- Email: [email protected]
Vapor transport of solids by vapor phase reactions
J. Phys. Chem. Solids
Pergamon Press 1962. Vol. 23, pp. 587-598.
Primed in Great Britain.
V A P O R T R A N S P O R T OF SOLIDS BY VAPOR PHASE R E A C T I O N S G. MANDEL
International Business Machines Corporation, Thomas J. Watson Research Center Yorktown Heights, New York (Received 4 October 1961 ; revised 15 November 1961)
A b s t r a c t - - A relatively rigorous theoretical analysis is developed for the quasi-equilibrium diffusion
controlled case of transport of solids by means of reaction with a "vapor solvent" in the presence of a temperature gradient. The treatment is based upon a combination of the usual forms of the equilibrium constant and the diffusion equations with the equation of continuity. An improvement over the method of $CHAFERet al is obtained, inasmuch as it is not necessary to assume an average diffusion coefficient applying to aU species in the vapor phase. It is further shown that the results of this analysis may be expressed in convenient analytical form for the limiting cases of very large or very small equilibrium constant. From these results, it is deduced that the variation of the rate of transport with temperature or pressure ought to show a maximum. The quasi-equilibrium theory is then extended to include the effects of finite surface reaction rates. This extension is essentially based only upon the law of mass action as it applies to surface reaction rates. The steady state rate of transport is shown to be given in terms of the quasi-equilibrium rate multiplied by a factor containing the surface reaction rate coefficients, in the limit of small temperature gradients. A discussion of the information obtainable from these results and various experimental measurements is also given. It is seen that determination of the temperature and pressure dependences of the steady state rate of transport can yield information about surface reaction rates. It is also shown that a measurement of the dependence of transport rate on substrate area allows a direct determination of the surface rate.
1. INTRODUCTION Recently, however, it has been shown by IT HAS been known for m a n y years that solid SCHAFER and co-workers (1) that various solid submaterials, crystalline or amorphous, may be trans- stances, e.g. Cu, CueO, Fe203, Ge, Si, etc. may be ported by sublimation through the vapor phase transported through the vapor phase in the u n d e r the influence of a temperature gradient. presence of a temperature gradient under relaSuch "direct" vapor transport has been used for tively mild conditions provided that a reactive gas, the growing of crystals and, under high vacuum such as a halogen or a hydrogen halide, is present. conditions, the deposition of films. T h e virtual It was suggested by these authors that the basis indifference of the vapor pressure of a substance for this transport was the rapid establishment of an equilibrium between the solid substance and to any outside influence except temperature, however, has placed m a n y chemical systems beyond the reactive gas, hereafter referred to as a "vapor the reach of the research worker and has rendered solvent", e.g. many others difficult to handle. Examples of such Cu20(s)+2HCl(g) = 2CuCl(g)+H2Ocg). (1) systems are gallium phosphide, which vaporizes incongruously, and cubic zinc sulfide, which is Diffusion of the gaseous reaction products into a unstable at temperatures at which the vapor temperature region in which the equilibrium conpressure of zinc sulfide reaches a reasonable value. stant for the above reaction was lower than at the 587
588
G. M A N D E L
starting point then results in deposition of the solid substance; Cu20 in the above example. Thus if the net enthalpy change for the reaction, AH, is positive, it would be expected that transport would proceed from the hotter zones to the colder ones and vice versa. This idea has been verified experimentally for several chemical systems. An approximate theory relating the rate of transport to various macroscopic parameters, e.g. the relative amount of vapor solvent, temperature, etc., was also developed by SCYIAFER and coworkers, (1) based upon a simple "two zone" model, i.e. it was assumed that a zone was maintained at equilibrium at a temperature T1 at a distance of one "diffusion length" from a zone at equilibrium at a temperature 7"2. It was further assumed that all the products of the reaction would diffuse at the same rate so that these products could be lumped together and considered as one, with a "combined" diffusion coefficient. Some experimental results were obtained which were in fair agreement with those predicted by the above energy. The purpose of Section I I of the following treatment is the development of a more rigorous calculation of the quasi-equilibrium rate of transport of a solid substance by a vapor solvent under the influence of a temperature gradient. The major improvement over the method of SCI{AFER(1) to be obtained is the elimination of the assumption of a "combined" diffusion coefficient. It will also be seen that the resulting rate of transport can be expressed in convenient analytic form in the limits of very large or very small equilibrium constant. In Section III, we will be concerned with the effect of finite surface reaction rates on the rate of transport, as opposed to the infinite surface reaction rates that are assumed in the quasiequilibrium theory. The resulting rate of transport will, in the limit of small temperature gradients, be expressed as the quasi-equilibrium rate multiplied by a factor containing the surface reaction rate coefficients. 2. QUASI-EQUILIBRIUM RATE OF TRANSPORT
Let us consider that we have added No moles of gaseous A to M0 moles of solid S at one end of a previously evacuated tube of constant crosssectional area C and length L. The temperature in the neighborhood of the charge is maintained at
7"1 while the temperature at the other end of the tube is maintained at 7"2. We write the following general equilibrium: S(s)+aA~q)= Bb(g)+cC(g)+dD(g)+ ...
(2)
in which S is the solid being transported. A is the gaseous vapor solvent and B, C, D ... are the gaseous reaction products. The numbers a, b, c, ... need not be integers but must be rational. We will assume throughout that the temperature is everywhere above the boiling point of all components except S. The equilibrium constant corresponding to equation (2) may, of course, be written K-
b
g
d
PBPcPD "'"
,
(3)
where PA, PB, PC . . . . are the pressures of A, B, C . . . . We have assumed all gases to be ideal and have ignored the very slight dependence of the vapor pressure of S upon external pressure. The value of K at any temperature may be calculated from thermodynamic data in the usual well-known fashion. (2) In this Section, the treatment of transport through the vapor phase will be based on several major assumptions. (1) The mean-free-path of molecules in the gas phase is small compared to the dimensions of the system. (2) The rates of surface reactions, i.e. condensation and evaporation, are very high, so that diffusion times are much greater than surface reaction times. (3) It will be assumed following SCHh~ER,(1) that the flux of molecules diffusing through the gas phase is proportional to the gradient of the partial pressure of those molecules. We write Di ~p~ F,-
R T Ox'
(4)
where F~ is the flux of molecules i, i.e. the number passing through a unit area per unit time D~ is a diffusion coefficient, R is the universal gas constant. T is the absolute temperature, p~ is the pressure of gas i and x is a position coordinate.
VAPOR T R A N S P O R T OF S O L I D S BY VAPOR PHASE R E A C T I O N S As a consequence of the first and second assumptions, it may be expected that a quasiequilibrium steady state will be established after a short transient time. All the following calculations will be based upon this steady state. It is expected that these calculations will at least indicate an upper limit on the deposition rate. The second assumption implies that the deposition rate is diffusion controlled and that surface rates may be ignored. Again, it is expected that the calculations will indicate a maximum deposition rate. (A more refined theory which includes surface limitations is developed in Section III.) The third assumption essentially gives us a mathematical basis upon which to compute the all important diffusion rates. Equation (4) is, of course, the ordinary Fick's law diffusion equation, as it applies to gases in the presence of a small temperature gradient. The appropriate diffusion coefficients can presumably be calculated from the kinetic theory of gases. It has been tacitly assumed that processes such as thermal diffusion, convection due to gravity and viscous flow ("streaming", according to SCHAFER(1)) may be ignored. We are thus restricted to low pressures, i.e. pressures below one atmosphere. Viscous flow effects will be considered in a future paper. Note that only one chemical reaction has been assumed to be occurring. We will see that equation (4) may be combined with the equation of continuity, the requirements of the steady state and the ordinary form of the equilibrium constant, equation (3), to determine the system completely. The familiar equation of continuity, in the absence of sources or sinks, would tell us that the time rate of change of density of any component in the gas phase is equal to the negative of the divergence of the flux, i.e., bF~
~n~
~x
bt
(5)
Under the steady-state conditions which apply in our case, however, 3n~/St is maintained at zero for each gaseous component by virtue of the chemical equilibrium process. Those molecules which would have gone towards increasing the density in the gas phase are absorbed by formation of solid. Those molecules which would have
589
diffused away, lowering the density in the gas phase, are replaced by molecules formed by the chemical equilibrium process. Thus, we have, finally,
~ns
1 ~ Da CgpA
Ot
a Ox R T ~x
1 0 DB OpB
1 0 Dc ~Pc
b ~x RT Ox
c Ox R T ~x
- -
-...
(6)
by combining equations (4) and (5). The quantity Ons/Ot is the net amount of solid produced in unit time per unit volume of container at any point, x. This combined equation contains ( N - 1 ) independent relations among the pressures, pA, PB, PC, . . . , p~r, where N is the total number of components in the gas phase, excluding S. In principle, we can integrate equation (6) to obtain each of the pressures in terms of any one of them, e.g. pB. Insertion of these results into equation (3), which relates the equilibrium constant to the various pressures, would give pB as a function of K. We will see that deposition rates and other quantities of interest can then be determined. We may integrate equation (6) once immediately to obtain
1 DA ~PA
1 DB ~PB
a R T ~x
b R T ~x
1 Dc ~Pc
1 DB OpB
c RT
b RT
(7)
~x
~x
where all integration constants must be zero in order that the steady state be maintained. Integration of equation (7), however, cannot be carried out inasmuch as the various diffusion coefficients vary with temperature and composition, and hence with x, in an unknown way. We may circumvent this difficulty by limiting our analysis to systems in which the temperature gradient is sufficiently small that the composition, and hence the diffusion coefficients, are sensibly constant. The results may be expected to be accurate to first order in [AK/K] where A K = /£2-/£1. We require
]AK/K I ~ 1
(8)
590
G. MANDEL
or, alternatively, IATI <
(9)
- -
where AH is the standard enthalpy change appropriate to equation (2). The usual equation relating 0 In K]OT to ZX/-/has been assumed. To a first approximation then, we define the following constants: DB
DB =
ct
I'A = i'A
cy
(15)
t'e = 1 ' c - - - : 0
DB
- -
DA
where T-is the average temperature and P0 is the average initial pressure of A, i.e. NoRT/CL. Equation (11) now reduces to
=
~,
- -
Dc
=
8
(10)
DD
d8
and integrate equation (7) to obtain act
while the conservation conditions, equations (12) and (13), may be written
PA = kA---ff PB
a
c~
(11)
Pc = kc + -~ PB d8
jba = P0--~ ihB C
/3c = -~3B
= k s + --:-pn,
(16)
o
where ka, kc, kD, ... are integration constants. The integration constants may be evaluated by means of the following conservation conditions:
lfpo
L 1
b
which may conveniently be combined with
L
PBC~
c
0
d
RT
0
L,
0
C~
-b "e "d
K = PnPePD ""
RT
(12)
Cdx = ...
L
l f pA Cdx a RT
No a
0
l fpB Cdx b
0
...,
(13)
RT
where the definitions of No, C and L have been given at the beginning of this Section. We may combine equations (11), (12) and (13) with the limitation of small temperature gradients and obtain --
L
RT f RPBT dx "= _PB b_
-bL -
0
P o - kA
kc
kD m
a(1--ct)
c O - y ) = d(1-8)
d
~
gO0
(14)
(17)
in the limit of small temperature gradients. Thus, we may initially evaluate j0A, j0B.... from thermodynamic data. From these results, combined with equation (3) and (15), we obtain a polynomial in pB which may be solved for any value of the equilibrium constant between K1, i.e. K at 7"1 and K~. In particular, we may evaluate PB at 7"1 and 7"2. According to assumptions (1) and (2), we expect a region between the source and deposit in which no chemical reaction is occurring. It may easily be shown that the equivalent flux of any component in this region multiplied by the cross-sectional area must be equal to the rate of transport of solid in order that the steady state be maintained. This statement is based upon the tacit assumption that the vapor pressure of Scs) is very low so that the flux of S(0) may be ignored. On the basis of the above remarks, we may state that the rate of transport of Sos) will be given by
VAPOR
TRANSPORT
OF SOLIDS
the product of the cross-sectional area, C, and the average equivalent flux of B(g), i.e.
Rt = -
1/~B
ffB2-pm)
b RT
L
BY V A P O R
where
IAHAT/RT1T2I .
We may further consider some simple approximations in general. The first is that in which the equilibrium constant is very large throughout the system so that the reaction given in equation (2) has gone virtually to completion. It is then possible to approximate equation (11) and solve equation (3) for pA. From equations (12), (13), (14) and the condition of large equilibrium constant, it may be shown that equation (11) becomes
K -lla dT ~ ~.-lla.
i f
AT
(18)
where PB2 is the pressure of B(g) in the neighborhood of the deposit, i.e. at x = L, while PB1 is the pressure of B(g) at x = 0. We have assumed that D B / R T does not vary appreciably with x and have used average quantities. This equation has been given essentially by SCHAFER.(1) Thus we see that the rate of transport may be evaluated by means of the above procedure for any set of initial conditions. Thermodynamic and diffusion data is required. The results may be expected to be accurate to first order in
REACTIONS
591
Tz+AT
K-11 a -
. C,
PHASE
(22)
T~
We have also used the relationship (2) K = Ko e x p ( - A H / R T ) .
(23)
We see, therefore, that equations (20) (21) would be expected to apply to the case in which the equilibrium constant, K, is sufficiently large that the reaction, equation (2), has virtually gone to completion. [AH I is assumed to be large, i.e. much greater than ~ aRT, and relatively temperature independent. The diffusion coefficient, DA, is assumed to be relatively constant. The temperature gradient is assumed small, equation (9). We may also attempt to consider the appropriate approximation for the situation in which the equilibrium constant is small everywhere. Simple results, however, cannot be obtained without further assumption. If, as is often the case, one of the products of the reaction given in equation (2) is much heavier and larger than any other product, we may label this product B. Under this further restriction, we may approximate equation (15) for small equilibrium constant as follows:
bp0 pB =
P A = PO
cp0 Pc = - -
Pc = ibc = ~ f B
gg
c
(19)
(24)
d
dpo pD = -
From equations (19), (3) and an analog of equation (18), we obtain immediately
where we have assumed, as per the above discussion,
y, a .... ~ 1 DA bb/ace/ad a:a ... poMIa(K;IIa-K~I lla) C Rt =---= .-RT a (Mla+l) L
(20) M ='b+c+d+
...
Under the condition imposed by equation (9), we may rewrite equation (20) as IAHAT[ DA bb/ace/ada/a ... K-1/aP~/a
D~ Rt =
(26)
C ( R"llb
(21)
(25)
We note that this is equivalent to the familiar assumption that the slowest step determines the over-all rate of a reaction. In this case, the slowest step is the diffusion of B. Combining equations (3), (18) and (24) we obtain
K(l/b~,
/,~/u "=i - ' = 2 , bblMcclMdalM ... [(K~Ib + K~Ib)12](1-blM)
O L
592
G. MANDEL
Under the condition imposed by equation (9), we may rewrite equation (26):
MIAT b~ R, =
p~JMK1/M
bRT z R T bb/mce/mda/m...'
(27)
where KIlM
Further examination of equations (21) and (27) indicates, upon realization that DA and DB will vary approximately as p o 1, that the rate of transport should also display a maximum in its variation with P0 at constant temperature, at least in the limit of low pressures.iS) It is expected that Rt will vary as
~ K IlM
p(~l l~--l)
is defined as in equation (22). We see, therefore, that equations (26) and (27) would be expected to apply to the case in which the equilibrium constant, K is sufficiently small that the reaction, equation (2) has virtually not gone at all. Further, product B is so heavy and has so large a collision cross-section that its diffusion coefficient, DB, is much smaller than that of any other product in the gas phase. IAH] is assumed to be much greater than ~ bRT and relatively temperature independent. The temperature gradient is assumed small, equation (9). We note that equations (21) and (27) can also be obtained from the treatment of SCHAFER,(1) except for the fact that the diffusion coefficients are here seen to be different for the two extreme cases. Neglect of this fact can obviously lead to error in certain chemical systems. The fact that the rate of deposition increases with increasing K for small values of K, as in equation (27), while it decreases with increasing K for large values of K, as in equation (21), must be a general result, i.e. it must be true for any system. For a system in which B is a very slowly diffusing molecule, equation (21) would be expected to apply when K has a high value while equation (27) would apply when K has a low value. The maximum possible deposition rate would occur approximately at the point at which equations (21) and (27) predict the same result, although this predicted deposition rate would be somewhat too high. From equations (21) and (27), we then obtain the following approximate condition under which a system with very slowly diffusing B would display the maximum deposition rate: ~.p(oa_M , = (__b ) M a / ( M + a , \acz]
bbc ` ... aM
(28)
The above discussion implies that the rate of transport will display a maximum in its variation with temperatureat constant initial pressure, P0.
at low pressures and as
p(~/M-1) at higher pressures, provided M > a. If M < a, these dependences are reversed. The approximate condition for the maximum is again given by equation (28). We conclude Section II by again pointing out a practical limitation of the above treatment. This is its restriction to low pressures and geometry such that convection does not enter. It appears unfortunately, that the conditions under which vapor transport is of interest in the laboratory are such that convection may be important in many cases. 3. SURFACE RATE EFFECTS
A reasonably rigorous theoretical calculation of the quasi-equilibrium steady-state rate of transport of solids through the vapor phase by means of chemical reactions in the presence of a temperature gradient has now been carried out. This, and prior analogous calculations, (1) have been based upon the assumption that the composition of the vapor phase above the source and deposit was that characteristic of chemical equilibrium at the appropriate temperatures. This is equivalent to the assumption that the surface reactions ("evaporation" and "condensation") are infinitely fast compared to diffusional transport through the vapor phase and, therefore, do not come into the calculation of over-all transport rates. We shall see that a simple extension of the previous calculation, to include both surface reaction rates and diffusional transport rates, may be carried out without significant further assumption. The diffusional rate of transport will be calculated in exactly the same manner as previously described except that the ratio 9 -
p b 4,,e,,,hd B Y C t ' D • •*
p~
(29)
VAPOR T R A N S P O R T OF S O L I D S BY VAPOR PHASE R E A C T I O N S will no longer be given by the equilibrium constant, K, appropriate to the chemical reaction, given as in equation (2), by means of which transport takes place. We will again be considering a cylindrical system in which transport takes place from x = 0, at temperature T1, to x = L, at temperature T2. The diffusional rate of transport will be shown to be proportional to the "apparent enthalpy change", AH*, in the limit of small AT. The "apparent enthalpy change", AH*, reflects the temperature dependence of the ratio Q, just as the actual standard enthalpy change, AH, appropriate to equation (2), reflects the temperature dependence of the equilibrium constant, K. The rates of surface reactions at the source and deposit will be considered simply on the basis of the law of mass action. It will be seen that these rates, in the usual manner, are related to the "undersaturation" in the neighborhood of the source and the "supersaturation" in the neighborhood of the deposit. Finally, it will be shown that, in the limit of small AT, the surface reaction rates are proportional to ( A H - A H * ) . Inasmuch as the steady state condition requires that the surface reaction rates and the diffusional transport rate be equal, we may combine the above described results with the elimination of AH*. This finally yields the over-all transport rate in terms of the quasi-equilibrium rate, which has been previously calculated, multiplied by an additional factor containing the surface rate coefficients. These results may also be rearranged to show that the reciprocal of the over-all rate is given in terms of the sum of the reciprocals of the surface rate and the diffusonal rate. This is the expected form for a system in which several rates are connected in series.
A. Surface rates From the law of mass action, we write the following relations for the rate of removal of solid ("evaporation"), R1 at the temperature T1, and the rate of deposition of solid ("condensation"), R2 at the temperature T2:
the reverse reaction at T1, k12 for the forward reaction at T2 and k22 for the reverse reaction at T2. The pressure of A(g) at the temperature T1, i.e. above the charge, is designatedpal, etc. No assumption has been made regarding the possible composition, pressure or temperature dependences of kn, k21, ... We obtain the equilibrium results, of course, by setting R1 and R2 equal to zero. Thus,
KI=
b
c
(PBlPc1...)
kll
P~I b
c
K2=(PB2PC2...
P~2
e.. = -~21
(32)
_ k12 ,
(33)
) e..
k22
where K1 and K2 are the equilibrium constants at the temperatures T1 and T2 respectively, as before. In transport experiments, R1 and R2 are both greater than zero. For the steady state, they are also equal. Thus, from equations (30), (31), (32) and (33),
(34) where b
c
PBlPcl "" 91 = pa a b
Q2 -
(35)
c
PB2Pc2 "'"
P~2
(36)
As has been pointed out by LEVER,(4) the quantity (1--Q1/K1) is a generalized undersaturation while the quantity (Q2/K2-1) is a generalized supersaturation. The quantity k n p ~ 1 is thus defined as the ratio of the surface reaction rate to the supersaturation and similarly for k12 P~i2As a consequence of equation (34), we have the following inequality
b c ... R1 = kllPA1--k21PB1Pc1
(30)
K2 < Q2 < Q1 < K1.
a + k22P~2Pc2 b c ..., R2 = -- k12PA2
(31)
We note that equation (37) is based upon the assumption that transport proceeds from T1 to T2. The positive quantity ( Q I - Q 2 ) , which serves as a measure of the diffusional rate of transport from
where kn is the rate coefficient for the forward reaction in equation (2) at temperature T1, k21 for 2q
593
(37)
594
G. MANDEL
Tlto 7"2,must be less than the quantity (K1-K2), according to equation (37). Therefore, the presence of finite surface rates lowers the rate of transport below its quasi-equilibrium value, as expected. In the same way that the ratio K2/K1 is given in terms of the enthalpy change, AH, for equation (2) (2), we define an "apparent enthalpy change", AH*, in terms of the ratio Q2/Q1, i.e. Q2 Q1
(1 1)]
exp
.
(38)
l R
We note that, according to equation (37), [An*[ < IAHI.
(39)
provided it is remembered that the quantity AHAT = AH(T2-T1) must be negative in order that transport from 7"1 to 7"2 occur. Combination of equations (34) and (38) finally allows evaluation of the steady-state deposition or transport rate, Rt, in terms of AH*:
shall now proceed to obtain another relationship between Rt and AH* by consideration of the diffusional rate of transport.
B. Diffusion rates Throughout the remainder of this treatment, we will confine our detailed analysis to systems in which the average equilibrium constant is either very large or very small. We limit ourselves in this manner since it is only for these eases that we have obtained simple analytical results for the quasi-equilibrium transport rate. For the case of large equilibrium constant, the quasi-equilibrium transport rate has been given as [(see equation (20)]: Req. =
.4(K~l/a-K~lla),
(43)
where .4 is a term that will be independent of surface effects. We obtain the desired more general
, _ (40)
Re = RI"~= R2 =
1
k11P.~l k12PaA2exp L This general result is based essentially only upon the form of the law of mass action. We may simplify equation (40) in the limit of small IAT[, equation (9). Equation (40) reduces to ( A H - A H * ) (1.
Rt=
R
f~l
1
1
kllP~ 1
kl2p~ 2
(41) 1
-R
( T2
1 -
Rs
- -
kllP~ll
diffusional transport rate by replacing/£2 by Q2 and/£1 by Q1. Thus,
Rt = A(O71/a- Q-~lla).
Rt = Req. [-K-fz)
1
T1 )"
1 - exp [ a R
In equation (41), we have also defined a .mean maximum surface rate", Rs, 1
111
/'2
(44)
i Q2 '~-11"
( A H - AH*)
= Rs
tT1
We may combine equations (38), (43) and (44) to obtain the more general transport rate in terms of the quasi-equilibrium transport rate:
1)
7"2
R
X
-
-
-
tl--exp
AH ~
I1 7"1 1 7"1
(45)
1 t
•
kl2p~4s
(42)
Equation (41) is the desired simple relation between the transport rate, Rt, and ( A H - A H * ) . We
Furthermore, from a combination of equations (37) and (38), it may be shown that, in the limit of small [ATI, i.e. under the condition of equation (9), the ratio Q2/K2 approaches unity. As a
595
VAPOR TRANSPORT OF SOLIDS BY VAPOR PHASE REACTIONS
consequence, in the limit of small [AT[, equation (45) reduces to
in which we see AH* to be independent of AT in the limit of small [AT I.
AH*
Rt = R e q . - - .
(46)
AH
4. EXPERIMENrAL DETERMINATION SURFACE RATES
OF
A similar analysis of the small equilibrium case based on equation (26) and the limit of small temperature gradients also leads to equation (46). We see, therefore, that equation (46) is valid for both the case of large and small equilibrium constant, in the limit of small ]AT].
We will now be concerned with the information on surface rates that can be indirectly obtained by measurement of the temperature and initial pressure dependence of the steady-state transport rate. We will again restrict our attention to experimental situations falling under the condition of equation (9) i.e. in the limit of small IATI.
C. Over-all steady-state rate
A. Temperature dependence
The maintenance of the steady state requires that the diffusional transport rate and the surface reaction rates be equal. We may, therefore, combine equations (41) and (46), with the elimination of AH*, and obtain the steady-state transport rate in the limit of small ]AT[. The result of such a procedure is
We are interested in the dependence of the steady-state transport rate, Rt, on the absolute temperature, T [or the average absolute temperature, T = ½(TI+T2)]. The experimental data would consist of measurements of Rt at different values of T, all other parameters, e.g. AT, remaining fixed. The theoretical analysis of this situation may be carried out most easily by consideration of equation (47). We obtain immediately:
1
1
1
Rt 7"2
(47)
S ln(RtT1T2)
T~
ST
where R0 is defined by the following relation: Req. = Ro ~
/'2
1) I
,,8,
so that R0 can be calculated from knowledge of gaseous diffusion coefficients and thermodynamic data, according to the quasi-equilibrium theory. In principle, we see from equation (47) that, inasmuch as Req. and, therefore, R0 can be calculated, experimental determination of the steadystate transport rate, Rt, gives information about the surface reaction rates. Such measurements must be made, of course, under the condition of equation (9), i.e. in the limit of small [AT I. In equation (47), we see, as usual, that rates in series add as reciprocals. We note also that M-/* may be evaluated by combination of equations (41) and (46). We obtain AH AH* = 1 + Ro/R-~s'
(49)
1 SlnR0
1 1 / R o + I/R
R0
-
ST -
1 SlnR8]
+ - R8 -
:"
(50)
The temperature dependence of R0 may be obtained directly from the quasi-equilibrium theory for the case of either very large or very small equilibrium constant [see equations (21) and (27)]. We may conveniently calculate the temperature dependence of the surface reaction rate, Rs, by rewriting equation (42) : R8 =
p) = 1/kn + 1/k12
kipS,
(51)
where we no longer distinguish between PAl and PA2 in the limit of small ]AT I. We have also defined a "mean" surface reaction rate coefficient, kl. The temperature dependence of the surface reaction rate coefficient, kl, may be deduced from
596
G. MANDEL
the absolute chemical reaction rate theory of EYRING et al. (5) We thus obtain
hi ST
(AH+Ea) RT 2
In
(52)
equilibrium theory will also allow an estimate of the "mean" surface rate coefficient, kl. Similarly, for the case of small equilibrium coefficient, we combine equations (50), (51), (52) and (54) to obtain
ln(RtT1T2) ST
where Ea is an "excess" potential energy barrier. We will assume, as a first approximation, that Ea is generally much less than Ajar. The temperature dependence of the pressure, PA, may be obtained directly from the quasiequilibrium theory. For the case of large equilibrium constant, we obtain
1 {~o AH F 1 (AH+Ea)} l/R0+ llRs MRT~ Rs RT 2
Sln(RtT1Tz) _ SIn PA
AH aRT 2"
ST
~(1/T)
(53) _
1 /~s( I+(R0
AH ( 1 + i R o / R s l =
AH
R0~H)
MR
Rs
AH* (1 +MR°I
For the case of small equilibrium constant, we have, of course, (56) In PA -
-
ST
=
O,
(54)
since PA is approximately equal to its initial pressure, p0, and constant. Thus, we may finally combine equations (50), (51), (52) and (53) to obtain, for the case of large equilibrium constant,
~3In(RtT1 T2)
B. Pressure dependence
ST
1 1 1/Ro+ l/R8 Ro I ( 1 +Ro/Rs S ln(RtT1T2) S0/T )
upon neglect of Ea relative to AH. We see, therefore, that, for the case of small equilibrium constant as well as for the case of large equilibrium constant, determination of the temperature dependence of the transport rate, Rt, allows estimation of the ratio Ro/Rs and, finally, of the surface reaction rate coefficient, kl.
leo}
-a--R-~ + - - - Rs ~ 2 " AH aRT 2
Ro Ea Rs RT 2
1 (AH)AH* = l+/~o/Rs ~ - - ~ - " (55)
The neglect of Ea relative to AH may not be generally valid. We see that determination of the temperature dependence of the transport rate, Rt, in the case of large equilibrium constant, may allow reasonable indirect estimate of the ratio Ro/Rs. Further calculation of R0 and pa by means of the quasi-
To obtain the dependence of the steady-state transport rate, Rt, on the initial pressure of A(o), P0, we may rewrite equation (47) as follows: Req.
Rt
Ro -
1 +--.
Rs
(57)
We note again that equation (47) and (57) apply only in the limit of small [AT[, i.e. under the condition of equation (9). In order to proceed further, we must evaluate the initial pressure dependence of the ratio Ro/Rt. For the case of large equilibrium constant, the quasi-equilibrium theory predicts that R0 will vary as p(oM/a-l), where we have taken into account the fact that the diffusion coefficient will vary as po I according to the kinetic theory of gases and assumed that M > a. For the case or large equilibrium constant, the quasi-equilibrium theory
VAPOR T R A N S P O R T OF S O L I D S BY VAPOR PHASE R E A C T I O N S also predicts t h a t p a will vary as pMla. Therefore, according to equation (51), Rs should vary as p ~ , providing we neglect any pressure or composition dependence in ks, which may not be generally valid. We see, as a consequence, that, for the case of large equilibrium constant, the ratio Req./Rt should vary linearly with p~U/a-l-M), all other parameters being held fixed. The ratio Ro.q./Rt is determined by measuring Rt and calculating Req. for each value of p0. We note, from equation (57), that the intercept of a plot of Req./Rt against p~oM/a-l-M) should be unity. If this is found not to be the case, it indicates that the calculated values of Req. are probably in error by a constant factor due to errors in our knowledge of diffusion coefficients. The results should then be multiplied by an appropriate correction factor such that the intercept becomes unity. We finally obtain, either from the slope of the plot or from an appropriate average of the values obtained at each point, a value for the ratio Ro/Rs at any particular value of P0. Since we may calculate R0 and pA from the quasi-equilibrium theory for this value of Po, we may finally estimate the surface rate coefficient, kl. In a manner similar to the above, we may ascertain, for the case of small equilibrium constant, that the ratio Req./Rt should vary as p~om/a-l-a), all other parameters being held fixed. The intercept of such a plot should again be unity, allowing correction of the calculated values of Req. The surface rate coefficient, ks may again be determined.
C. Substrate area dependence The simplest determination of surface rates can be carried out by measurement of the rates of transport of a powdered source to single crystal substrates of well defined but different areas. The expected effects are most easily understood on the basis of equations (42) and (47). We expect the surface rate coefficients, kn and k12, to be directly proportional to the areas of the surfaces involved. Since we are considering a powdered source (kn) and a single crystal substrate (kse), we see from equation (42) that the major contribution to Rs is due to k12. If we now measure the rates of transport, Rt (1) and Rt (2), in two experiments in which the substrate areas are
597
different while all other parameters are held constant, we see from equation (47) that
1
1 Rt(2) Rt(1)
1 IAH/R (l/T2-1/T1)[
1)
(1 _ 1/r~ × R~2) R¢~) ]±H/R (1/T2-1/Tz)]
(2 1)
x
) 2~i)
(58)
'
where rs is the specific surface rate (rate per unit area) and A (2) and A (1) are the substrate areas. We see that rs, Rs and ks2 are thus immediately determined.
rs=
Rs(S)
Rs(2)
Aa)
A(2)
-
kl2(1)p~2 - - A(1)
-
ks2(2)p,~ 2 A(2)
(59)
CONCLUSIONS It has been shown that the quasi-equilibrium theory for the transport of solids through the vapor phase by means of chemical reactions may be extended to include the effects of finite surface reaction rates. It has been assumed that (1) diffusion coefficients are relatively independent of position and temperature although different components in the gas phase are allowed to have different diffusion coefficients; (2) the enthalpy change for the reaction in equation (2) is large compared to RT and relatively temperature independent; (3) The only chemical reaction occurring is that given in equation (2); (4) diffusion is the only mechanism of transport through the gas phase; (5) the equilibrium constant for equation (2) is either very large or very small. In this analysis, the restriction of small [AT[ has also been required. Results have been obtained in very simple form. The surface rates are found to add in a reciprocal manner with the previously obtained diffusion limited rate, as expected. It has been shown that experimental determination of the temperature, initial pressure and
598
G. M A N D E L
substrate area dependences of the steady-state transport rate can yield information about the surface reaction rates.
Acknowledgements--The author wishes to express his gratitude for many helpful discussions with Dr. A. REISMAN, Dr. F. P. JONA, Mr. V. J. LYoNs, Mr. R. F. LEYEa, Dr. S. P. KELL~.aand Dr. R. W. KEvm.
REFERENCES
1. SUHAFERH., Z..4norg. Chem. 286, 27 (1956). 2. MOOREW. J., Physical Chemistry, Chap. 4. Prentice Hall, New York (1955). 3. KEYESR. W., private communication. 4. LEVERR. F., private communication. 5. HILL T. L., Introduction to Statistical Thermodynamics p. 194 if. Addison-Wesley, Reading, Mass. (1960).
Pergamon Press 1962. Vol. 23, pp. 587-598.
Primed in Great Britain.
V A P O R T R A N S P O R T OF SOLIDS BY VAPOR PHASE R E A C T I O N S G. MANDEL
International Business Machines Corporation, Thomas J. Watson Research Center Yorktown Heights, New York (Received 4 October 1961 ; revised 15 November 1961)
A b s t r a c t - - A relatively rigorous theoretical analysis is developed for the quasi-equilibrium diffusion
controlled case of transport of solids by means of reaction with a "vapor solvent" in the presence of a temperature gradient. The treatment is based upon a combination of the usual forms of the equilibrium constant and the diffusion equations with the equation of continuity. An improvement over the method of $CHAFERet al is obtained, inasmuch as it is not necessary to assume an average diffusion coefficient applying to aU species in the vapor phase. It is further shown that the results of this analysis may be expressed in convenient analytical form for the limiting cases of very large or very small equilibrium constant. From these results, it is deduced that the variation of the rate of transport with temperature or pressure ought to show a maximum. The quasi-equilibrium theory is then extended to include the effects of finite surface reaction rates. This extension is essentially based only upon the law of mass action as it applies to surface reaction rates. The steady state rate of transport is shown to be given in terms of the quasi-equilibrium rate multiplied by a factor containing the surface reaction rate coefficients, in the limit of small temperature gradients. A discussion of the information obtainable from these results and various experimental measurements is also given. It is seen that determination of the temperature and pressure dependences of the steady state rate of transport can yield information about surface reaction rates. It is also shown that a measurement of the dependence of transport rate on substrate area allows a direct determination of the surface rate.
1. INTRODUCTION Recently, however, it has been shown by IT HAS been known for m a n y years that solid SCHAFER and co-workers (1) that various solid submaterials, crystalline or amorphous, may be trans- stances, e.g. Cu, CueO, Fe203, Ge, Si, etc. may be ported by sublimation through the vapor phase transported through the vapor phase in the u n d e r the influence of a temperature gradient. presence of a temperature gradient under relaSuch "direct" vapor transport has been used for tively mild conditions provided that a reactive gas, the growing of crystals and, under high vacuum such as a halogen or a hydrogen halide, is present. conditions, the deposition of films. T h e virtual It was suggested by these authors that the basis indifference of the vapor pressure of a substance for this transport was the rapid establishment of an equilibrium between the solid substance and to any outside influence except temperature, however, has placed m a n y chemical systems beyond the reactive gas, hereafter referred to as a "vapor the reach of the research worker and has rendered solvent", e.g. many others difficult to handle. Examples of such Cu20(s)+2HCl(g) = 2CuCl(g)+H2Ocg). (1) systems are gallium phosphide, which vaporizes incongruously, and cubic zinc sulfide, which is Diffusion of the gaseous reaction products into a unstable at temperatures at which the vapor temperature region in which the equilibrium conpressure of zinc sulfide reaches a reasonable value. stant for the above reaction was lower than at the 587
588
G. M A N D E L
starting point then results in deposition of the solid substance; Cu20 in the above example. Thus if the net enthalpy change for the reaction, AH, is positive, it would be expected that transport would proceed from the hotter zones to the colder ones and vice versa. This idea has been verified experimentally for several chemical systems. An approximate theory relating the rate of transport to various macroscopic parameters, e.g. the relative amount of vapor solvent, temperature, etc., was also developed by SCYIAFER and coworkers, (1) based upon a simple "two zone" model, i.e. it was assumed that a zone was maintained at equilibrium at a temperature T1 at a distance of one "diffusion length" from a zone at equilibrium at a temperature 7"2. It was further assumed that all the products of the reaction would diffuse at the same rate so that these products could be lumped together and considered as one, with a "combined" diffusion coefficient. Some experimental results were obtained which were in fair agreement with those predicted by the above energy. The purpose of Section I I of the following treatment is the development of a more rigorous calculation of the quasi-equilibrium rate of transport of a solid substance by a vapor solvent under the influence of a temperature gradient. The major improvement over the method of SCI{AFER(1) to be obtained is the elimination of the assumption of a "combined" diffusion coefficient. It will also be seen that the resulting rate of transport can be expressed in convenient analytic form in the limits of very large or very small equilibrium constant. In Section III, we will be concerned with the effect of finite surface reaction rates on the rate of transport, as opposed to the infinite surface reaction rates that are assumed in the quasiequilibrium theory. The resulting rate of transport will, in the limit of small temperature gradients, be expressed as the quasi-equilibrium rate multiplied by a factor containing the surface reaction rate coefficients. 2. QUASI-EQUILIBRIUM RATE OF TRANSPORT
Let us consider that we have added No moles of gaseous A to M0 moles of solid S at one end of a previously evacuated tube of constant crosssectional area C and length L. The temperature in the neighborhood of the charge is maintained at
7"1 while the temperature at the other end of the tube is maintained at 7"2. We write the following general equilibrium: S(s)+aA~q)= Bb(g)+cC(g)+dD(g)+ ...
(2)
in which S is the solid being transported. A is the gaseous vapor solvent and B, C, D ... are the gaseous reaction products. The numbers a, b, c, ... need not be integers but must be rational. We will assume throughout that the temperature is everywhere above the boiling point of all components except S. The equilibrium constant corresponding to equation (2) may, of course, be written K-
b
g
d
PBPcPD "'"
,
(3)
where PA, PB, PC . . . . are the pressures of A, B, C . . . . We have assumed all gases to be ideal and have ignored the very slight dependence of the vapor pressure of S upon external pressure. The value of K at any temperature may be calculated from thermodynamic data in the usual well-known fashion. (2) In this Section, the treatment of transport through the vapor phase will be based on several major assumptions. (1) The mean-free-path of molecules in the gas phase is small compared to the dimensions of the system. (2) The rates of surface reactions, i.e. condensation and evaporation, are very high, so that diffusion times are much greater than surface reaction times. (3) It will be assumed following SCHh~ER,(1) that the flux of molecules diffusing through the gas phase is proportional to the gradient of the partial pressure of those molecules. We write Di ~p~ F,-
R T Ox'
(4)
where F~ is the flux of molecules i, i.e. the number passing through a unit area per unit time D~ is a diffusion coefficient, R is the universal gas constant. T is the absolute temperature, p~ is the pressure of gas i and x is a position coordinate.
VAPOR T R A N S P O R T OF S O L I D S BY VAPOR PHASE R E A C T I O N S As a consequence of the first and second assumptions, it may be expected that a quasiequilibrium steady state will be established after a short transient time. All the following calculations will be based upon this steady state. It is expected that these calculations will at least indicate an upper limit on the deposition rate. The second assumption implies that the deposition rate is diffusion controlled and that surface rates may be ignored. Again, it is expected that the calculations will indicate a maximum deposition rate. (A more refined theory which includes surface limitations is developed in Section III.) The third assumption essentially gives us a mathematical basis upon which to compute the all important diffusion rates. Equation (4) is, of course, the ordinary Fick's law diffusion equation, as it applies to gases in the presence of a small temperature gradient. The appropriate diffusion coefficients can presumably be calculated from the kinetic theory of gases. It has been tacitly assumed that processes such as thermal diffusion, convection due to gravity and viscous flow ("streaming", according to SCHAFER(1)) may be ignored. We are thus restricted to low pressures, i.e. pressures below one atmosphere. Viscous flow effects will be considered in a future paper. Note that only one chemical reaction has been assumed to be occurring. We will see that equation (4) may be combined with the equation of continuity, the requirements of the steady state and the ordinary form of the equilibrium constant, equation (3), to determine the system completely. The familiar equation of continuity, in the absence of sources or sinks, would tell us that the time rate of change of density of any component in the gas phase is equal to the negative of the divergence of the flux, i.e., bF~
~n~
~x
bt
(5)
Under the steady-state conditions which apply in our case, however, 3n~/St is maintained at zero for each gaseous component by virtue of the chemical equilibrium process. Those molecules which would have gone towards increasing the density in the gas phase are absorbed by formation of solid. Those molecules which would have
589
diffused away, lowering the density in the gas phase, are replaced by molecules formed by the chemical equilibrium process. Thus, we have, finally,
~ns
1 ~ Da CgpA
Ot
a Ox R T ~x
1 0 DB OpB
1 0 Dc ~Pc
b ~x RT Ox
c Ox R T ~x
- -
-...
(6)
by combining equations (4) and (5). The quantity Ons/Ot is the net amount of solid produced in unit time per unit volume of container at any point, x. This combined equation contains ( N - 1 ) independent relations among the pressures, pA, PB, PC, . . . , p~r, where N is the total number of components in the gas phase, excluding S. In principle, we can integrate equation (6) to obtain each of the pressures in terms of any one of them, e.g. pB. Insertion of these results into equation (3), which relates the equilibrium constant to the various pressures, would give pB as a function of K. We will see that deposition rates and other quantities of interest can then be determined. We may integrate equation (6) once immediately to obtain
1 DA ~PA
1 DB ~PB
a R T ~x
b R T ~x
1 Dc ~Pc
1 DB OpB
c RT
b RT
(7)
~x
~x
where all integration constants must be zero in order that the steady state be maintained. Integration of equation (7), however, cannot be carried out inasmuch as the various diffusion coefficients vary with temperature and composition, and hence with x, in an unknown way. We may circumvent this difficulty by limiting our analysis to systems in which the temperature gradient is sufficiently small that the composition, and hence the diffusion coefficients, are sensibly constant. The results may be expected to be accurate to first order in [AK/K] where A K = /£2-/£1. We require
]AK/K I ~ 1
(8)
590
G. MANDEL
or, alternatively, IATI <
(9)
- -
where AH is the standard enthalpy change appropriate to equation (2). The usual equation relating 0 In K]OT to ZX/-/has been assumed. To a first approximation then, we define the following constants: DB
DB =
ct
I'A = i'A
cy
(15)
t'e = 1 ' c - - - : 0
DB
- -
DA
where T-is the average temperature and P0 is the average initial pressure of A, i.e. NoRT/CL. Equation (11) now reduces to
=
~,
- -
Dc
=
8
(10)
DD
d8
and integrate equation (7) to obtain act
while the conservation conditions, equations (12) and (13), may be written
PA = kA---ff PB
a
c~
(11)
Pc = kc + -~ PB d8
jba = P0--~ ihB C
/3c = -~3B
= k s + --:-pn,
(16)
o
where ka, kc, kD, ... are integration constants. The integration constants may be evaluated by means of the following conservation conditions:
lfpo
L 1
b
which may conveniently be combined with
L
PBC~
c
0
d
RT
0
L,
0
C~
-b "e "d
K = PnPePD ""
RT
(12)
Cdx = ...
L
l f pA Cdx a RT
No a
0
l fpB Cdx b
0
...,
(13)
RT
where the definitions of No, C and L have been given at the beginning of this Section. We may combine equations (11), (12) and (13) with the limitation of small temperature gradients and obtain --
L
RT f RPBT dx "= _PB b_
-bL -
0
P o - kA
kc
kD m
a(1--ct)
c O - y ) = d(1-8)
d
~
gO0
(14)
(17)
in the limit of small temperature gradients. Thus, we may initially evaluate j0A, j0B.... from thermodynamic data. From these results, combined with equation (3) and (15), we obtain a polynomial in pB which may be solved for any value of the equilibrium constant between K1, i.e. K at 7"1 and K~. In particular, we may evaluate PB at 7"1 and 7"2. According to assumptions (1) and (2), we expect a region between the source and deposit in which no chemical reaction is occurring. It may easily be shown that the equivalent flux of any component in this region multiplied by the cross-sectional area must be equal to the rate of transport of solid in order that the steady state be maintained. This statement is based upon the tacit assumption that the vapor pressure of Scs) is very low so that the flux of S(0) may be ignored. On the basis of the above remarks, we may state that the rate of transport of Sos) will be given by
VAPOR
TRANSPORT
OF SOLIDS
the product of the cross-sectional area, C, and the average equivalent flux of B(g), i.e.
Rt = -
1/~B
ffB2-pm)
b RT
L
BY V A P O R
where
IAHAT/RT1T2I .
We may further consider some simple approximations in general. The first is that in which the equilibrium constant is very large throughout the system so that the reaction given in equation (2) has gone virtually to completion. It is then possible to approximate equation (11) and solve equation (3) for pA. From equations (12), (13), (14) and the condition of large equilibrium constant, it may be shown that equation (11) becomes
K -lla dT ~ ~.-lla.
i f
AT
(18)
where PB2 is the pressure of B(g) in the neighborhood of the deposit, i.e. at x = L, while PB1 is the pressure of B(g) at x = 0. We have assumed that D B / R T does not vary appreciably with x and have used average quantities. This equation has been given essentially by SCHAFER.(1) Thus we see that the rate of transport may be evaluated by means of the above procedure for any set of initial conditions. Thermodynamic and diffusion data is required. The results may be expected to be accurate to first order in
REACTIONS
591
Tz+AT
K-11 a -
. C,
PHASE
(22)
T~
We have also used the relationship (2) K = Ko e x p ( - A H / R T ) .
(23)
We see, therefore, that equations (20) (21) would be expected to apply to the case in which the equilibrium constant, K, is sufficiently large that the reaction, equation (2), has virtually gone to completion. [AH I is assumed to be large, i.e. much greater than ~ aRT, and relatively temperature independent. The diffusion coefficient, DA, is assumed to be relatively constant. The temperature gradient is assumed small, equation (9). We may also attempt to consider the appropriate approximation for the situation in which the equilibrium constant is small everywhere. Simple results, however, cannot be obtained without further assumption. If, as is often the case, one of the products of the reaction given in equation (2) is much heavier and larger than any other product, we may label this product B. Under this further restriction, we may approximate equation (15) for small equilibrium constant as follows:
bp0 pB =
P A = PO
cp0 Pc = - -
Pc = ibc = ~ f B
gg
c
(19)
(24)
d
dpo pD = -
From equations (19), (3) and an analog of equation (18), we obtain immediately
where we have assumed, as per the above discussion,
y, a .... ~ 1 DA bb/ace/ad a:a ... poMIa(K;IIa-K~I lla) C Rt =---= .-RT a (Mla+l) L
(20) M ='b+c+d+
...
Under the condition imposed by equation (9), we may rewrite equation (20) as IAHAT[ DA bb/ace/ada/a ... K-1/aP~/a
D~ Rt =
(26)
C ( R"llb
(21)
(25)
We note that this is equivalent to the familiar assumption that the slowest step determines the over-all rate of a reaction. In this case, the slowest step is the diffusion of B. Combining equations (3), (18) and (24) we obtain
K(l/b~,
/,~/u "=i - ' = 2 , bblMcclMdalM ... [(K~Ib + K~Ib)12](1-blM)
O L
592
G. MANDEL
Under the condition imposed by equation (9), we may rewrite equation (26):
MIAT b~ R, =
p~JMK1/M
bRT z R T bb/mce/mda/m...'
(27)
where KIlM
Further examination of equations (21) and (27) indicates, upon realization that DA and DB will vary approximately as p o 1, that the rate of transport should also display a maximum in its variation with P0 at constant temperature, at least in the limit of low pressures.iS) It is expected that Rt will vary as
~ K IlM
p(~l l~--l)
is defined as in equation (22). We see, therefore, that equations (26) and (27) would be expected to apply to the case in which the equilibrium constant, K is sufficiently small that the reaction, equation (2) has virtually not gone at all. Further, product B is so heavy and has so large a collision cross-section that its diffusion coefficient, DB, is much smaller than that of any other product in the gas phase. IAH] is assumed to be much greater than ~ bRT and relatively temperature independent. The temperature gradient is assumed small, equation (9). We note that equations (21) and (27) can also be obtained from the treatment of SCHAFER,(1) except for the fact that the diffusion coefficients are here seen to be different for the two extreme cases. Neglect of this fact can obviously lead to error in certain chemical systems. The fact that the rate of deposition increases with increasing K for small values of K, as in equation (27), while it decreases with increasing K for large values of K, as in equation (21), must be a general result, i.e. it must be true for any system. For a system in which B is a very slowly diffusing molecule, equation (21) would be expected to apply when K has a high value while equation (27) would apply when K has a low value. The maximum possible deposition rate would occur approximately at the point at which equations (21) and (27) predict the same result, although this predicted deposition rate would be somewhat too high. From equations (21) and (27), we then obtain the following approximate condition under which a system with very slowly diffusing B would display the maximum deposition rate: ~.p(oa_M , = (__b ) M a / ( M + a , \acz]
bbc ` ... aM
(28)
The above discussion implies that the rate of transport will display a maximum in its variation with temperatureat constant initial pressure, P0.
at low pressures and as
p(~/M-1) at higher pressures, provided M > a. If M < a, these dependences are reversed. The approximate condition for the maximum is again given by equation (28). We conclude Section II by again pointing out a practical limitation of the above treatment. This is its restriction to low pressures and geometry such that convection does not enter. It appears unfortunately, that the conditions under which vapor transport is of interest in the laboratory are such that convection may be important in many cases. 3. SURFACE RATE EFFECTS
A reasonably rigorous theoretical calculation of the quasi-equilibrium steady-state rate of transport of solids through the vapor phase by means of chemical reactions in the presence of a temperature gradient has now been carried out. This, and prior analogous calculations, (1) have been based upon the assumption that the composition of the vapor phase above the source and deposit was that characteristic of chemical equilibrium at the appropriate temperatures. This is equivalent to the assumption that the surface reactions ("evaporation" and "condensation") are infinitely fast compared to diffusional transport through the vapor phase and, therefore, do not come into the calculation of over-all transport rates. We shall see that a simple extension of the previous calculation, to include both surface reaction rates and diffusional transport rates, may be carried out without significant further assumption. The diffusional rate of transport will be calculated in exactly the same manner as previously described except that the ratio 9 -
p b 4,,e,,,hd B Y C t ' D • •*
p~
(29)
VAPOR T R A N S P O R T OF S O L I D S BY VAPOR PHASE R E A C T I O N S will no longer be given by the equilibrium constant, K, appropriate to the chemical reaction, given as in equation (2), by means of which transport takes place. We will again be considering a cylindrical system in which transport takes place from x = 0, at temperature T1, to x = L, at temperature T2. The diffusional rate of transport will be shown to be proportional to the "apparent enthalpy change", AH*, in the limit of small AT. The "apparent enthalpy change", AH*, reflects the temperature dependence of the ratio Q, just as the actual standard enthalpy change, AH, appropriate to equation (2), reflects the temperature dependence of the equilibrium constant, K. The rates of surface reactions at the source and deposit will be considered simply on the basis of the law of mass action. It will be seen that these rates, in the usual manner, are related to the "undersaturation" in the neighborhood of the source and the "supersaturation" in the neighborhood of the deposit. Finally, it will be shown that, in the limit of small AT, the surface reaction rates are proportional to ( A H - A H * ) . Inasmuch as the steady state condition requires that the surface reaction rates and the diffusional transport rate be equal, we may combine the above described results with the elimination of AH*. This finally yields the over-all transport rate in terms of the quasi-equilibrium rate, which has been previously calculated, multiplied by an additional factor containing the surface rate coefficients. These results may also be rearranged to show that the reciprocal of the over-all rate is given in terms of the sum of the reciprocals of the surface rate and the diffusonal rate. This is the expected form for a system in which several rates are connected in series.
A. Surface rates From the law of mass action, we write the following relations for the rate of removal of solid ("evaporation"), R1 at the temperature T1, and the rate of deposition of solid ("condensation"), R2 at the temperature T2:
the reverse reaction at T1, k12 for the forward reaction at T2 and k22 for the reverse reaction at T2. The pressure of A(g) at the temperature T1, i.e. above the charge, is designatedpal, etc. No assumption has been made regarding the possible composition, pressure or temperature dependences of kn, k21, ... We obtain the equilibrium results, of course, by setting R1 and R2 equal to zero. Thus,
KI=
b
c
(PBlPc1...)
kll
P~I b
c
K2=(PB2PC2...
P~2
e.. = -~21
(32)
_ k12 ,
(33)
) e..
k22
where K1 and K2 are the equilibrium constants at the temperatures T1 and T2 respectively, as before. In transport experiments, R1 and R2 are both greater than zero. For the steady state, they are also equal. Thus, from equations (30), (31), (32) and (33),
(34) where b
c
PBlPcl "" 91 = pa a b
Q2 -
(35)
c
PB2Pc2 "'"
P~2
(36)
As has been pointed out by LEVER,(4) the quantity (1--Q1/K1) is a generalized undersaturation while the quantity (Q2/K2-1) is a generalized supersaturation. The quantity k n p ~ 1 is thus defined as the ratio of the surface reaction rate to the supersaturation and similarly for k12 P~i2As a consequence of equation (34), we have the following inequality
b c ... R1 = kllPA1--k21PB1Pc1
(30)
K2 < Q2 < Q1 < K1.
a + k22P~2Pc2 b c ..., R2 = -- k12PA2
(31)
We note that equation (37) is based upon the assumption that transport proceeds from T1 to T2. The positive quantity ( Q I - Q 2 ) , which serves as a measure of the diffusional rate of transport from
where kn is the rate coefficient for the forward reaction in equation (2) at temperature T1, k21 for 2q
593
(37)
594
G. MANDEL
Tlto 7"2,must be less than the quantity (K1-K2), according to equation (37). Therefore, the presence of finite surface rates lowers the rate of transport below its quasi-equilibrium value, as expected. In the same way that the ratio K2/K1 is given in terms of the enthalpy change, AH, for equation (2) (2), we define an "apparent enthalpy change", AH*, in terms of the ratio Q2/Q1, i.e. Q2 Q1
(1 1)]
exp
.
(38)
l R
We note that, according to equation (37), [An*[ < IAHI.
(39)
provided it is remembered that the quantity AHAT = AH(T2-T1) must be negative in order that transport from 7"1 to 7"2 occur. Combination of equations (34) and (38) finally allows evaluation of the steady-state deposition or transport rate, Rt, in terms of AH*:
shall now proceed to obtain another relationship between Rt and AH* by consideration of the diffusional rate of transport.
B. Diffusion rates Throughout the remainder of this treatment, we will confine our detailed analysis to systems in which the average equilibrium constant is either very large or very small. We limit ourselves in this manner since it is only for these eases that we have obtained simple analytical results for the quasi-equilibrium transport rate. For the case of large equilibrium constant, the quasi-equilibrium transport rate has been given as [(see equation (20)]: Req. =
.4(K~l/a-K~lla),
(43)
where .4 is a term that will be independent of surface effects. We obtain the desired more general
, _ (40)
Re = RI"~= R2 =
1
k11P.~l k12PaA2exp L This general result is based essentially only upon the form of the law of mass action. We may simplify equation (40) in the limit of small IAT[, equation (9). Equation (40) reduces to ( A H - A H * ) (1.
Rt=
R
f~l
1
1
kllP~ 1
kl2p~ 2
(41) 1
-R
( T2
1 -
Rs
- -
kllP~ll
diffusional transport rate by replacing/£2 by Q2 and/£1 by Q1. Thus,
Rt = A(O71/a- Q-~lla).
Rt = Req. [-K-fz)
1
T1 )"
1 - exp [ a R
In equation (41), we have also defined a .mean maximum surface rate", Rs, 1
111
/'2
(44)
i Q2 '~-11"
( A H - AH*)
= Rs
tT1
We may combine equations (38), (43) and (44) to obtain the more general transport rate in terms of the quasi-equilibrium transport rate:
1)
7"2
R
X
-
-
-
tl--exp
AH ~
I1 7"1 1 7"1
(45)
1 t
•
kl2p~4s
(42)
Equation (41) is the desired simple relation between the transport rate, Rt, and ( A H - A H * ) . We
Furthermore, from a combination of equations (37) and (38), it may be shown that, in the limit of small [ATI, i.e. under the condition of equation (9), the ratio Q2/K2 approaches unity. As a
595
VAPOR TRANSPORT OF SOLIDS BY VAPOR PHASE REACTIONS
consequence, in the limit of small [AT[, equation (45) reduces to
in which we see AH* to be independent of AT in the limit of small [AT I.
AH*
Rt = R e q . - - .
(46)
AH
4. EXPERIMENrAL DETERMINATION SURFACE RATES
OF
A similar analysis of the small equilibrium case based on equation (26) and the limit of small temperature gradients also leads to equation (46). We see, therefore, that equation (46) is valid for both the case of large and small equilibrium constant, in the limit of small ]AT].
We will now be concerned with the information on surface rates that can be indirectly obtained by measurement of the temperature and initial pressure dependence of the steady-state transport rate. We will again restrict our attention to experimental situations falling under the condition of equation (9) i.e. in the limit of small IATI.
C. Over-all steady-state rate
A. Temperature dependence
The maintenance of the steady state requires that the diffusional transport rate and the surface reaction rates be equal. We may, therefore, combine equations (41) and (46), with the elimination of AH*, and obtain the steady-state transport rate in the limit of small ]AT[. The result of such a procedure is
We are interested in the dependence of the steady-state transport rate, Rt, on the absolute temperature, T [or the average absolute temperature, T = ½(TI+T2)]. The experimental data would consist of measurements of Rt at different values of T, all other parameters, e.g. AT, remaining fixed. The theoretical analysis of this situation may be carried out most easily by consideration of equation (47). We obtain immediately:
1
1
1
Rt 7"2
(47)
S ln(RtT1T2)
T~
ST
where R0 is defined by the following relation: Req. = Ro ~
/'2
1) I
,,8,
so that R0 can be calculated from knowledge of gaseous diffusion coefficients and thermodynamic data, according to the quasi-equilibrium theory. In principle, we see from equation (47) that, inasmuch as Req. and, therefore, R0 can be calculated, experimental determination of the steadystate transport rate, Rt, gives information about the surface reaction rates. Such measurements must be made, of course, under the condition of equation (9), i.e. in the limit of small [AT I. In equation (47), we see, as usual, that rates in series add as reciprocals. We note also that M-/* may be evaluated by combination of equations (41) and (46). We obtain AH AH* = 1 + Ro/R-~s'
(49)
1 SlnR0
1 1 / R o + I/R
R0
-
ST -
1 SlnR8]
+ - R8 -
:"
(50)
The temperature dependence of R0 may be obtained directly from the quasi-equilibrium theory for the case of either very large or very small equilibrium constant [see equations (21) and (27)]. We may conveniently calculate the temperature dependence of the surface reaction rate, Rs, by rewriting equation (42) : R8 =
p) = 1/kn + 1/k12
kipS,
(51)
where we no longer distinguish between PAl and PA2 in the limit of small ]AT I. We have also defined a "mean" surface reaction rate coefficient, kl. The temperature dependence of the surface reaction rate coefficient, kl, may be deduced from
596
G. MANDEL
the absolute chemical reaction rate theory of EYRING et al. (5) We thus obtain
hi ST
(AH+Ea) RT 2
In
(52)
equilibrium theory will also allow an estimate of the "mean" surface rate coefficient, kl. Similarly, for the case of small equilibrium coefficient, we combine equations (50), (51), (52) and (54) to obtain
ln(RtT1T2) ST
where Ea is an "excess" potential energy barrier. We will assume, as a first approximation, that Ea is generally much less than Ajar. The temperature dependence of the pressure, PA, may be obtained directly from the quasiequilibrium theory. For the case of large equilibrium constant, we obtain
1 {~o AH F 1 (AH+Ea)} l/R0+ llRs MRT~ Rs RT 2
Sln(RtT1Tz) _ SIn PA
AH aRT 2"
ST
~(1/T)
(53) _
1 /~s( I+(R0
AH ( 1 + i R o / R s l =
AH
R0~H)
MR
Rs
AH* (1 +MR°I
For the case of small equilibrium constant, we have, of course, (56) In PA -
-
ST
=
O,
(54)
since PA is approximately equal to its initial pressure, p0, and constant. Thus, we may finally combine equations (50), (51), (52) and (53) to obtain, for the case of large equilibrium constant,
~3In(RtT1 T2)
B. Pressure dependence
ST
1 1 1/Ro+ l/R8 Ro I ( 1 +Ro/Rs S ln(RtT1T2) S0/T )
upon neglect of Ea relative to AH. We see, therefore, that, for the case of small equilibrium constant as well as for the case of large equilibrium constant, determination of the temperature dependence of the transport rate, Rt, allows estimation of the ratio Ro/Rs and, finally, of the surface reaction rate coefficient, kl.
leo}
-a--R-~ + - - - Rs ~ 2 " AH aRT 2
Ro Ea Rs RT 2
1 (AH)AH* = l+/~o/Rs ~ - - ~ - " (55)
The neglect of Ea relative to AH may not be generally valid. We see that determination of the temperature dependence of the transport rate, Rt, in the case of large equilibrium constant, may allow reasonable indirect estimate of the ratio Ro/Rs. Further calculation of R0 and pa by means of the quasi-
To obtain the dependence of the steady-state transport rate, Rt, on the initial pressure of A(o), P0, we may rewrite equation (47) as follows: Req.
Rt
Ro -
1 +--.
Rs
(57)
We note again that equation (47) and (57) apply only in the limit of small [AT[, i.e. under the condition of equation (9). In order to proceed further, we must evaluate the initial pressure dependence of the ratio Ro/Rt. For the case of large equilibrium constant, the quasi-equilibrium theory predicts that R0 will vary as p(oM/a-l), where we have taken into account the fact that the diffusion coefficient will vary as po I according to the kinetic theory of gases and assumed that M > a. For the case or large equilibrium constant, the quasi-equilibrium theory
VAPOR T R A N S P O R T OF S O L I D S BY VAPOR PHASE R E A C T I O N S also predicts t h a t p a will vary as pMla. Therefore, according to equation (51), Rs should vary as p ~ , providing we neglect any pressure or composition dependence in ks, which may not be generally valid. We see, as a consequence, that, for the case of large equilibrium constant, the ratio Req./Rt should vary linearly with p~U/a-l-M), all other parameters being held fixed. The ratio Ro.q./Rt is determined by measuring Rt and calculating Req. for each value of p0. We note, from equation (57), that the intercept of a plot of Req./Rt against p~oM/a-l-M) should be unity. If this is found not to be the case, it indicates that the calculated values of Req. are probably in error by a constant factor due to errors in our knowledge of diffusion coefficients. The results should then be multiplied by an appropriate correction factor such that the intercept becomes unity. We finally obtain, either from the slope of the plot or from an appropriate average of the values obtained at each point, a value for the ratio Ro/Rs at any particular value of P0. Since we may calculate R0 and pA from the quasi-equilibrium theory for this value of Po, we may finally estimate the surface rate coefficient, kl. In a manner similar to the above, we may ascertain, for the case of small equilibrium constant, that the ratio Req./Rt should vary as p~om/a-l-a), all other parameters being held fixed. The intercept of such a plot should again be unity, allowing correction of the calculated values of Req. The surface rate coefficient, ks may again be determined.
C. Substrate area dependence The simplest determination of surface rates can be carried out by measurement of the rates of transport of a powdered source to single crystal substrates of well defined but different areas. The expected effects are most easily understood on the basis of equations (42) and (47). We expect the surface rate coefficients, kn and k12, to be directly proportional to the areas of the surfaces involved. Since we are considering a powdered source (kn) and a single crystal substrate (kse), we see from equation (42) that the major contribution to Rs is due to k12. If we now measure the rates of transport, Rt (1) and Rt (2), in two experiments in which the substrate areas are
597
different while all other parameters are held constant, we see from equation (47) that
1
1 Rt(2) Rt(1)
1 IAH/R (l/T2-1/T1)[
1)
(1 _ 1/r~ × R~2) R¢~) ]±H/R (1/T2-1/Tz)]
(2 1)
x
) 2~i)
(58)
'
where rs is the specific surface rate (rate per unit area) and A (2) and A (1) are the substrate areas. We see that rs, Rs and ks2 are thus immediately determined.
rs=
Rs(S)
Rs(2)
Aa)
A(2)
-
kl2(1)p~2 - - A(1)
-
ks2(2)p,~ 2 A(2)
(59)
CONCLUSIONS It has been shown that the quasi-equilibrium theory for the transport of solids through the vapor phase by means of chemical reactions may be extended to include the effects of finite surface reaction rates. It has been assumed that (1) diffusion coefficients are relatively independent of position and temperature although different components in the gas phase are allowed to have different diffusion coefficients; (2) the enthalpy change for the reaction in equation (2) is large compared to RT and relatively temperature independent; (3) The only chemical reaction occurring is that given in equation (2); (4) diffusion is the only mechanism of transport through the gas phase; (5) the equilibrium constant for equation (2) is either very large or very small. In this analysis, the restriction of small [AT[ has also been required. Results have been obtained in very simple form. The surface rates are found to add in a reciprocal manner with the previously obtained diffusion limited rate, as expected. It has been shown that experimental determination of the temperature, initial pressure and
598
G. M A N D E L
substrate area dependences of the steady-state transport rate can yield information about the surface reaction rates.
Acknowledgements--The author wishes to express his gratitude for many helpful discussions with Dr. A. REISMAN, Dr. F. P. JONA, Mr. V. J. LYoNs, Mr. R. F. LEYEa, Dr. S. P. KELL~.aand Dr. R. W. KEvm.
REFERENCES
1. SUHAFERH., Z..4norg. Chem. 286, 27 (1956). 2. MOOREW. J., Physical Chemistry, Chap. 4. Prentice Hall, New York (1955). 3. KEYESR. W., private communication. 4. LEVERR. F., private communication. 5. HILL T. L., Introduction to Statistical Thermodynamics p. 194 if. Addison-Wesley, Reading, Mass. (1960).