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Interpretation of interfacial phenomena using schematic density profiles
Interpretation of Interfacial Phenomena Using Schematic Density Profiles PAUL D. FLEMING III Research and Development Department, Phillips Petroleum Company, Bartlesville, Oklahoma 74004 Received August 12, 1977; accepted October 17, 1977 The treatment of interfacial phenomena using schematic density profiles is discussed. It is seen, for example, that the width and tension of a single-component liquid-vapor interface can be characterized using schematic profiles. When critical point scaling theory is applied to the approximate expression for the surface tension obtained from a schematic profile, a simple scaling relation results. This relation says that the surface tension is proportional to k T / ~ 2, where s¢ is the correlation length. This relation, which is implied by the "universality" concept to be more general than the context from which it was derived, indicates that interfaciai tensions are low whenever a large scaling length exists. For the Fisk-Widom theory the approximate expression for the tension scales properly and is in error by only 8%. Schematic profiles also lend themselves easily to discussion of surface activity. No surface excess occurs for a one-component system, of course, but for multicomponent systems manifestations of surface absorption should emerge. The unstable equilibrium associated with spherical interfaces in a pure system is also discussed in terms of schematic profiles. It is seen that the spherical droplets are well defined so long as the interfacial energy exceeds thermal energy. A similar discussion of spherical interfaces in multicomponent systems should be applicable to emulsion and foam stability. I. INTRODUCTION
The statistical mechanical theory of fluid interfaces, pioneered by van der Waals (1), has been advanced significantly in the past two decades (2-19). Most of the discussions (4-16, 18) have concentrated on liquidvapor surface tensions in a pure component system. Some authors (2, 3, 17), however, have considered liquid-liquid interfacial tensions in multicomponent systems. In a previous paper (16) (hereinafter referred to as I) a discussion of the van der Waals ("square gradient") theory has been presented in a form which makes no reference to the detailed geometry of the interfacial region. A local free energy density was seen to arise naturally from a rigorous expansion of the Helmholtz free energy in powers of the density gradient. The three-dimensional theory yielded mi-
croscopic interpretations of both Bernouli's principle and the Young-Laplace (20) equations for the pressure variation across a curved interface. In a more recent article (17) (hereinafter referred to as II) the development of I was generalized to multicomponent systems. For the particular case of binary systems the theory of II is a generalization of that of Cahn and Hilliard (2). Most recently (18) (in an article hereinafter referred to as III) the effects of gravity on the liquid-vapor interface have been discussed in detail. The effects of gravity have also been discussed recently by Wertheim (19). In the development of I a schematic density profile such as that shown in Fig. 1 proved useful in characterizing the interfacial width and interfacial tension of the liquid-vapor interface in a single-component
46 0021-9797/78/0651-0046502.00/0 Copyright© 1978by AcademicPress, Inc. All rightsof reproductionin any formreserved.
Journalof Colloidand InterfaceScience, Vol.65. No. 1, June 1, 1978
INTERFACIAL PHENOMENA nk
47
L2 is the position of the right-hand boundary. The free energy density is given by (16)
n(x)
~b(x) = to(n(x)) _~
0
+ ½ A ( n ( x ) ) n ' ( x ) 2,
FIG. 1. Schematic density profile for a liquid-vapor interface. system. Here we wish to demonstrate further the utility of such schematic profiles in extracting the dominant contributions to interfacial quantities. Here we only discuss surface tension in a single-component system. Many of the results will apply to and can easily be explicitly generalized, using the formulation of II, to multicomponent systems. The effects of gravity and other important external fields, such as those observed, for example, in centrifugation, can be discussed using a schematic profile based on the asymptotic solution discussed in III. In Section II we use a schematic density profile along with critical point scaling theory to obtain a simple expression for the surface tension in the critical region. In Section III the treatment of surface activity using schematic profiles is discussed. In Section IV spherical interfaces are discussed using schematic profiles. In particular, fluctuation phenomena and nucleation are considered. II. C H A R A C T E R I Z A T I O N OF INTERFACIAL TENSION WITH A SCHEMATIC DENSITY PROFILE
We briefly review the approximate calculation of the tension and width of the liquid-vapor interface in a one-component system. We begin with the Helmholtz free energy in terms of the local van der Waals free energy density. For an isothermal system in which the density varies in only one dimension we have (16) F = M
IL2 dxto(x),
[11.2]
where t0(n) is the free energy density of a uniform system having number density n and A ( n ) is related to the direct correlation function by (16) A(n) = ~
~rr2C(r,n).
We assume a schematic density profile of the form (Fig. I) n ( x ) = nL, -
-
X<
--~/2,
nL + nV - - + x
nv
-- n L
2
xE [ = nv,
~:2 ' 2~:]'
[II.3]
x > U2.
With this profile VL = L~A is the volume of the liquid phase and Vv = LzA is the volume of the vapor phase. The free energy (II. 1) then becomes F
=
Fv + Fs,
E L +
[II.4]
where E L =
Fv = VvO(nv)
VLto(nL) ,
and F s = ~fs A ds to -
= ~ J--1f2
-
2
+ s(nv
-
nL))
- ½(to(nL) + to(nv)) + ½ A ( n L + nv - + s(nv \ 2
-
nL))
[II.1]
--L1
where M is the cross-sectional area, - L t is the position of the left-hand boundary, and
With this profile, x = 0 corresponds to the position of the Gibbs surface since the total number of molecules is Journal o f Colloid and Interface Science, Vol. 65, No. 1, June 1. 1978
48
PAUL D. FLEMING III [II.5]
N=NL+Nv+Ns,
with NL = nL VL,
From (II.4) and (11.5) we have F - /xN = (~b(nL) -- IZnL)VL
N v = nvVv,
and
+ (~(nv) - /znv)Vv
Ns
~-
-
rs
(112
[ + nv dx nL J-l/2 T
=
+ ~f~a.
X
nL + nv]
~:
2
+--(nv--nL)
Variation with respect to nL and nv yields
J
=0.
As seen in I, the density profile is obtained by functional variation of F - IzN, where/z is the chemical potential, with respect to the density n(x). Therefore, for the approximate profile [II.3], the values of nL, nv, and ~: are obtained by requiring that the derivatives of F - / z N with respect to these variables vanish.
,?
(+nv
requiring that the chemical potential be that of the uniform liquid and vapor phases. Variation with respect to ~: yields, as before,
-- n v ) 2
d s A nL
2
J--l/2
[II.6]
r[(ds qs -+nv ) ~ + s(nv - nL) "
=
V2(qJ(nL)+ q~(nv))]
[II.83
J--112 or I1/2
dsA
-~;2
~ 2 ~___
(
+nv
--
nL--L------ + s(nv 2
[II. 10]
~:2 = 3 ~ / ~ ~ ,
=
-
(nL
J-V~
dsA ~
-
-
dnA(n) nv
nL
v
and (112 ~b" = 3
[ (112 ds
J--l/2
+
LaS
ds'
JSt
ds"
fl ; }( 1/2
ds . . . .
1/2
nv)2
[ (nL+nv 2~
2
~-112
_
3
(nL -- nvP
In~L v
dn[2qJ(n)
- qJ(nL) -- tk(nv)].
(112 ds'
ds
+s(nv- nL))- t0(nL)- tk(nv)]
+ s(nv - nL)
2
nv) 2
- qJ(nL)-- qJ(nv) ]
3
-
where
--
[I1.93
f~;~2 ds[2q~( "nL ~+ nv + S ( n v - n 0 )
We write Eq. (I1.9) in the abbreviated form
)
hE) (nL
nL + nv 2
+ s"(nv - nL) l /
Journalof ColloidandInterfaceScience,Vol. 65, No. 1, June 1, 1978
The relationships between the different expressions for tb" were shown in I. The approximate expressions for the surface tension can be obtained either by direct differentiation with respect to M or by substitution of the approximate density
INTERFACIAL PHENOMENA profile into the more general van der Waals expression for the surface tension (Eq. [II.18] or I). That expression is Ii/2
YI a ~--" ~:fs =
dsA
( n L q- nv 2
d --1/2
-- nv) 2 __ (nL --
nv) 2 ~ _
(nL --
s~
nv) z ~d/---7
3
In the neighborhood of the critical point the expression for the tension can be further characterized by making use of some scaling arguments originally suggested by Widom (5, 21). Recall that q/'(n) is related to the isothermal compressibility, and hence the mean square density fluctuation, by
0. qt"(n) = (--~n)T=
±(0p t
n\OnJT
=kr/[darG(r),
[II. 12]
where G(r) = ((n(r) -
n)(n(O) - n)).
According to scaling theory (5, 21) and as a fundamental assumption of renormalization group theory (22),' the integral of G(r) is of the form
49 ]lia= ckTlf2.
[11.15]
This expression, which was suggested by Widom (5), has a simple physical interpretation. It says that, in the critical region, the surface tension, which is an energy per unit area, is proportional to the fundamental unit of energy, kT, divided by the fundamental unit of area, the square of the correlation length. This expression implies, of course, that the interfacial tension critical exponent,/z, is twice that of the correlation length, u (~ = (d - 1)u in d dimensions). In view of the "universality" of fluidfluid critical points, we would expect [II.15] to be valid in the neighborhood of any fluid-fluid critical point. It also suggests, solely from dimensional considerations, that interfacial tensions may be low if a large scaling length exists, even when that scaling length is not due to the onset of long-range order. Miller et al. (23) recently suggested that such a large scaling length, i.e., the "micelle size," is responsible for interfacial tension lowering by surface active agents. Let us return to the approximate expression for the tension, [II.11]. We recall that the equilibrium pressure, the same in both phases, is P = ~nL -- 0(nL) = /znv -- tO(nv).
[II.16]
Thus ~b" can be rewritten as 3 (nL -- nv)a
I d3rG(r)a~3(nL -
nv) 2.
[II. 13]
~: is usually taken to be the correlation length, but as argued previously (5, 16) the correlation length and interfacial width are proportional. Thus, upon averaging over density, we expect that sufficiently near the critical point,
(0"= 3ckT/~a(nL-
nv) 2,
[11.14]
where c is a dimensionless proportionality constant of order of magnitude unity. Then [II. 11] becomes ' For a comprehensive review, see Ref. (22b).
fl ~Ldn2(O(n) + P - tzn).
[II. 17]
v
Thus [II. 11] can be rewritten as
Tla = [2 I[[ dna(n) I][ dn(~b(n) ] 1/2
+ P - /xn)l ,j
.
[II.18]
This should be compared with the corresponding expression based on the exact solution to the van der Waals differential equation. Recall that the pressure is given by (see Eq. [I1.15] of I) Journal of Colloidand InterfaceScience, Vol. 65, No. I, June 1, 1978
50
PAUL D. FLEMING II!
P = I.¢n(x) - $ ( n ( x ) ) + V2A(n(x))n'(x) 2
[11.19]
or equivalently
n'(x) [ O(n(x)) + P-
= --+ 2
~4~n~
tzn(x)]1/2 .
[II.20]
Equation [II.20] expresses n ' ( x ) as a function ofn. Thus we have by a variable change ")/I =
dxA(n(x))n'(x)
pression, [11.15], for the surface tension in the critical region. In order to compare the numerical values of Eq. [II.18] and [II.21], we calculate 3' from both expressions using the Fisk-Widom (5) (FW) theory. It is easy to see from the FW equation of state that 0(n)-/zn + e = (Tc - T)~ Ii v d n ' ( n ' - nc)
2
L-7
J --L1
= +-- f n .L d n A ( n ) n ' ( n )
= (T~ -
v
=
I?
dn[2A(n)(O(n)
T) ~'
x (1
fl .Ld n ' ( n '
aln_'-- ~llZ° 1
v
Tc+ P - lzn)] ltd.
[II.21]
The approximate expression, [II.18], is very similar to the more rigorous one, [II.21], The two should be close to one another in numerical value. Expression [II.18] is not so useful in itself; [II.21] is just as easy to evaluate. The utility of the schematic profile method is that from [II.9] it is possible to characterize the important features of the interfacial density profile without solving the differential equation? That [II.18] and [II.21] must surely scale the same way in the critical region does lend further credibility to the scaling exz Note that for the one-component case the solution to the differential equation can be reduced to quadrature. We see this because [11.20] can be integrated to yield x = fix) dn [ v
a(n) ~(n) + P -
This results because there is only one density variable to go along with the single-position variable. For multicomponent systems such a representation of the solution to the van der Waals differential equations is not possible because there are more density variables. Even in the pure system it is simpler to (numerically) perform the two integrations in Eq. [II.9] to obtain ~: than it is to numerically integrate the above equation for every x. Journal of Colloid and Interface Science, Vol. 65, No. l, June 1, 1978
T
/
× s (aln'~re--_ - nol" I ~. ] ,
[II.22]
where n¢ is the critical density, T~ is the critical temperature, y is the compressibility exponent, and /3 is the coexistence curve exponent. With a change of integration variable [II.22] becomes O ( n ) - /zn + P 1 ~ 2tJj(1)wl = / 3 ( T o - T) ~,+2~/[--~-) --~- ~xX,,
[II.23]
where w(x) = 2
f~
dx'(x)2e-l(1
" Xt ) - x ' ) J( j(1)
and x = a l n - ncIX/a/(Tc -
T).
Then [II. 18] and [11.21] become
]1/2. Ixn
- nc)
yl a = 3/3z(nL -- n v ) 2 ( A / X ) v~
[11.24]
"/i = C/32(nL -- n v ) 2 ( A / X ) v2,
[11.25]
and respectively, where
'Io
C --- "~
c=2 _
dxxa-lwll2(x),
dxxa-lw(x
,
INTERFACIAL PHENOMENA
nL
L
n(x)
¥
and j(1)
X=
/3 (To -
T) y = 0"(nL) = tO"(nv).
FW assumed a j ( x ) of the form (1 - x ) j ( x ) -
%(1
-
x 4/a) [II.26]
j(1) along with/3 = 1/3. They therefore found w ( x ) = 3A(1 - x*/a)(2 + x2/3).
[II.27]
As FW showed 9(3 v2) In (1 + 31/2) --
-
-
8
/[
/33
dxx~-lw(x)
.
[II.33]
For the particular case [I1.27] this is
FIG. 2. Schematic density profile which allows for a deviation from monatonic behavior.
C
?:/32 = 2
2
51
~/L
= (42) v2 = 6.48 . . . .
Thus, the schematic profile method is one which is easily applied and yields a numerically accurate approximation for the interfacial tension. More importantly, the method allows the interfacial width to be characterized by direct calculation and as seen from [11.24] or [11.32] the corresponding surface tension has the correct scaling behavior in the critical region. This correct scaling behavior lends credence to the scaling expression, [II.15], for the surface tension. Because of the universality of fluid critical phenomena this expression may have validity outside the context from which it was obtained.
21/2 III. SURFACE ACTIVITY
= 1.283 . . . .
[II.28]
As discussed in the previous section, a schematic density profile, such as that depicted in Fig. 1, is useful in characteriz?: = 3/2(6/7)1/2 --- 1.389 . . . . [11.29] ing the interfacial tension and interfacial This constitutes an error of only 8% from width of the liquid-vapor interface in a the more correct expression. single-component system. Such a monotonic We recall that the tension in the FW profile will be less useful, however, in theory can be written in the alternative form systems where surface active materials are present. A modified profile, such as that Tl = c/32A/L(nL - nv) z, [11.30] shown in Fig. 2, should enable quantification where L is the equilibrium correlation of surface activity. The equation for such a profile is of the form length given by ?: is seen by direct integration to be
L = (AX) 1/2.
[II.31]
Similarly we have for the approximate expression yl a = (/32A/L(nL - nv) 2.
[II.32]
n ( x ) = nL,
X<
--~/2,
= ½(nL + nv) + (nv -- nL) +h(1
x
2[xl),
Comparison with [II.l 1] yields 3 3 The expression given in footnote 22 of I is in error. If the integrals in [D. 12] of I are properly evaluated, [II.33] will be obtained.
x~( ~2 ' 2 ~] ' = nv,
x > s¢/2.
[III.1]
Journal o f Colloid and Interface Science, V o l . 65, N o . 1, J u n e 1, 1978
52
PAUL D. FLEMING III
h is a measure of the amount of matter concentrated at the interface. We will see, however, that the value of h, determined by requiring that the Helmholtz free energy be a minimum, is always such that a Gibbs surface can be chosen so that there is no surface excess of matter in a one-component system. In fact, we will find that h is never large enough that a nonmonotonic profile results. The surface excess of matter, Fs, for profile [Ili.1] is given by
Fs =
f2
dx[n(x)
-
-
F A
nL]
#/2
+
~
-- (t0(nL) -- l~nL)Li
12
dx[n(x) - nv],
where D is the position of the dividing surface. The particular choice of the Gibbs surface is the one which makes Fs = 0. Using [III.1], we see, with a change of integration variable, that - ½(nL + nv) + h - nL(d + ½) - nv(½ - d) - 4h
I
+ (th(nv) - nv)L~ + ~f~,
[Ill.2]
.)D
Fs
In order to see that h does indeed satisfy [111.5], we consider the free energy [II.2]. As before, the equilibrium density profile in the van der Waals theory is the one which renders F - / z N a minimum. An approximate density profile is obtained by substituting [III.1] into [11.2] and requiring that F-/zN be a minimum with respect to variations of nL, nv, ~:, and h. Here we need only be concerned with variations with respect to h. For profile [III. 1] the free energy is of the form
[III.6]
where ,f/2
(fs =
dxr(x) a -~/2
- ~:[~b(nL)(d + ½)
+ qJ(nv)(½ - d)].
We see that
ll2
J-el2 dxq,(x) = - I = ,o
dss
ds q,
Jo =
½f'l
nv),
-- d(n L -
[III.3]
×
nL +
nv
2
where
+ s(nv
-
t/L)
+
n(l
--
2s))
d = D/~. Thus, the position of the Gibbs surface is given by d = ½n/(nL
--
nv).
[III.4]
This positioning is physical only when [d[ -< ½; i.e., the Gibbs surface occurs in the interfacial region. This requires that --(nL -- Ilv) --< h --< (ilL -- nv).
[III.5]
+ ½ A ( nL +2 nv + S ( n v - nL)+ h ( 1 - 2s)) X (nv--nL--2h)2]
ds[ 1/2
X
nv + nL ) ~ + S(nv-- nL) + h(1 + 2S) 2
+ ½A(nL~-4-2 n v
Thus if h satisfies [III.5] there will be no surface excess of matter provided d is given by [III.4].
I
X
+ S(nv
(nv--nL+
-- BL)+
2 h ) 2]
n(1 +
.
2s))
[III.7]
Thus we have
0f~ Oh
0/ Oh
1 qJ(nL) -- qJ(nv) 2 IlL - nv
Journal of Colloid and Interface Science, Vol. 65, No. 1, June 1, 1978
[III.8]
INTERFACIAL PHENOMENA
53
F r o m [II. 16] the second term in [Ill.8] becomes/z/2. W h e n [111.7] is differentiated, [Ill.8] becomes
Oh
Jo
ds (1 - 2s) ~b' ne + n v + s ( n v - nL) + h(1 - 2s 2
+ ½A'( nL+nv
+ s(nv - nL) + h(1 - 2s) )(nv
2
- nL - 2h)2J ~:
+ s ( n v - nL) + h(1 - 2s) (n L - nv + 2h)
2
+ f~
- ~b'(nv)
ds[(l + 2s)[q/( -nL -+ nv + s(nv - nL) + h(1 + 2 s ) ) - ~b'(nL) ~/2
2
+ ½a'( nL+nv
2
+ s(nv
-
nL) + h(1 + 2s)
+ 2 A ( nL - - ++s ( nnvv - n L ) 2
)(
nv
- nL + 2h)2J
~:
+h(1 + 2 s ) ) ( n v -
nL + 2h) ] .
[III.9]
We observe that
~b'( 'nL -~ + nv +
s(nv - nL) + h(1 - 2s)
fl ds' Iz
= (nv-
ds'
)
(k'(nv)
-
+nv
df -nL -+s'(nv--ne)+h(1-
2s'))l
2
nL-- 2h)
/2 ds'd/' -nL ' - 7+" -nv + s(nv -- he) + h(1 - 2s)
. [III.lO]
When the qJ' terms in the second integral in [III.9] are manipulated in a similar manner, we obtain Of~ _ (nv - nL -- 2h) ds (1 - 2s) Oh Jo
12
ds'tk" ( -nL -+ nv + s'(nv - nL) + h(1 - 2 s ' ) ) 2
+ V 2 A , ( n L + n v + s'(nv - nL) + h(1 - 2s') ) n v - nL-- 2hi 2 ~ -
2a(nL+ nv + s'(nv ~ 2
ds (1 + 2s) I/2
l
f
- ha) + h(1 - 2s')
)}
+ (nv - nL + 2fi)
ds'O" nL - -+ nv + s'(nv - nL) + h(1 + 2s')) 1/2
(
2
+ V2A,(.nL + nv + s'(nv - n~) + h(1 - 2s')) nv - nL + 2h \ 2 ~+s'(nv-nL)+h(1 2
+ 2s')
.
[III.11]
Journal of Colloid and Interface Science, Vo|. 65, No. I, June 1, 1978
54
PAUL D. FLEMING III
The equilibrium value of h is obtained by setting the right-hand side of [III.11] equal to zero. This yields a complicated nonlinear equation for h. We can, however, obtain bounds on h by examining the asymptotic behavior of [III.1 1]. As shown in III, $" diverges at both high and low densities because of hard-core repulsions at high densities and because t k " ~ kT/n (ideal gas limit) at low densities. Thus for large values of h the second integral will be expected to dominate [III.11] and we must have h
"~" ½ ( n L -
nv).
[III. 12]
Likewise for h negative and large in absolute value, the first integral will dominate and we must have h ~ ½(nv - nL).
[III.13]
For a curved interface we need to consider a schematic profile of the form (Fig. 3) n(r) = n2,
r >t R + ~/2,
nl + n2 + n ~ - nl ( r 2 rE
=nl,
R),
R--~,R
r~
+
,
[IV.l]
where nl denotes the internal phase density and n~ denotes the external phase density. It will be unnecessary to specify which phase is liquid and which is vapor. As we see below, R is essentially the position of the interface. For profile [IV.1] the total number of molecules is
Since these should correspond to asymptotic values, h should satisfy ½(nv - nL) -< h -< ½(nL - nv).
[III.14]
Not only does h satisfy an inequality more stringent than [III.5], but the actual inequality [III.14] assures that the density profile [III.1] will always be monotonic. Thus there will be no surface excess and the Gibbs surface will always exist. Thus we find, as should be expected, that a single-component system can have no surface excess and that the density profile is indeed monotonic. However, when an additional component is added, there will be two independent " h ' s " (one for each component) so that surface excess of one of the components may then be possible. The conditions for such an excess, within the context of the multicomponent theory of II, should enable a molecular interpretation of surface activity of surfactants to be obtained. IV.
CURVED
INTERFACES
Here we show that schematic profiles are also useful in discussing curved interfaces and associated questions of nucleation. Journalof CoUoidandInterfaceScience,Vol. 65, No. 1, June 1, 1978
+ iR+el2 dr [ nl + n2 JR--#I2 2
+ n2 - n__________(r 51 - R)]47rr 2. J
[IV.2]
The integral in [IV.2] can easily be evaluated and we find that N can be written as N = nlV1 + n2V2,
where V, = (4"rr/3)R z,
V2 = V -
(4~'/3)k ~,
with k = R(1 + (UZR)2) "~.
[IV.3]
Thus, if the actual position of the interface is taken as k there will be no surface excess at the interface. However, since discussion of interfacial phenomena at curved interfaces is meaningless unless ~ / R ,~ 1 we can take R as the position of the interface.
INTERFACIAL PHENOMENA
55
The discussion of Section II suggests that f~ should be expressed in the concise form Y6~b(n2
-
nl) 2 +
V2,~ nl
T
n2
where
n(r)
74 = nI > n2
n2
>n 1
[v2
J--l/2
dsA
( na + n~ 2
+ (n2 - nOs) [IV.5]
\N
/
nI < n2
/
and
\~__
n2 < n 1
3
11/2 J--l/2
Fro. 3. Schematic density profile [IV.l] for a spherical interface.The solid line denotes the schematic profile for the case in which the vapor is the internal phase (nl < n2), while for the dotted line the liquid is the internal phase (n2 < nl).
[-fI/2 11f2 d s l l as' as" LJs Js'
+
ds'
I/2
ds"
, nl + n2
1/2
2
+ s"(n2 - nz)) /
If [IV.l] is substituted into [II.1] we obtain
=
(
6
)2~112
'rll
n2
[(
ds qJ nl
J-1/2
+n2 -2
47r
+ qJ(n2)(V- 4 ~ ( R + - - ~ ) +
47rr2dr
In order to obtain the equilibrium values of nl, n2, ~:, and R we consider variations of F - IxN with respect to these variables. From [IV.3] and [IV.4] we see that
Io(n +n 2
J -~t2
+ ( n 2 - nl) r - R )
+ 1/2A( nl + n2 2
+ (n2_no_~R)(n~
+ 4~rR2~ Ofs [IV.6a] Onl
-~ n 1 )2]
= qJ(nl)V1 + qJ(n2)V~ + 4~rRZ~f~,
[IV.4]
where
fs =
O(f - IxN) - Ok'(nO - Ix)V~ Onl
and
O ( F - IxN)
r [0(- ds
J-l/2
nl q- n2 + (n2 2
-
hi)S)
tO(n1) + qJ(n2) 2 + ½ A (.nl + n2 \ 2
On2
- (tO'(nz) - Ix)V2 + 47rR2~:~
0/12
-4- ( n 2 - -
nOs)
. [IV.6b]
Requiring that these derivatives vanish yields ~b'(nl) = IX(1 - 3 ~ 1 0 f s ]
[IV.7a]
R tx On1] Journal of Colloid and Interface Science,
Vol.65,No. 1,June1, 1978
56
PAUL D. FLEMING III
and
where
0'(n2) = ft(1
y, = Cfs = A(n2 - nl)2/~.
- 3 ~:
R
(R/~) 3 0f~] /z(1 --- (-~-~)3) O n , } '
×
[IV.7b]
where =
(3V/47r)I/3
is the radius of the sphere having volume V. Thus for sufficiently large systems and ~/R ~ 1, n~ and n2 are densities of phases having chemical potential/~. We also see that O(F - I.tN) _ 47rR2{£
Of~]
[IV.8]
+~: o~:/
-
lzN) - -
02( F - ~ N ) -
~"(nl)V~ >i 0
[IV.14a]
On12
and
[IV.14b]
We also see that 02( F -
and since f~ is independent of R O(F
We see that the expression for the tension is identical in form to that obtained in the flat interface case. Thus near the critical point [II. 13] and [II. 15] should also be valid for spherical interfaces. Next we examine the stability of the equilibrium. For sufficiently large systems and ~ / R ~ 1 we see that
02(F - i~N) _ ~"(n2)V2 >i O. On22
o#
[IV.13]
tzN)
O~2
4 7 r R 2 ( e 2 -- P1) + 8~'Rsrf~,
OR
= 47rg 2 2 0~: + ¢ ' ~ - ~
" [IV.15]
[IV.9] where
From [IV. 10] we have
P1 = ftnl - t0(n0
and
P2
=
/-tn2
-- 0(n,)
are the pressures of the respective phases. From [IV.5] we see that 0fs _
,3, .(n2 - nO 2
0~
[IV.10]
~3
-- nl)2
=.
1/2.~(n' -- n l ) 2
02(F-
izN)
47rR 2 ~2 Yx ~> 0.
- - -
[IV.12]
As in Section II, [IV. 11] actually provides an equation for ¢', an equation identical in form to [II. 10]. From requiring that O(F - l z N ) / O R vanish we have the Y o u n g - L a p l a c e (20) equation P~ - P2 = 2 T I / R ,
Vol. 65, No. 1, June 1, 1978
[IV.17]
From [IV.9] we have OR z
- 8~rR(P2 - P1) + 81r~f~
= -87ry~ ~< 0.
f~ = ~ (n2 - n 0 ' _ V3Jf'(n, - nO z.
[IV.16]
~4
Hence we have
02( F - t~N)
[IV.11]
and therefore that
Journal of CoUoid and Interface Science,
0~2
0~2
Thus, from setting the right-hand side of [IV.8] equal to zero we have l/6~.f'(n2
02fs - 3,'~ (n2 - nO 2
[IV.18]
Thus, as is well known (24) but often ignored in elementary discussions (see, e.g., Ref. (25)), the mechanical equilibrium of the spherical interface is unstable with respect to the radius. The radius defined by [IV. 13] plays the role of a nucleation size. Internal phase fluctuations of radius less than R shrink, while those of larger radius grow. Finally, we examine the surface free
57
1NTERFACIAL PHENOMENA
energy Fs = 47rR2~:fs = 47rR2%.
[IV. 19]
We see that Fs/kT = 4~rR2y,/kT.
[IV.20]
If we assume [II.15] then this becomes F J k T = 4rcc(RZ/sez).
[IV.21]
This says that the surface energy is significant, i.e., FJkT>> 1, whenever the spherical interface is well defined. Conversely when the interfacial energy is comparable to thermal energy, i.e., when F J k T ~ 1, the interface is no longer well defined; i.e., its width is the order of its radius. V. DISCUSSION
We have seen that schematic density profiles are of great utility in the characterization of interfacial phenomena. The schematic density profile allows the interfacial width to be characterized in terms of molecular quantities. In the vicinity of the critical point of a one component system a very simple scaling expression for the surface tension was obtained from a schematic density profile and Widom scaling theory. This expression, inversely proportional to the square of the correlation length, suggests from dimensional considerations that interfacial tensions will be low whenever a large scaling length exists. Such an interpretation of low interfacial tensions may prove useful in understanding the lowering of the interfacial tension by surface active agents. The approximate expression for the surface tension obtained from the schematic density profile is in error by only 8% for the Fisk-Widom theory. More important than the detailed numerical agreement is the fact that the expression scales correctly near the critical point. Schematic density profiles also prove useful in discussing surface activity. We have seen, as expected, that no surface excess occurs in a one-component system
but the expected excess in multicomponent systems should be describable in terms of schematic profiles. Schematic density profiles should be of special utility in multicomponent systems where solutions to the differential equations may not be feasible. The resulting simplifications should facilitate the understanding of the microscopic basis for interfacial tension lowering by surfactants. We have also seen that spherical interfaces and nucleation can be readily treated in terms of schematic profiles. The instability of the curved interface equilibrium with respect to the radius emerges directly from the analysis. The interfacial energy was seen to exceed the thermal energy (kT) whenever the interfacial radius is larger than the interfacial width. The interface is not well defined because the interfacial width is of the order of the radius when the interfacial energy is about equal to the thermal energy. A generalization of the curved interfacial theory should provide a description of emulsion and foam stability. In summary, we have developed a method of approximate treatment of interfacial phenomena which retains the essential features of more rigorous treatments, while providing the simplifications necessary to discuss complex problems. The method should be especially applicable to discussions of interfacial phenomena in multicomponent systems containing surfactant species. A molecular interpretation of surface activity and emulsion coalescence should be obtainable from such discussions. REFERENCES 1. van der Waals, J. D., Z. Phys. Chem. 13, 657 (1894). 2. Cahn, J. W., and Hilliard, J. E., J. Chem. Phys. 28, 258 (1958). 3. Rice, 0. K., J. Phys. Chem. 64, 976 (1960). 4. Buff, F. P., Lovett, R. A., and Stillinger, F. H., Phys. Rev. Lett. 15, 621 (1965), Lovett, R. A., Ph.D. Thesis, University of Rochester, 1965; Vieceli, J. J., Ph.D. Thesis, University of Rochester, 1971; Lovett, R. A., DeHaven, P. W., Vieceli, J. J., and Buff, F. P., J. Chem. Phys. 50, 1880 (1973). Journal of Colloidand Interface Science, Vol. 65, No. 1. June 1, 1978
58
PAUL D. FLEMING III
5. Widom, B., J. Chem Phys. 43, 3892 (1965); and in "Phase Transitions and Critical Phenomena," (C. Domb and M. S. Green, Eds.), Academic Press, New York, 1972; Fisk, S., and Widom, B., Vol. 2, Chap. 3. J. Chem. Phys. 50, 3219 (1969). 6. Felderhof, B. U., Physica 48, 541 (1970). 7. Toxvaerd, S., J. Chem. Phys. 55, 3116 (1971); 62, 1589 (1975); 64, 2863 (1976); and in "Statistical Mechanics" (K. Singer, Ed.), Vol. 2. The Chemical Society of London, London, 1975. 8. Triezenberg, D. G., and Zwanzig, R., Phys. Rev. Lett. 28, 1183 (1972). 9. Lee, J. K., Barker, J. A., and Pound, G. M., J. Chem. Phys. 60, 1976 (1974). 10. Woodbury, G. W.,J. Chem. Phys. 60, 3674 (1974). 11. Gray, C. G., and Gubbins, K. E., Mol. Phys. 30, 179 (1975); Hally, J. M., Gray, C. G., and Gubbins, K. E., J. Chem. Phys. 64, 1852, 2569 (1976). 12. Davis, H. T., J. Chem. Phys. 62, 3412 (1975), Salter, S. J., and Davis, H. T., J. Chem. Phys. 63, 3295 (1975); Bongiorno, V., and Davis, H. T., Phys. Rev. A 12, 2213 (1975).
Journal of Colloidand Interface Science, Vol.65, No. l, June 1, 1978
13. Sarkies, K. W., and Frankel, N. E., Phys. Rev. A 11, 1724 (1975). 14. Abraham, F. F., J. Chem. Phys. 63, 157, 1316 (1975). 15. Mulholland, G. W.,J. Chem. Phys. 64, 862 (1976). 16. Yang, A. J. M., Fleming, P. D., and Gibbs, J. H., J. Chem. Phys. 64, 3732 (1976). 17. Fleming, P. D., Yang, A. J. M., and Gibbs, J. H., J. Chem. Phys. 65, 7 (1976). 18. Yang, A. J. M., Fleming, P. D., and Gibbs, J. H., J. Chem. Phys. 67, 74 (1976). 19. Wertheim, M. S.,J. Chem. Phys. 65, 2377 (1976). 20. Gibbs, J. W., "Scientific Papers," Vol. 1, p. 229, Eq. 500. Dover, New York, 1961. 21. Widom, B., J. Chem. Phys. 43, 3892 (1965); Kadanoff, L. P., Physics 2, 263 (1966). 22. (a) Wilson, K. G., Phys. Rev. B 4, 3174, 3184 (1971). (b) Wilson, K. G., and Kogut, J., Phys. Rep. C 12, 76 (1974). 23. Miller, C. A., Hwan, R., Benton, W. J., and Fort, T., J. Colloid lnterface Sci. 61,554 (1977). 24. Reiss, H., "Methods of Thermodynamics," pp. 202-203. Blaisdell, New York, 1965. 25. Moore, W. J., "Physical Chemistry." PrenticeHall, Englewood Cliffs, N. J.
The statistical mechanical theory of fluid interfaces, pioneered by van der Waals (1), has been advanced significantly in the past two decades (2-19). Most of the discussions (4-16, 18) have concentrated on liquidvapor surface tensions in a pure component system. Some authors (2, 3, 17), however, have considered liquid-liquid interfacial tensions in multicomponent systems. In a previous paper (16) (hereinafter referred to as I) a discussion of the van der Waals ("square gradient") theory has been presented in a form which makes no reference to the detailed geometry of the interfacial region. A local free energy density was seen to arise naturally from a rigorous expansion of the Helmholtz free energy in powers of the density gradient. The three-dimensional theory yielded mi-
croscopic interpretations of both Bernouli's principle and the Young-Laplace (20) equations for the pressure variation across a curved interface. In a more recent article (17) (hereinafter referred to as II) the development of I was generalized to multicomponent systems. For the particular case of binary systems the theory of II is a generalization of that of Cahn and Hilliard (2). Most recently (18) (in an article hereinafter referred to as III) the effects of gravity on the liquid-vapor interface have been discussed in detail. The effects of gravity have also been discussed recently by Wertheim (19). In the development of I a schematic density profile such as that shown in Fig. 1 proved useful in characterizing the interfacial width and interfacial tension of the liquid-vapor interface in a single-component
46 0021-9797/78/0651-0046502.00/0 Copyright© 1978by AcademicPress, Inc. All rightsof reproductionin any formreserved.
Journalof Colloidand InterfaceScience, Vol.65. No. 1, June 1, 1978
INTERFACIAL PHENOMENA nk
47
L2 is the position of the right-hand boundary. The free energy density is given by (16)
n(x)
~b(x) = to(n(x)) _~
0
+ ½ A ( n ( x ) ) n ' ( x ) 2,
FIG. 1. Schematic density profile for a liquid-vapor interface. system. Here we wish to demonstrate further the utility of such schematic profiles in extracting the dominant contributions to interfacial quantities. Here we only discuss surface tension in a single-component system. Many of the results will apply to and can easily be explicitly generalized, using the formulation of II, to multicomponent systems. The effects of gravity and other important external fields, such as those observed, for example, in centrifugation, can be discussed using a schematic profile based on the asymptotic solution discussed in III. In Section II we use a schematic density profile along with critical point scaling theory to obtain a simple expression for the surface tension in the critical region. In Section III the treatment of surface activity using schematic profiles is discussed. In Section IV spherical interfaces are discussed using schematic profiles. In particular, fluctuation phenomena and nucleation are considered. II. C H A R A C T E R I Z A T I O N OF INTERFACIAL TENSION WITH A SCHEMATIC DENSITY PROFILE
We briefly review the approximate calculation of the tension and width of the liquid-vapor interface in a one-component system. We begin with the Helmholtz free energy in terms of the local van der Waals free energy density. For an isothermal system in which the density varies in only one dimension we have (16) F = M
IL2 dxto(x),
[11.2]
where t0(n) is the free energy density of a uniform system having number density n and A ( n ) is related to the direct correlation function by (16) A(n) = ~
~rr2C(r,n).
We assume a schematic density profile of the form (Fig. I) n ( x ) = nL, -
-
X<
--~/2,
nL + nV - - + x
nv
-- n L
2
xE [ = nv,
~:2 ' 2~:]'
[II.3]
x > U2.
With this profile VL = L~A is the volume of the liquid phase and Vv = LzA is the volume of the vapor phase. The free energy (II. 1) then becomes F
=
Fv + Fs,
E L +
[II.4]
where E L =
Fv = VvO(nv)
VLto(nL) ,
and F s = ~fs A ds to -
= ~ J--1f2
-
2
+ s(nv
-
nL))
- ½(to(nL) + to(nv)) + ½ A ( n L + nv - + s(nv \ 2
-
nL))
[II.1]
--L1
where M is the cross-sectional area, - L t is the position of the left-hand boundary, and
With this profile, x = 0 corresponds to the position of the Gibbs surface since the total number of molecules is Journal o f Colloid and Interface Science, Vol. 65, No. 1, June 1. 1978
48
PAUL D. FLEMING III [II.5]
N=NL+Nv+Ns,
with NL = nL VL,
From (II.4) and (11.5) we have F - /xN = (~b(nL) -- IZnL)VL
N v = nvVv,
and
+ (~(nv) - /znv)Vv
Ns
~-
-
rs
(112
[ + nv dx nL J-l/2 T
=
+ ~f~a.
X
nL + nv]
~:
2
+--(nv--nL)
Variation with respect to nL and nv yields
J
=0.
As seen in I, the density profile is obtained by functional variation of F - IzN, where/z is the chemical potential, with respect to the density n(x). Therefore, for the approximate profile [II.3], the values of nL, nv, and ~: are obtained by requiring that the derivatives of F - / z N with respect to these variables vanish.
,?
(+nv
requiring that the chemical potential be that of the uniform liquid and vapor phases. Variation with respect to ~: yields, as before,
-- n v ) 2
d s A nL
2
J--l/2
[II.6]
r[(ds qs -+nv ) ~ + s(nv - nL) "
=
V2(qJ(nL)+ q~(nv))]
[II.83
J--112 or I1/2
dsA
-~;2
~ 2 ~___
(
+nv
--
nL--L------ + s(nv 2
[II. 10]
~:2 = 3 ~ / ~ ~ ,
=
-
(nL
J-V~
dsA ~
-
-
dnA(n) nv
nL
v
and (112 ~b" = 3
[ (112 ds
J--l/2
+
LaS
ds'
JSt
ds"
fl ; }( 1/2
ds . . . .
1/2
nv)2
[ (nL+nv 2~
2
~-112
_
3
(nL -- nvP
In~L v
dn[2qJ(n)
- qJ(nL) -- tk(nv)].
(112 ds'
ds
+s(nv- nL))- t0(nL)- tk(nv)]
+ s(nv - nL)
2
nv) 2
- qJ(nL)-- qJ(nv) ]
3
-
where
--
[I1.93
f~;~2 ds[2q~( "nL ~+ nv + S ( n v - n 0 )
We write Eq. (I1.9) in the abbreviated form
)
hE) (nL
nL + nv 2
+ s"(nv - nL) l /
Journalof ColloidandInterfaceScience,Vol. 65, No. 1, June 1, 1978
The relationships between the different expressions for tb" were shown in I. The approximate expressions for the surface tension can be obtained either by direct differentiation with respect to M or by substitution of the approximate density
INTERFACIAL PHENOMENA profile into the more general van der Waals expression for the surface tension (Eq. [II.18] or I). That expression is Ii/2
YI a ~--" ~:fs =
dsA
( n L q- nv 2
d --1/2
-- nv) 2 __ (nL --
nv) 2 ~ _
(nL --
s~
nv) z ~d/---7
3
In the neighborhood of the critical point the expression for the tension can be further characterized by making use of some scaling arguments originally suggested by Widom (5, 21). Recall that q/'(n) is related to the isothermal compressibility, and hence the mean square density fluctuation, by
0. qt"(n) = (--~n)T=
±(0p t
n\OnJT
=kr/[darG(r),
[II. 12]
where G(r) = ((n(r) -
n)(n(O) - n)).
According to scaling theory (5, 21) and as a fundamental assumption of renormalization group theory (22),' the integral of G(r) is of the form
49 ]lia= ckTlf2.
[11.15]
This expression, which was suggested by Widom (5), has a simple physical interpretation. It says that, in the critical region, the surface tension, which is an energy per unit area, is proportional to the fundamental unit of energy, kT, divided by the fundamental unit of area, the square of the correlation length. This expression implies, of course, that the interfacial tension critical exponent,/z, is twice that of the correlation length, u (~ = (d - 1)u in d dimensions). In view of the "universality" of fluidfluid critical points, we would expect [II.15] to be valid in the neighborhood of any fluid-fluid critical point. It also suggests, solely from dimensional considerations, that interfacial tensions may be low if a large scaling length exists, even when that scaling length is not due to the onset of long-range order. Miller et al. (23) recently suggested that such a large scaling length, i.e., the "micelle size," is responsible for interfacial tension lowering by surface active agents. Let us return to the approximate expression for the tension, [II.11]. We recall that the equilibrium pressure, the same in both phases, is P = ~nL -- 0(nL) = /znv -- tO(nv).
[II.16]
Thus ~b" can be rewritten as 3 (nL -- nv)a
I d3rG(r)a~3(nL -
nv) 2.
[II. 13]
~: is usually taken to be the correlation length, but as argued previously (5, 16) the correlation length and interfacial width are proportional. Thus, upon averaging over density, we expect that sufficiently near the critical point,
(0"= 3ckT/~a(nL-
nv) 2,
[11.14]
where c is a dimensionless proportionality constant of order of magnitude unity. Then [II. 11] becomes ' For a comprehensive review, see Ref. (22b).
fl ~Ldn2(O(n) + P - tzn).
[II. 17]
v
Thus [II. 11] can be rewritten as
Tla = [2 I[[ dna(n) I][ dn(~b(n) ] 1/2
+ P - /xn)l ,j
.
[II.18]
This should be compared with the corresponding expression based on the exact solution to the van der Waals differential equation. Recall that the pressure is given by (see Eq. [I1.15] of I) Journal of Colloidand InterfaceScience, Vol. 65, No. I, June 1, 1978
50
PAUL D. FLEMING II!
P = I.¢n(x) - $ ( n ( x ) ) + V2A(n(x))n'(x) 2
[11.19]
or equivalently
n'(x) [ O(n(x)) + P-
= --+ 2
~4~n~
tzn(x)]1/2 .
[II.20]
Equation [II.20] expresses n ' ( x ) as a function ofn. Thus we have by a variable change ")/I =
dxA(n(x))n'(x)
pression, [11.15], for the surface tension in the critical region. In order to compare the numerical values of Eq. [II.18] and [II.21], we calculate 3' from both expressions using the Fisk-Widom (5) (FW) theory. It is easy to see from the FW equation of state that 0(n)-/zn + e = (Tc - T)~ Ii v d n ' ( n ' - nc)
2
L-7
J --L1
= +-- f n .L d n A ( n ) n ' ( n )
= (T~ -
v
=
I?
dn[2A(n)(O(n)
T) ~'
x (1
fl .Ld n ' ( n '
aln_'-- ~llZ° 1
v
Tc+ P - lzn)] ltd.
[II.21]
The approximate expression, [II.18], is very similar to the more rigorous one, [II.21], The two should be close to one another in numerical value. Expression [II.18] is not so useful in itself; [II.21] is just as easy to evaluate. The utility of the schematic profile method is that from [II.9] it is possible to characterize the important features of the interfacial density profile without solving the differential equation? That [II.18] and [II.21] must surely scale the same way in the critical region does lend further credibility to the scaling exz Note that for the one-component case the solution to the differential equation can be reduced to quadrature. We see this because [11.20] can be integrated to yield x = fix) dn [ v
a(n) ~(n) + P -
This results because there is only one density variable to go along with the single-position variable. For multicomponent systems such a representation of the solution to the van der Waals differential equations is not possible because there are more density variables. Even in the pure system it is simpler to (numerically) perform the two integrations in Eq. [II.9] to obtain ~: than it is to numerically integrate the above equation for every x. Journal of Colloid and Interface Science, Vol. 65, No. l, June 1, 1978
T
/
× s (aln'~re--_ - nol" I ~. ] ,
[II.22]
where n¢ is the critical density, T~ is the critical temperature, y is the compressibility exponent, and /3 is the coexistence curve exponent. With a change of integration variable [II.22] becomes O ( n ) - /zn + P 1 ~ 2tJj(1)wl = / 3 ( T o - T) ~,+2~/[--~-) --~- ~xX,,
[II.23]
where w(x) = 2
f~
dx'(x)2e-l(1
" Xt ) - x ' ) J( j(1)
and x = a l n - ncIX/a/(Tc -
T).
Then [II. 18] and [11.21] become
]1/2. Ixn
- nc)
yl a = 3/3z(nL -- n v ) 2 ( A / X ) v~
[11.24]
"/i = C/32(nL -- n v ) 2 ( A / X ) v2,
[11.25]
and respectively, where
'Io
C --- "~
c=2 _
dxxa-lwll2(x),
dxxa-lw(x
,
INTERFACIAL PHENOMENA
nL
L
n(x)
¥
and j(1)
X=
/3 (To -
T) y = 0"(nL) = tO"(nv).
FW assumed a j ( x ) of the form (1 - x ) j ( x ) -
%(1
-
x 4/a) [II.26]
j(1) along with/3 = 1/3. They therefore found w ( x ) = 3A(1 - x*/a)(2 + x2/3).
[II.27]
As FW showed 9(3 v2) In (1 + 31/2) --
-
-
8
/[
/33
dxx~-lw(x)
.
[II.33]
For the particular case [I1.27] this is
FIG. 2. Schematic density profile which allows for a deviation from monatonic behavior.
C
?:/32 = 2
2
51
~/L
= (42) v2 = 6.48 . . . .
Thus, the schematic profile method is one which is easily applied and yields a numerically accurate approximation for the interfacial tension. More importantly, the method allows the interfacial width to be characterized by direct calculation and as seen from [11.24] or [11.32] the corresponding surface tension has the correct scaling behavior in the critical region. This correct scaling behavior lends credence to the scaling expression, [II.15], for the surface tension. Because of the universality of fluid critical phenomena this expression may have validity outside the context from which it was obtained.
21/2 III. SURFACE ACTIVITY
= 1.283 . . . .
[II.28]
As discussed in the previous section, a schematic density profile, such as that depicted in Fig. 1, is useful in characteriz?: = 3/2(6/7)1/2 --- 1.389 . . . . [11.29] ing the interfacial tension and interfacial This constitutes an error of only 8% from width of the liquid-vapor interface in a the more correct expression. single-component system. Such a monotonic We recall that the tension in the FW profile will be less useful, however, in theory can be written in the alternative form systems where surface active materials are present. A modified profile, such as that Tl = c/32A/L(nL - nv) z, [11.30] shown in Fig. 2, should enable quantification where L is the equilibrium correlation of surface activity. The equation for such a profile is of the form length given by ?: is seen by direct integration to be
L = (AX) 1/2.
[II.31]
Similarly we have for the approximate expression yl a = (/32A/L(nL - nv) 2.
[II.32]
n ( x ) = nL,
X<
--~/2,
= ½(nL + nv) + (nv -- nL) +h(1
x
2[xl),
Comparison with [II.l 1] yields 3 3 The expression given in footnote 22 of I is in error. If the integrals in [D. 12] of I are properly evaluated, [II.33] will be obtained.
x~( ~2 ' 2 ~] ' = nv,
x > s¢/2.
[III.1]
Journal o f Colloid and Interface Science, V o l . 65, N o . 1, J u n e 1, 1978
52
PAUL D. FLEMING III
h is a measure of the amount of matter concentrated at the interface. We will see, however, that the value of h, determined by requiring that the Helmholtz free energy be a minimum, is always such that a Gibbs surface can be chosen so that there is no surface excess of matter in a one-component system. In fact, we will find that h is never large enough that a nonmonotonic profile results. The surface excess of matter, Fs, for profile [Ili.1] is given by
Fs =
f2
dx[n(x)
-
-
F A
nL]
#/2
+
~
-- (t0(nL) -- l~nL)Li
12
dx[n(x) - nv],
where D is the position of the dividing surface. The particular choice of the Gibbs surface is the one which makes Fs = 0. Using [III.1], we see, with a change of integration variable, that - ½(nL + nv) + h - nL(d + ½) - nv(½ - d) - 4h
I
+ (th(nv) - nv)L~ + ~f~,
[Ill.2]
.)D
Fs
In order to see that h does indeed satisfy [111.5], we consider the free energy [II.2]. As before, the equilibrium density profile in the van der Waals theory is the one which renders F - / z N a minimum. An approximate density profile is obtained by substituting [III.1] into [11.2] and requiring that F-/zN be a minimum with respect to variations of nL, nv, ~:, and h. Here we need only be concerned with variations with respect to h. For profile [III. 1] the free energy is of the form
[III.6]
where ,f/2
(fs =
dxr(x) a -~/2
- ~:[~b(nL)(d + ½)
+ qJ(nv)(½ - d)].
We see that
ll2
J-el2 dxq,(x) = - I = ,o
dss
ds q,
Jo =
½f'l
nv),
-- d(n L -
[III.3]
×
nL +
nv
2
where
+ s(nv
-
t/L)
+
n(l
--
2s))
d = D/~. Thus, the position of the Gibbs surface is given by d = ½n/(nL
--
nv).
[III.4]
This positioning is physical only when [d[ -< ½; i.e., the Gibbs surface occurs in the interfacial region. This requires that --(nL -- Ilv) --< h --< (ilL -- nv).
[III.5]
+ ½ A ( nL +2 nv + S ( n v - nL)+ h ( 1 - 2s)) X (nv--nL--2h)2]
ds[ 1/2
X
nv + nL ) ~ + S(nv-- nL) + h(1 + 2S) 2
+ ½A(nL~-4-2 n v
Thus if h satisfies [III.5] there will be no surface excess of matter provided d is given by [III.4].
I
X
+ S(nv
(nv--nL+
-- BL)+
2 h ) 2]
n(1 +
.
2s))
[III.7]
Thus we have
0f~ Oh
0/ Oh
1 qJ(nL) -- qJ(nv) 2 IlL - nv
Journal of Colloid and Interface Science, Vol. 65, No. 1, June 1, 1978
[III.8]
INTERFACIAL PHENOMENA
53
F r o m [II. 16] the second term in [Ill.8] becomes/z/2. W h e n [111.7] is differentiated, [Ill.8] becomes
Oh
Jo
ds (1 - 2s) ~b' ne + n v + s ( n v - nL) + h(1 - 2s 2
+ ½A'( nL+nv
+ s(nv - nL) + h(1 - 2s) )(nv
2
- nL - 2h)2J ~:
+ s ( n v - nL) + h(1 - 2s) (n L - nv + 2h)
2
+ f~
- ~b'(nv)
ds[(l + 2s)[q/( -nL -+ nv + s(nv - nL) + h(1 + 2 s ) ) - ~b'(nL) ~/2
2
+ ½a'( nL+nv
2
+ s(nv
-
nL) + h(1 + 2s)
+ 2 A ( nL - - ++s ( nnvv - n L ) 2
)(
nv
- nL + 2h)2J
~:
+h(1 + 2 s ) ) ( n v -
nL + 2h) ] .
[III.9]
We observe that
~b'( 'nL -~ + nv +
s(nv - nL) + h(1 - 2s)
fl ds' Iz
= (nv-
ds'
)
(k'(nv)
-
+nv
df -nL -+s'(nv--ne)+h(1-
2s'))l
2
nL-- 2h)
/2 ds'd/' -nL ' - 7+" -nv + s(nv -- he) + h(1 - 2s)
. [III.lO]
When the qJ' terms in the second integral in [III.9] are manipulated in a similar manner, we obtain Of~ _ (nv - nL -- 2h) ds (1 - 2s) Oh Jo
12
ds'tk" ( -nL -+ nv + s'(nv - nL) + h(1 - 2 s ' ) ) 2
+ V 2 A , ( n L + n v + s'(nv - nL) + h(1 - 2s') ) n v - nL-- 2hi 2 ~ -
2a(nL+ nv + s'(nv ~ 2
ds (1 + 2s) I/2
l
f
- ha) + h(1 - 2s')
)}
+ (nv - nL + 2fi)
ds'O" nL - -+ nv + s'(nv - nL) + h(1 + 2s')) 1/2
(
2
+ V2A,(.nL + nv + s'(nv - n~) + h(1 - 2s')) nv - nL + 2h \ 2 ~+s'(nv-nL)+h(1 2
+ 2s')
.
[III.11]
Journal of Colloid and Interface Science, Vo|. 65, No. I, June 1, 1978
54
PAUL D. FLEMING III
The equilibrium value of h is obtained by setting the right-hand side of [III.11] equal to zero. This yields a complicated nonlinear equation for h. We can, however, obtain bounds on h by examining the asymptotic behavior of [III.1 1]. As shown in III, $" diverges at both high and low densities because of hard-core repulsions at high densities and because t k " ~ kT/n (ideal gas limit) at low densities. Thus for large values of h the second integral will be expected to dominate [III.11] and we must have h
"~" ½ ( n L -
nv).
[III. 12]
Likewise for h negative and large in absolute value, the first integral will dominate and we must have h ~ ½(nv - nL).
[III.13]
For a curved interface we need to consider a schematic profile of the form (Fig. 3) n(r) = n2,
r >t R + ~/2,
nl + n2 + n ~ - nl ( r 2 rE
=nl,
R),
R--~,R
r~
+
,
[IV.l]
where nl denotes the internal phase density and n~ denotes the external phase density. It will be unnecessary to specify which phase is liquid and which is vapor. As we see below, R is essentially the position of the interface. For profile [IV.1] the total number of molecules is
Since these should correspond to asymptotic values, h should satisfy ½(nv - nL) -< h -< ½(nL - nv).
[III.14]
Not only does h satisfy an inequality more stringent than [III.5], but the actual inequality [III.14] assures that the density profile [III.1] will always be monotonic. Thus there will be no surface excess and the Gibbs surface will always exist. Thus we find, as should be expected, that a single-component system can have no surface excess and that the density profile is indeed monotonic. However, when an additional component is added, there will be two independent " h ' s " (one for each component) so that surface excess of one of the components may then be possible. The conditions for such an excess, within the context of the multicomponent theory of II, should enable a molecular interpretation of surface activity of surfactants to be obtained. IV.
CURVED
INTERFACES
Here we show that schematic profiles are also useful in discussing curved interfaces and associated questions of nucleation. Journalof CoUoidandInterfaceScience,Vol. 65, No. 1, June 1, 1978
+ iR+el2 dr [ nl + n2 JR--#I2 2
+ n2 - n__________(r 51 - R)]47rr 2. J
[IV.2]
The integral in [IV.2] can easily be evaluated and we find that N can be written as N = nlV1 + n2V2,
where V, = (4"rr/3)R z,
V2 = V -
(4~'/3)k ~,
with k = R(1 + (UZR)2) "~.
[IV.3]
Thus, if the actual position of the interface is taken as k there will be no surface excess at the interface. However, since discussion of interfacial phenomena at curved interfaces is meaningless unless ~ / R ,~ 1 we can take R as the position of the interface.
INTERFACIAL PHENOMENA
55
The discussion of Section II suggests that f~ should be expressed in the concise form Y6~b(n2
-
nl) 2 +
V2,~ nl
T
n2
where
n(r)
74 = nI > n2
n2
>n 1
[v2
J--l/2
dsA
( na + n~ 2
+ (n2 - nOs) [IV.5]
\N
/
nI < n2
/
and
\~__
n2 < n 1
3
11/2 J--l/2
Fro. 3. Schematic density profile [IV.l] for a spherical interface.The solid line denotes the schematic profile for the case in which the vapor is the internal phase (nl < n2), while for the dotted line the liquid is the internal phase (n2 < nl).
[-fI/2 11f2 d s l l as' as" LJs Js'
+
ds'
I/2
ds"
, nl + n2
1/2
2
+ s"(n2 - nz)) /
If [IV.l] is substituted into [II.1] we obtain
=
(
6
)2~112
'rll
n2
[(
ds qJ nl
J-1/2
+n2 -2
47r
+ qJ(n2)(V- 4 ~ ( R + - - ~ ) +
47rr2dr
In order to obtain the equilibrium values of nl, n2, ~:, and R we consider variations of F - IxN with respect to these variables. From [IV.3] and [IV.4] we see that
Io(n +n 2
J -~t2
+ ( n 2 - nl) r - R )
+ 1/2A( nl + n2 2
+ (n2_no_~R)(n~
+ 4~rR2~ Ofs [IV.6a] Onl
-~ n 1 )2]
= qJ(nl)V1 + qJ(n2)V~ + 4~rRZ~f~,
[IV.4]
where
fs =
O(f - IxN) - Ok'(nO - Ix)V~ Onl
and
O ( F - IxN)
r [0(- ds
J-l/2
nl q- n2 + (n2 2
-
hi)S)
tO(n1) + qJ(n2) 2 + ½ A (.nl + n2 \ 2
On2
- (tO'(nz) - Ix)V2 + 47rR2~:~
0/12
-4- ( n 2 - -
nOs)
. [IV.6b]
Requiring that these derivatives vanish yields ~b'(nl) = IX(1 - 3 ~ 1 0 f s ]
[IV.7a]
R tx On1] Journal of Colloid and Interface Science,
Vol.65,No. 1,June1, 1978
56
PAUL D. FLEMING III
and
where
0'(n2) = ft(1
y, = Cfs = A(n2 - nl)2/~.
- 3 ~:
R
(R/~) 3 0f~] /z(1 --- (-~-~)3) O n , } '
×
[IV.7b]
where =
(3V/47r)I/3
is the radius of the sphere having volume V. Thus for sufficiently large systems and ~/R ~ 1, n~ and n2 are densities of phases having chemical potential/~. We also see that O(F - I.tN) _ 47rR2{£
Of~]
[IV.8]
+~: o~:/
-
lzN) - -
02( F - ~ N ) -
~"(nl)V~ >i 0
[IV.14a]
On12
and
[IV.14b]
We also see that 02( F -
and since f~ is independent of R O(F
We see that the expression for the tension is identical in form to that obtained in the flat interface case. Thus near the critical point [II. 13] and [II. 15] should also be valid for spherical interfaces. Next we examine the stability of the equilibrium. For sufficiently large systems and ~ / R ~ 1 we see that
02(F - i~N) _ ~"(n2)V2 >i O. On22
o#
[IV.13]
tzN)
O~2
4 7 r R 2 ( e 2 -- P1) + 8~'Rsrf~,
OR
= 47rg 2 2 0~: + ¢ ' ~ - ~
" [IV.15]
[IV.9] where
From [IV. 10] we have
P1 = ftnl - t0(n0
and
P2
=
/-tn2
-- 0(n,)
are the pressures of the respective phases. From [IV.5] we see that 0fs _
,3, .(n2 - nO 2
0~
[IV.10]
~3
-- nl)2
=.
1/2.~(n' -- n l ) 2
02(F-
izN)
47rR 2 ~2 Yx ~> 0.
- - -
[IV.12]
As in Section II, [IV. 11] actually provides an equation for ¢', an equation identical in form to [II. 10]. From requiring that O(F - l z N ) / O R vanish we have the Y o u n g - L a p l a c e (20) equation P~ - P2 = 2 T I / R ,
Vol. 65, No. 1, June 1, 1978
[IV.17]
From [IV.9] we have OR z
- 8~rR(P2 - P1) + 81r~f~
= -87ry~ ~< 0.
f~ = ~ (n2 - n 0 ' _ V3Jf'(n, - nO z.
[IV.16]
~4
Hence we have
02( F - t~N)
[IV.11]
and therefore that
Journal of CoUoid and Interface Science,
0~2
0~2
Thus, from setting the right-hand side of [IV.8] equal to zero we have l/6~.f'(n2
02fs - 3,'~ (n2 - nO 2
[IV.18]
Thus, as is well known (24) but often ignored in elementary discussions (see, e.g., Ref. (25)), the mechanical equilibrium of the spherical interface is unstable with respect to the radius. The radius defined by [IV. 13] plays the role of a nucleation size. Internal phase fluctuations of radius less than R shrink, while those of larger radius grow. Finally, we examine the surface free
57
1NTERFACIAL PHENOMENA
energy Fs = 47rR2~:fs = 47rR2%.
[IV. 19]
We see that Fs/kT = 4~rR2y,/kT.
[IV.20]
If we assume [II.15] then this becomes F J k T = 4rcc(RZ/sez).
[IV.21]
This says that the surface energy is significant, i.e., FJkT>> 1, whenever the spherical interface is well defined. Conversely when the interfacial energy is comparable to thermal energy, i.e., when F J k T ~ 1, the interface is no longer well defined; i.e., its width is the order of its radius. V. DISCUSSION
We have seen that schematic density profiles are of great utility in the characterization of interfacial phenomena. The schematic density profile allows the interfacial width to be characterized in terms of molecular quantities. In the vicinity of the critical point of a one component system a very simple scaling expression for the surface tension was obtained from a schematic density profile and Widom scaling theory. This expression, inversely proportional to the square of the correlation length, suggests from dimensional considerations that interfacial tensions will be low whenever a large scaling length exists. Such an interpretation of low interfacial tensions may prove useful in understanding the lowering of the interfacial tension by surface active agents. The approximate expression for the surface tension obtained from the schematic density profile is in error by only 8% for the Fisk-Widom theory. More important than the detailed numerical agreement is the fact that the expression scales correctly near the critical point. Schematic density profiles also prove useful in discussing surface activity. We have seen, as expected, that no surface excess occurs in a one-component system
but the expected excess in multicomponent systems should be describable in terms of schematic profiles. Schematic density profiles should be of special utility in multicomponent systems where solutions to the differential equations may not be feasible. The resulting simplifications should facilitate the understanding of the microscopic basis for interfacial tension lowering by surfactants. We have also seen that spherical interfaces and nucleation can be readily treated in terms of schematic profiles. The instability of the curved interface equilibrium with respect to the radius emerges directly from the analysis. The interfacial energy was seen to exceed the thermal energy (kT) whenever the interfacial radius is larger than the interfacial width. The interface is not well defined because the interfacial width is of the order of the radius when the interfacial energy is about equal to the thermal energy. A generalization of the curved interfacial theory should provide a description of emulsion and foam stability. In summary, we have developed a method of approximate treatment of interfacial phenomena which retains the essential features of more rigorous treatments, while providing the simplifications necessary to discuss complex problems. The method should be especially applicable to discussions of interfacial phenomena in multicomponent systems containing surfactant species. A molecular interpretation of surface activity and emulsion coalescence should be obtainable from such discussions. REFERENCES 1. van der Waals, J. D., Z. Phys. Chem. 13, 657 (1894). 2. Cahn, J. W., and Hilliard, J. E., J. Chem. Phys. 28, 258 (1958). 3. Rice, 0. K., J. Phys. Chem. 64, 976 (1960). 4. Buff, F. P., Lovett, R. A., and Stillinger, F. H., Phys. Rev. Lett. 15, 621 (1965), Lovett, R. A., Ph.D. Thesis, University of Rochester, 1965; Vieceli, J. J., Ph.D. Thesis, University of Rochester, 1971; Lovett, R. A., DeHaven, P. W., Vieceli, J. J., and Buff, F. P., J. Chem. Phys. 50, 1880 (1973). Journal of Colloidand Interface Science, Vol. 65, No. 1. June 1, 1978
58
PAUL D. FLEMING III
5. Widom, B., J. Chem Phys. 43, 3892 (1965); and in "Phase Transitions and Critical Phenomena," (C. Domb and M. S. Green, Eds.), Academic Press, New York, 1972; Fisk, S., and Widom, B., Vol. 2, Chap. 3. J. Chem. Phys. 50, 3219 (1969). 6. Felderhof, B. U., Physica 48, 541 (1970). 7. Toxvaerd, S., J. Chem. Phys. 55, 3116 (1971); 62, 1589 (1975); 64, 2863 (1976); and in "Statistical Mechanics" (K. Singer, Ed.), Vol. 2. The Chemical Society of London, London, 1975. 8. Triezenberg, D. G., and Zwanzig, R., Phys. Rev. Lett. 28, 1183 (1972). 9. Lee, J. K., Barker, J. A., and Pound, G. M., J. Chem. Phys. 60, 1976 (1974). 10. Woodbury, G. W.,J. Chem. Phys. 60, 3674 (1974). 11. Gray, C. G., and Gubbins, K. E., Mol. Phys. 30, 179 (1975); Hally, J. M., Gray, C. G., and Gubbins, K. E., J. Chem. Phys. 64, 1852, 2569 (1976). 12. Davis, H. T., J. Chem. Phys. 62, 3412 (1975), Salter, S. J., and Davis, H. T., J. Chem. Phys. 63, 3295 (1975); Bongiorno, V., and Davis, H. T., Phys. Rev. A 12, 2213 (1975).
Journal of Colloidand Interface Science, Vol.65, No. l, June 1, 1978
13. Sarkies, K. W., and Frankel, N. E., Phys. Rev. A 11, 1724 (1975). 14. Abraham, F. F., J. Chem. Phys. 63, 157, 1316 (1975). 15. Mulholland, G. W.,J. Chem. Phys. 64, 862 (1976). 16. Yang, A. J. M., Fleming, P. D., and Gibbs, J. H., J. Chem. Phys. 64, 3732 (1976). 17. Fleming, P. D., Yang, A. J. M., and Gibbs, J. H., J. Chem. Phys. 65, 7 (1976). 18. Yang, A. J. M., Fleming, P. D., and Gibbs, J. H., J. Chem. Phys. 67, 74 (1976). 19. Wertheim, M. S.,J. Chem. Phys. 65, 2377 (1976). 20. Gibbs, J. W., "Scientific Papers," Vol. 1, p. 229, Eq. 500. Dover, New York, 1961. 21. Widom, B., J. Chem. Phys. 43, 3892 (1965); Kadanoff, L. P., Physics 2, 263 (1966). 22. (a) Wilson, K. G., Phys. Rev. B 4, 3174, 3184 (1971). (b) Wilson, K. G., and Kogut, J., Phys. Rep. C 12, 76 (1974). 23. Miller, C. A., Hwan, R., Benton, W. J., and Fort, T., J. Colloid lnterface Sci. 61,554 (1977). 24. Reiss, H., "Methods of Thermodynamics," pp. 202-203. Blaisdell, New York, 1965. 25. Moore, W. J., "Physical Chemistry." PrenticeHall, Englewood Cliffs, N. J.