J. Chem. Phys. 112, 6023 (2000); https://doi.org/10.1063/1.481175 112, 6023 © 2000 American Institute of Physics.
Correspondence between the pressure
expressions and van der Waals theory for a
curved surface
Cite as: J. Chem. Phys. 112, 6023 (2000); https://doi.org/10.1063/1.481175
Submitted: 01 September 1999 . Accepted: 05 January 2000 . Published Online: 17 March 2000 Edgar M. Blokhuis, H. N. W. Lekkerkerker, and Igal Szleifer
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Correspondence between the pressure expressions and van der Waals
theory for a curved surface
Edgar M. Blokhuisa)
Colloid and Interface Science, Leiden Institute of Chemistry, Gorlaeus Laboratories, P.O. Box 9502, 2300 RA Leiden, The Netherlands
H. N. W. Lekkerkerker
Department for Physical and Colloid Chemistry, Debye Research Institute, H. R. Kruytgebouw, P.O. Box 80000, 3508 TA Utrecht, The Netherlands
Igal Szleifer
Department of Chemistry, Purdue University, West Lafayette, Indiana 47907-1393 共Received 1 September 1999; accepted 5 January 2000兲
We investigate the apparent contradiction between the pressure expressions, or ‘‘mechanical expressions,’’ and the van der Waals squared-gradient expressions for the curvature coefficients
k/R0, k, and k¯ . We show that, in the context of the mean-field theory discussed, both types of
expression are indeed equivalent, with the differences only being caused by the thermodynamic conditions used to vary the curvature. © 2000 American Institute of Physics.
关S0021-9606共00兲50713-2兴
I. INTRODUCTION
The introduction of the Helfrich expression for the cur-vature free energy1has led to an important advancement in the theoretical understanding of complex interfaces.2,3 In terms of two elasticity or rigidity constants, k and k¯ , as well as the radius of spontaneous curvature R0, the Helfrich free energy has been used to describe the shape, fluctuations, and free energy of membranes, vesicles, microemulsions, etc.2It has the following form:
FH⫽
冕
dA冋
⫺ 2k R0 J⫹k 2J 2⫹k¯K册
, 共1.1兲where is the surface tension of the planar surface, J
⫽1/R1⫹1/R2 is the total curvature, K⫽1/(R1R2) is the
Gaussian curvature, and R1and R2 are the principal radii of
curvature at a certain point on the surface A.
The Helfrich expression is, however, phenomenological in nature: no information is provided on the value of the parameters , k/R0, k, and k¯ and a lot of theoretical work
has therefore been devoted to the determination of these pa-rameters in the context of a more microscopic theory.4–8We should mention that Eq.共1.1兲 can be viewed in two, equiva-lent, ways. In the approach by Romero-Rochı´n, Varea and Robledo,7 Eq. 共1.1兲 is the expression for an arbitrarily shaped surface with curvature dependent coefficients ,
k/R0, k, and k¯ . In the approach considered here, Eq.共1.1兲 is
viewed as an expansion in curvature with the coefficients,
k/R0, k, and k¯ curvature independent.
Inspection of the form of the Helfrich free energy in Eq.
共1.1兲 shows that the theoretical determination of k/R0, k,
and k¯ requires the variation of the free energy with
curva-ture. The coefficients are then related to the first and second
derivatives of the free energy with respect to the curvature. The way one varies the curvature depends very much on the system at hand. In the following we discuss three types of interfaces; those made of surfactant monolayers or bilayers, the solid–liquid interface, and the liquid–vapor interface.
We first discuss the case in which the surface is made of surfactant 共or lipid兲 bilayers or monolayers. This is the sys-tem for which Helfrich originally wrote down his free en-ergy, i.e., for the interface between two immiscible fluids where the specific molecules at the interface 共e.g., the sur-factant or lipid molecules兲 are the ones responsible for the strength of the rigidity constants. In this case one may change, for instance, the amount (⌫surf) or chemical potential
(surf) of the component that is predominantly adsorbed at the interface in order to vary the curvature. In fact, Porte and Ligoure9 investigated the influence of either changing the chemical potential or the composition on the value of the rigidity constants for these systems.
The result of changing only the properties of the surfac-tant molecules at the interface is that the thermodynamic state of the bulk regions away from the surface is unaffected. For such a system Helfrich supplied formulas for the calcu-lation of the curvature coefficients. Using ‘‘mechanical’’ ar-guments he derived expressions for k/R0 and k¯ in terms of
moments of the transverse pressure profile,4 ⌸0(z), of the
planar surface, analogous to Buff’s10‘‘mechanical’’ expres-sion for the surface tenexpres-sion, , as the zeroth moment of the transverse pressure profile 关see Eq. 共1.2兲 below兴. He added, however, that the expression for k/R0 is only valid for a
tensionless interface (⫽0) and the expression for k¯ is only
valid for a tensionless, symmetric interface (⫽0, k/R0
⫽0). Later, Szleifer and coworkers5extended the analysis of
Helfrich, in the context of mean-field theory, to go beyond the mechanical arguments by Helfrich and showed that
Hel-a兲Electronic mail: e.blokhuis@chem.Leidenuniv.nl
6023
frich’s expressions are correct also when ⫽0 and k/R0
⫽0. Furthermore, Szleifer and coworkers provided an
ex-pression for the rigidity constant k. The exex-pressions by Helfrich4and Szleifer et al.5read
⫽⫺
冕
⫺⬁ ⬁ dz⌸0共z兲, 2k R0⫽冕
⫺⬁ ⬁ dz z⌸0共z兲, 共1.2兲 k ¯⫽⫺冕
⫺⬁ ⬁ dz z2⌸ 0共z兲, k⫽⫺ 1 2冕
⫺⬁ ⬁ dz z⌸s,1共z兲,where z is the coordinate normal to the interface. Since these expressions are derived using arguments beyond the me-chanical arguments by Helfrich, we will refer to these ex-pressions as the pressure exex-pressions11 共rather than the ‘‘mechanical expressions’’兲. The expression for k features
⌸s,1(z), the first order coefficient of the lateral pressure pro-file of a spherical interface, ⌸s(r;R), in an expansion in 1/R, the reciprocal radius,
⌸s共r;R兲⫽⌸0共z兲⫹
1
R⌸s,1共z兲⫹¯ , 共1.3兲
where the radial distance r⬅R⫹z. ⌸s,1is the leading order change in the lateral pressure profile due to bending of the interface. Here it is expressed in terms of the lateral pressure profile of the spherical interface, but we should keep in mind that we could equally well have expressed it in terms of the lateral pressure profile of a cylindrical interface replacing
⌸s,1by 2⌸c,1, or in general replace⌸s,1by 2(⌸/J). In the following the subscript s refers to the spherical surface and the subscript c to the cylindrical surface. The additional number to s and c 共e.g., s,1兲 refers to the coefficient in an expansion in 1/R to that order.
A second class of systems, for which the Helfrich free energy has been used, are those in contact with a solid curved wall. Several authors12–15have calculated the electro-static contribution to the curvature coefficients using double layer theory for a charged solid wall,16while recently Clem-ent and Joanny17 calculated the curvature energy associated with polymer adsorption onto a curved substrate. In these systems the curvature of the interface is varied simply by changing the radius of the solid wall. As in the case de-scribed above, the thermodynamic state of the system away from the surface is unaffected by the variation of the curva-ture of the interface. The result is that the curvacurva-ture coeffi-cients can be calculated using the pressure expressions in Eq.
共1.2兲 with the only difference that the integration over z runs
from the hard wall共at z⫽0兲 to infinity.
A third route to the calculation of the curvature coeffi-cients has used van der Waals’ squared-gradient expression for the surface free energy8,18,19 of a simple liquid–vapor interface,
F关兴⫽
冕
drជ关m兩ⵜជ 共rជ兲兩2⫹ f共兲⫺⌬共rជ兲兴, 共1.4兲where m is the usual coefficient of the squared-gradient term,
f () is the free energy density for a fluid constrained to have uniform density , and⌬ is the chemical potential differ-ence between the chemical potential of the curved surface and that of the planar surface (⌬⫽⫺coex). Using the
above expression for the free energy, Gompper and Zschocke8 and Blokhuis and Bedeaux19 derived the follow-ing formulas for the curvature coefficients:
⫽2m
冕
⫺⬁ ⬁ dz关0⬘
兴2, 2k R0⫽⫺2m冕
⫺⬁ ⬁ dz z关0⬘
兴2 ⫹1 2⌬s,1冕
⫺⬁ ⬁ dz关0⫺0,bulk兴, 共1.5兲 k ¯⫽2m冕
⫺⬁ ⬁ dz z2关0⬘
兴2 ⫹共4 ⌬c,2⫺⌬s,2兲冕
⫺⬁ ⬁ dz关0⫺0,bulk兴, k⫽⫺m冕
⫺⬁ ⬁ dz0⬘
s,1⫺ 1 4⌬s,1冕
⫺⬁ ⬁ dz关s,1⫺s,1,bulk兴 ⫺⌬s,1冕
⫺⬁ ⬁ dzz关0⫺0,bulk兴 ⫺2 ⌬c,2冕
⫺⬁ ⬁ dz关0⫺0,bulk兴.In the above formulas is0⫽0(z) the density profile of the
planar interface and s,1(z) the first order correction to the density profile of the spherical interface in an expansion in 1/R. The subscript ‘‘bulk’’ refers to the bulk value extrapo-lated to the surface at z⫽0 so that for example 0,bulk
⫽0,l(⫺z)⫹0,v(z).
Comparing the expressions in Eqs.共1.2兲 and 共1.5兲 it is not obvious that the pressure expressions and the expressions derived from van der Waals theory are in agreement. Yet the validity of both expressions seems well established. The van der Waals expressions in Eq. 共1.5兲 were derived indepen-dently by Gompper and Zschocke8 and by Blokhuis and Bedeaux.19 Furthermore the result for k/R0 agrees with an
earlier expression by Fisher and Wortis.18Also, these expres-sions can be derived from the virial expresexpres-sions for the cur-vature coefficients6by making a mean-field approximation to the pair density.19,20
On the other hand, it is well established by various authors12–15that the expressions for the electrostatic contri-bution to the curvature coefficients in double layer theory calculated directly via the free energy are the same as when the pressure expressions are used. This was shown by Win-terhalter and Helfrich12 using the Debye–Hu¨ckel theory, by Lekkerkerker13 and by Mitchell and Ninham14 using Poisson–Boltzmann theory in excess salt, and by Fogden, Daicic, and coworkers15 using Poisson–Boltzmann theory with an arbitrary amount of added salt. Our object in this article is to establish in more detail the correspondence be-tween the expressions in Eqs. 共1.2兲 and 共1.5兲. It will be shown that, keeping the thermodynamic conditions under which the surface is bend in mind, that Eqs.共1.2兲 and 共1.5兲 are indeed equivalent.
When we compare Eqs.共1.2兲 and 共1.5兲 some similarities are apparent. Suppose that we set ⌬s⫽⌬c⫽0 so that
⌬s,1⫽⌬c,1⫽0 and ⌬s,2⫽⌬c,2⫽0, and, furthermore, identify ⌸0(z) as⌸0(z)⫽⫺2m(0
⬘
)2, then already we see
that the pressure expressions for , k/R0, and k¯ reduce
ex-actly to the squared-gradient expressions. The correspon-dence between the two expressions for k is less clear but still, this is very encouraging and we now first wonder whether we can understand why there is no contribution from the change in chemical potential in the pressure expres-sions. In order to do this we first need to consider the ther-modynamic circumstances.
II. THERMODYNAMIC CONDITIONS
In the van der Waals squared-gradient theory as used by Gompper and Zschocke,8Fisher and Wortis,18and Blokhuis and Bedeaux,19a spherical共cylindrical兲 drop of liquid is con-sidered surrounded by a vapor phase. The two phases are not in coexistence when the radius R⬍⬁, and the distance to two-phase coexistence is determined by the chemical poten-tial difference ⌬s⬅s⫺coex 共or ⌬c for the cylindrical interface兲. The droplet radius is determined by ⌬s and when we consider the variation of the free energy with the radius we really are considering the variation of the free energy with the chemical potential. The Laplace equation enables us to relate directly the expansion coefficients of the chemical potential (⌬s,1,⌬s,2,...) to the radius. In order to show this a bit more explicitly, we need to consider the Laplace equation for a spherical and cylindrical interface:
⌬p⫽2R⫺4kR 0 1 R2, spherical interface, ⌬p⫽R⫺2k R13, cylindrical interface, 共2.1兲
which are both derived from the generalized Laplace equa-tion, ⌬p⫽J⫺4k R0 K⫺k 2J 3⫹2kJK⫺k⌬ sJ. 共2.2兲 In this equation ⌬p⬅pl⫺pv is the pressure difference
be-tween the liquid inside the droplet 共cylinder兲 and the vapor outside, and ⌬s is the surface Laplacian which is important when the curvature varies from point to point on the surface. For the spherical and cylindrical interface we consider in this analysis, the curvature is constant along the surface so that
⌬sJ⫽0.
Since the pressure difference is directly related to the chemical potential, one can show that19
⌬s,1⫽2 ⌬c,1⫽ 2 ⌬0 , ⌬s,2⫽⫺ 共⌬0兲2 ⌬s,1⫺ 4k R0 1 ⌬0 , 共2.3兲 ⌬c,2⫽⫺ 1 4 共⌬0兲2 ⌬s,1,
with ⌬⫽l⫺v the density difference between the liquid
and the vapor phase.
To summarize, in the derivation of the van der Waals expressions the radius R is varied by varying the chemical potential. The variation of the chemical potential leads to the presence of additional terms in the squared-gradient expres-sions in Eq.共1.5兲 with the explicit value of the coefficients of these terms given in Eq. 共2.3兲.
Next, we now consider the thermodynamic conditions that are used in the calculation of the curvature coefficients of a fluid in contact with a hard wall. Here the situation is somewhat simpler. A rigid sphere共colloidal particle兲 is con-sidered with a certain fixed radius R. The variation of the radius does not influence the thermodynamic state of the fluid outside with fixed density l and pressure pl. Terms connected with the expansion of the chemical potential in the radius are therefore not present so that as a consequence,
⌬s,1⫽⌬c,1⫽0,
共2.4兲 ⌬s,2⫽⌬c,2⫽0.
To show the correspondence between the van der Waals ex-pressions and pressure exex-pressions in more detail, we calcu-late the curvature coefficients with the only assumption that the free energy density is some function of the density (rជ) and the gradient of the density ⵜជ (rជ) 共to keep the calcula-tion as general as possible兲. This is done under the conditon that the chemical potential is varied to vary the curvature
共Sec. III兲 and under the condition that the chemical potential
is constant共Sec. IV兲.
III. VARIABLE CHEMICAL POTENTIAL
The surface free energy is written in the following gen-eral way:
F关兴⫽
冕
drជ关⫺⌸共,ⵜជ 兲⫺⌬ 共rជ兲兴, 共3.1兲 where it is supposed that the free energy in the bulk region is subtracted so that the above free energy is an excess free energy. Furthermore ⌸(,ⵜជ ) is the 共grand兲 free energy density and is some function of(rជ) andⵜជ (rជ). In van der Waals’ squared-gradient theory, for instance, it has the fol-lowing form关cf. Eq. 共1.4兲兴:⌸共,ⵜជ 兲⫽⫺m兩ⵜជ 共rជ兲兩2⫺ f共兲⫹coex共rជ兲, 共3.2兲
but we leave it unspecified in the remainder of this section. Several authors7,8,19 have included besides a squared-gradient term a squared Laplacian term to the above free energy. The inclusion of such a term in Eq.共3.1兲 leads to the presence of additional terms in the Euler–Lagrange equa-tions below with the consequence that the identification in Eqs. 共3.8兲 and 共3.10兲 below needs to be modified. In the following it should therefore be realized that the conclusions drawn only apply to mean-field theories whose free energy is of the form of Eq.共3.1兲.20,21
⌸共,ⵜជ 兲⫽ⵜជ•
ⵜជ ⌸共,ⵜជ 兲⫺⌬. 共3.3兲
In the following, we expand the surface free energy around the planar interface and compare the results with the surface free energy expression by Helfrich 关Eq. 共1.1兲兴 which for a spherical and cylindrical geometry read
Fs A ⫽⫺ 4k R0 1 R⫹共2k⫹k¯兲 1 R2, 共3.4兲 Fc A ⫽⫺ 2k R0 1 R⫹ k 2 1 R2.
Expanding the surface free energy for the spherical and cy-lindrical interface to second order in 1/R gives
Fs A ⫽
冕
⫺⬁ ⬁ dz冉
1⫹z R冊
2冋
⫺⌸0⫺ 1 R⌸1s,1⫺ 1 R⌸2s,1⬘
⫺ 1 R2⌸1s,2⫺ 1 R2⌸2s,2⬘
⫺ 1 2R2⌸11s,1 2 ⫺R12⌸12s,1s,1⬘
⫺ 1 2R2⌸22共s,1⬘
兲 2⫺1 R⌬s,10 ⫺R12⌬s,1s,1⫺ 1 R2⌬s,20⫺B.T.册
, 共3.5兲 Fc A ⫽冕
⫺⬁ ⬁ dz冉
1⫹ z R冊冋
⫺⌸0⫺ 1 R⌸1c,1⫺ 1 R⌸2c,1⬘
⫺R12⌸1c,2⫺ 1 R2⌸2c,2⬘
⫺ 1 2R2⌸11c,1 2 ⫺ 1 R2⌸12c,1c,1⬘
⫺ 1 2R2⌸22共c,1⬘
兲 2⫺1 R⌬c,10 ⫺R12⌬c,1c,1⫺ 1 R2⌬c,20⫺B.T.册
,where the subscripts 1 and 2 to ⌸ refer to a differentiation with respect to the first or second argument evaluated at the planar interface, e.g.,⌸1⬅ (/0)⌸(0,0
⬘
). ⌸0 is simply defined as⌸0⬅⌸(0,0⬘
). The abbreviation B.T. stands forthe boundary terms at z⫽⫾⬁ which are to be subtracted. The Euler–Lagrange equation 共3.3兲 is expanded in the curvature for the spherical and cylindrical interface. To first order the following equations result:
⌸1⫽ z⌸2, ⌸1s,1⫺⌸2s,1
⬘
⫽2⌸2⫹ z共⌸12s,1⫺⌸22s,1⬘
兲⫺⌬s,1, 共3.6兲 ⌸1c,1⫺⌸2c,1⬘
⫽⌸2⫹ z共⌸12c,1⫺⌸22c,1⬘
兲⫺⌬c,1.The first equation determines the profile0(z) of the planar
interface while the latter two determine the profiles s,1(z) and c,1(z). One immediately notices that since ⌬s,1
⫽2 ⌬c,1one has thats,1(z)⫽2c,1(z).
Inserting the above expressions for⌸1 into the surface
free energy and integrating by parts gives, after some alge-bra, the following results for the surface free energy of the sphere and the cylinder:
Fs A ⫽
冕
⫺⬁ ⬁ dz关⫺⌸0兴 ⫹1R冕
⫺⬁ ⬁ dz关⫺2z ⌸0⫺⌬s,1共0⫺0,bulk兲兴 ⫹ 1 R2冕
⫺⬁ ⬁ dz冋
⫺z2⌸0⫹⌸2s,1⫺2 ⌬s,1z共0⫺0,bulk兲 ⫺12⌬s,1共s,1⫺s,1,bulk兲⫺⌬s,2共0⫺0,bulk兲册
, 共3.7兲 Fc A ⫽冕
⫺⬁ ⬁ dz关⫺⌸0兴 ⫹1 R冕
⫺⬁ ⬁ dz关⫺z ⌸0⫺⌬c,1共0⫺0,bulk兲兴 ⫹R12冕
⫺⬁ ⬁ dz冋
12⌸2c,1⫺⌬c,1z共0⫺0,bulk兲 ⫺12⌬c,1共c,1⫺c,1,bulk兲⫺⌬c,2共0⫺0,bulk兲册
.Comparing Eqs. 共3.7兲 and 共3.4兲 allows us to identify the curvature coefficients as ⫽⫺
冕
⫺⬁ ⬁ dz⌸0, 2k R0 ⫽冕
⫺⬁ ⬁ dz冋
z⌸0⫹1 2⌬s,1共0⫺0,bulk兲册
, 共3.8兲 k ¯⫽冕
⫺⬁ ⬁ dz关⫺z2⌸0⫹共4 ⌬c,2⫺⌬s,2兲 共0⫺0,bulk兲兴, k⫽冕
⫺⬁ ⬁ dz冋
1 2⌸2s,1⫺⌬s,1z共0⫺0,bulk兲 ⫺14⌬s,1共s,1⫺s,1,bulk兲⫺2 ⌬c,2共0⫺0,bulk兲册
,where we have used that ⌬s,1⫽2 ⌬c,1 and s,1⫽2c,1. As a final step we rewrite the expression for k somewhat by using that
冕
⫺⬁ ⬁ dz z⌸s,1⫽冕
⫺⬁ ⬁ dz关z⌸s,1s,1⫹z⌸2s,1⬘
兴 ⫽冕
⫺⬁ ⬁ dz关⫺⌸2s,1兴, 共3.9兲 so thatk⫽
冕
⫺⬁ ⬁ dz冋
⫺z 2⌸s,1⫺⌬s,1z共0⫺0,bulk兲 ⫺1 4⌬s,1共s,1⫺s,1,bulk兲⫺2 ⌬c,2共0⫺0,bulk兲册
. 共3.10兲We have thus derived in Eqs. 共3.8兲 and 共3.10兲 the pressure expressions for the curvature coefficients in the case that the chemical potential is varied to change the curvature of the interface. The pressure ⌸ was identified as the 共grand兲 free energy density defined by Eq. 共3.1兲. One may verify that these pressure expressions are equivalent to the van der Waals squared-gradient expressions in Eq. 共1.5兲 when one inserts the explicit expressions for ⌸0 and⌸s,1,
⌸0⫽⫺m 共0
⬘
兲 2⫺ f共 0兲⫹coex0⫽⫺2m 共0⬘
兲 2, 共3.11兲 ⌸s,1⫽⫺2m0⬘
s,1⬘
⫺ f⬘
共0兲s,1⫹coexs,1 ⫽⫺2m0⬘
s,1⬘
⫺2m0⬙
s,1,derived using the form for⌸(,ⵜជ ) in Eq.共3.2兲.
An important issue that we have not addressed thus far is the fact that a certain arbitrariness exists in locating the exact position of the dividing surface and therefore in the exact value of the radius R.22 The consequences hereof for the curvature coefficients can be read off from the expressions in Eq. 共3.8兲 which are derived without specifying the location of the dividing surface. One finds that the surface tension of the planar interfacedoes not depend on the location of the dividing surface. Also, the spontaneous curvature k/R0 is
independent of the location of the dividing surface. This can
be checked by shifting the dividing surface over a distance
⌬. The first contribution to k/R0 is then changed by an
amount ⌬, while the second contribution changes by an amount⫺1/2(⌬0)⌬. Use of Eq. 共2.3兲 then yields that the
net change in k/R0 of shifting the dividing surface is zero.
The rigidity constants, however, do depend on the location of the dividing surface, and when numerical values are supplied for k and k¯ this can be done only after a certain choice for the location of the dividing surface has been made. For in-stance, in the derivation of the pressure equations for mono-layers and bimono-layers by Szleifer et al.,5 the location of the dividing surface was chosen to be the ‘‘surface of inexten-sion’’ or ‘‘neutral surface.’’ This is the surface whose area is unchanged during the variation of the curvature. We refer to Ref. 5 for a more elaborate discussion of this point共see also Ref. 3兲. For the evaluation of the curvature coefficients in the van der Waals theory for a liquid–vapor droplet, the dividing surface was located at Gibb’s equimolar surface.19 In the next section we investigate the case of a fluid in contact with a hard spherical 共cylindrical兲 wall. In that case the dividing surface is chosen at the hard wall.
It is noteworthy that Eqs.共3.8兲 and 共3.10兲 reduce exactly to the pressure expressions in Eq. 共1.2兲 when one sets
⌬s,1⫽⌬c,1⫽0 and ⌬s,2⫽⌬c,2⫽0. Therefore, we next look in more detail into the situation where the chemical potential is kept constant.
IV. CONSTANT CHEMICAL POTENTIAL
We now investigate the situation in which the chemical potential is fixed (⌬⫽0) and the radius of curvature is varied independent of the thermodynamic state. Therefore we consider the surface free energy of a fluid in contact with a hard spherical 共cylindrical兲 wall with fixed radius R,
F关兴⫽⫺
冕
drជ关⌸共,ⵜជ 兲 兴⫹A⌽w共w兲, 共4.1兲 where⌽w(w) is the interaction of the fluid with the wall atz⫽0 and is assumed to depend only on the density at the
wall,w. The form of the above free energy is quite general and in the Appendix we give two examples where the free energy indeed has this form.
The Euler–Lagrange equation to the surface free energy in Eq.共4.1兲 reads
⌸共,ⵜជ 兲⫽ⵜជ•
ⵜជ ⌸共,ⵜជ 兲, 共4.2兲
with the boundary condition at the wall,
w⌽w共
w兲⫽nˆ•
ⵜជ w⌸共
w,ⵜជ w兲. 共4.3兲
In this expression nˆ is the unit vector in the direction normal to the interface. Expanding the surface free energy for the spherical and cylindrical interface to second order in 1/R now gives Fs A ⫽
冕
0 ⬁ dz冉
1⫹ z R冊
2冋
⫺⌸0⫺ 1 R⌸1s,1⫺ 1 R⌸2s,1⬘
⫺R12⌸1s,2⫺ 1 R2⌸2s,2⬘
⫺ 1 2R2⌸11s,1 2 ⫺R12⌸12s,1s,1⬘
⫺ 1 2R2⌸22共s,1⬘
兲 2⫺B.T.册
, ⫹⌽w,0⫹ 1 R⌽w,0⬘
s,1 w⫹ 1 R2⌽w,0⬘
s,2 w ⫹ 1 2R2⌽w,0⬙
共s,1 w 兲2, 共4.4兲 Fc A ⫽冕
0 ⬁ dz冉
1⫹z R冊冋
⫺⌸0⫺ 1 R⌸1c,1⫺ 1 R⌸2c,1⬘
⫺R12⌸1c,2⫺ 1 R2⌸2c,2⬘
⫺ 1 2R2⌸11c,1 2 ⫺ 1 R2⌸12c,1c,1⬘
⫺ 1 2R2⌸22共c,1⬘
兲 2⫺B.T.册
⫹⌽w,0⫹ 1 R⌽w,0⬘
c,1 w ⫹ 1 R2⌽w,0⬘
c,2 w ⫹ 1 2R2⌽w,0⬙
共c,1 w 兲2,where we have defined⌽w,0⬅⌽w(0
w
⌸1⫽ z⌸2, ⌸1s,1⫺⌸2s,1
⬘
⫽2⌸2⫹ z共⌸12s,1⫺⌸22s,1⬘
兲, 共4.5兲 ⌸1c,1⫺⌸2c,1⬘
⫽⌸2⫹ z共⌸12c,1⫺⌸22c,1⬘
兲,and so is the boundary condition in Eq.共4.3兲,
⫺⌸2 w⫽⌽ w,0
⬘
, ⌸12 w s,1 w ⫺⌸ 22 w 共 s,1 w 兲⬘
⫽⌽ w,0⬙
s,1 w , 共4.6兲 ⌸12 w c,1 w ⫺⌸ 22 w 共 c,1 w 兲⬘
⫽⌽ w,0⬙
c,1 w .Again it is noted thats,1(z)⫽2c,1(z). Using Eqs.共4.5兲 and
共4.6兲, the surface free energy of the sphere and cylinder can
now be written as Fs A ⫽
冕
0 ⬁ dz关⫺⌸0兴⫹⌽w,0⫹1 R冕
0 ⬁ dz关⫺2z ⌸0兴 ⫹ 1 R2冕
0 ⬁ dz关⫺z2⌸0⫹⌸2s,1兴, 共4.7兲 Fc A ⫽冕
0 ⬁ dz关⫺⌸0兴⫹⌽w,0⫹ 1 R冕
0 ⬁ dz关⫺z ⌸0兴 ⫹ 1 R2冕
0 ⬁ dz冋
1 2⌸2c,1册
,so that one may identify the curvature coefficients as
⫽⫺
冕
0 ⬁ dz关⌸0兴⫹⌽w,0, 2k R0 ⫽冕
0 ⬁ dz关z⌸0兴, 共4.8兲 k ¯⫽冕
0 ⬁ dz关⫺z2⌸0兴, k⫽冕
0 ⬁ dz冋
1 2⌸2s,1册
.Notice that all the terms involving the interaction with sub-strate,⌽w, have dropped out of the expressions for k/R0, k,
and k¯ . As a final step we, again, rewrite the expression for k somewhat by using that
冕
0 ⬁ dz z⌸s,1⫽冕
0 ⬁ dz关z⌸1s,1⫹z⌸2s,1⬘
兴 ⫽冕
0 ⬁ dz关⫺⌸2s,1兴, 共4.9兲 so that k⫽冕
0 ⬁ dz冋
⫺z 2⌸s,1册
. 共4.10兲Apart from the presence of ⌽w,0 in the expression for the surface tension and the fact that the integration runs from z
⫽0 instead of z⫽⫺⬁, the expressions in Eqs. 共4.8兲 and 共4.10兲 are exactly equal to the pressure expressions in Eq. 共1.2兲. Starting with the general expression for the surface
free energy in Eq.共4.1兲 we have thus rederived all the pres-sure expressions for the system in which the chemical poten-tial is fixed.
V. SUMMARY
We have investigated the apparent contradiction between the pressure expressions and van der Waals expressions for the curvature coefficients k/R0, k, and k¯ . In the context of
the mean-field theory expressed by the similar Eqs.共3.1兲 and
共4.1兲, we showed that the origin of the difference between
the two types of expression lies solely in the thermodynamic conditions used to vary the curvature. As we have seen, the appropriate thermodynamic conditions depend very much on the system at hand. To study, for instance, the nucleation of liquid droplets, the curvature depends on the thermodynamic distance to coexistence (⌬⫽⫺coex) via the Laplace
equation, and the analysis in Sec. III is the appropriate one, while for the description of the electric double layer of a colloidal particle with fixed radius or the description of the adsorption of a polymer onto a curved wall, the analysis in Sec. IV is more suited. The latter analysis is also closely related to the investigation of microemulsion systems and systems containing membrane bilayers. In these cases one may, for instance, change the chemical potential of the com-ponent that is predominantly adsorbed at the interface共e.g., the surfactant or lipid molecules兲 in order to change the cur-vature. As in the case of a system in contact with a curved wall, the thermodynamic state of the system away from the surface23,24is unaffected.
With these two thermodynamic conditions, expressions for the curvature coefficients were derived. In Sec. III the curvature was varied by varying the chemical potential, while in Sec. IV the chemical potential is kept constant and the curvature is varied by varying the radius of the spherical or cylindrical substrate that is in contact with the system. The resulting expressions from the former analysis were shown to reduce to those obtained in van der Waals theory, while the results from the latter analysis were shown to be equal to the pressure expressions identifying the lateral pressure pro-file as the excess共grand兲 free energy density.
ACKNOWLEDGMENTS
E.M.B. would like to thank Theo Odijk for discussions on the electrostatic double layer theory, Tinus Oversteegen for discussions concerning the derivation of the pressure ex-pressions, and Dick Bedeaux for continued inspiration and encouragement. The research of E.M.B. has been made pos-sible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.
APPENDIX: EXAMPLES OF THE FREE ENERGY AS GIVEN IN EQ.„4…
We now discuss two recent examples from the literature in which the surface free energy has the form of Eq.共4.1兲. In the first example, the adsorption of polymers onto a curved surface is considered, while in the second example the elec-tric double layer theory is discussed.
1. Polymer adsorption
In the recent description of polymer adsorption onto a curved substrate by Clement and Joanny,17 the surface free energy is a functional of , which is related to the local monomer concentration c by 2⫽c. It has the following form: F关兴⫽
冕
drជ冋
兩ⵜជ 共r兲兩2⫹1 2共 2⫺ b 2兲2册
⫺A1 d共 w兲2, 共A1兲where d is the extrapolation length which measures the strength with which polymers are adsorbed at the wall, is the excluded volume parameter, and b
2⫽c
b, the local monomer concentration in the bulk. Lengths are scaled with
a/
冑
6, with a the monomer size, and energies by kBT, withkB Boltzmann’s constant and T the absolute temperature. When Eq.共A1兲 is compared to Eq. 共4.1兲 we can identify
⌸ and ⌽w as ⌸共,ⵜជ 兲⫽⫺兩ⵜជ 共r兲兩2⫺ 12共 2⫺ b 2兲2, 共A2兲 ⌽w共w兲⫽⫺ 1 d共 w兲2.
The Euler–Lagrange equation, Eq.共4.2兲, and boundary con-dition, Eq.共4.3兲, now read as
⌬⫽共2⫺ b 2兲, 共A.3兲 nˆ•ⵜជ w⫽⫺1 d w.
With the identification in Eq. 共A2兲 and after solving the Euler–Lagrange equation in Eq. 共A3兲, one is then able to calculate the curvature coefficients using the expressions in Eqs. 共4.8兲 and 共4.10兲.
2. Electrostatic double layer
The electrostatic contribution to the free energy of the double layer immersed in a 1–1 electrolyte has the following form:13,25 Fel⫽
冕
drជ再
⫺ 1 2⑀0⑀r兩ⵜជ 共rជ兲兩 2 ⫺2nelkBT冋
cosh冉
e kBT冊
⫺1册冎
⫹Aw, 共A4兲 where⑀r is the dielectric of the aqueous medium,⑀0 is thepermittivity of vacuum, nelis the electrolyte number density,
e is the elementary charge, and 共not to be confused with
the surface tension兲 is the surface charge density.
The electrical free energy above is written in terms of the electrostatic potential(rជ), which is to be determined by solving the Poisson–Boltzmann equation,
⌬⌿⫽2sinh共⌿兲, 共A5兲
where the dimensionless potential ⌿⬅ e/kBT and inverse Debye length⬅(2e2nel/⑀0⑀rkBT)1/2have been introduced. The requirement of constant surface charge density leads to the following boundary condition to the Poisson–Boltzmann equation:
nˆ•ⵜជ⌿w⫽⫺2p, 共A6兲 where p⬅ 2Q/e, with Q⬅ e2/4⑀0⑀rkBT the Bjerrum length.
In this case, the differential equation determining the profile of the order parameter, the Poisson–Boltzmann equa-tion, is derived from electrostatics 共the Poisson equation兲 rather than from the minimization of the free energy in Eq.
共A4兲. In fact, in the derivation of the electrical free energy in
Eq. 共A4兲 one has already made use of the Poisson– Boltzmann equation.25 However, the Poisson–Boltzmann equation, Eq. 共A4兲, and boundary condition, Eq. 共A5兲, both
do result from the Euler–Lagrange equation treating Felas if it were a functional of; Fel⫽Fel关兴. The result is that if we
now identify ⌸共,ⵜជ 兲⫽1 2⑀0⑀r兩ⵜជ 共rជ兲兩 2⫹2 n elkBT
冋
cosh冉
e kBT冊
⫺1册
, 共A7兲 ⌽w共w兲⫽w,we can again calculate the curvature coefficients using the expressions in Eqs.共4.8兲 and 共4.10兲 as noted by Winterhalter and Helfrich,12 Lekkerkerker,13 and by Fogden, Daicic, and coworkers.15
1W. Helfrich, Z. Naturforsch. C 28, 693共1973兲.
2For reviews see Micelles, Membranes, Microemulsions, and Monolayers,
edited by W. M. Gelbart, A. Ben-Shaul, and D. Roux共Springer-Verlag, New York, 1994兲; Statistical Mechanics of Membranes and Surfaces, ed-ited by D. Nelson, T. Piran, and S. Weinberg共World Scientific, Singapore, 1988兲.
3S. A. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and
Membranes共Addison-Wesley, Reading, 1994兲.
4
W. Helfrich in Physics of Defects, Les Houches, edited by R. Balian et al.
共North-Holland, Amsterdam, 1981兲.
5I. Szleifer, Ph.D. thesis, Hebrew University of Jerusalem, 1988; I. Szleifer,
D. Kramer, A. Ben-Shaul, W. M. Gelbart, and S. A. Safran, J. Chem. Phys. 92, 6800共1990兲; I. Szleifer, D. Kramer, A. Ben-Shaul, D. Roux, and W. M. Gelbart, Phys. Rev. Lett. 60, 1966共1988兲.
6E. M. Blokhuis and D. Bedeaux, Physica A 184, 42共1992兲.
7V. Romero-Rochı´n, C. Varea, and A. Robledo, Phys. Rev. A 44, 8417 共1991兲; C. Varea and A. Robledo, Mol. Phys. 85, 477 共1995兲.
8
G. Gompper and S. Zschocke, Europhys. Lett. 16, 731共1991兲; G. Gomp-per and S. Zschocke, Phys. Rev. A 46, 4386共1992兲.
9G. Porte and C. Ligoure, J. Chem. Phys. 102, 4290共1995兲. 10F. P. Buff, J. Chem. Phys. 23, 419共1955兲.
11Only in the context of the mean-field theory that is considered here can we
identify⌸ as the lateral pressure profile. In general the pressure tensor has two lateral components and cannot be uniquely defined. For a discussion see, e.g., S. Ono and S. Kondo, Handbook Phys. 10, 134共1960兲; E. M. Blokhuis and D. Bedeaux, J. Chem. Phys. 97, 3576共1992兲.
12M. Winterhalter and W. Helfrich, J. Phys. Chem. 96, 327共1992兲. 13
H. N. W. Lekkerkerker, Physica A 159, 319 共1989兲; H. N. W. Lek-kerkerker, ibid. 167, 384共1990兲.
14D. J. Mitchell and B. W. Ninham, Langmuir 5, 1121共1989兲.
15J. Daicic, A. Fogden, I. Carlsson, H. Wennerstro¨m, and B. Jo¨nsson, Phys.
Rev. E 54, 3984共1996兲; A. Fogden, I. Carlsson, and J. Daicic, ibid. 57,
5694共1998兲.
16It should be recognized that the results for the double layer theory共Refs.
12–15兲 extend beyond the situation of a system in contact with a charged solid wall. As long as the curvature is fixed independent of, say, surface charge or added salt, the calculated formulas then give the electrostatic
contribution to the curvature coefficients.
17F. Clement and J.-F. Joanny, J. Phys. II 7, 973共1997兲. 18M. P. A. Fisher and M. Wortis, Phys. Rev. B 29, 6252共1984兲. 19
E. M. Blokhuis and D. Bedeaux, Mol. Phys. 80, 705共1993兲.
20
E. M. Blokhuis and D. Bedeaux, Heterog. Chem. Rev. 1, 55共1994兲.
3732共1976兲; C. Varea and A. Robledo, Physica A 233, 132 共1996兲.
22Shifting the position of the interface R→R⫹⌬R leads to a change in the
bulk contribution to the free energy, ⫺4R2⌬p ⌬R 共for a spherical
interface兲. The reason that this has no consequence to the excess free energy is that the change in free energy is cancelled by the change in free energy due to surface tension, 8R⌬R, via the Laplace equation.
23See G. Gompper and M. Schick, ‘‘Self-assembling amphiphilic system,’’
Phase Transitions and Critical Phenomena edited by C. Domb and J.
Lebowitz共Academic, London, 1994兲, Vol. 16.
24E. M. Blokhuis, Ber. Bunsenges. Phys. Chem. 100, 313共1996兲. 25E. J. W. Verwey and J. Th. G. Overbeek, Theory of the Stability of
Lyo-phobic Colloids共Elsevier, Amsterdam, 1948兲, Eq. 共37a兲 on p. 79.
