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Chemical potential shifts due to capillarity-unary systems
Pergamon PII:
Acta mater. Vol. 45, No. 12, Pp. 4963-4968, 1997 0 1997 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/97 $17.00 + 0.00 s1359-6454(97)00199-7
CHEMICAL POTENTIAL CAPILLARITY-UNARY
SHIFTS DUE SYSTEMS
TO
N. KULKARNI and R. T. DEHOFF Department
of Materials
Science & Engineering, 253A Rhines Fl 32611, U.S.A.
Hall, University
of Florida,
Gainesville,
(Received I7 September 1996; accepted 22 May 1997)
Abstract-The temperature,
effect of curvature on the chemical potential for two-phase unary systems is usually given in the literature by the GibbssThomson equation [l-6],
at constant
yB(H) = /_?(H = 0) + 2yVBH where H is the local mean curvature of a /l particle in equilibrium with an a matrix and @, the molar volume of the fi phase. It can be shown, based on the above equation, that the corresponding shift in chemical potential for the CIphase is [I]: pa(H) = pLOI(H= 0) Thus the usual form of the Gibbs-Thomson equation that appears in the literature is not rigorous and hence results in a violation of the condition for chemical equilibrium for a curved interface (g(H) # #(H)) assuming incompressibility and isotropic stresses in the case of the solid phases [7-91. A rigorous derivation of the Gibbs-Thomson equation for unary two-phase systems at constant temperature is provided, that gives the correct equation for the chemical potential shift: /_&H) = #(H)
= &H
= 0) + 2yH[ V” VB/( V” - VP)]
While the above equation can be approximated (Ip = ‘-Vfl~p) in the case of a condensed-gas (C) system to give the usual form of the Gibbs-Thomson equation, a similar approximation for a condensed&condensed system can result in a significant error in both the sign and magnitude of the shift. The corrected expression for the chemical potential shift has an inverse dependence on the difference in the molar volumes (p-p) between the two condensed phases. For solid (q-liquid (L) equilibrium in certain systems the shift may be negative (e.g.V” = L < v8 = ’ for Bi, Si, Ge) or even approach infinity (e.g. v” = ‘z VP = ’ for carbon). A similar dependence is obtained for the pressure shifts in each phase. The present formulation is consistent with the Clausius+Clapeyron equation used to represent unary phase diagrams, which has the same dependence on the relative molar volume. 0 2997 Acta A4etallurgica Inc.
1. INTRODUCTION The assumption of local equilibrium across an interface has been invoked frequently in studying many basic processes in condensed systems such as solidification, growth, diffusion, sintering, coarsening, etc. Gibbs was the first to derive the conditions for equilibrium for planar and curved interfaces [7]. While the condition for thermal equilibrium is identical to that for a planar interface, the conditions for chemical and mechanical equilibrium are in genera1 different [7-91. The conditions for chemical equilibrium for curved interfaces are modified from the planar case due to the presence of non-isotropic stresses in case of interfaces where one or both of the phases are solid. They are however always identical to the planar case for liquid-liquid interfaces [7-91. In case of interfaces consisting of solid phases, if it is assumed that the surface stresses v> present in solid phases are isotropic and equal to y, the specific interfacial free energy, the conditions
for chemical equilibrium for curved and planar interfaces become the same. The conditions for mechanical equilibrium across a curved interface differ from those across a planar interface by a factor which depends on the product of the local mean curvature (H) and the specific interfacial free energy (y). These conditions of equilibrium are used in deriving an expression relating the local chemical potential shifts in a system with curvature of the interface, at constant temperature. This expression is classically known as the Thomson-Freundlich or the Gibbs-Thomson relation [l-6]. Although the original Gibbs-Thomson relation applied to liquid-
4963
gas equilibria in unary and binary systems P, 31,it has been assumed to be valid for condensed phase equilibria in unary, binary and multicomponent systerns. Some of these formulations contain inaccuraties and inconsistencies, which can result in significant errors in the calculated chemical potential shifts due to curvature. This presentation
4964
KULKARNI
and DEHOFF:
CHEMICAL POTENTIAL SHIFTS
focuses on corrections to the Gibbs-Thomson equation applied to unary condensed systems. It is to be noted that Gibb’s association with the GibbsThomson relation is only in context with the conditions for equilibrium and not the erroneous expression appearing in the literature. The assumptions of isotropic stresses and incompressibility of solids which are inherent in some of the earlier formulations [l, 61 are also valid in this work, although as pointed out by Cahn [8,9] hardly practical. Nevertheless it is hoped that a rigorous treatment of the simplest case would lead to adaptation to more general cases based on the framework of the present treatment. The Gibbs-Thomson relation at constant temperature is usually stated as [l-6]: #(H)
= &H /f(H)
= 0) + 2yVPH
= p”(H = 0)
(1) (2)
where V@is the molar volume of the p phase. It will be demonstrated later that the relationship between the corresponding chemical potentials for the CI phase, predicted from this equation, is that the chemical potential of the c(phase does not shift with curvature [2]. Since the condition for chemical equilibrium in the bulk system requires p’(H = 0) = #(H = 0), comparison of Equations (1) and (2) shows that p”(H) # #(H). This violates the condition for chemical equilibrium [7]. In this discussion, a self-consistent procedure to evaluate the effects of curvature on thermodynamic properties in unary systems is given, which results in a form of the Gibbs-Thomson equation for which p*(H) = ,u”(H). The significance of this result is briefly explored.
2. PROCEDURE
The variables system are:
H’;(K,+KZ) For each of the principal normal curvatures (K, or K~), a vector is defined that is normal to the surface element and points towards the center of curvature. Choose a convention such that the curvature is positive if this vector points into the /I phase and negative if it points into the CI phase. The local mean curvature H is positive for surface elements that are convex with respect to the p phase, negative for concave elements, and may be either positive or negative for saddle surface elements depending upon the relative values of the curvatures [lo].
to specify
this two phase
Ta Pa, Tp, PB and H. TM, Tp are the temperatures and P*, PB are the pressures in the c( and /J’phases, respectively. An additional variable (H) is required to describe the local mean curvature for this system. The conditions for equilibrium for this two-phase system with a curved interface are according to Gibbs [7]: Ta = 7-b
(4)
Ps(H) = Pa(H) + 2y H
(5)
/f(H)
= p!(H)
(6)
p’(H) = $(H) are the chemical potentials of the c( and p phases in two-phase equilibrium with an interface of curvature H. It is assumed in this treatment that y, the specific interfacial energy, is independent of curvature, thereby implying incompressibility of c( and p. As mentioned earlier if either CIor /I or both are solid phases it is necessary to assume that their surface stresses are isotropic and equal to y, the specific interfacial free energy, for equation (6) to be valid [7-91. Since there are five variables for this system and three relations among them [equations (4)-(6)], the number of degrees of freedom or independent variables for this system is equal to two. 2.2. Calculation of phase boundary shifts in wary systems Consider an incremental change in the state of a two-phase system with a curved interface. The change in chemical potential for each of the phases is
2.1. Conditions for equilibrium Consider a two phase (a + /I) unary system. Let r1 and r2 be the principal radii of curvature for an a-j3 surface element. The principal normal curvatures are given by ~~ = l/r1 and ~~ = 1/r2. Let the local mean curvature across an a-/I interfacial element be H where
required
dp’ = -S”dTa
+ VadPa
(7)
d#
+ VfldPfl
(8)
= -SBdTP
S”, V” and Sp, VP are the molar entropies and volumes of the o! and /I phases, respectively. If during this change, the system remains in two-phase equilibrium, then from the conditions for equilibrium [equations (4X6)] dTfi = dTa = dT
(9)
dPfi = dP” + d(2yH) = dP” + 2y dH d#
= dpa
y is assumed to be independent Substituting for dPP from equation (8) can be written as d# Equations
= -SBdTB
(11) of curvature. (lo), equation
+ Vp(dPa + 2ydH)
(7) and (8) are related
(10)
(12)
by the condition
KULKARNI for chemical -SadTa
equilibrium
[equation
+ V”dP”L = -SfldTfl
and DEHOFF: (1 l)]. Hence
+ I’p(dPa + 2y dH) (13)
d7” and d@ are equal due to the condition thermal equilibrium [equation (9)]. equation becomes
for (13)
-SadT+l’“dP=-SfldT+Vfl(dPa+2YdH)
(14)
-(S”-@)dT+(l’=-
(15)
I’O)dPa-2y@dH=O
4965
CHEMICAL POTENTIAL SHIFTS
equation (21) relates the pressure in the a phase in a finely divided system (P”(H)) to that in the same phase in a bulk system (P”(H = 0)) at the same temperature. The corresponding shift in the p phase is obtained by utilizing the condition for mechanical equilibrium across a curved interface [equation
(lO>l. dl”=dP’+27dH=(2$+ZY)dH Hence the pressure
(22)
shift in the b phase is
or dPp = -ASdT
+ A VdP” - 2y VPdH = 0
(16) The integrated
where AS=S’-Sp
and
AV=
Va-
VP.
ppVO
(17)
equation (16) is the general form of the ClausiusClapeyron relation for a unary two phase system with curved interfaces. Since it represents a single equation among three variables, the unary two phase system with a curved interface has two degrees of freedom, as discussed earlier. Systematic exploration of equation (16) is carried out by comparing equilibrium states in unary two-phase systems in which one of the variables (T or P”) is constant. This leaves one degree of freedom. This is usually assigned to the curvature H, since one is normally interested in the effect of curvature on other properties. Hence, if P’ is constant in this one can obtain the dependence of comparison, temperature on curvature [3,10, II]; if T is constant, the dependence of P” on curvature is obtained.
2.3. Effect of curvature on pressure To evaluate the effect of curvature on pressure in a unary two-phase system, compare two systems that are at the same temperature T, but differ in their local equilibrium states on account of a difference in mean curvature H. equation (16) can be written as
Hence the pressure
(dT=O)
2Yva
Pp(H) = Pp(H = 0) + Bv Subtracting
equation
Pfl(H) -P(H)
dP”= .IPa(H=O)
2y VP p”(H) = poL(H = 0) + =H
(25)
= PB(H = 0) - PGL(H= 0)
Thus the condition for mechanical equilibrium across a curved interface [equation (lo)] is recovered. equation (21) along with equation (25) gives the correct expression for the pressure shift across a curved interface.
2.4. Effect of curvature on chemical potential The chemical potential in a unary system is a function of the temperature and pressure for that phase. Since in the comparison between the two states of the system with different curvatures the temperature of the system is constant, the change in chemical potential is given by: dpa = V”dP” - SadTa = V”dP” for dP” from equation
dp” = V’(~dH)
(19)
H 2YVfldH ~ s HE,, AV
(21) from equation
(25)
(dTa = 0)
(27)
(19)
(18)
If TVand /? are both condensed phases, v*, fl and A V can be assumed to be independent of the curvature (H). The above equation can be integrated: p(H)
H
+ 2yH -=2yH ‘” - I/p AV
shift in the CIphase is dP”=gdH
(24)
@(H=O)
Substituting AVdP”-2yVPdH=0
form is given by
(20)
(21)
(28)
=qdH
Similarly the chemical potential shift for the p phase can be obtained by substituting for dPp from equation (23). d$
= VpdPfl - SBdTB = VfidPfl
d#
= Vfi(2$dH)
=qd,
(dTfl = 0)
(29)
(30)
Equations (28) and (30) are identical. Thus the condition for chemical equilibrium is recovered demonstrating the self-consistency of the procedure.
walsk se%pasuapuo3 :H 30 luapuadapu! aq 01 d/l 8yuInssE ‘~~03 01 QUO alqmgdde s! ‘saldpuyd pmoy+~en Ou!sn [r] Iegualagp B u! passaldxa aq um (I) uogenba moymba uoswoyJ-sqq!g ayl30 w.103 $3al.ro3 ayl suylnw pm [v] OuulaH dq paylap uaaq w?y qq~ ‘(SE) uogenba .fym JOUalo3alayl SFuogdurnsw sg~_ IOU s! (I) uogmba ]t?ql umoys ,$Sea aq ue3 11 ‘[(LE) ~amwlz.m 01 anp sygs rvpualod pr+uay~ ayl .IOJ uogenba] anoqt! paauap auo ayl u.1o.131uaIagp s! [(I) uogenba] swalsds pasuapuo3 103 uoymba l[nsaJ ~~a~.xoy UB 01 spsal ‘o=dp )eql uogdurnsse uoswoyL-sqqg ayl 30 mo3 pasn LIap!M aqL ayl snyL wn!~qg!nba pm!uIay3 103 uoypuo~ ayl 30 uogeloy u! air! ‘3ag.wa passnmp se Ypq~ ‘IClay NOISSll3SIa ‘E -padsa ‘sawyd x) pm d ayl ~03 amlm.xm 01 anp s13!ys Iegualod pm!uIaq:, aql ‘(sp pw 1) suogmba paylap uogdurnsw !PaA!‘.L ‘(0 = zdP) W .a.xnle.xaduIal ~ue~suo~ ie seal pmoy!ppe s!yl 8u~sn ‘o= ,,dp put2 0 = up .a.! -sl(s pasuapuo3-pasuapuo3 103 uogmba uosuroy~ ‘JUE~SUOD aq 01 pawnsm ale aseyd x1ayl u! amssaJd -sqq!g aq 30 1~1103lDallo:, ayl SF (LE) uogenba ayl pm amwadmal ayj ~104 stualsh 30 uos!md -u103 sg u! ~t?ql slear\al 13~9spzgualod pzyuay3 ayl (Sl.mlSliS pasuapuo3-pasuapuo3) (LE) 103 uoye+ap sg 30 uoyu!uwxa uv ‘[I] !paAnL Aq pa@ap uaaq amq (sp) pm (I) suoynba = H)dti = (H)# = (H)ad H s+(O ‘[L] acw3lay paam:, ‘I?sso_m tumq -ynba Iw!way 103 uog!puo~ ayl 30 uo,zzv~o~nJea13 (O=H)a’f (0=ffL$d v s! sw ‘CMgfl # oi),$ say8 (SP pue I) suoynba (9E) HP& o=H = s”p I= ,arlp 1 %uyduro3 ams ‘(0 = H)# = (0 = HW d/l.i’i‘k Hs (H)yd CHW (SP)
(0 = H)$
= @A’~ sagdw! q3q~
(PP)
0 = zdP&l = JfP
aq 01 pap!pa.rd s! aseyd Bydpz ayl 103 am1eA.m Q!M 13ys pzgualod pm!wayD ay$ amaH ~ammm~ 30 uoFpun3 B IOU s! aseyd r, aq$ u! amssa.rd aql leyl sp!pald ‘(g&) uogenba ‘uo!lelaJ Ieguajod pguxay3 ayl 30 ~~03 ayl snqL (EP)
(ZP)
0=&P HPAZ = HPLZ + zdP = gdP
.aee3lalu! paam ayl ssom urn!lqg!nba pm!uvq3aw 103 uog!puo~ ayl dq paufuualap s! 13ys amssald ayl ‘aseyd x) ayl UI (IP)
30 suogmn3 8~0~~s $0~ a.w puw%ay ayl u! seal ayl am!s ‘palwflalu! LpDa.up aq utm (1 E) uogmba ‘smaisk pasuapuo+pasuapuo3 JOT .a.mlwalg ayl u! auop uaaq wy SE ‘suralsds pasuapuo3-pasuapuo3 01 pagddv uayM pyv~ lai3uoI ou s! (EE) uoyenba 30 ~olmauxnu ayl UT D = J slam23 Qua!uaa -uo3 leyl [(ZQ uo!lenba] nv .103 uogmurxolddc sum ayL w,ualsAs sGpasuapuo3 01 Quo saydde ‘a.mlwalg ayl u! ua@ uoymba l! ‘.IaAaMOH uostuoyJ-sqq!D ayl 30 ~103 pmsn ayl ‘(I) uogsnba se auws aql s! 33ys Iylualod pm!uraqD ayl 103 uogenba ayl 30 1~1103pale&talu! aAoqe ayL (SE)
O=H
HPLZ = gdP
(tiE)
HP d,‘i‘Z
= H)drl
I
= d’p
H
= (H)dd
= (Hj9d
(O=H)dd (O=H)$ = #P (H),q”
I
(H),d
J’
aq UBDuogmba aAoqe ayl ‘amlm.m ayl 30 luap -uadapu! aq 01 (4) aseyd pasuapuo3 ayl30 aum[oA ~efotu ayl pur! (X) k%aua a2e3ms ayl %uFurnssv
‘(6~ pm 8~) suogenba S?u~.wduro3 (Ed s! aseyd d aql u! 13ys amssald aqL
(0 = gP = .JP)
(dn < S/j :sul2lsxs sv8-pasuapuo2) Hdzd @+(o
H*Z + (0 = H>gd = (H&d
:am3lalu! leg e qj!~ walk awes aqj 01 aA!lvlaJ aDtz3lalu! parl.m B ~J!M malsl(s e 103 awyd 8’ ayl u! 13ys amssaId aql say% uogmba aaoqE ayl30 uoge.SaluI (OP) Lq pauplqo (6E)
H?‘&Z=HPL= ,&‘f+‘i ‘k
drlP= r~‘p amaH
s’dP+l
(zd
9A=gA
- 9=aA = AV
= &pdS - ddPdA = (H)d+ .a.mlwadural JUEJSUOD 1~ apmu SFsaml -mm3 Syagp ypm smalsk oM1 ayl uaarnlaq uos -!.wdwoD ayl ams .amssald pw amltmdural aql30 uoyun3 B s! aseyd d ayl 30 Iyualod p+uay ayL (8E)
.aseyd pasuapuog ayl uayl Jai?.wI d~1um~u8~s s! aseyd SE%?ayl 30 auyoa lelotu ayl acys ‘apm aq um uoym!xo~ddv UE ‘awyd w8 ayl s! xi alayh4 ‘malsh ~2%pasuapuo3 I? 30 asm ayl UI
(Id
HpdA xZ = d’p SUIHS WI1NXLOd W3IWBH3
HP-=
AV
gA xl/l kZ
d’P = 1o’P
:JrlOHEICl PuE INlIVTIflX
996P
KULKARNI
It is seen from equation (16) the general form of the Clausius-Clapeyron relation for a unary two phase system with curvature, that there are two independent variables needed to describe the state of the system. If the temperature is maintained constant (dT = 0), then equation (16) gives the relation between the change in pressure (dP”) and curvature (dH). If it is assumed, that both the temperature as well as the pressure are maintained constant (dT = 0 and dP=O), the two degrees of freedom are assigned. The state of the system is fixed and the variation with H cannot be explored. The role played by A V in shifting phase boundaries in unary condensed systems may be illustrated by considering the effect of curvature on the state of solid-liquid interfaces as exposed in the modified Clausius-Clapeyron equation (16). If two systems with differing curvature are compared at the same pressure in the CIphase, -ASdT
- ~YV’=~ dH = 0
Thomson’s equation ing point is obtained
(dP=L
for the depression [3, 10, 111.
= 0)
(46)
of the melt-
dT=-gdH+
T(H)=T(H=O)-%H
2y vs (47)
This result is independent of A V. Thus the melting point always decreases as the curvature of the solid surface increases. If this result is applied at a series of pressures it may be visualized as a shift in the melting curves to lower temperatures at all pressures, Figs 1 and 2 [3, 10-121. In a manner similar to that employed in deriving equation (30) the chemical potential shift at constant pressure (dP”’ L=O) can be shown to be dpL = dpS =
2y vs&!z dH As
4967
CHEMICAL POTENTIAL SHIFTS
and DEHOFF:
(48)
Temperature (K) Fig. 1. Capillarity shift for nickel as a function of particle radius. Since the slope (dP/dT) is positive, the pressure shift is positive while the temperature shift is negative.
in chemical potential for the solid phase is the same since the two phases are in equilibrium, see equation (31). For the few cases for which A V is negative and the slope of the melting curve is negative, the pressure in the liquid phase decreases as the curvature of the solid-liquid interface increases, see equation (21). Accordingly, for this case the chemical potential for the system decreases with increasing curvature at constant temperature, equation (37). The magnitude of the chemical potential shift with curvature at constant temperature varies with l/AI’. There exist examples of unary phase boundaries for which the slope changes sign at some particular state. For example the graphite-liquid phase boundary for carbon has a vertical slope at approximately 4700 K [13]. For states near this condition, the chemical potential shift with curvature is very large, approaching infinity as the slope changes sign and A V becomes zero. Table 1 lists the correction factors for solidliquid equilibria for a few elements. The correction
Since the terms in the integrand are not strong functions of curvature the above equation may be integrated to give pL=ps=pS(H
2y vss = 0) + PHAs
(dPL =0)
(49)
If the corresponding comparison was made at fixed temperatures, then it is seen that the direction of shift depends upon the sign of the slope of the phase boundary. The sign of the slope in turn depends upon the difference in molar volume between the two phases, A V = VL-Vs. If A V and hence the slope is positive, which is the most common case, then the pressure in the liquid phase increases with curvature of the solid-liquid interface, see equation (21). Since ApL= VLAP, the chemical potential of the liquid phase increases with curvature of the interface at constant temperature, as is also the deduced in equation (37). The change
50
Temperature (K) Fig. 2. Capillarity shift for bismuth as a function of particle radius. Since the slope (dP/dT) is negative, the pressure and temperature shifts are both negative.
4968
KULKARNI Table 1. Correction
Element
Melting point (K)
Al Fe Ni CU So Ag Pb Si Zll Ti Bi Sb In
933.4 1811 1726 1358 505 1235 600 1683 693 1941 544 904 430
and DEHOFF:
CHEMICAL POTENTIAL SHIFTS
factors for chemical potential
shifts in unary systems (solid-liquid
Molar volume of solid (V”) at 298 K (cm3jmole)
Molar volume of
9.99 7.1 6.6 7.09 16.26 10.27 17.74 12 9.16 10.64 21.32 18.21 15.73
10.45 7.5 6.99 7.48 16.5 10.83 18.21 12.39 9.50 11.12 21.53 18.51 15.88
Molar volume
of
solid (@) at
liquid at melting
melting point
point (V)
factor is in the range from 10 to 50 for most elements. Examination of Figs 1 and2 shows that the chemical potential and pressure shifts become significant for particle sizes smaller than 1 micron. It should be noted that the chemical potentials of both the solid and liquid phases at a solid-liquid interface are changed by the same amount. The implications on solidification behavior needs greater review in the light of this analysis. 4. CONCLUSIONS A rigorous formulation for the Gibbs-Thomson effect in unary condensed systems [equation (37)] has revealed a significant error in the relation widely used in the literature. The correction involves a factor which is inversely proportional to the difference in the molar volumes between the two condensed phases. The pressure and chemical potential shifts are positive for solid-liquid equilibria in most elements at atmospheric pressure, but in a few instances where A V is negative, the shifts are negative (e.g. solid-liquid equilibrium in Bi, Si, Ge). Elements for which A I/ changes sign as a function of pressure, are predicted to have an infinite shift where A V goes to zero (e.g. graphiteliquid equilibrium in carbon). A similar correction applicable to chemical potential shifts in binary systems will be the focus of an upcoming paper.
11.31 7.96 7.43 7.94 16.96 11.54 19.4 11.19 9.94 II.65 20.75 18.77 16.35
equilibrium)
Relative
molar
volume AV=
v-c” 0.86 0.46 0.44 0.46 0.46 0.71 1.19 -1.2 0.43 0.53 -0.78 0.26 0.47
Correction
factor for
chemical potential shift rP/AV 13.1 17.2 16.9 17.4 36.9 16.3 16.4 -9.4 22.7 22.1 -26.6 72.2 34.8
REFERENCES I.
2. 3.
4. 5.
6.
7. 8. 9. 10.
11. 12.
13.
Trivedi, R. K., in Lectures on the Theory of Phase Tran
Acta mater. Vol. 45, No. 12, Pp. 4963-4968, 1997 0 1997 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/97 $17.00 + 0.00 s1359-6454(97)00199-7
CHEMICAL POTENTIAL CAPILLARITY-UNARY
SHIFTS DUE SYSTEMS
TO
N. KULKARNI and R. T. DEHOFF Department
of Materials
Science & Engineering, 253A Rhines Fl 32611, U.S.A.
Hall, University
of Florida,
Gainesville,
(Received I7 September 1996; accepted 22 May 1997)
Abstract-The temperature,
effect of curvature on the chemical potential for two-phase unary systems is usually given in the literature by the GibbssThomson equation [l-6],
at constant
yB(H) = /_?(H = 0) + 2yVBH where H is the local mean curvature of a /l particle in equilibrium with an a matrix and @, the molar volume of the fi phase. It can be shown, based on the above equation, that the corresponding shift in chemical potential for the CIphase is [I]: pa(H) = pLOI(H= 0) Thus the usual form of the Gibbs-Thomson equation that appears in the literature is not rigorous and hence results in a violation of the condition for chemical equilibrium for a curved interface (g(H) # #(H)) assuming incompressibility and isotropic stresses in the case of the solid phases [7-91. A rigorous derivation of the Gibbs-Thomson equation for unary two-phase systems at constant temperature is provided, that gives the correct equation for the chemical potential shift: /_&H) = #(H)
= &H
= 0) + 2yH[ V” VB/( V” - VP)]
While the above equation can be approximated (Ip = ‘-Vfl~p) in the case of a condensed-gas (C) system to give the usual form of the Gibbs-Thomson equation, a similar approximation for a condensed&condensed system can result in a significant error in both the sign and magnitude of the shift. The corrected expression for the chemical potential shift has an inverse dependence on the difference in the molar volumes (p-p) between the two condensed phases. For solid (q-liquid (L) equilibrium in certain systems the shift may be negative (e.g.V” = L < v8 = ’ for Bi, Si, Ge) or even approach infinity (e.g. v” = ‘z VP = ’ for carbon). A similar dependence is obtained for the pressure shifts in each phase. The present formulation is consistent with the Clausius+Clapeyron equation used to represent unary phase diagrams, which has the same dependence on the relative molar volume. 0 2997 Acta A4etallurgica Inc.
1. INTRODUCTION The assumption of local equilibrium across an interface has been invoked frequently in studying many basic processes in condensed systems such as solidification, growth, diffusion, sintering, coarsening, etc. Gibbs was the first to derive the conditions for equilibrium for planar and curved interfaces [7]. While the condition for thermal equilibrium is identical to that for a planar interface, the conditions for chemical and mechanical equilibrium are in genera1 different [7-91. The conditions for chemical equilibrium for curved interfaces are modified from the planar case due to the presence of non-isotropic stresses in case of interfaces where one or both of the phases are solid. They are however always identical to the planar case for liquid-liquid interfaces [7-91. In case of interfaces consisting of solid phases, if it is assumed that the surface stresses v> present in solid phases are isotropic and equal to y, the specific interfacial free energy, the conditions
for chemical equilibrium for curved and planar interfaces become the same. The conditions for mechanical equilibrium across a curved interface differ from those across a planar interface by a factor which depends on the product of the local mean curvature (H) and the specific interfacial free energy (y). These conditions of equilibrium are used in deriving an expression relating the local chemical potential shifts in a system with curvature of the interface, at constant temperature. This expression is classically known as the Thomson-Freundlich or the Gibbs-Thomson relation [l-6]. Although the original Gibbs-Thomson relation applied to liquid-
4963
gas equilibria in unary and binary systems P, 31,it has been assumed to be valid for condensed phase equilibria in unary, binary and multicomponent systerns. Some of these formulations contain inaccuraties and inconsistencies, which can result in significant errors in the calculated chemical potential shifts due to curvature. This presentation
4964
KULKARNI
and DEHOFF:
CHEMICAL POTENTIAL SHIFTS
focuses on corrections to the Gibbs-Thomson equation applied to unary condensed systems. It is to be noted that Gibb’s association with the GibbsThomson relation is only in context with the conditions for equilibrium and not the erroneous expression appearing in the literature. The assumptions of isotropic stresses and incompressibility of solids which are inherent in some of the earlier formulations [l, 61 are also valid in this work, although as pointed out by Cahn [8,9] hardly practical. Nevertheless it is hoped that a rigorous treatment of the simplest case would lead to adaptation to more general cases based on the framework of the present treatment. The Gibbs-Thomson relation at constant temperature is usually stated as [l-6]: #(H)
= &H /f(H)
= 0) + 2yVPH
= p”(H = 0)
(1) (2)
where V@is the molar volume of the p phase. It will be demonstrated later that the relationship between the corresponding chemical potentials for the CI phase, predicted from this equation, is that the chemical potential of the c(phase does not shift with curvature [2]. Since the condition for chemical equilibrium in the bulk system requires p’(H = 0) = #(H = 0), comparison of Equations (1) and (2) shows that p”(H) # #(H). This violates the condition for chemical equilibrium [7]. In this discussion, a self-consistent procedure to evaluate the effects of curvature on thermodynamic properties in unary systems is given, which results in a form of the Gibbs-Thomson equation for which p*(H) = ,u”(H). The significance of this result is briefly explored.
2. PROCEDURE
The variables system are:
H’;(K,+KZ) For each of the principal normal curvatures (K, or K~), a vector is defined that is normal to the surface element and points towards the center of curvature. Choose a convention such that the curvature is positive if this vector points into the /I phase and negative if it points into the CI phase. The local mean curvature H is positive for surface elements that are convex with respect to the p phase, negative for concave elements, and may be either positive or negative for saddle surface elements depending upon the relative values of the curvatures [lo].
to specify
this two phase
Ta Pa, Tp, PB and H. TM, Tp are the temperatures and P*, PB are the pressures in the c( and /J’phases, respectively. An additional variable (H) is required to describe the local mean curvature for this system. The conditions for equilibrium for this two-phase system with a curved interface are according to Gibbs [7]: Ta = 7-b
(4)
Ps(H) = Pa(H) + 2y H
(5)
/f(H)
= p!(H)
(6)
p’(H) = $(H) are the chemical potentials of the c( and p phases in two-phase equilibrium with an interface of curvature H. It is assumed in this treatment that y, the specific interfacial energy, is independent of curvature, thereby implying incompressibility of c( and p. As mentioned earlier if either CIor /I or both are solid phases it is necessary to assume that their surface stresses are isotropic and equal to y, the specific interfacial free energy, for equation (6) to be valid [7-91. Since there are five variables for this system and three relations among them [equations (4)-(6)], the number of degrees of freedom or independent variables for this system is equal to two. 2.2. Calculation of phase boundary shifts in wary systems Consider an incremental change in the state of a two-phase system with a curved interface. The change in chemical potential for each of the phases is
2.1. Conditions for equilibrium Consider a two phase (a + /I) unary system. Let r1 and r2 be the principal radii of curvature for an a-j3 surface element. The principal normal curvatures are given by ~~ = l/r1 and ~~ = 1/r2. Let the local mean curvature across an a-/I interfacial element be H where
required
dp’ = -S”dTa
+ VadPa
(7)
d#
+ VfldPfl
(8)
= -SBdTP
S”, V” and Sp, VP are the molar entropies and volumes of the o! and /I phases, respectively. If during this change, the system remains in two-phase equilibrium, then from the conditions for equilibrium [equations (4X6)] dTfi = dTa = dT
(9)
dPfi = dP” + d(2yH) = dP” + 2y dH d#
= dpa
y is assumed to be independent Substituting for dPP from equation (8) can be written as d# Equations
= -SBdTB
(11) of curvature. (lo), equation
+ Vp(dPa + 2ydH)
(7) and (8) are related
(10)
(12)
by the condition
KULKARNI for chemical -SadTa
equilibrium
[equation
+ V”dP”L = -SfldTfl
and DEHOFF: (1 l)]. Hence
+ I’p(dPa + 2y dH) (13)
d7” and d@ are equal due to the condition thermal equilibrium [equation (9)]. equation becomes
for (13)
-SadT+l’“dP=-SfldT+Vfl(dPa+2YdH)
(14)
-(S”-@)dT+(l’=-
(15)
I’O)dPa-2y@dH=O
4965
CHEMICAL POTENTIAL SHIFTS
equation (21) relates the pressure in the a phase in a finely divided system (P”(H)) to that in the same phase in a bulk system (P”(H = 0)) at the same temperature. The corresponding shift in the p phase is obtained by utilizing the condition for mechanical equilibrium across a curved interface [equation
(lO>l. dl”=dP’+27dH=(2$+ZY)dH Hence the pressure
(22)
shift in the b phase is
or dPp = -ASdT
+ A VdP” - 2y VPdH = 0
(16) The integrated
where AS=S’-Sp
and
AV=
Va-
VP.
ppVO
(17)
equation (16) is the general form of the ClausiusClapeyron relation for a unary two phase system with curved interfaces. Since it represents a single equation among three variables, the unary two phase system with a curved interface has two degrees of freedom, as discussed earlier. Systematic exploration of equation (16) is carried out by comparing equilibrium states in unary two-phase systems in which one of the variables (T or P”) is constant. This leaves one degree of freedom. This is usually assigned to the curvature H, since one is normally interested in the effect of curvature on other properties. Hence, if P’ is constant in this one can obtain the dependence of comparison, temperature on curvature [3,10, II]; if T is constant, the dependence of P” on curvature is obtained.
2.3. Effect of curvature on pressure To evaluate the effect of curvature on pressure in a unary two-phase system, compare two systems that are at the same temperature T, but differ in their local equilibrium states on account of a difference in mean curvature H. equation (16) can be written as
Hence the pressure
(dT=O)
2Yva
Pp(H) = Pp(H = 0) + Bv Subtracting
equation
Pfl(H) -P(H)
dP”= .IPa(H=O)
2y VP p”(H) = poL(H = 0) + =H
(25)
= PB(H = 0) - PGL(H= 0)
Thus the condition for mechanical equilibrium across a curved interface [equation (lo)] is recovered. equation (21) along with equation (25) gives the correct expression for the pressure shift across a curved interface.
2.4. Effect of curvature on chemical potential The chemical potential in a unary system is a function of the temperature and pressure for that phase. Since in the comparison between the two states of the system with different curvatures the temperature of the system is constant, the change in chemical potential is given by: dpa = V”dP” - SadTa = V”dP” for dP” from equation
dp” = V’(~dH)
(19)
H 2YVfldH ~ s HE,, AV
(21) from equation
(25)
(dTa = 0)
(27)
(19)
(18)
If TVand /? are both condensed phases, v*, fl and A V can be assumed to be independent of the curvature (H). The above equation can be integrated: p(H)
H
+ 2yH -=2yH ‘” - I/p AV
shift in the CIphase is dP”=gdH
(24)
@(H=O)
Substituting AVdP”-2yVPdH=0
form is given by
(20)
(21)
(28)
=qdH
Similarly the chemical potential shift for the p phase can be obtained by substituting for dPp from equation (23). d$
= VpdPfl - SBdTB = VfidPfl
d#
= Vfi(2$dH)
=qd,
(dTfl = 0)
(29)
(30)
Equations (28) and (30) are identical. Thus the condition for chemical equilibrium is recovered demonstrating the self-consistency of the procedure.
walsk se%pasuapuo3 :H 30 luapuadapu! aq 01 d/l 8yuInssE ‘~~03 01 QUO alqmgdde s! ‘saldpuyd pmoy+~en Ou!sn [r] Iegualagp B u! passaldxa aq um (I) uogenba moymba uoswoyJ-sqq!g ayl30 w.103 $3al.ro3 ayl suylnw pm [v] OuulaH dq paylap uaaq w?y qq~ ‘(SE) uogenba .fym JOUalo3alayl SFuogdurnsw sg~_ IOU s! (I) uogmba ]t?ql umoys ,$Sea aq ue3 11 ‘[(LE) ~amwlz.m 01 anp sygs rvpualod pr+uay~ ayl .IOJ uogenba] anoqt! paauap auo ayl u.1o.131uaIagp s! [(I) uogenba] swalsds pasuapuo3 103 uoymba l[nsaJ ~~a~.xoy UB 01 spsal ‘o=dp )eql uogdurnsse uoswoyL-sqqg ayl 30 mo3 pasn LIap!M aqL ayl snyL wn!~qg!nba pm!uIay3 103 uoypuo~ ayl 30 uogeloy u! air! ‘3ag.wa passnmp se Ypq~ ‘IClay NOISSll3SIa ‘E -padsa ‘sawyd x) pm d ayl ~03 amlm.xm 01 anp s13!ys Iegualod pm!uIaq:, aql ‘(sp pw 1) suogmba paylap uogdurnsw !PaA!‘.L ‘(0 = zdP) W .a.xnle.xaduIal ~ue~suo~ ie seal pmoy!ppe s!yl 8u~sn ‘o= ,,dp put2 0 = up .a.! -sl(s pasuapuo3-pasuapuo3 103 uogmba uosuroy~ ‘JUE~SUOD aq 01 pawnsm ale aseyd x1ayl u! amssaJd -sqq!g aq 30 1~1103lDallo:, ayl SF (LE) uogenba ayl pm amwadmal ayj ~104 stualsh 30 uos!md -u103 sg u! ~t?ql slear\al 13~9spzgualod pzyuay3 ayl (Sl.mlSliS pasuapuo3-pasuapuo3) (LE) 103 uoye+ap sg 30 uoyu!uwxa uv ‘[I] !paAnL Aq pa@ap uaaq amq (sp) pm (I) suoynba = H)dti = (H)# = (H)ad H s+(O ‘[L] acw3lay paam:, ‘I?sso_m tumq -ynba Iw!way 103 uog!puo~ ayl 30 uo,zzv~o~nJea13 (O=H)a’f (0=ffL$d v s! sw ‘CMgfl # oi),$ say8 (SP pue I) suoynba (9E) HP& o=H = s”p I= ,arlp 1 %uyduro3 ams ‘(0 = H)# = (0 = HW d/l.i’i‘k Hs (H)yd CHW (SP)
(0 = H)$
= @A’~ sagdw! q3q~
(PP)
0 = zdP&l = JfP
aq 01 pap!pa.rd s! aseyd Bydpz ayl 103 am1eA.m Q!M 13ys pzgualod pm!wayD ay$ amaH ~ammm~ 30 uoFpun3 B IOU s! aseyd r, aq$ u! amssa.rd aql leyl sp!pald ‘(g&) uogenba ‘uo!lelaJ Ieguajod pguxay3 ayl 30 ~~03 ayl snqL (EP)
(ZP)
0=&P HPAZ = HPLZ + zdP = gdP
.aee3lalu! paam ayl ssom urn!lqg!nba pm!uvq3aw 103 uog!puo~ ayl dq paufuualap s! 13ys amssald ayl ‘aseyd x) ayl UI (IP)
30 suogmn3 8~0~~s $0~ a.w puw%ay ayl u! seal ayl am!s ‘palwflalu! LpDa.up aq utm (1 E) uogmba ‘smaisk pasuapuo+pasuapuo3 JOT .a.mlwalg ayl u! auop uaaq wy SE ‘suralsds pasuapuo3-pasuapuo3 01 pagddv uayM pyv~ lai3uoI ou s! (EE) uoyenba 30 ~olmauxnu ayl UT D = J slam23 Qua!uaa -uo3 leyl [(ZQ uo!lenba] nv .103 uogmurxolddc sum ayL w,ualsAs sGpasuapuo3 01 Quo saydde ‘a.mlwalg ayl u! ua@ uoymba l! ‘.IaAaMOH uostuoyJ-sqq!D ayl 30 ~103 pmsn ayl ‘(I) uogsnba se auws aql s! 33ys Iylualod pm!uraqD ayl 103 uogenba ayl 30 1~1103pale&talu! aAoqe ayL (SE)
O=H
HPLZ = gdP
(tiE)
HP d,‘i‘Z
= H)drl
I
= d’p
H
= (H)dd
= (Hj9d
(O=H)dd (O=H)$ = #P (H),q”
I
(H),d
J’
aq UBDuogmba aAoqe ayl ‘amlm.m ayl 30 luap -uadapu! aq 01 (4) aseyd pasuapuo3 ayl30 aum[oA ~efotu ayl pur! (X) k%aua a2e3ms ayl %uFurnssv
‘(6~ pm 8~) suogenba S?u~.wduro3 (Ed s! aseyd d aql u! 13ys amssald aqL
(0 = gP = .JP)
(dn < S/j :sul2lsxs sv8-pasuapuo2) Hdzd @+(o
H*Z + (0 = H>gd = (H&d
:am3lalu! leg e qj!~ walk awes aqj 01 aA!lvlaJ aDtz3lalu! parl.m B ~J!M malsl(s e 103 awyd 8’ ayl u! 13ys amssaId aql say% uogmba aaoqE ayl30 uoge.SaluI (OP) Lq pauplqo (6E)
H?‘&Z=HPL= ,&‘f+‘i ‘k
drlP= r~‘p amaH
s’dP+l
(zd
9A=gA
- 9=aA = AV
= &pdS - ddPdA = (H)d+ .a.mlwadural JUEJSUOD 1~ apmu SFsaml -mm3 Syagp ypm smalsk oM1 ayl uaarnlaq uos -!.wdwoD ayl ams .amssald pw amltmdural aql30 uoyun3 B s! aseyd d ayl 30 Iyualod p+uay ayL (8E)
.aseyd pasuapuog ayl uayl Jai?.wI d~1um~u8~s s! aseyd SE%?ayl 30 auyoa lelotu ayl acys ‘apm aq um uoym!xo~ddv UE ‘awyd w8 ayl s! xi alayh4 ‘malsh ~2%pasuapuo3 I? 30 asm ayl UI
(Id
HpdA xZ = d’p SUIHS WI1NXLOd W3IWBH3
HP-=
AV
gA xl/l kZ
d’P = 1o’P
:JrlOHEICl PuE INlIVTIflX
996P
KULKARNI
It is seen from equation (16) the general form of the Clausius-Clapeyron relation for a unary two phase system with curvature, that there are two independent variables needed to describe the state of the system. If the temperature is maintained constant (dT = 0), then equation (16) gives the relation between the change in pressure (dP”) and curvature (dH). If it is assumed, that both the temperature as well as the pressure are maintained constant (dT = 0 and dP=O), the two degrees of freedom are assigned. The state of the system is fixed and the variation with H cannot be explored. The role played by A V in shifting phase boundaries in unary condensed systems may be illustrated by considering the effect of curvature on the state of solid-liquid interfaces as exposed in the modified Clausius-Clapeyron equation (16). If two systems with differing curvature are compared at the same pressure in the CIphase, -ASdT
- ~YV’=~ dH = 0
Thomson’s equation ing point is obtained
(dP=L
for the depression [3, 10, 111.
= 0)
(46)
of the melt-
dT=-gdH+
T(H)=T(H=O)-%H
2y vs (47)
This result is independent of A V. Thus the melting point always decreases as the curvature of the solid surface increases. If this result is applied at a series of pressures it may be visualized as a shift in the melting curves to lower temperatures at all pressures, Figs 1 and 2 [3, 10-121. In a manner similar to that employed in deriving equation (30) the chemical potential shift at constant pressure (dP”’ L=O) can be shown to be dpL = dpS =
2y vs&!z dH As
4967
CHEMICAL POTENTIAL SHIFTS
and DEHOFF:
(48)
Temperature (K) Fig. 1. Capillarity shift for nickel as a function of particle radius. Since the slope (dP/dT) is positive, the pressure shift is positive while the temperature shift is negative.
in chemical potential for the solid phase is the same since the two phases are in equilibrium, see equation (31). For the few cases for which A V is negative and the slope of the melting curve is negative, the pressure in the liquid phase decreases as the curvature of the solid-liquid interface increases, see equation (21). Accordingly, for this case the chemical potential for the system decreases with increasing curvature at constant temperature, equation (37). The magnitude of the chemical potential shift with curvature at constant temperature varies with l/AI’. There exist examples of unary phase boundaries for which the slope changes sign at some particular state. For example the graphite-liquid phase boundary for carbon has a vertical slope at approximately 4700 K [13]. For states near this condition, the chemical potential shift with curvature is very large, approaching infinity as the slope changes sign and A V becomes zero. Table 1 lists the correction factors for solidliquid equilibria for a few elements. The correction
Since the terms in the integrand are not strong functions of curvature the above equation may be integrated to give pL=ps=pS(H
2y vss = 0) + PHAs
(dPL =0)
(49)
If the corresponding comparison was made at fixed temperatures, then it is seen that the direction of shift depends upon the sign of the slope of the phase boundary. The sign of the slope in turn depends upon the difference in molar volume between the two phases, A V = VL-Vs. If A V and hence the slope is positive, which is the most common case, then the pressure in the liquid phase increases with curvature of the solid-liquid interface, see equation (21). Since ApL= VLAP, the chemical potential of the liquid phase increases with curvature of the interface at constant temperature, as is also the deduced in equation (37). The change
50
Temperature (K) Fig. 2. Capillarity shift for bismuth as a function of particle radius. Since the slope (dP/dT) is negative, the pressure and temperature shifts are both negative.
4968
KULKARNI Table 1. Correction
Element
Melting point (K)
Al Fe Ni CU So Ag Pb Si Zll Ti Bi Sb In
933.4 1811 1726 1358 505 1235 600 1683 693 1941 544 904 430
and DEHOFF:
CHEMICAL POTENTIAL SHIFTS
factors for chemical potential
shifts in unary systems (solid-liquid
Molar volume of solid (V”) at 298 K (cm3jmole)
Molar volume of
9.99 7.1 6.6 7.09 16.26 10.27 17.74 12 9.16 10.64 21.32 18.21 15.73
10.45 7.5 6.99 7.48 16.5 10.83 18.21 12.39 9.50 11.12 21.53 18.51 15.88
Molar volume
of
solid (@) at
liquid at melting
melting point
point (V)
factor is in the range from 10 to 50 for most elements. Examination of Figs 1 and2 shows that the chemical potential and pressure shifts become significant for particle sizes smaller than 1 micron. It should be noted that the chemical potentials of both the solid and liquid phases at a solid-liquid interface are changed by the same amount. The implications on solidification behavior needs greater review in the light of this analysis. 4. CONCLUSIONS A rigorous formulation for the Gibbs-Thomson effect in unary condensed systems [equation (37)] has revealed a significant error in the relation widely used in the literature. The correction involves a factor which is inversely proportional to the difference in the molar volumes between the two condensed phases. The pressure and chemical potential shifts are positive for solid-liquid equilibria in most elements at atmospheric pressure, but in a few instances where A V is negative, the shifts are negative (e.g. solid-liquid equilibrium in Bi, Si, Ge). Elements for which A I/ changes sign as a function of pressure, are predicted to have an infinite shift where A V goes to zero (e.g. graphiteliquid equilibrium in carbon). A similar correction applicable to chemical potential shifts in binary systems will be the focus of an upcoming paper.
11.31 7.96 7.43 7.94 16.96 11.54 19.4 11.19 9.94 II.65 20.75 18.77 16.35
equilibrium)
Relative
molar
volume AV=
v-c” 0.86 0.46 0.44 0.46 0.46 0.71 1.19 -1.2 0.43 0.53 -0.78 0.26 0.47
Correction
factor for
chemical potential shift rP/AV 13.1 17.2 16.9 17.4 36.9 16.3 16.4 -9.4 22.7 22.1 -26.6 72.2 34.8
REFERENCES I.
2. 3.
4. 5.
6.
7. 8. 9. 10.
11. 12.
13.
Trivedi, R. K., in Lectures on the Theory of Phase Tran