Creator:J. Smiltens Date Created:July 1958 Place Created:Bedford, Massachusetts Keywords:silicon carbide,gaseous equilibria Context:article from conference on silicon carbide ************************************************** AFCRC -TR-58-I6I ASTIA Document No. ADI52446 THE GASEOUS EQUILIBRIA IN THE CARBON-SILICON BINARY SYSTEM J. SMILTENS JULY 1958 ELECTRONICS RESEARCH DIRECTORATE AIR FORCE CAMBRIDGE RESEARCH CENTER AIR RESEARCH AND DEVELOPMENT COMMAND UNITED STATES AIR FORCE BEDFORD MASSACHUSETTS Requests {or additional copies by Agencies of the Department of Defense, their contractors, and other government agencies should be directed to the: Armed Services Technical Information Agency Arlington Hall Station Arlington 12, Virginia Department of Defense contractors must be established for ASTIA services, or have their 'need-to-know' certified by the cognizant military agency of their project or contract. All other persons and organizations should apply to the: U. S. DEPARTMENT OF COMMERCE OFFICE OF TECHNICAL SERVICES, WASHINGTON 25, D. C. ELECTRONICS RESEARCH DIRECTORATE SILICON CARBIDE AIR FORCE CAMBRIDGE RESEARCH CENTER 2-3 APRIL 1959 CONFERENCE ON SILICON CARBIDE AIR FORCE CAMBRIDGE RESEARCH CENTER L. G. HANSCOM FIELD BEDFORD, MASSACHUSETTS 2-3 APRIL 1959 TENTATIVE PROGRAM Growing of Silicon Carbide Single Crystals A. Growth by Sublimation B. Growth by Gaseous Cracking C. Growth from Melts D. The Binary System Silicon-Carbon Silicon Carbide as a Solid A. Crystal Structure B. Stacking Faults C. Growth Spirals and Other Dislocations Silicon Carbide as a Semiconductor A. Energy Band Structure B. Electronic Properties: Resistivity, Carrier Mobility, Carrier Lifetime, etc. C. Optical Properties: Photoconductivity, Electroluminescence, etc. Silicon Carbide Devices A. Diodes, Unipolar Transistors, etc. B. Thermoelectric Generators C. Luminescent Sources, Photoconductors, etc. CONFERENCE ON SILICON CARBIDE AIR FORCE CAMBRIDGE RESEARCH CENTER L. G. HANSCOM FIELD BEDFORD, MASSACHUSETTS 2 - 3 APRIL 1959 Date _ To assist in Conference planning, it is requested that you furnish information in the following categories: 1. Name(s) of those planning to attend. 2. Topics of interest in addition to those listed in the tentative program. 3. Papers, if any, to be presented. Title Author, Organization, Address Please return this participation notice to: Mr. J. R. O'Connor, CRRCSS Electronics Research Directorate Air Force Cambridge Research Center L. G. Hanscom Field Bedford, Massachusetts CONFERENCE ON SILICON CARBIDE AIR FORCE CAMBRIDGE RESEARCH CENTER L. G. HANSCOM FIELD BEDFORD, MASSACHUSETTS 2 - 3 APRIL 1959 The Electronics Research Directorate of the Air Force Cambridge Research Center will sponsor a Conference on the new high temperature semiconductor, silicon carbide. This Conference will be held in Boston, Massachusetts, on 2 - 3 April 1959. The purpose of this Conference is to further the exchange of information among scientists interested in research and development of silicon carbide. Emphasis will be placed upon theoretical and experimental aspects of this material as well as its various applications. It is hoped that the Conference will provide a focal point for all scientific work carried out on this relatively new semiconductor. To help achieve such a goal many leading workers on silicon carbide in this country and abroad have been invited. As a group or individual actively interested in silicon carbide you are invited to attend or to send technical representatives to this Conference. It is requested that you complete and return the enclosed participation notice by 1 November 1958. A tentative agenda illustrates the range of material to be covered. The chairman will welcome suggestions for additional topics and comments on the overall program; suggestions may be made on the participation notice. Papers on one or more of the topics of interest are invited. These papers will be limited to approximately 30 minutes, including discussion period. An abstract of about 200 words should be submitted to the chairman before 1 January 1959. The Conference also encourages presentation of shorter "recent news" type papers of about ten minutes in length; abstracts of these should be submitted not later than 1 March 1959. Although this announcement is being distributed as widely as possible, some workers in the field of silicon carbide may have been unintentionally overlooked. Please feel free to circulate this invitation to others having an interest, or potential interest, in this new semiconductor material. AFCRC -TR-58-I6I ASTIA Document No. ADI52446 THE GASEOUS EQUILIBRIA IN THE CARBON-SILICON BINARY SYSTEM J. SMILTENS PROJECT 5620 TASK 56202 JULY 1958 ELECTRONIC MATERIAL SCIENCES LABORATORY ELECTRONICS RESEARCH DIRECTORATE AIR FORCE CAMBRIDGE RESEARCH CENTER AIR RESEARCH AND DEVELOPMEN%.COMMAND UNITED STATES AIR FORCE BEDFORD MASSACHUSETTS ABSTRACT A method is presented, for calculating the pressure and composition of the vapor phase in the carbon-silicon binary system. The following quantities are necessary: partition functions of the various molecular species present in the vapor, vapor pressures of graphite and liquid silicon, and the standard free-energy increment for the change C(graphite) + Si(liquid) = SiC(solid) . A study is made of the equilibrium: vapor and solution of carbon in liquid silicon. iii ACKNOWLEDGMENTS The author is grateful for the encouragement and support received from R. F. Cornelissen, Chief of the Semiconductor Section, and from J. R. O'Connor, Chief of the Materials Research Unit. It is his desire to thank Miss H. E. Quirk of the Computer Laboratory for the numerical calculation of the AgJ^^ function in Appendix 2 and Miss 0. Denniston for suggestions on the written presentation of this report. v TABLE OF CONTENTS Abstract .....................................iii Acknowledgments.......................... . v 1. Introduction ............................1 2. Tentative Sketch of the Phase Diagram .......................1 3- Preliminaries .................... ..... 5 1+. The Conditions Derived from the Five Equilibrium Boxes ..... 11 Appendix 1: The Present Status of the Vapor Pressure of Liquid Silicon 29 Appendix 2: The Present Status of the AG * Function..............31 vii THE GASEOUS EQUILIBRIA IN THE CARBON-SILICON BINARY SYSTEM 1. INTRODUCTION This report is prepared with a primary purpose: To arouse the interest of colleagues working in molecular and mass spectroscopy, thermochemistry, and general physical chemistry in the carbon-silicon binary system. With the new semiconductor silicon carbide rapidly emerging, the practical importance of the carbon-silicon system has greatly increased. In order to perform numerical calculations, we still need unquestionable values for the vapor pressure of liquid silicon and the heat of formation of silicon carbide, together with, possibly, a large set of molecular constants for the species present in the binary vapor phase. It is hoped that with more workers interested in the problem, these data will gradually become available. Since the calculations involve sums of terms with two running indices, some of the derivations (especially towards the end of the work) contain large, though not complicated, equations. With one exception, nevertheless, the final results are brief forms. It may appear that some elementary steps and repetitions in the derivations could have been omitted. However, in view of the preliminary nature of the report, one should not be too concerned about this, inasmuch as it is our purpose to familiarize the reader with the mathematical device used. 2. TENTATIVE SKETCH OF THE PHASE DIAGRAM At the outset it is clear that pressure must be considered along with composition and temperature. Thus the complete phase diagram should be visualized as a prismatic body with composition, temperature, and pressure coordinate axes as its edges. We have studied an isothermal composition-pressure section of this body. Figure 1 shows the initial sketch of such a section, supposedly as projected. The sketch is based on the following considerations: M T = const. 0 0.5 Atomic fraction of carbon, X Fig. 1. A tentative isothermal cross section of the carbon-silicon phase diagram. (1) The positions of points a and h are given by the vapor pressures of pure carbon and silicon, respectively. (2) At very low pressure, i.e., corresponding to the top region of Fig. 1, there will be ohly binary vapor phase. (3) As the pressure is increased, the first condensed phase which appears is graphite. Then the other two condensed phases, silicon carbide and liquid silicon appear. (4) According to the studies of Ruff and Konschak, "Shere exist no other silicon carbides than SiC. (5) It is assumed that silicon does not substitute for carbon in graphite. 2 Silicon carbide is considered to be stoichiometric only. At temperatures around the melting point of silicon, the solubility of carbon in liquid silicon is negligible. Thus we can draw three vertical lines: ab and extension; ce and extension; also hg and extension for the three condensed phases: graphite, silicon carbide, and liquid silicon. At higher temperatures, the solubility of carbon in liquid silicon becomes appreciable. The extension of the problem is treated in the final part of this work. (6) In application for present purposes, Gibbs1 phase rule is formulated as follows: the maximal number of coexistent phases equals the number of components plus two. There are two components: silicon and carbon. Hence the maximal number of phases is four. This corresponds to a quadruple point - a singular pair of temperature and pressure. On the line bed, three coexistent phases have been indicated: graphite b, silicon carbide c, and vapor d. This combination of phases being short of the maximal number of phases by one phase, therefore, has one degree of freedom. Analytically, pressure at the bed level is a function of temperature. Above the line bed, we have a combination of only two phases: graphite ab and vapor ad. Here there are two degrees of freedom. Analytically, at a given temperature this combination of phases can exist at a range of pressures from a to b. Analogously, the line efg has been established. 1. 0. Ruff and M. Konschak, Z. Elektrochem. 32, 515 (1926). 2. J. A. Lely, Ber. deut. keram. Ges., 32, 229 (1955)- k Peq. box Si i rSi. Si2 i Si, Si, i 3 | ^CSi eq.box. C3,----, Si,Si2,Si3,....., CSi, C2Si, CSi2,. - "Condensed. Phase CgSi Fig. 2. Van't Hoff's equilibrium box. An extensive use of the thermodynamical concept of van't Hoff's equina 4 librium box ' has been made here (Fig. 2). It is an ideal vessel supplied with a set of hypothetical membranes, each of which is permeable to only one of the molecular species present in the binary vapor in the box. P^ , Pg^j P , ... are the partial pressures of the respective molecular species, and 2 ^eq box total pressure of the mixture. By reversible operation of the pistons, certain species can be withdrawn or introduced into the box. From these mental experiments, the free-energy increments for changes involving molecular species within the box can be calculated. In Fig. 1, the areas I, III, and V, and the lines II and IV can be thought to correspond to van't Hoff's equilibrium boxes. As already mentioned, pressure of the boxes I, III, and V is variable within a certain range. In the boxes II and IV, it is constant. After some preliminary arrangements that will be made in the next chapter, we will again return to the equilibrium boxes, deriving from them information regarding curves ad, df, and fh. 3. PRELIMINARIES Let us denote a general molecule present in the binary vapor phase by CLSiy Further, consider a volume V cm of vapor phase, and let the number of C.Si. molecules present in this volume be N... At this point it should 1 J iJ be mentioned that the quantity V will cancel out in the derivation, and therefore we do not have to be concerned about its magnitude. Our treatment is confined only to the low-density vapor cases when the forces of cohesion between the molecules can be neglected. Then, according to statistical 3. J. H. van't Hoff, Z. physik. Chem., 1, 48l (1887). b. S. Glasstone, Textbook of Physical Chemistry (New York: D. Van Nostrand Co., 19W), cf. p. 818"! mechanics, w. . ij N. . Q. . ij ij Nl,0 N0,l , (D where Slj* 0' anC^ Sd 1 are functions of the C^Si . molecule, monatomic carbon, and monatomic silicon, respectively; w. . is the work of formation, in ergs, of the molecule C.Si.. 1J ' S ' i J It is the work which the world's work bank gains when i atoms of carbon, separated at infinitely large distances, at rest and at ground state of excitation, come together with j atoms of silicon, initially at the same conditions, and form a C.Si. molecule, also at rest and at ground state. This work is usually 1 J 0 called the dissociation energy D^ , and is given in electron volts. The relationship between w.. and . is: ij 0 ij w. . = 1.60186-10"12 D_°. ij 0 IJ' l 6 = Boltzmann1s constant = 1.38026-10" , T = temperature in °K 5- G. S. Rushbrooke, Introduction to Statistical Mechanics (Oxford: Clarendon Press, 19^9), cf. p. 102, Eq. (37"). 6. R. Fowler and E. A. Guggenheim, Statistical Thermodynamics (Cambridge: University Press, 19^9), cf. p. 165, Eqs. (506,7), (507,1)- 7- R. H. Fowler, Statistical Mechanics (Cambridge: University Press, 1955), cf. p. 164, Eqo (4797; Equation (l) can "be rewritten: w. . M 1J \,J W 1J We shall denote and ^ E a (3) Qi,o = X - CO 1 Then (2) "becomes w. . N. . = a1 Q. . e*^ . (5) ij ij Also, either from the definitive equations (3) and (k), or from (5), considering that both w^ ^ and w^ ^ are zero, we obtain: and Ni,o = a Ql,0 ' (6) N0,1 = x V • (7) The partition function Q.. can be rigorously decomposed into two factors J the translational partition function Q. . . and the internal partition xrans ij function Q. , . .. mt ij Q. . = Q. . . Q. , . . • (8) ij trans ij mt ij The translational partition function is /2jt M. At\ | Vans ij " ) 2 " V < (9) where M. . .the molecular weight of the C.Si. , ij 1 J h = Planck's constant = 6.6238*10"2? , NAv = Avogadro's number = 6.0254.102^ . The internal partition function ^ has to be calculated from the energy levels of the C^Si^. molecule, due to internuclear vibration, rotation, and electronic excitation of the molecule. When Q^ . . in (8) is replaced by the right-hand side of (9), and the urails ij new expression for Q. . thus, obtained is substituted into (5), we get J v N. . = ( T 2 V a1 \j M. .2 Q. , . . e"^ . (10) ij J 1J int 1J Av The product of the last three factors on the right-hand side of (10) is a characteristic of the C.Si. molecule alone. Let us denote it by k. .: 1 J w.. 1J 3 ^ kij ^ V «int ij 6 ' (ID Then, upon calculating the numerical value of the aggregate of universal constants, (10) becomes 3 N . = I08789.IO20 T 2 V a1 k. . . (12) -'-J 1J Each CiSi . molecule'contains i atoms of carbon. Therefore, the total i J number of carbon atoms, free and combined, in the volume V, is I u 1HiJ ' (13) and the total number of silicon atoms is I ^ • (14) The ratio of carbon to silicon atoms in the vapor then is iN,. T-. i a1 ^ k. . x = 3-J = _il . (15) T. j N. . j a1 V3 k. . ij ij ij ij To indicate composition in Fig 1, we used the atomic fraction of carbon X, which is the conventional usage. However, for our present purposes, x as defined by (15), and which also corresponds to x in SiC^jWill be a more convenient variable. For pure carbon, x = , and the line ab in Fig. 1 is removed to infinity. However, it will be shown that the curve ad when plotted as log10 ^ versus x reaches infinity at zero slope, which is a simplifying feature. Also, the new variable x is convenient in the treatment of the solution of carbon in liquid silicon. The total number of all kinds of particles in the volume V is I N. = I.8789 • 1020 T 2 V Z. Ci1 k. . . (16) ij IJ ij iJ At any instant, the pressure, volume, and temperature of the vapor will be connected by the gas law: In,. PV = 1J • RT , (17) Av 10 where R = 82.079, when P is measured in atmospheres. In (17), when N. . is substituted by the right-hand side of (16), V on x j 1 j both sides of the equation cancels out, and upon multiplying and dividing of the constants, we get P = 2.5595.10"2 T 2 X. a1 k.. . (18) ij ij -2 * The product 2.5595*10 T 2 , which is a temperature-dependent constant, we shall denote by a. Then p = a £ a1 k. . , (19) ij and the partial pressure of the species C^Si^. is P. . = a a1 k. . . (20) ij ij 9 In the. case of pure carbon vapor (l$) becomes p ^ = a J a£ k._ , (21) carbon 1 9 1Q0 ' ' the subscript 'Jo" indicating absence of silicon; and for pure silicon vapor P . = a Z k . , (22) silicon j o ©g the subscript "o" indicating absence of carbon. The quantities Q^ and formally are temperature-dependent constants; we shall need them later. They can be calculated from the experimentally measured vapor pressures of carbon and silicon by means of (21) and (22), respectively, provided that a sufficient number of k^ and k^ values is available. 1 4. THE CONDITIONS DERIVED FROM THE FIVE EQUILIBRIUM BOXES The First Equilibrium Box. Curve ad. - In the first equilibrium box, the condensed phase is graphite, and the pressure of this box can be any fixed value in the range ab. By means of this box, we can find the Gibbs1 free-energy increment Ag^"8" for the change Q- C(graphite, 1 atm,T) + Si(monatomic gas, 1 atm,T) + AG^ = SiC(gas consisting of SiC molecules only, 1 atm,T) (23) as AG^ = RT m V b 1 • (24) p 1,1 eq b I In order to calculate Ag^"9" in calories, R = 1.9^773 mus"l: used. . Q According to thermodynamics, the value AG^ is independent of the pressure, which has been chosen in the range ab, of the box. Therefore, if in (24) P„ , , T and P-, , _ are substituted by a\ , - k- _ and aa , _. 0,1 eq b I 1,1 eq b I J eq b I 0,1 eq b I \ k , respectively, we get eq D 1 1,1 -©. „ l^i (25) eq b I k 1 j -L which is a constant for given temperature. The condition (25) subsists along the whole extent of the curve ad; consequently, also at the terminal point a. Therefore a , _ = const = a , (26) eq b I o v which expresses in our way the fact that the pressure of carbon vapor in this box, on account of graphite being present as a condensed phase, is constant. By means of (26), (19), and (15), the curve ad can be calculated. We need only find the \ value for the terminal point d; this will be possible from the second equilibrium box. 12 Before leaving, the first equilibrium box, we shall furnish the proof that lim d 1Og10 P" x —> ©o dx d losio P dx o = 0; (27) a = a x = 0 i.e., that the curve ad, when the abscissa is x, instead of X, reaches the point a (now at infinity) at zero slope. We shall proceed as follows : d 1Og10 P d lQgio P dx a = a dX dx dX (28) a = a. d lQg10 p dX a..= a _ 0.4343 i ^ 0 0 P dX (29) a = a From (19) dP dX a. = a = a I. j a1 x^-1 k. . ij o ij (30) At X = 0 in (39), all terms which contain X vanish, and we get dP dX a = a = a (k0,l + % o ' ' X = 0 () 1 Then (29) becomes d log 1 10 P dX a=oc c X=0 - 0.4343 a (k0A + ao k1A) i carbon From (15) dx dX I ij ij a£ XJ_1 k. IjJCg a1 XJ k..\ ° ij) <1 .2 i , j-1 . - J "o^ kij)(£ __J I a1 \J k. o ij) la=a which at X = 0 becomes dx dX (k + a k ) . 0 - (k + a k_ .) X i a1 0,1 o 1,1' 0,1 o 1,1' 1 o 1O —- = _ CX). a=a o X=0 Therefore from (28), (29), (3l), and (34), indeed, d logio P dx a=a = 0 x=o We can arrive at the same result by a simpler method: As X—> 0 in the P and x functions, all terms which contain X with exponents > 1 vanish as small of higher order. Thus (19) becomes P = a X ^(k.Q + , il (36) and (15) by a still more drastic simplification becomes X i^So 1 x = --- , il (ft) which at a = ^ gives dP d\ = a a=a y cc1 k. o il (38) and dx dX Therefore, Qt=a Y i a1 k i— r> o iO Y_ a1 k . o il (39) dP dx (V a1 k.J2 L- o 11 ' y i a1 k. n 1 x2, o iO a=a (40) which at \ = 0 becomes zero. This result when substituted in (29), in which \ previously has been formally replaced by x, gives (35)* The Second Equilibrium Box. Point d. - The condensed phases in this box are graphite and silicon carbide. The pressure is a single value, P^. By means Q of this box, we can find the free-energy increment A G^ for the change C(graphite) + Si(monatomic gas) + A = SiC(solid). (M) We have omitted the "1 atm, T" in the parentheses at the chemical symbols, since it is understood that in the future the superscript -e- at the free-energy increment (the standard free-energy increment) will be a sufficient indication of this. 1 It can be shown that AGir - RT ln V eo II • m Q On the other hand, AGjj can also be found from the two changes: first, C(graphite) + Si(liquid) + AG-j.-^ = SiC(solid) ; (43) and second, Si (liquid) + AG^.^ = Si(monatomic gas). (44) Q AGjj ^ is the standard free-energy increment for a change with all phases in condensed state. This increment can be calculated by conventional methods from the heat of formation of silicon carbide, the heat of fusion of silicon, and the heat capacities of graphite, silicon (solid and liquid), and silicon carbide. Since this quantity is important throughout the whole work, we shall change the notation AGtt^ SAG"0" '. (45) II 1 cond x The standard free-energy increment for the second change (44), can be shown as AG* = - RT In P . , (46) II 2 0,1 liq silicon where P- , n . ., . is the partial pressure of monatomic silicon above 0,1 liq silicon pure liquid silicon at the temperature under consideration. Eq. (44) can be transposed to Si.(liquid) = Si (monatomic gas) - . (47) When (47) is substituted for Si(liquid) in (43), we shall get (4l), and Q consequently AG^^ must be AGtt®" = AG*, - AGTT#0 . (48) II cond II 2 16 or, Q When in (48), AGn is substituted by (42) and AGn 2 by (46) , we get RT In P , TT = AG * , + RT In P. . . . .. . (49) 0,1 eq b II cond 0,1 liq silicon AG^, p cond Q'1 11- = e RT , (50) p 0,1 liq silicon cond x tt = X e RT . (51) eq b II o ' Therefore, the coordinates of the point d are or, and AG^, cond = Xq e RT , (52) a= a . (53) d o The Third Equilibrium Box. Curve df. - By analogous reasoning, we can find that the equation of the curve df is cond QX = ao \o e RT . (54) The Fourth Equilibrium Box. Point f. - Here f o and cond 0 in the P and x functions, all terms which contain a with exponents ^>1 vanish as small of higher order. Thus (19) becomes P = a I N! (k01 + a klj} ' (65) and (15), by a still more drastic approximation becomes I x = a 4- X k.. _i_0 t j ^ K- j o Oj (66) Therefore, dP da X = X c CtJ ~ 0 dP da and dx da X = X dx da = a I k , X = Xq j o Ij a = 0 I xJ kT . J O Ij 4- j X k . J J o Oj X = X c a = 0 (67) (68) Then dP dx . = a Y j \J k . X = X h- o Oj ° J a = 0 (69) When the right-hand side of (69) is substituted for dP/dx in (29), in which previously X has been formally replaced by x, a = a^ replaced by a = 0, and l/P by l/pli(1 silicon' We again o^3-1-11 (64). What relationships can be extracted from the present mathematical device when the solubility of carbon in liquid silicon cannot be neglected? 0 Let us consider the change in^which x is a positive fractional number, e.g., 0.2 : (l-x) C(monatomic gas) + SiC (liquid) + A& * = SiC(gas consisting of x SiC molecules only) (70) Again, the superscript -e- at the Gibbs1 free-energy increment stands for a change when all reactants and products are at 1 atm and at the same temperature Tj we therefore, as previously agreed, do not indicate these conditions in the parentheses at the symbol of the substance concerned. The formula SiC^ is used only as a shorthand symbol to denote a solution of carbon in silicon, in which on every atom of silicon there are x (a fraction) atoms of carbon. It implies no structural notions. It can be shown that p l-x AG* = RT In - , (71) 1,1 where P and P are the partial pressures of the species C and SiC above 1,0 1,1 the solution SiC . x Let us consider another change: x C(monatomic gas) + Si(liquid) + Agv = SiC (liquid) . (72) V X It can be shown that /»X Ag = RT r In P dx . (73) JO ' By adding (70) and (72), we find that the standard free-energy increment for the change C(monatomic gas) + Si(liquid) + Ag^ = SiC(gas consistihg of SiC mole- * cules only) (74) must be Agv" =AGv i + AGV 2 . (75) 1 Upon substituting in (75) for AGy 1 the right-hand side of (71), and for AGy"^ the right-hand side of (73), we obtain l-x P, n n x AG * = RT In --+ RT / In P, _ dx . (76) v- 3 p1?1 J 0 Q Now, as can be seen from (74), the AGV ^ is independent of x. When P . is replaced by aa Is and P by aa \ k , (76) can be simplified: a. j jl _l j _l rx AGv * ki i x In a - / In a dx . + In \ = --p- In • (77) J9 RT \o The limit for (77) as x —>■ 0 is: A G % k lim (x lntn) + In XQ = -^ In ^^ • (78) x-vO 1,0 Therefore (77) can be rewritten as x In Q! — f In a dx + In X = lim (x In a) + In x (79) J0 x-»0 or, rx x In a - lim (x In a) - I In a dx =-ln X + In X (80) x->0 J 0 or, ~x . /l x x din a = - din \ . (8l) Jo Jo When the integral signs of (8l) are dropped, we obtain din X = - x . , (82) d m a soln g which can be recognized as the Gibbs-Duhem equation. We have attached the subscript "soln" at x in order to emphasize that x is the atomic ratio of carbon to silicon in the solution, whereas a and \ are the effective concentrations of monatomic carbon and silicon in the gas phase equilibrating with the solution. We shall rewrite (82) as da = " a xsoln ' By means of (83), we can approach the problem of the slope d log^^l/P)/dx. By differentiation, As the case is general, all we can write for dP/dx is + ax dP da _ dx \2aJ\ \3Kja da From (19) and . a I i c^. ^ k±J (86, 1 J y 1J 8. G.N. Lewis and M. Randall, Thermodynamics and the Free Energy of Chemical Substances (New York: McGraw-Hill Book Co., 1923), cf. pp. 207-210. When dA/aa in the numerator of (85) is substituted by the right-hand side of (83), we obtain a /y i a1 k. . - X n V j a1 k. .) • (88) f-. ij soln A ° xj/ a ,xj " xj When in (88) we carry X j ^ X"' k. , before the parentheses, we get ij - (x - x n ) I j a1 k. . . (89) CO. vap soln Jj J ij K For the first term in the denominator of (85), we get from (15) L \ (l iV"V k. .) X j c*V k. . - ll ij Q^-Vk. .) I. i a1 k. = JiJ_xj/ 13 _ij \xj _ij/ xj_x. /V . . \2 (90) X -j a1 k. A xj * xj/ which when divided in the numerator and denominator by X j a^ ^ k. . ij simplifies to U \ T. i2 a1'1 k. . - x X ij a1_V k. . = ^J_ij vap xj _xj________^ j ~t>oc \ X j a1 XJ k ij 1J For the in the second term of the denominator of (85), we get ^ . (x ij a1 x^1 k. .) f. j a1 A. .- (X j2 aVS. .) I i k. . _ Xxj _xj/ IJ _XJ \lj _xJ IJ_x^ ^ fe j^xJk.J ■/a /y . . \2 X j a1 k. . XJ XJ 24 which when divided in the numerator and denominator by ) j a1 k. . ij gives />v\ £ ij a1 k. . - x T. j2 a1 7J'1 k. . HE ] = _xj vap ij _ij Xb\h y . i ] A a \J k. . ij ij When in the denominator of (85) the factor dfy%2is substituted by the right-hand side of (83), we get 1 a a X £ j ax k. . ij ij Then (9^) becomes (93) t" - fflf xsoln] ' (*> From (91), T-. i2 a1 k. . - x £ ij a1 k ft*) = _ij vap ij 0 * ij , , V i i ' ™ f. j a k. . ij and from (93) £ i.i a1 k. . - x £ i2 a1 ^ H ij a k. . - x j a k. . - _ ij vap ij _ij r :- " ^ , £ i2 afV k. . - X T-. ij aV k. . - x £ ij aVk. .+ x . x £j2a\jk. . 1 _ij_ij vap ij _ij soln ij _ij soln vap ij"_ij . ^7) T-. j a1 k. . ij ij The numerator of (97) can be factored to £ a1 k. .(i-j x ) (i-j x , ) • (98) ij ij vap'v u soln' ' Thus we finally obtain for (94) X n1 x-J f-. a k. . (i-j x ) (i-j x n ) ij_ij ^ 0 vap \ 0 soln a [ j a1 \J k. • ■ i, ij From (89) and (99) we get — (x - x _ ) A j a1 k. . dp a vap soln' ij 0 xj dx X a1 k. . (i-j x )(i-j x _ ) £j 1j vap' 0 soln' a X j a1 k.. ij which simplifies to (I J a1 k, y (99) (100) f- = a (x - x . ) -U3LI-—-----(101) dx vap soln' 7 a1 \J k. .(i-j x j(i-J x n ) i-. ij vap'v 0 soln' ij When in (84) the factor dP/dx is substituted by the right-hand side of (101), and P is substituted by the right-hand side of (19), we finally obtain: d loS10 f (X j a1 k )2 - - -1* J = - 0.4343 (x - x . ) ua-n cnl n ' ij dx vaP soln' __ 1 1 . , , _ (I ct-A ) V aVk. ,(i-j x ) (i-j-x 1 )1 XQ/ L- 0 vap'v ° soln-1 i.i 1 i J 1J (102) When x = x the slope becomes zero, in agreement with the Gibbs-vap soln 9 Konovalov theorem. (64) It remains only to be proved that at xsolr = 0 and- <2 = 0, (102) becomes The proof follows: In (102) we remove x and replace x by the v ' soln * vap right-hand side of (15)• Accordingly, , , 1 y ja1^ k.. Y i a1 k. . d loSin B h IJ h XJ 10P _ _ 0.1,31,3 iJ . .---ij---(103) V a1 k. . y i a1\J k. ,-x V ij a1 :k. . f-. ij h ij vap f-. 0 ij ij ij ij At a = 0 the first fractional factor in (103) becomes I J ^ k0i y k_. I— o O.i O Oj (104) - ) o "Oj J and the second fractional factor becomes an indeterminate form. Let us resolve it. By differentiating both the numerator and the denominator with respect to a , we get I i^-Vk.. ij--- . (105) I i3 c/-Vk..- I ijaVk. ,-x y iVVk.. ij \2>a )\ f- d ij vap f ^ 9- I. Prigogine and E, Defay, Chemical Thermodynamics (New York: Longmans, Green and Co., 1954), cf. p. 282. When a = 0, the numerator and the first term in the denominator of (105) become / k, . . The second term in the denominator becomes zero because, i o lj d _ 4 from (91), the first factor in this term is ( kij)/( ^ j and the J J second factor is zero. The third term in the denominator is also zero because the first factor in this term is zero,,and the second factor is ) j k .. ° U Therefore, since (105) is unity, (103) reduces to (64). J APPENDIX 1 THE PRESENT STATUS OF THE VAPOR PRESSURE OF LIQUID SILICON In Fig. 3, data are assembled on the vapor pressure of liquid silicon as given by Ruff and Konschak,1 Baur and Brunner,10 Grieger,11 and Honig.12 In their experiments, Ruff and Konschak, and also Grieger, used a silicon carbide crucible as a container for the liquid silicon. Therefore, their measurements actually represent the pressure for the level efg in Fig.l. Calculating the separation hg for 1700°K, the temperature at which it is safe to assume no solubility of carbon in liquid silicon, one finds a very small value. This is due to the vapor pressure of graphite being much lower than the vapor pressure of liquid silicon. Hence the silicon carbide container is permissible. At temperatures above 1700°K, this discrepancy may increase considerably. Baur and Brunner have used an aluminum oxide crucible, which may have distorted their results. Honig developed his function from mass spectroscopic measurements and from k, comparison with germanium. In the range of low 10 /T, we have extrapolated somewhat beyond Honig's original limit (lO^/T = 3-6). Since his function is lightly curved, this may have affected the slope of our plot. It is believed that the error thus introduced is small. 10. E. Baur and R. Brunner, Helv. Chim. Acta 17, 958 (1934). 11. 0. Ruff, Trans. Electrochem. Soc. 68, 87 Tl935)> curve 3 in Fig. 6. 12. R. E. Honig, RCA Rev. 18, 195 (1957)• 29 APPENDIX 1 Fig. 3. The vapor pressure of liquid silicon. APPENDIX 2 Equation (107) has been calculated by Humphrey and his co-workers. It represents the average function for both the hexagonal and cubic modifications their difference being small. Equation (108) has been calculated in this work The numerical values of thermochemical data used are given in Table 1. APPENDIX 2 TABLE 1 THERMOCHEMICAL DATA USED FOR CALCULATION OF EQUATION (108) uJ ■p- SUBSTANCE c 298.16 CP = a + b • 10"3T + c • lO^T"2 Form Formula cal Refer- cal Refer- Weight fwt deg ence fvt deg ence a b c Range of Validity OK Deviation ± percent Graphite 12.010 1.3609 15 4.10 1.02: I -2.10 298 - 2300 2-5 ! 16 Silicon, solid 28.06 4.47 15 5-79 0.56: t -1.09 298 - 1200 2 ; 16 Silicon, liquid not given .. 6.12 ' 0 ! 1 0 i I 1 <1825 0.4; 17 SiC, solid, hex.(?) 40.07 3-935 15 8.93 3.00 i -3.07 j 1 298 - 1700 ■ 1 3 16 Si(-solid, 1 atm, Tf) + AeJ*~ = Si{liquid, 1 atm, Tf) Tf = 1685 ± 2°K (ref. 17) Ah^ = 12095 ± 100 cal (ref. 17). C(graphite, 1 atm, 29b.l6°K) + Si(solid, 1 atm, 298.l6°K) + AH^q l6 = SiC(solid, hex.(?), 1 atm, 298.l6°K) AH^g ^ = -26700 ± 2100 cal (ref. 14). 15. F. D. Rossini et al., Selected Values of Chemical Thermodynamic Properties (Circular of the National Bureau of Standards 500, Ser. I [Washington, D. C., 1952 3)• 16. K. K. Kelley, Contributions to the Data on Theoretical Metallurgy (U. S. Department of the Interior, Bureau of Mines, Bulletin 476 [Washington, D. C., 19491 )• 17. M. Olette, Cornpt. rend. 244, 1033"(1957)- 33 APPENDIX 2 Equation (107) has been calculated by Humphrey and his co-workers. It represents the average function for both the hexagonal and cubic modifications their difference being small. Equation (108) has been calculated in this work The numerical values of thermochemical data used are given in Table 1. APPENDIX 2 TABLE 1 THERMOCHEMICAL DATA USED FOR CALCULATION OF EQUATION (108) T SUBSTANCE a ^ 298.16 CP - a + b • 10"3T + c • 105T-2 Form Formula cal Refer- cal Refer- Weight fwt deg ence fwt deg ence a b c Range of Validity °K Deviation ± percent Graphite 12.010 1.3609 15 4.10 1.02! 1 -2.10 298 - 2300 2.5 ! 16 Silicon, solid 28.06 4.47 li 15 5-79 O.56 | -1.09 298 - 1200 2 16 Silicon, liquid not given ...... * 6.12 ' 0 0 <1825 0.4 17 SiC, solid, hex.(?) 40.07 3-935 i. 15 8.93 3-00 : -3.07 j i 298 - 1700 i 3 16 Si (-solid, 1 atm, Tf) + AHfe"= Si(liquid, 1 atm, Tf) Tf = 1685 ± 2°K (ref. 17) Ah^ = 12095 ± 100 cal (ref. 17). C(graphite, 1 atm, 29b.l6°K) + Si(solid, 1 atm, 298.l6°K) + AH^g^g = SiC(solid, hex.(?), 1 atm, 298.l6°K) AH2^8. incl. illus. (Proj. 5620; Task 56202) (AFCRC-TR-58-161) Unclassified report A method is presented for calculating the pressure and composition of the vapor phase in the carbon-silicon binary system. The following quantities are necessary: partition functions of the various molecular species present in the vapor, vapor pressures of graphite and liquid silicon, and the standard free-energy increment for the change C(graphite) + Sl(liquid) = SiC(solld) . vapor and so- A study is made of the equilibrium lution of carbon il liquid silicon UNCLASSIFIED Carbon-silicon -phase diagram Silicon-carbon -phase diagram Silicon carbide -phase diagram Smiltens, J. UWCLASSIFIED AD-152 AF Cambridge Research Center, Bedford, Mass THE GASEOUS EQUILIBRIA IN THE CARBON-SILICON BINARY SYSTEM, by J. Smiltens. July 1958. 3Up. incl. illus. (Proj. 5620; Task 56202) (AFCRC-TR-58-161) Unclassified report A method is presented for calculating the pressure and composition of the vapor phase in the carbon-silicon binary system. The following quantities are necessary: partition functions of the various molecular species present in the vapor, vapor pressures of graphite and liquid silicon, and the standard free-energy increment for the change C(graphite) + Si(liquid) = SiC(solld) . A study is made of the equilibrium: vapor and solution of carbon in liquid silicon. UNCLASSIFIED 1 Carbon-silicon - phase diagram 2- Silicon-carbon - phase diagram 3. Silicon carbide phase diagram I. Smiltens, J. AD-15.2 W6 AF Cambridge Research Center, Bedford, Mass. THE GASEOUS EQUILIBRIA IN THE CARBON-SILICON,BINARY SYSTEM, by J. Smiltens. July 1958. 3%>. incl.' illus. (Proj. 562O; Task 56202) (AFCRC-TR-58-161) Unclassified report A method is presented for calculating the pressure and composition of the vapor phase in the carbon-silicon binary system. The following quantities are necessary: partition functions of the various molecular species present in the vapor,.vapor pressures of graphite and liquid silicon, and the standard free-energy increment for the change C(graphite) + Si(liquid) = SiC(solid) . UNCLASSIFIED 1. Carbon-silicon -phase diagram 2. Silicon-carbon -phase diagram 3. Silicon carbide -phase diagram I. Smiltens, J. A study is made of the equilibrium: lution of carbon in liquid silicon. vapor and so- UNCLASSIFIED UNCLASSIFIED AD "152 hk6 UNCLASSIFIED AD-152 kkG UNCLASSIFIED AF Cambridge Research Center, Bedford, Mass. AF Cambridge Research Center, i Bedford, Mass. c. | 1 UNCLASSIFIED UNCLASSIFIED AD -152 446 UNCLASSIFIED AD-152 kb6 UNCLASSIFIED AF Cambridge Research Center, Bedford, Mass. AF Cambridge Research Center, Bedford, Mass. UNCLASSIFIED UNCLASSIFIED . AD-152 446 AF Cambridge Research Center, Bedford, Mass. THE GASEOUS EQUILIBRIA IN THE CARBON-SILICON BINARY SYSTEM, by J. Smlltens. July 1958. 34p. incl. illus. (ProJ. 5620; Task 56202) (AFCRC-TR-58-161) Unclassified report A method Is presented for calculating the pressure and composition of the vapor phase in the carbon-silicon binary system. The following quantities are necessary: partition functions of the various molecular species present in the vapor, vapor pressures of graphite and liquid silicon, and the standard free-energy increment for the change C(graphite) + Si(liquid) = SiC(solid) . A study is made of the equilibrium: vapor and solution of carbon in liquid silicon. 2. 3. I. UNCLASSIFIED Carbon-silicon -phase diagram Silicon-carbon -phase diagram Silicon carbide phase diagram Smiltens, J. UNCLASSIFIED AD-152 446 AF Cambridge Research Center, Bedford, Mass. THE GASEOUS EQUILIBRIA IN THE CARBON-SILICON BINARY SYSTEM, by J. Smiltens. July 1958. 34p. incl. illus. (Proj. 562O; Task 56202) (AFCRC-TR-58-161) Unclassified report A method is presented for calculating the pressure and composition of the vapor phase in the carbon-silicon binary system. The following quantities are necessary: partition functions of the various molecular species present in the vapor, vapor pressures of graphite and liquid silicon, and the standard, free-energy increment for the change ' C(graphite) + Si(liquid) = SiC(solid) . A study is made of the equilibrium: vapor and solution of carbon in liquid silicon. 2. I. UNCLASSIFIED Carbon-silicon -phase diagram Silicon-carbon -phase diagram Silicon carbide -phase diagram Smiltens, J. UNCLASSIFIED AD-152 446 AF Cambridge Research Center, Bedford, Mass. THE GASEOUS EQUILIBRIA IN THE CARBON-SILICON BINARY SYSTEM, by J. Smiltens. July 1958. 34p. incl. illus. (Proj. 5620; Task 56202) (AFCRC-TR-58-161) Unclassified report A method is presented for calculating the pressure and composition of the vapor phase in the carbon-silicon binary system. The following quantities are necessary: partition functions of the various molecular species present in the vapor, vapor pressures of graphite and liquid silicon, and the standard free-energy increment for the change C(graphite) + Si(liquid) = SiC(solid) . A study is made of the equilibrium: vapor and solution of carbon in liquid silicon. UNCLASSIFIED 1 Carbon-silicon - phase diagram 2- Silicon-carbon - phase diagram ,3- Silicon carbide phase diagram Smiltens, J. AD-152 446 AF Cambridge Research Center, Bedford, Mass. THE GASEOUS EQUILIBRIA IN THE CARBON-SILICON,BINARY SYSTEM, by J. Smiltens. July 1958. 34p. incl. illus. (Proj. 5620; Task 56202) (AFCRC-TR-58-161) Unclassified report A method is presented for calculating the pressure and composition of the vapor phase in the carbon-silicon binary system. The following quantities are necessary: partition functions of the various molecular species present in the vapor,,vapor pressures of graphite and liquid silicon, and the standard free-energy increment for the change C(graphite) + Si(liquid) = SiC(solid) . A study is made of the equilibrium: vapor and solution of carbon in liquid silicon. UNCLASSIFIED 1. Carbon-silicon -phase diagram 2. Silicon-carbon -phase diagram 3. Silicon carbide -phase diagram I. Smiltens, J. UNCLASSIFIED UNCLASSIFIED AD -152 446 UNCLASSIFIED AD-152 446 UNCLASSIFIED i AF Cambridge Reoearch Center, Bedford, Mass. AF Cambridge Research Center, Bedford, Mass. - UNCLASSIFIED UNCLASSIFIED AD-152 446 UNCLASSIFIED AD-152 446 UNCLASSIFIED AF Cambridge Research Center, Bedford, Mass. AF Cambridge Research Center, Bedford, Mass. t UNCLASSIFIED UNCLASSIFIED . « « • » - ---- -- -- AFCRC-TR-58-l6l ASTIA Document No. AD152446 SUPPLEMENT To THE GASEOUS EQUILIBRIA IN THE CARBON-SILICON BINARY SYSTEM By J. SMILTENS APPENDIX 2 Let us denote by primes all quantities measured at the level bed (Fig. l) and by double primes those measured at efg: jAG^ cond P! . - as,1 V* = ^ ^ e HT ^ (gl) and, at temperatures somewhat above the melting point of silicon, iAG"®" „ cond P! 1 = aa"1 « j > . = aa1 e RT V3 k,, . . (S2) ij ij o o ij On dividing (Si) by (S2), we obtain: pii=e M . (S3) ij Thus the Ag0" , can also be determined from the pressure ratios, cond