Cf. Eq. A-5. 11) x = Distance variable, cm x = Mole fraction of compound in solution y = Temperature function (Cf. Eq. 2. 4)

173 z = Temperature function (Cf. Eq. 2. 5) a = A defined function of s; with subscript, an eigenvalue a = A function of the equivalent conductances for radiation (Cf. Eq. 3. 9) a = Temperature coefficient of electrical resistance a = Absorptivity p = A defined function of s; with subscript, an eigenvalue p = A function of the equivalent conductances for radiation (Cf. Eq. 3.8) $ = Real number in Equation A-5. 32 y = Heating rate, K/min y = Activity coefficient E = Dimensionless parameter (Cf. Eq. 2. 56 ff.) E = Emissivity = Dimensionless space variable, r/R or x/R Tn = Dimensionless parameter, Ri /R. 9 = Differential temperature, T- TR K v = Dimensionless parameter, T/t5 (Cf. Eq. 2.58 ff.) = 3.14159... 3 p = Density, g/cm p = Reflectivity p = Residue -= Stefan-Boltzmann constant ( = Standard deviation c = Entropy factor (Cf. Eq. 4. 2)

174 o = Electrical conductivity T = Time constant, sec or min T = Transmissivity = Dimensionless parameter (Cf. Eq. 1.3) = Thermal conductivity ratio (Cf. Eq. 2. 10 ff.) = Function defined by Equation 5. 3

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