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Localized adsorption in one layer on a crystalmdash;Solution interface
Journal of Crystal Growth 46(1979) 495—503 © North-Holland Publishing Company
LOCALIZED ADSORPTION IN ONE LAYER ON A CRYSTAL—SOLUTION INTERFACE H.E. LUNDAGER MADSEN Chemistry Department, Royal Veterinary and Agricultural University, Thorvaldsensve/ 40, DK-1871 Copenhagen V, Denmark Received 30 May 1978; manuscript received in final form 25 October 1978
Deviations from ideality in the adlayer are discussed in terms of a Bragg—Williams monolayer model with empty sites. Equations for equilibrium and nonequilibrium surface concentrations of growth units are derived together with an equation for the variation of surface diffusion coefficient with surface supersaturation. A generalized form of the BCF differential equation for surface diffusion is established, and curves of crystal growth rate as a function of supersaturation are calculated from the results of numerical integration of this equation. It is shown that the BCF theory involves rather crude approximations, but the growth curves obtained from the more realistic model do not deviate very much from the BCF growth curve.
1. Introduction
which is inconsistency defined [I]. The reason why no precise formulation exists is that measurements of adsorption on a crystal surface of the crystal’s own molecules are extremely difficult, if not impossible, notably during growth of the crystal; even the distinction between adsorption and surface roughness may be troublesome. Hence, no empirical data exist which could provide some indication of the proper way to treat the problem theoretically. In this paper a model is proposed with the aim of overcoming some of the difficulties outlined above. It is a simple one-layer model of localized adsorption, based on the Bragg—Williams approximation, and it is
In a previous paper [1] it was shown that currently used models for the adsorbed layer on a crystal—solution interface are not quite adequate from a thermodynamic point of view. The Langmuir and Bragg—Williams models fail to account for the interface tension as an independent variable. The reason for this is that only two states of an adsorption site are considered in these models: either it is occupied by a growth unit or by solvent. The number of independent variables is thus limited to three, e.g. T, number of sites M, and number of growth units adsorbed N2. The number of degrees of freedom of a crystal—
not supposed to yield quantitatively correct result except in simple cases, but merely intended as a means for analyzing the adsorption problem of a crystal—solution system.
solution system at equilibrium (2 components, 2 phases) is 2, according to Gibb’s phase rule, and just one additional independent variable is insufficient for a discussion of all possible deviations from equilibrium. In the case of a BCF growth mechanism [2,3], for instance, it is clear that the state of the adsorbed layer at a given supersaturation of the solution depends on kinetic and other factors not directly related to the adsorption problem. Several attempts have been made to express the equilibrium activity (or, for an ideal system, the concentration) of growth units in the adsorbed layer in terms of a standard free enthalpy increment [3—5], but these expressions are difficult to apply, due to either an ill-defined standard state of a parameter
2. The partition function and its partial derivatives The fundamental thermodynamic equation for a plane interface in a two-component system is [6—8] dU
— —
TdS +y da+ji1 dn1
~P2
dn2
(1)
where a is the interface area, and the superscript a denotes interface excess quantities. In connection with a statistical treatment it is easier to use an alternative equation, valid for the total quantities of the 495
496
11./i. Lundat’cr Madsen
/ Localized adsorption
lai’er (~iicr,’sral
iii ui/i’
-solution interface
elc’5i iV~+
adsorbed monolayer, vit~.
+
+ 2~)A’, iV
+
2
Xexp
dL/ = T dS ~ d~l1+ p, diV1 + P2 rh’\’2 (2) where M is the number of sites, N, and N2 the nuni-
~
2 lILT
(6)
.
ber of molecLiles adsorbed, and 1 is the interface pressure; in (2) it has the dimension of energy per site. We shall later discuss the relationship between (1) and (2). It should be noticed that N, + N2 is not assumed equal to M. dA is obtained from dU using A = U — 15. and A is related to the canonical partition function Q(T. M, A’,, It2) of the adsorbed layer by [9]
or. ~isingStirling’s approximation (or the laclorials,
Q
In
M in M N1
iV, in ,V1 ‘~‘2) iii(41
(ill
A’, Iii ~li
+
+ ~~‘2 in ~/2
+
(.~5J\J
iV2 in N~ IV, iV~)
-
+ (~I + ~2 + 2ç5’) /V,IV,
~~5j\
--~jT~
-
(7)
-
We now have. Irom (2), (3), and (7), A
kTlnQ. (3) 5, he the energy of interaction of two niolecuies Letofc type I on neighbouring sites, let ~2 be the corresponding energy of a 22 pair, and let the energy of a 12 pair he ~, + ~2) + ~‘. If the adsorbed molecules are arranged such tlìat the number of 11 pairs is N, ~, the number of 22 pairs N 22. and of 12 pairs N, 2. the total energy of lateral interactions in the layer is I:’,
=
N, ~
+
N~5~+ ~iVi ~(c~i + -
The partition (unction Q(T, 41, iVy, A’2)
=
is
A’1
~
hT~, /dlnQ\ dIll
— -
!-~ ~
+
~
0) +
~
+
i)A’ / i~A2 kThn q1 + kT In -
(4)
02
~/~1q~2
A
~T(
=
12)
+~+2~1 (5)
where q, and ~/2 are molecular partition functions of single adsorbed molecules, g is the number of arrangements with the given values of the arguments, and the sunis extend over all possible values ofN1 . N22. and 1V,7.
(c~+
+
+ I~TIn
hi
~
Q
=
~‘\‘,!
M! N2! (31
-—
‘~‘i
12)!
q~’Iq~’2 -
+
kT in x2
“~
~__~
+
r
kO --In 0
+
A’
‘V
)~
-
x, (in q ÷T ~_hi~~
L
(10)
site
‘‘‘‘‘ —
(9)
,
+2c5’)x1j 0.
lIT /~ In 0 =
In
x,
dT ,
Tue crucial probleni in lattice statistics is tile cal-
Q
0 --~--~
~2
and the entropy Pin
~M
j 0
~
+~cI2~2x2 ~
d in q~
cuiation of g: when this has been done, Q may be obtained with the aid of the maximum term method
[91.In tile Bragg—Williams approximation N, 1 . N2 2’ and N, 2 in the exponential factor are replaced by their mean values for all possible arrangements. . neglecting any preferential ciusterrng. This. leads to the followmg expression for Q:
-
+ 2ç5’) x2
~2
Q
ki In ~/2
-
+~V22~2 ~~!Y,2~l
(5)
-
+ kTln x1
1 + ~cI2~,xi
‘
A’,
02
~
=
‘
din
,,
I-’ 2
-
IT in( 1
-
then
‘~22A I 2
XexP[-
2d’)
-
g(M. A,, A~,1V,
X
+
~,
-
(j)
+ X2
~ln q2
k( I
-
where
0
+
T
0) ln( 1
in
—~-~—
0)
A,
(1 I)
.
=
.
-
(N, + N2)/M is the fraction of sites occu-
.
.
-
-
pied. As- we are dealing with a solid—solution interface. 0 is close to I , and we approximate eqs. (8)—-
(Il) di
-
+
by
RT ln( I
---
0
~e(ç~1x~ + c~2~2+ 2ç~’x~A2) -
(12)
II.F. Lundager Madsen
—kTln q, kTln(l —0) + kTln x, + ~c[20,x, + (0, =
—
Pz
=
+
÷20’) x2]
(13)
÷20’) x,},
(14)
+ 02
--kTlnq2 —kTln(l —0) kT ln x2 + ~c [202x2 + (0~÷02
+ SM
/ Localized adsorption in one layer on
kLxi (ln q, +
dlnq, dT T~
—
x 2(ln q2
+
Td In q2 dT
--
In
x2)1
(15)
.
—
02
~
--
kTln q,
+ ct~+
~c02
—
kTinq2
+ 1
kTln x,
=
and y~,the activity coefficient, is given by 2, i = 1, 2. (23) kTln y~ = cO’(l x1) Equations analogous to (19)--(23) have been derived
+ c0’x~,
+kTlnx2 ±c0’x~ .
for the surface of a liquid solution of molecules of similar size ([101, ch. 12).
4. Equilibrium considerations In the following, component 1 will be the solvent and component 2 the growth units. Except at very high growth or dissolution rates, there will be equilibrium between adlayer and solution with respect to
(16)
solvent. Then Pi may be treated as a constant, and we may eliminate 1 between (16) and (17):
(17)
02=0,
An ordinary solution is termed ideal if p1 kT ln x, is constant at constant temperature and pressure for all components /. For a binary solution, this condition is equivalent with —
kT/x1,
i = 1, 2,
(18)
where, in fact, one of the relations may be deduced from the other with the .aid of the Gibbs—Duhem equation. By analogy, an ideal adsorbed layer may be defined as one, for which
(öpIIöxl)T,~= kT/x1,
=
1,2.
(19)
If we look at (16) and (17), we see that the condition of ideality is 0’ = 0. In this case the energy of interaction of a pair is the mean of the 11 and 22 interaction energies. In the nonideal case we may write
kTin(q2/q,)
—
x2 x2 +cO’(l
—
—
2x2).
(24)
We now introduce the parameters n0, the number of sites per unit area of interface, and n~,tile number of adsorbed growth units per unit area. Then, for 0 =
=
+~c(02 — Oi)
+kTln 1
3. Ideal and nonideal layers
(apl/axl)T,P
(22)
,
—
The last equation shows that the entropy is equal to that of an ideal mixture; this is the usual result of the Bragg—Williams approximation. The singularity of c1, ji,, and 02 for 0 = 1 results from the use of Stirling’s approximation for the logarithm of (M —N, —N2)!; it is not present in Q. The term kT ln(1 0) may be eliminated from p5 and ~2 by substitution: =
497
which is the chemical potential of an adsorbed layer of pure component i at zero interface pressure (a bypothetical state). a1, the activity, may be written a1
in x, /
crystal—solution interface
(25)
1l~/n,~ ,
and (24) becomes 02 =
+
p, ÷~c(02 kT ln
—
ns —
ns
0,) +
kTln(q2/q1)
—
cO
,
n0
-——-----~
(26)
Eq. (24) may also be written, using eqs. (20)---(23), 02
=
o,
—
p~+
+
kTIn(a2/a1)
‘
(27)
If the standard state of solvent in solution is pure liquid solvent with chemical potential p~,then the activity of solvent is the same in the adlayer and in solution at equilibrium, and p~ = ‘D0, the interface pressure of an adlayer in equilibrium with pure —
=
i = 1, 2,
+ ~ + kTln a~,
(20)
with the standard chemical potential =
—
kTlnq1,
i
=
1,2,
liquid solvent. Then (27) becomes (21)
02
=
b0
÷p~+ kTlna2.
(28)
IlL’. Lundager Madsen / Localized adsorption in n,ic la,’er on cri’stal---solulio,, interface
498
If finally, the adlayer is in equilibrium with the crystal. then 02 = p~,the chemical potential of the crystal, amid we have
We now wish to calculate the surface saturation ratio~
*
a,(sat)
=
exp!(u2
02
-~
(l)0)/Jcfl
5defined by [IJ
(29)
-
-
p~=p~+kTlnd5.
For an adlayer a2(sat) = x2(sat) = nso/no. where n50 is the value of n5 at equilibrium. Eq. (29) is thus equiva=
n
1)
0
Fr/un (33) and (26) tve imave ~ In ~ I I 2e0’
(30)
,
~
~
F~I
0 which yields, upon integration.
p~+
dI0
-
Fig. I shows the dependence of ~
(31) Ofl ~6ks
= ,
//~ 11s0
r
‘1s0
11ii
p~
(34)
lt’Tii
11~
exp(- ~GkS/k
if tve make the assignment =
(33)
-
lent with the Kaischew equation [4,5] 11s0
Surface supersaturation
5.
expl ~
-
2c0’p,,
L
~s
-
I.
-~ -—-~~—--~
kT,i
J
11
(eqs. (29) and (31)) for a square lattice (c = 4), with a2 given by (22) and (23). The temperature dependence of a2(sat) is given by in a9~at)\ ~
Jl~--
— -
+
-- -
H~- H~ —
- -~
~
-)
-
where the Gibbs—Flelmholtz equation has been used; H, H~,H~,and H~are the molecular euthalpies of the same standard states as those used for the chenncal potentials.
(35)
The last factor is equal to (y,(sat)/y, )(72/72(sat)). For a dilute adlayer (n5 ~ n0). the second factor is close to 1. and so is the ratio y,/’y,(sat), whereas y2/y2(sat) may still be substantially different from I, if we are dealing with, a highly nonideal system. Thus, the currently uised relation [2.3] ~s
=
(~6)
~s/h1sQ
is valid for a dilute adlayer unless the deviation from ideality is strong. For a concentrated adlayer, (36) cannot be expected to hold, not even for a perfectly ideal system; the error may be of the order of niagnitude of,05 — I.
nsa/no n5
~\7kT~
1
2
3
/
5
5
~
AG 4kT k~
I’11s0’ ig. I 5~hhu1ctmoi~ - liqu,hbr,um0f ~Gks surtacefurconcentration units. various values of of growth ~S.
~
12
12
13
14
17
16
17
18 19
I“so 1g. =2.Ohio ii~/i,~~as a fi,nctinn ot’ surface sopersatoration /3~i’or and vario~isvalues of ~‘ . Dashed line: ,i~/ii~~ =
HE. Lundager Madsen / Localized adsorption in one layer on crystal—solution interface
Fig. 2 shows how n~/n5odepends on !3, for c = 4 and n~o/no= 0.1 (~Gk.= 2.3kT if 0’ = 0). Even in the ideal case (0’ = 0), n5/n~ois not equal to 13~,except at low supersaturations. The best agreement between n~/n~oand ~3,,, is found for 0’ = (5/36)kT or, in the general case, 0’ =
kTno/[2c(no
--
n50)] -
0’ = 0.5kT (generally 0’ = (2/c)kT) is a critical value, which yeidls a vertical tangent at n5 = 0.5n0. For 0’> 0.5kT we find an S-shaped curve, part of which represents unstable states. This means that, at a certain value of ~ a phenomenon analogous to a firstorder phase transition takes place; ~ rises abruptly from a low to a high value close to n0. This value of !3~corresponds to the one found for n~= 0.5n0 from (35),viz. = 9 exp(—3.2~’/kfl for n~o/no= 0.1.
499
adlayer (172 ~ F~)in equilibrium with saturated solution d In r2 F’?a~~~des + = R (41)
~
—
where L~J1?des is the enthalpy of desorption of solvent, and Zxll~,is the enthalpy of transfer of growth units from the crystal (kthk) to the surface. In the present case F~a = 1, so that ~J~des+
(~~)F~ ~,
=
—
~ (42)
—~-~---~.
In a dilute solution c2 is proportional to x2, if we neglect the thermal expansion of the solution, hence, for saturated solution a in 12 (a(1/n)~0~’ (43) where ~~)I is the molar enthalpy of solution in dilute solution. From (40) we then find, using (42) and (43),
(ad(l/fl In
nso),,,,,,
6. Connection with interface thermodynamics
=
--
TC2
L~H~)1 + [‘2k,i~’ (n—”? des +
~.HkS)
In order to be able to relate the results of the
(44)
above derivations with those of the previously pub-
Comparison of (44) with (32) for the ideal case a2(sat.) = n50/no leads to TC2 11s0 H~ + H~ H~= + (~H~ dos + z~H~). (45)
lished thermodynamic analysis [1], we must work
out the connection between I and y as well as between n~and excess [‘2.isGuggenheim’s modification [7]theofinterface Gibbs’ formalism useful in this respect. Let r be the thickness of the adlayer. —ddM represents the work performed on the interface layer
when the number of sites is increased by dM without changing r or n0. This quantity equals —pdV°+yda, where V° is the volume of the interface layer. We
then have =
a
=
rM/i,0
M/no
(38)
,
hence, under the condition of constant
(J)
=
(pr
—
(37)
,
7)/n,) .
r
and n0 (39)
Similarly, let c, be the molar concentration in solution of component i and F~= ny/a the surface excess of this component. We then have =
IVA(rc2 + [‘2)
(40)
It was shown previously [11 that for a dilute ideal
--
—
~so
In this equation, the terms n 50,H~,andH~refer to the adsorbed monolayer as a whole, whereas ~‘2, M1~d~, and L~II~ are interface excess quantities [6—81. The equatiomi shows how the total enthalpy increment for transfer of one growth unit from the crystal to the adsorbed layer is partitioned between the interface excess enthalpy and the solution enthalpy. The fraction NAT2/nSo of the growth units transferred represents an increase in interface excess and gives rise to the last term of (45), whereas the rest, NATc2/nSo, may be considered as simply going into solution and therefore contributing the first term of
the From right hand side of(45). the above considerations, the thermodynamic consistency of our model, at least in so far as the ideal dilute layer is concerned, should be clear.
!1L. Lundager Madsen / Localized adsorption in one layer on cr,’stal—solution interface
500
7. Density of surface vacancies As we know the relation between dI and y, eq.
4~1O~
(39), we may calculate the fraction of vacant sites, —0, for typical values of y from eq. (17). We shall use the following values: 2, p = I atm = 1.01325 X l0~N/m = 5 A = 5 X lo-’°m, = 4 X 1018 n’2 (square lattice where each site is
5A
x
/
S A),
T= 25°C= 298.15 K, c = 4,
0,
0’
—4kT (typical value for a volatile solvent which follows Trouton’s rule for the entropy of vaporization), =
=
0.
Instead of using x 2 as an independent variable, we
prefer 02 —system p, with by (26). a low value of If the is p~given in equilibrium, 02 —02 will as a rule, correspond to a low solubility, though there is no one-to-one relationship between the two.
.01 .02 .Oi~ 01. 11g. 4. I
.07
.05 - 75
0 :isafu,nction of -y for 02
2 09 g. Jim SkTand v:iri-
08 02
=
0115 values of 5~2.Other parameters as in fig. 3.
1
—0
occurs for x
2 given by
Fig. 3 shows the dependence of 1 0 on y for 02 = —8kT and different values of 02 — p, and fig. 4 shows I 0 as a function of y for 112 112 = 5kT
There will thus be no maximum unless
amid different values of
~ckT.
—
*
—
—
02.
The maximum value of
=
2kT/c(çS
--
(46)
02).
01
-— 02 >
Froni the results we see that, except for low values of --. p~and relatively valuesdensity of y, 0found 1 is ina safe02approximation. The high molecular the adlayer is substantially higher than in solution, where it is normally of the order of 90% of the crys-
1-83 x10
9 8
~2
~
tal density, if the molecules are of similar size. This is characteristic for the localized adlayer model;
*
-10
Spaepen and Meyer [11] have found, for the solidliquid interface in a one-component monatomic system, a density in the randomly packed adlayer which
6 5
is not very different from the liquid density. /.
3
8. Surface diffusion and the BCF theory
-5
2
In order to see the consequences of ouir results for crystal growth theory we must analyze in detail the 01
02 03
01.
05
I’ig. 3. Fraction of empty Sites 1
05 --
07
08
09
~,
Jim2
0 as a fund ion of inter-
facial tension ‘y for various valujes of 02 Values are given in the text.
-
probleni of surface diffusion. By analogy with Eyring’s theory for volume diffusion [12], we may
express the surface diffusion coefficient by
u~. Parameter = (a2/rsd)(
1
---
0) .
(47)
HE. Lundager Madsen / Localized adsorption in one layer on crystal—solution interface where a is the distance between neighbouring sites, and TSd is the time constant for the jump of a molecule to a neighbouring empty site. As 1 — 0 and rS~
depend on f3~,so does D~.We shall work out the relationship between D~and I3~. We have from (13), (14) and (33), with x1 = (no n~)/noandx2 = n8/n0: —.
~,
—
ln(1
—
0) =
+ ~ in
+
—
—
——--—
—
2kT
~ in )3~
where q~is the partition function of a growth unit at the saddle point between two neighbouring sites.
Both q2 and q~depend on the types of molecules adsorbed on neighbouring sites. In the Bragg—Williams theory as well as in several other lattice theories this dependence is neglected; we shall therefore, for the sake of consistency, treat Tsd as a constant, the more so as the variations ofq2 and q~tendto cancel. We thus have, using (47),
n~(no— n~)
D5
D50
~
c
(48)
2kTn
=
10
0s0
=
I —0~ n0
—
xexp[f~-0’
+~~~_~s0)1
0
N, dp, + N2 dp2
=
M d’I),
(49)
where D~ois the value of D~at saturation. The dependence of D8 on i3~is given by (56) in connection with (35). Fig. 5 shows how D~varies with i3~for n~o= 0.1n0, c = 4,01 = —4kT, and 02 = —8kT. The surface flux is given by Fick’s first law
which yields n~ (~)
—
+
0~
(arl~
(50)
(ap~ aflS/T n0 beti~found an5 Tfromn0 (17)n5 with an5!T 11)/aflS)T may the proper (a substitutions for x, and x —
—
2: T
J~= —D~Vn~
(57) towards the surface is usually written [3] and the flux of growth units from the solution = (nS~,/r~,,~)(~3 i3~) (58) where Tdes is the time constant for desorption. In (58) the adlayer is assumed to be ideal; a more gen—
(a02) — kT
/a~\
(56)
2kTn0
To eliminate p~we make use of the Gibbs—Duhem equation with constant T:
=
501
+
~s
2c0’
—
.
(51)
,
When this is inserted in (50) we obtain, using (33) ( api)
kT(~-~-~
T=
a~
~
kTn0 + n~(no— n~ n0 ~
—
(52) .9
Using this result, we find (aIn(l—o)~
I
c 2kTn0
(02
—
Oi + 20’),
,5L (53)
which yields, upon integration ln
1—0 =
—
l—0~
in
0.4
.5
0.6
no—na n0—n50
(kT/h) q~/q2
05
2 .1
2kTn0 0o is the value of 0 at saturation. where Tsd may be written [9] =
0
.4
___________________
l/Tsd
=
05
n0—n5
aflS/T
+
ø’/kT
8~
_________
______________________
1.1 ~l~213
1.4 1.5 1.6 1.7 1.8 1.9
Fig. 5. Surface diffusion coefficient D
(55)
5 as a function of surface supersaturation j3~for n~o= 0.lno and various values of 4’. Other parameters as in fig. 3.
IlL’. I.undager Madsen / Localized adsorption in one lacer on cri’stal -solution interface
502
eral form is, by analogy with chemical reaction rates [13] 11~ =
—-——
I~
rI exp (02(soO L
—
.
-
—-
kT
—~
I I J
i
(59)
,
where 02(501) is the chemical potential of growth units in 5011,00,). Using (33) amid the analogous equation for the solution, we find ~)/~ -
= (11~Tdes)(~
(60)
I~and J.~are related through the equation of continuity
w~
(61)
~
If D5 were independent of,,5, arid n5 were equal to we should find, from (57), (60) and (61), the BCF diffusiomi equation [2,3] 20 ~5= 0 , (62) x~V where ~i = — ~ amid x 2. However, we 5 = (Dsees)” have
VJ,
=
--D
(65)
is symmetric around
t’
=
0 (steps are at s’
±v0/2),and a~,5/a3’= 0 for v
=
0. We consider the
=
primary law only, with n~= n50 for v = ±v0/2;thus means a rapid incorporation of the growth units once they have arrived at the step.
Tbte calculations are carried out as follows: We choose a value of,,5 fury
0 and integrate (65) from this gives ,‘~/2for the given starting value of n~and the given value of~. According to the BCF theory [2,3], ,v0 is propor-
.i’ =
=
0 to the point where n~= ~
tiomial to the radius of the critical two-dimensional miucleus which, in turn, is inversely proportional to In ~, according to the two-dimensional Gibbs—Kelvin equation. Hence we select, by trial and error, corresponding values of ~ and n5 for v = 0 to yield a constant value of,y0 In ~. The velocity of advancement of a step v is equal to v Jstep/no , (66) where ‘step is the flux of growth units into the step, given by
2,i, — VD 5V
5 V,i,,
(63)
,
VD5 Vn~+
=
—2D50(a~~5Iay)~~012 -
The growth rate R
so that, instead of(62). +
Jst0,,
uS --—
Tdes
d ____
=
(
(67)
velocity of advancement of the
(64)
0 -
~s
If we introduce D5 from (56)2 and f3~from (35), we obtain, rwith I X~0 = (‘~sO~des)” I V2n = -. —---— (0~-- 0 + 20) (Vt,
R 4
~
r
BCF/7~
Li? 0 1s0 .2 ‘-~sO
+
/‘s ~‘(i
-—
uI~
r—
2kTn0
-
)2
0 517
~-
L~kThn(0,
‘I’~
‘lso)I
20’)(n~
--
02 +
--~
20’)(~ “sO) H
-
‘tsO
11n11s
X~ 0
r X expi (0, L2kT,o —----—
—
02
]
(65)
This equation may be integrated numerically, if u~amid Vn 5 are known at a given point nmi the surface. We shall consider the one-dimensional problem of a series of equidistant parallel steps along the x direc(ion with spacingb -v the z axis beimig perpendicular to the surface. If the growth units have equal probability of being incorporated in the crystal when arriving from either side of the step, then the solution of
~J1Q5kT
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8 P
l’ig. 6. of Calculated curves as forgo and growth various values 4’. Othergrowth parameters in l’ig.= x50/ln 5. The ~BCI-’ curve, with the same parameter values, is shown for comparison.
HE. Lundager Madsen / Localized adsorption in one layer on crystal—solution interface
crystal face) is equal to R
=
v(t/)’o
where
d
=
~step/Yo
(68)
,
is the thickness of a layer of growth units in
the crystal, and ~2 is the volume occupied by a growth unit in the crystal. R may thus be calculated
fromyo and the value of an5/ay aty =y~/2. 2FlflsQ/Tdes as a function R inç3;units of Fig. i3 for 6y,,shows = x~o/1n this of corresponds to the region of transition from parabolic to linear growth law. As before, we have n 5o = 0.1n0, Ot = —4kT, and 02 = —8kT. The curve R = ((3 1)2 tanh[l/2(O 1)1 obtained from the BCF theory with the same parameter values, is shown for comparison. The difference between the BCF growth curve and the other curves —
—
503
(ems, but for the growth of paraffin crystals from solutions with solvents of low molecular weight [5], for instance, the present model is not of much use.
The similarity between the BCF growth curve and the growth curves obtained from the present model shows that the errors introduced by the approxima-
tions n~/n~o = ~ ln f3 ~e13 — 1, D5 tenda to cancel, and indicates that weand mayconstant safely analyze linear or parabolic growth curve in terms of the BCF theory as long as we do not expect highly precise results of the calculations. In view of the present state of crystal growth theory, this is no serious limitation; in some cases, the difference between the curves may even lie within experimental errors.
References
is remarkably small in view of the approximations
involved in the BCF theory. In the linear region,y0 x~o/ln(3, the BCF curve turns out to be virtually coincident with the other curves. ‘~
6. Discussion A number of relations for quantities frequently encountered in crystal growth theory, O~’~ etc., have been derived from our simple model of a localized adlayer at a crystal—solution interface. As was stated in the introduction, no quantitative agreement with experimental results can be expected, but a clear indication of the meaning and range of validity of some currently used relations has been obtained. Apart from the assumption of random distribution of solvent and solute molecules in the adlayer, inherent in the Bragg—Williams approximation, the applicability of the model is limited in two other respects: it is a model of a localized layer, and solvent and solute molecules occupy one site each. The latter will, indeed, be the case for a considerable number of sys-
[11 HE. Lundager Madsen, J. Crystal Growth 39 (1977) 250. [21W.K. Burton, N. Cabrera and F.C. Frank, Phil. Trans. Roy. Soc. (London) 243 (1951) 299. [31 P. Bennma and G.H. Gilmer, Kinetics of Crystal Growth; in: Crystal Growth, an Introduction, Ed. P. Hartman (North-Holland, Amsterdam, 1973). [41 R. Kaischew, Acta Phys. Hung. 8 (1957) 75. 151 R. Boistelle and A. Doussoulin, J. Crystal Growth 33
(1976) 335. [61 J.W. Gibbs, Collected Works, Vol. 1
(Yale Univ. Press,
New Haven, CT, 1948) pp. 219-328.
171 181 191
E.A. Guggenheim, Thermodynamics (North-Holland, Amsterdam, 1957 and later editions) ch. 5.
J.G. Kirkwood and 1. Oppenheim, Chemical Thermodynamics (McGraw-Hill, New York, 1961) ch. 10. T.L. Hill, Introduction to Statistical Thermodynamuics (Addison-Wesley, Reading, MA, 1960). [101R. Defay, I. Prigogine, A. Bellemans and DII. Everett, Surface Tension and Adsorption (Longmuans, London, 1966). [111F. Spaepen and R.B. Meyer, Scrmpta Met. 10 (1976) 257. [121Fl. Eyring, J. Chem. Phys. 4(1936) 283. [131I. Prigogine, Thermodynamics of Irreversible Processes (Interscienee, New York, 1967).
LOCALIZED ADSORPTION IN ONE LAYER ON A CRYSTAL—SOLUTION INTERFACE H.E. LUNDAGER MADSEN Chemistry Department, Royal Veterinary and Agricultural University, Thorvaldsensve/ 40, DK-1871 Copenhagen V, Denmark Received 30 May 1978; manuscript received in final form 25 October 1978
Deviations from ideality in the adlayer are discussed in terms of a Bragg—Williams monolayer model with empty sites. Equations for equilibrium and nonequilibrium surface concentrations of growth units are derived together with an equation for the variation of surface diffusion coefficient with surface supersaturation. A generalized form of the BCF differential equation for surface diffusion is established, and curves of crystal growth rate as a function of supersaturation are calculated from the results of numerical integration of this equation. It is shown that the BCF theory involves rather crude approximations, but the growth curves obtained from the more realistic model do not deviate very much from the BCF growth curve.
1. Introduction
which is inconsistency defined [I]. The reason why no precise formulation exists is that measurements of adsorption on a crystal surface of the crystal’s own molecules are extremely difficult, if not impossible, notably during growth of the crystal; even the distinction between adsorption and surface roughness may be troublesome. Hence, no empirical data exist which could provide some indication of the proper way to treat the problem theoretically. In this paper a model is proposed with the aim of overcoming some of the difficulties outlined above. It is a simple one-layer model of localized adsorption, based on the Bragg—Williams approximation, and it is
In a previous paper [1] it was shown that currently used models for the adsorbed layer on a crystal—solution interface are not quite adequate from a thermodynamic point of view. The Langmuir and Bragg—Williams models fail to account for the interface tension as an independent variable. The reason for this is that only two states of an adsorption site are considered in these models: either it is occupied by a growth unit or by solvent. The number of independent variables is thus limited to three, e.g. T, number of sites M, and number of growth units adsorbed N2. The number of degrees of freedom of a crystal—
not supposed to yield quantitatively correct result except in simple cases, but merely intended as a means for analyzing the adsorption problem of a crystal—solution system.
solution system at equilibrium (2 components, 2 phases) is 2, according to Gibb’s phase rule, and just one additional independent variable is insufficient for a discussion of all possible deviations from equilibrium. In the case of a BCF growth mechanism [2,3], for instance, it is clear that the state of the adsorbed layer at a given supersaturation of the solution depends on kinetic and other factors not directly related to the adsorption problem. Several attempts have been made to express the equilibrium activity (or, for an ideal system, the concentration) of growth units in the adsorbed layer in terms of a standard free enthalpy increment [3—5], but these expressions are difficult to apply, due to either an ill-defined standard state of a parameter
2. The partition function and its partial derivatives The fundamental thermodynamic equation for a plane interface in a two-component system is [6—8] dU
— —
TdS +y da+ji1 dn1
~P2
dn2
(1)
where a is the interface area, and the superscript a denotes interface excess quantities. In connection with a statistical treatment it is easier to use an alternative equation, valid for the total quantities of the 495
496
11./i. Lundat’cr Madsen
/ Localized adsorption
lai’er (~iicr,’sral
iii ui/i’
-solution interface
elc’5i iV~+
adsorbed monolayer, vit~.
+
+ 2~)A’, iV
+
2
Xexp
dL/ = T dS ~ d~l1+ p, diV1 + P2 rh’\’2 (2) where M is the number of sites, N, and N2 the nuni-
~
2 lILT
(6)
.
ber of molecLiles adsorbed, and 1 is the interface pressure; in (2) it has the dimension of energy per site. We shall later discuss the relationship between (1) and (2). It should be noticed that N, + N2 is not assumed equal to M. dA is obtained from dU using A = U — 15. and A is related to the canonical partition function Q(T. M, A’,, It2) of the adsorbed layer by [9]
or. ~isingStirling’s approximation (or the laclorials,
Q
In
M in M N1
iV, in ,V1 ‘~‘2) iii(41
(ill
A’, Iii ~li
+
+ ~~‘2 in ~/2
+
(.~5J\J
iV2 in N~ IV, iV~)
-
+ (~I + ~2 + 2ç5’) /V,IV,
~~5j\
--~jT~
-
(7)
-
We now have. Irom (2), (3), and (7), A
kTlnQ. (3) 5, he the energy of interaction of two niolecuies Letofc type I on neighbouring sites, let ~2 be the corresponding energy of a 22 pair, and let the energy of a 12 pair he ~, + ~2) + ~‘. If the adsorbed molecules are arranged such tlìat the number of 11 pairs is N, ~, the number of 22 pairs N 22. and of 12 pairs N, 2. the total energy of lateral interactions in the layer is I:’,
=
N, ~
+
N~5~+ ~iVi ~(c~i + -
The partition (unction Q(T, 41, iVy, A’2)
=
is
A’1
~
hT~, /dlnQ\ dIll
— -
!-~ ~
+
~
0) +
~
+
i)A’ / i~A2 kThn q1 + kT In -
(4)
02
~/~1q~2
A
~T(
=
12)
+~+2~1 (5)
where q, and ~/2 are molecular partition functions of single adsorbed molecules, g is the number of arrangements with the given values of the arguments, and the sunis extend over all possible values ofN1 . N22. and 1V,7.
(c~+
+
+ I~TIn
hi
~
Q
=
~‘\‘,!
M! N2! (31
-—
‘~‘i
12)!
q~’Iq~’2 -
+
kT in x2
“~
~__~
+
r
kO --In 0
+
A’
‘V
)~
-
x, (in q ÷T ~_hi~~
L
(10)
site
‘‘‘‘‘ —
(9)
,
+2c5’)x1j 0.
lIT /~ In 0 =
In
x,
dT ,
Tue crucial probleni in lattice statistics is tile cal-
Q
0 --~--~
~2
and the entropy Pin
~M
j 0
~
+~cI2~2x2 ~
d in q~
cuiation of g: when this has been done, Q may be obtained with the aid of the maximum term method
[91.In tile Bragg—Williams approximation N, 1 . N2 2’ and N, 2 in the exponential factor are replaced by their mean values for all possible arrangements. . neglecting any preferential ciusterrng. This. leads to the followmg expression for Q:
-
+ 2ç5’) x2
~2
Q
ki In ~/2
-
+~V22~2 ~~!Y,2~l
(5)
-
+ kTln x1
1 + ~cI2~,xi
‘
A’,
02
~
=
‘
din
,,
I-’ 2
-
IT in( 1
-
then
‘~22A I 2
XexP[-
2d’)
-
g(M. A,, A~,1V,
X
+
~,
-
(j)
+ X2
~ln q2
k( I
-
where
0
+
T
0) ln( 1
in
—~-~—
0)
A,
(1 I)
.
=
.
-
(N, + N2)/M is the fraction of sites occu-
.
.
-
-
pied. As- we are dealing with a solid—solution interface. 0 is close to I , and we approximate eqs. (8)—-
(Il) di
-
+
by
RT ln( I
---
0
~e(ç~1x~ + c~2~2+ 2ç~’x~A2) -
(12)
II.F. Lundager Madsen
—kTln q, kTln(l —0) + kTln x, + ~c[20,x, + (0, =
—
Pz
=
+
÷20’) x2]
(13)
÷20’) x,},
(14)
+ 02
--kTlnq2 —kTln(l —0) kT ln x2 + ~c [202x2 + (0~÷02
+ SM
/ Localized adsorption in one layer on
kLxi (ln q, +
dlnq, dT T~
—
x 2(ln q2
+
Td In q2 dT
--
In
x2)1
(15)
.
—
02
~
--
kTln q,
+ ct~+
~c02
—
kTinq2
+ 1
kTln x,
=
and y~,the activity coefficient, is given by 2, i = 1, 2. (23) kTln y~ = cO’(l x1) Equations analogous to (19)--(23) have been derived
+ c0’x~,
+kTlnx2 ±c0’x~ .
for the surface of a liquid solution of molecules of similar size ([101, ch. 12).
4. Equilibrium considerations In the following, component 1 will be the solvent and component 2 the growth units. Except at very high growth or dissolution rates, there will be equilibrium between adlayer and solution with respect to
(16)
solvent. Then Pi may be treated as a constant, and we may eliminate 1 between (16) and (17):
(17)
02=0,
An ordinary solution is termed ideal if p1 kT ln x, is constant at constant temperature and pressure for all components /. For a binary solution, this condition is equivalent with —
kT/x1,
i = 1, 2,
(18)
where, in fact, one of the relations may be deduced from the other with the .aid of the Gibbs—Duhem equation. By analogy, an ideal adsorbed layer may be defined as one, for which
(öpIIöxl)T,~= kT/x1,
=
1,2.
(19)
If we look at (16) and (17), we see that the condition of ideality is 0’ = 0. In this case the energy of interaction of a pair is the mean of the 11 and 22 interaction energies. In the nonideal case we may write
kTin(q2/q,)
—
x2 x2 +cO’(l
—
—
2x2).
(24)
We now introduce the parameters n0, the number of sites per unit area of interface, and n~,tile number of adsorbed growth units per unit area. Then, for 0 =
=
+~c(02 — Oi)
+kTln 1
3. Ideal and nonideal layers
(apl/axl)T,P
(22)
,
—
The last equation shows that the entropy is equal to that of an ideal mixture; this is the usual result of the Bragg—Williams approximation. The singularity of c1, ji,, and 02 for 0 = 1 results from the use of Stirling’s approximation for the logarithm of (M —N, —N2)!; it is not present in Q. The term kT ln(1 0) may be eliminated from p5 and ~2 by substitution: =
497
which is the chemical potential of an adsorbed layer of pure component i at zero interface pressure (a bypothetical state). a1, the activity, may be written a1
in x, /
crystal—solution interface
(25)
1l~/n,~ ,
and (24) becomes 02 =
+
p, ÷~c(02 kT ln
—
ns —
ns
0,) +
kTln(q2/q1)
—
cO
,
n0
-——-----~
(26)
Eq. (24) may also be written, using eqs. (20)---(23), 02
=
o,
—
p~+
+
kTIn(a2/a1)
‘
(27)
If the standard state of solvent in solution is pure liquid solvent with chemical potential p~,then the activity of solvent is the same in the adlayer and in solution at equilibrium, and p~ = ‘D0, the interface pressure of an adlayer in equilibrium with pure —
=
i = 1, 2,
+ ~ + kTln a~,
(20)
with the standard chemical potential =
—
kTlnq1,
i
=
1,2,
liquid solvent. Then (27) becomes (21)
02
=
b0
÷p~+ kTlna2.
(28)
IlL’. Lundager Madsen / Localized adsorption in n,ic la,’er on cri’stal---solulio,, interface
498
If finally, the adlayer is in equilibrium with the crystal. then 02 = p~,the chemical potential of the crystal, amid we have
We now wish to calculate the surface saturation ratio~
*
a,(sat)
=
exp!(u2
02
-~
(l)0)/Jcfl
5defined by [IJ
(29)
-
-
p~=p~+kTlnd5.
For an adlayer a2(sat) = x2(sat) = nso/no. where n50 is the value of n5 at equilibrium. Eq. (29) is thus equiva=
n
1)
0
Fr/un (33) and (26) tve imave ~ In ~ I I 2e0’
(30)
,
~
~
F~I
0 which yields, upon integration.
p~+
dI0
-
Fig. I shows the dependence of ~
(31) Ofl ~6ks
= ,
//~ 11s0
r
‘1s0
11ii
p~
(34)
lt’Tii
11~
exp(- ~GkS/k
if tve make the assignment =
(33)
-
lent with the Kaischew equation [4,5] 11s0
Surface supersaturation
5.
expl ~
-
2c0’p,,
L
~s
-
I.
-~ -—-~~—--~
kT,i
J
11
(eqs. (29) and (31)) for a square lattice (c = 4), with a2 given by (22) and (23). The temperature dependence of a2(sat) is given by in a9~at)\ ~
Jl~--
— -
+
-- -
H~- H~ —
- -~
~
-)
-
where the Gibbs—Flelmholtz equation has been used; H, H~,H~,and H~are the molecular euthalpies of the same standard states as those used for the chenncal potentials.
(35)
The last factor is equal to (y,(sat)/y, )(72/72(sat)). For a dilute adlayer (n5 ~ n0). the second factor is close to 1. and so is the ratio y,/’y,(sat), whereas y2/y2(sat) may still be substantially different from I, if we are dealing with, a highly nonideal system. Thus, the currently uised relation [2.3] ~s
=
(~6)
~s/h1sQ
is valid for a dilute adlayer unless the deviation from ideality is strong. For a concentrated adlayer, (36) cannot be expected to hold, not even for a perfectly ideal system; the error may be of the order of niagnitude of,05 — I.
nsa/no n5
~\7kT~
1
2
3
/
5
5
~
AG 4kT k~
I’11s0’ ig. I 5~hhu1ctmoi~ - liqu,hbr,um0f ~Gks surtacefurconcentration units. various values of of growth ~S.
~
12
12
13
14
17
16
17
18 19
I“so 1g. =2.Ohio ii~/i,~~as a fi,nctinn ot’ surface sopersatoration /3~i’or and vario~isvalues of ~‘ . Dashed line: ,i~/ii~~ =
HE. Lundager Madsen / Localized adsorption in one layer on crystal—solution interface
Fig. 2 shows how n~/n5odepends on !3, for c = 4 and n~o/no= 0.1 (~Gk.= 2.3kT if 0’ = 0). Even in the ideal case (0’ = 0), n5/n~ois not equal to 13~,except at low supersaturations. The best agreement between n~/n~oand ~3,,, is found for 0’ = (5/36)kT or, in the general case, 0’ =
kTno/[2c(no
--
n50)] -
0’ = 0.5kT (generally 0’ = (2/c)kT) is a critical value, which yeidls a vertical tangent at n5 = 0.5n0. For 0’> 0.5kT we find an S-shaped curve, part of which represents unstable states. This means that, at a certain value of ~ a phenomenon analogous to a firstorder phase transition takes place; ~ rises abruptly from a low to a high value close to n0. This value of !3~corresponds to the one found for n~= 0.5n0 from (35),viz. = 9 exp(—3.2~’/kfl for n~o/no= 0.1.
499
adlayer (172 ~ F~)in equilibrium with saturated solution d In r2 F’?a~~~des + = R (41)
~
—
where L~J1?des is the enthalpy of desorption of solvent, and Zxll~,is the enthalpy of transfer of growth units from the crystal (kthk) to the surface. In the present case F~a = 1, so that ~J~des+
(~~)F~ ~,
=
—
~ (42)
—~-~---~.
In a dilute solution c2 is proportional to x2, if we neglect the thermal expansion of the solution, hence, for saturated solution a in 12 (a(1/n)~0~’ (43) where ~~)I is the molar enthalpy of solution in dilute solution. From (40) we then find, using (42) and (43),
(ad(l/fl In
nso),,,,,,
6. Connection with interface thermodynamics
=
--
TC2
L~H~)1 + [‘2k,i~’ (n—”? des +
~.HkS)
In order to be able to relate the results of the
(44)
above derivations with those of the previously pub-
Comparison of (44) with (32) for the ideal case a2(sat.) = n50/no leads to TC2 11s0 H~ + H~ H~= + (~H~ dos + z~H~). (45)
lished thermodynamic analysis [1], we must work
out the connection between I and y as well as between n~and excess [‘2.isGuggenheim’s modification [7]theofinterface Gibbs’ formalism useful in this respect. Let r be the thickness of the adlayer. —ddM represents the work performed on the interface layer
when the number of sites is increased by dM without changing r or n0. This quantity equals —pdV°+yda, where V° is the volume of the interface layer. We
then have =
a
=
rM/i,0
M/no
(38)
,
hence, under the condition of constant
(J)
=
(pr
—
(37)
,
7)/n,) .
r
and n0 (39)
Similarly, let c, be the molar concentration in solution of component i and F~= ny/a the surface excess of this component. We then have =
IVA(rc2 + [‘2)
(40)
It was shown previously [11 that for a dilute ideal
--
—
~so
In this equation, the terms n 50,H~,andH~refer to the adsorbed monolayer as a whole, whereas ~‘2, M1~d~, and L~II~ are interface excess quantities [6—81. The equatiomi shows how the total enthalpy increment for transfer of one growth unit from the crystal to the adsorbed layer is partitioned between the interface excess enthalpy and the solution enthalpy. The fraction NAT2/nSo of the growth units transferred represents an increase in interface excess and gives rise to the last term of (45), whereas the rest, NATc2/nSo, may be considered as simply going into solution and therefore contributing the first term of
the From right hand side of(45). the above considerations, the thermodynamic consistency of our model, at least in so far as the ideal dilute layer is concerned, should be clear.
!1L. Lundager Madsen / Localized adsorption in one layer on cr,’stal—solution interface
500
7. Density of surface vacancies As we know the relation between dI and y, eq.
4~1O~
(39), we may calculate the fraction of vacant sites, —0, for typical values of y from eq. (17). We shall use the following values: 2, p = I atm = 1.01325 X l0~N/m = 5 A = 5 X lo-’°m, = 4 X 1018 n’2 (square lattice where each site is
5A
x
/
S A),
T= 25°C= 298.15 K, c = 4,
0,
0’
—4kT (typical value for a volatile solvent which follows Trouton’s rule for the entropy of vaporization), =
=
0.
Instead of using x 2 as an independent variable, we
prefer 02 —system p, with by (26). a low value of If the is p~given in equilibrium, 02 —02 will as a rule, correspond to a low solubility, though there is no one-to-one relationship between the two.
.01 .02 .Oi~ 01. 11g. 4. I
.07
.05 - 75
0 :isafu,nction of -y for 02
2 09 g. Jim SkTand v:iri-
08 02
=
0115 values of 5~2.Other parameters as in fig. 3.
1
—0
occurs for x
2 given by
Fig. 3 shows the dependence of 1 0 on y for 02 = —8kT and different values of 02 — p, and fig. 4 shows I 0 as a function of y for 112 112 = 5kT
There will thus be no maximum unless
amid different values of
~ckT.
—
*
—
—
02.
The maximum value of
=
2kT/c(çS
--
(46)
02).
01
-— 02 >
Froni the results we see that, except for low values of --. p~and relatively valuesdensity of y, 0found 1 is ina safe02approximation. The high molecular the adlayer is substantially higher than in solution, where it is normally of the order of 90% of the crys-
1-83 x10
9 8
~2
~
tal density, if the molecules are of similar size. This is characteristic for the localized adlayer model;
*
-10
Spaepen and Meyer [11] have found, for the solidliquid interface in a one-component monatomic system, a density in the randomly packed adlayer which
6 5
is not very different from the liquid density. /.
3
8. Surface diffusion and the BCF theory
-5
2
In order to see the consequences of ouir results for crystal growth theory we must analyze in detail the 01
02 03
01.
05
I’ig. 3. Fraction of empty Sites 1
05 --
07
08
09
~,
Jim2
0 as a fund ion of inter-
facial tension ‘y for various valujes of 02 Values are given in the text.
-
probleni of surface diffusion. By analogy with Eyring’s theory for volume diffusion [12], we may
express the surface diffusion coefficient by
u~. Parameter = (a2/rsd)(
1
---
0) .
(47)
HE. Lundager Madsen / Localized adsorption in one layer on crystal—solution interface where a is the distance between neighbouring sites, and TSd is the time constant for the jump of a molecule to a neighbouring empty site. As 1 — 0 and rS~
depend on f3~,so does D~.We shall work out the relationship between D~and I3~. We have from (13), (14) and (33), with x1 = (no n~)/noandx2 = n8/n0: —.
~,
—
ln(1
—
0) =
+ ~ in
+
—
—
——--—
—
2kT
~ in )3~
where q~is the partition function of a growth unit at the saddle point between two neighbouring sites.
Both q2 and q~depend on the types of molecules adsorbed on neighbouring sites. In the Bragg—Williams theory as well as in several other lattice theories this dependence is neglected; we shall therefore, for the sake of consistency, treat Tsd as a constant, the more so as the variations ofq2 and q~tendto cancel. We thus have, using (47),
n~(no— n~)
D5
D50
~
c
(48)
2kTn
=
10
0s0
=
I —0~ n0
—
xexp[f~-0’
+~~~_~s0)1
0
N, dp, + N2 dp2
=
M d’I),
(49)
where D~ois the value of D~at saturation. The dependence of D8 on i3~is given by (56) in connection with (35). Fig. 5 shows how D~varies with i3~for n~o= 0.1n0, c = 4,01 = —4kT, and 02 = —8kT. The surface flux is given by Fick’s first law
which yields n~ (~)
—
+
0~
(arl~
(50)
(ap~ aflS/T n0 beti~found an5 Tfromn0 (17)n5 with an5!T 11)/aflS)T may the proper (a substitutions for x, and x —
—
2: T
J~= —D~Vn~
(57) towards the surface is usually written [3] and the flux of growth units from the solution = (nS~,/r~,,~)(~3 i3~) (58) where Tdes is the time constant for desorption. In (58) the adlayer is assumed to be ideal; a more gen—
(a02) — kT
/a~\
(56)
2kTn0
To eliminate p~we make use of the Gibbs—Duhem equation with constant T:
=
501
+
~s
2c0’
—
.
(51)
,
When this is inserted in (50) we obtain, using (33) ( api)
kT(~-~-~
T=
a~
~
kTn0 + n~(no— n~ n0 ~
—
(52) .9
Using this result, we find (aIn(l—o)~
I
c 2kTn0
(02
—
Oi + 20’),
,5L (53)
which yields, upon integration ln
1—0 =
—
l—0~
in
0.4
.5
0.6
no—na n0—n50
(kT/h) q~/q2
05
2 .1
2kTn0 0o is the value of 0 at saturation. where Tsd may be written [9] =
0
.4
___________________
l/Tsd
=
05
n0—n5
aflS/T
+
ø’/kT
8~
_________
______________________
1.1 ~l~213
1.4 1.5 1.6 1.7 1.8 1.9
Fig. 5. Surface diffusion coefficient D
(55)
5 as a function of surface supersaturation j3~for n~o= 0.lno and various values of 4’. Other parameters as in fig. 3.
IlL’. I.undager Madsen / Localized adsorption in one lacer on cri’stal -solution interface
502
eral form is, by analogy with chemical reaction rates [13] 11~ =
—-——
I~
rI exp (02(soO L
—
.
-
—-
kT
—~
I I J
i
(59)
,
where 02(501) is the chemical potential of growth units in 5011,00,). Using (33) amid the analogous equation for the solution, we find ~)/~ -
= (11~Tdes)(~
(60)
I~and J.~are related through the equation of continuity
w~
(61)
~
If D5 were independent of,,5, arid n5 were equal to we should find, from (57), (60) and (61), the BCF diffusiomi equation [2,3] 20 ~5= 0 , (62) x~V where ~i = — ~ amid x 2. However, we 5 = (Dsees)” have
VJ,
=
--D
(65)
is symmetric around
t’
=
0 (steps are at s’
±v0/2),and a~,5/a3’= 0 for v
=
0. We consider the
=
primary law only, with n~= n50 for v = ±v0/2;thus means a rapid incorporation of the growth units once they have arrived at the step.
Tbte calculations are carried out as follows: We choose a value of,,5 fury
0 and integrate (65) from this gives ,‘~/2for the given starting value of n~and the given value of~. According to the BCF theory [2,3], ,v0 is propor-
.i’ =
=
0 to the point where n~= ~
tiomial to the radius of the critical two-dimensional miucleus which, in turn, is inversely proportional to In ~, according to the two-dimensional Gibbs—Kelvin equation. Hence we select, by trial and error, corresponding values of ~ and n5 for v = 0 to yield a constant value of,y0 In ~. The velocity of advancement of a step v is equal to v Jstep/no , (66) where ‘step is the flux of growth units into the step, given by
2,i, — VD 5V
5 V,i,,
(63)
,
VD5 Vn~+
=
—2D50(a~~5Iay)~~012 -
The growth rate R
so that, instead of(62). +
Jst0,,
uS --—
Tdes
d ____
=
(
(67)
velocity of advancement of the
(64)
0 -
~s
If we introduce D5 from (56)2 and f3~from (35), we obtain, rwith I X~0 = (‘~sO~des)” I V2n = -. —---— (0~-- 0 + 20) (Vt,
R 4
~
r
BCF/7~
Li? 0 1s0 .2 ‘-~sO
+
/‘s ~‘(i
-—
uI~
r—
2kTn0
-
)2
0 517
~-
L~kThn(0,
‘I’~
‘lso)I
20’)(n~
--
02 +
--~
20’)(~ “sO) H
-
‘tsO
11n11s
X~ 0
r X expi (0, L2kT,o —----—
—
02
]
(65)
This equation may be integrated numerically, if u~amid Vn 5 are known at a given point nmi the surface. We shall consider the one-dimensional problem of a series of equidistant parallel steps along the x direc(ion with spacingb -v the z axis beimig perpendicular to the surface. If the growth units have equal probability of being incorporated in the crystal when arriving from either side of the step, then the solution of
~J1Q5kT
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8 P
l’ig. 6. of Calculated curves as forgo and growth various values 4’. Othergrowth parameters in l’ig.= x50/ln 5. The ~BCI-’ curve, with the same parameter values, is shown for comparison.
HE. Lundager Madsen / Localized adsorption in one layer on crystal—solution interface
crystal face) is equal to R
=
v(t/)’o
where
d
=
~step/Yo
(68)
,
is the thickness of a layer of growth units in
the crystal, and ~2 is the volume occupied by a growth unit in the crystal. R may thus be calculated
fromyo and the value of an5/ay aty =y~/2. 2FlflsQ/Tdes as a function R inç3;units of Fig. i3 for 6y,,shows = x~o/1n this of corresponds to the region of transition from parabolic to linear growth law. As before, we have n 5o = 0.1n0, Ot = —4kT, and 02 = —8kT. The curve R = ((3 1)2 tanh[l/2(O 1)1 obtained from the BCF theory with the same parameter values, is shown for comparison. The difference between the BCF growth curve and the other curves —
—
503
(ems, but for the growth of paraffin crystals from solutions with solvents of low molecular weight [5], for instance, the present model is not of much use.
The similarity between the BCF growth curve and the growth curves obtained from the present model shows that the errors introduced by the approxima-
tions n~/n~o = ~ ln f3 ~e13 — 1, D5 tenda to cancel, and indicates that weand mayconstant safely analyze linear or parabolic growth curve in terms of the BCF theory as long as we do not expect highly precise results of the calculations. In view of the present state of crystal growth theory, this is no serious limitation; in some cases, the difference between the curves may even lie within experimental errors.
References
is remarkably small in view of the approximations
involved in the BCF theory. In the linear region,y0 x~o/ln(3, the BCF curve turns out to be virtually coincident with the other curves. ‘~
6. Discussion A number of relations for quantities frequently encountered in crystal growth theory, O~’~ etc., have been derived from our simple model of a localized adlayer at a crystal—solution interface. As was stated in the introduction, no quantitative agreement with experimental results can be expected, but a clear indication of the meaning and range of validity of some currently used relations has been obtained. Apart from the assumption of random distribution of solvent and solute molecules in the adlayer, inherent in the Bragg—Williams approximation, the applicability of the model is limited in two other respects: it is a model of a localized layer, and solvent and solute molecules occupy one site each. The latter will, indeed, be the case for a considerable number of sys-
[11 HE. Lundager Madsen, J. Crystal Growth 39 (1977) 250. [21W.K. Burton, N. Cabrera and F.C. Frank, Phil. Trans. Roy. Soc. (London) 243 (1951) 299. [31 P. Bennma and G.H. Gilmer, Kinetics of Crystal Growth; in: Crystal Growth, an Introduction, Ed. P. Hartman (North-Holland, Amsterdam, 1973). [41 R. Kaischew, Acta Phys. Hung. 8 (1957) 75. 151 R. Boistelle and A. Doussoulin, J. Crystal Growth 33
(1976) 335. [61 J.W. Gibbs, Collected Works, Vol. 1
(Yale Univ. Press,
New Haven, CT, 1948) pp. 219-328.
171 181 191
E.A. Guggenheim, Thermodynamics (North-Holland, Amsterdam, 1957 and later editions) ch. 5.
J.G. Kirkwood and 1. Oppenheim, Chemical Thermodynamics (McGraw-Hill, New York, 1961) ch. 10. T.L. Hill, Introduction to Statistical Thermodynamuics (Addison-Wesley, Reading, MA, 1960). [101R. Defay, I. Prigogine, A. Bellemans and DII. Everett, Surface Tension and Adsorption (Longmuans, London, 1966). [111F. Spaepen and R.B. Meyer, Scrmpta Met. 10 (1976) 257. [121Fl. Eyring, J. Chem. Phys. 4(1936) 283. [131I. Prigogine, Thermodynamics of Irreversible Processes (Interscienee, New York, 1967).