Effective Hamiltonian of Liquid-Vapor Curved Interfaces in Mean Field
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Effective Hamiltonian of Liquid-Vapor CurvedInterfaces in Mean Field
Jos´e G. Segovia-L´opez ∗ , Adolfo Zamora † and Jos´e Antonio Santiago ‡ Divisi´on Acad´emica de Ciencias B´asicas,Universidad Ju´arez Aut´onoma de Tabasco,Km 1 Carretera Cunduac´an-Jalpa, Apartado Postal 24,86690, Cunduac´an, Tabasco, Mexico Departamento de Matem´aticas Aplicadas y Sistemas,Universidad Aut´onoma Metropolitana – Cuajimalpa,M´exico D.F. 01120, Mexico
Abstract
We analyze a one-component simple fluid in a liquid-vapor coexistencestate, which forms an arbitrarily curved interface. By using an approachbased on density functional theory, we obtain an exact and simple expres-sion for the grand potential at the level of mean field approximation thatdepends on the density profile and the short-range interaction potential.By introducing the step-function approximation for the density profile,and using general geometric arguments, we expand the grand potential inpowers of the principal curvatures of the surface and find consistency withthe Helfrich phenomenological model in the second order approximation.
PACS numbers: 23.23.+x, 56.65.Dy ∗ [email protected] † [email protected] ‡ [email protected] Introduction
Although the description of a one-component fluid in a liquid-vapor coexistencestate has been studied since long ago, it still remains as a topic of interest forinhomogeneous fluids [1, 2, 3, 4]. Not just open questions are yet to be answeredwithin this problem, but also its appropriate description may be useful for theanalysis of more complex systems such as colloids, polymers, mixtures of these,etc. Particularly, investigation of the microscopic expression for the free energythat represents an arbitrarily curved interface remains a topic of interest forthe one-component fluid in a coexistence state. In order to analyze this sys-tem different approximation schemes, rigorous derivations, and many genuineshortcuts have been implemented [4, 5, 6, 7, 8]. Associated to the free energythere is the calculation of microscopic expressions for interfacial properties suchas surface tension, spontaneous curvature, and rigidity coefficients. The mainobjective of this work is to derive the free energy and all these quantities in arigorous and fully general form.One of the approximation schemes that has provided a first principles descrip-tion is the stress tensor theory, implemented initially by Romero-Roch´ın, Varea,and Robledo [9]. In this theory the authors identify the normal component ofthe stress tensor as the fundamental quantity to calculate the grand potential,which represents the energetic cost to maintain the interface having a given ge-ometry. Construction of the most general form of this stress tensor is due toPercus and Romero-Roch´ın, and represents one of the important achievementswithin this approximation scheme [10, 11]. The microscopic expression for thisstress tensor has been used to describe the interfacial region, within the vander Waals model, when the interface has planar, spherical, and cylindrical ge-ometries. For each of these interfaces, the exact microscopic expression for thegrand potential has been constructed considering drops of arbitrary size [12].Nevertheless, in order to compare these results, it is convenient to approximatethe grand potential as a limiting case; which is, when the radius of the Gibbsdividing surface is much larger than the range of the interaction potential. Thishas lead to a microscopic expression for the grand potential as a power series ofthe inverse radii of curvature. At this level, we establish an equivalence betweenthe microscopic expression for the grand potential and the Helfrich phenomeno-logical model [13] for each of the corresponding geometries.Despite the Helfrich model is widely used to describe a variety of systems [7,14], no first-principles derivation of the model is known. Several efforts havebeen made aimed at achieving the task [15]. The usual strategy to get to thephenomenological form of the Helfrich free energy is based on the assumptionthat the interface is a bidimensional elastic continuous-medium [16]. Accordingto this model, the free energy of the interface is a function of the principalcurvatures of the system, which can be written asΩ S = Z dS [ γ − κc H + κH + ¯ κK ] , (1)2here H = (1 /R ) + (1 /R ) and K = 1 /R R are the mean and Gaussian cur-vature respectively, with R and R being the principal radii of curvature. Thecoefficients γ , c , κ , and ¯ κ are equilibrium surface properties.As a second part of this work, we carry out a rigorous derivation of the Hel-frich model corresponding to a fluid interface, Eq. (1), calculate the surfaceproperties, and show that it is a legitimate model to describe arbitrarily curvedinterfaces. We follow an essentially different program as compared to previousapproaches. Instead of analyzing diverse particular geometries, which is nota simple task, we consider the general case of an arbitrarily curved interfacialregion and study it using a semi-orthogonal coordinate system. A notable factthat appears in the description of curved interfaces concerns the localization ofthe Gibbs dividing surface. As the physical properties of the system must beindependent of this choice, a displacement in the localization of the interfacein microscopic distances shall not modify the value of the surface tension, butwill alter the value of the rigidity coefficients. Thus, different localizations ofthe Gibbs dividing surface give rise to different values of the rigidity coefficients,which results in an arbitrariness of their values. Explanation of the origin of thisarbitrariness is one of the questions yet to be addressed. In this study, we fix theradius of the Gibbs dividing surface within the approximation introduced forthe density profile. Although this criterion is not unique, the results obtainedare consistent with previous works.This paper is organized as follows. In Section 2 we briefly outline the stresstensor formalism that describes a smooth, but otherwise general, interface. Sec-tion 3 is devoted to the calculation of the microscopic expression for the grandpotential of a curved interface within the van der Waals approximation. Next,the limit of large radii of curvature, as compared to the range of the interac-tion potential, is introduced in Section 4 and a further approximation to thegrand potential is obtained, which coincides with the Helfrich phenomenologicalmodel. Finally, some concluding remarks are drawn in Section 5. According to density functional theory, the grand potential can be written inthe form [1, 2] Ω[ ρ ( ~r )] = F [ ρ ( ~r )] + Z d~r [ µ − V ext ( ~r )] ρ ( ~r ) , (2)where F [ ρ ( ~r )] is the intrinsic Helmholtz free energy, µ is the chemical potential,and V ext is the external potential. The equilibrium value of the density profileis obtained by minimizing the grand potential; that is by solving equation δF [ ρ ( ~r )] δρ (cid:12)(cid:12)(cid:12)(cid:12) ρ − [ µ − V ext ( ~r )] = 0 . (3)3n absence of exact analytic solutions, the usual approach is through numericalmethods, which provide estimate values. However, in this work we insist in ananalytic solution and follow the route of the stress tensor to obtain it. Next webriefly outline the general formalism, which can be read in detail in Ref. [9].In an inhomogeneous fluid the condition for mechanical equilibrium implies theexistence of a conservation equation, obtained from (3), which may be expressedas ∇ · σ = ρ ∇ V ext , (4)with σ being the stress tensor and ρ V ext ( ~r ) the external force per unit area.The stress tensor is not unique. As it can be seen from (4), a term with zerodivergence may always be added. Nature of the system suggests a separation ofthe stress tensor in two contributions σ = σ + σ S , (5)where σ is the homogeneous part and σ S is the inhomogeneous one. The ho-mogeneous part describes the bulk phases, where the density profile is uniform,whereas the inhomogeneous region is that where ∇ ρ ( ~r ) = 0.The free energy of the system is obtained by integrating the normal componentof the stress tensor over the whole space. Once again, we separate this normalcomponent in two contributions, one associated to the homogeneous region andthe other from the inhomogeneous one [9]Ω[ ρ ( ~r )] = Z d~rσ N ( ~r ) = Z d~rσ N ( ~r ) − Z d~rσ NS ( ~r ) . (6)As we are interested in the inhomogeneous region, we neglect the contributionto the energy arising from the bulk phases. From now on we concentrate inobtaining the microscopic expression for Ω S , which is given byΩ S [ ρ ( ~r )] = − Z d~rσ NS ( ~r ) . (7)We observe that σ NS is the fundamental quantity needed to obtain microscopicexpressions for the properties of the surface. For the system under study themicroscopic expression for the stress tensor of the inhomogeneous region, withinthe van der Waals approximation, is [11, 12] σ αβS ( ~r ) = − Z d~r ′ Z dλρ ( ~r − (1 − λ ) ~r ′ )˜ ω ( | ~r ′ | ) r ′ α ∇ β ρ ( ~r + λ~r ′ ) − ∇ ν Z d~r ′ Z dλρ ( ~r − (1 − λ ) ~r ′ )˜ ω ( | ~r ′ | ) × r ′ β [ r ′ α ∇ ν ρ ( ~r + λ~r ′ ) − r ′ ν ∇ α ρ ( ~r + λ~r ′ )] . (8)4his quantity depends exclusively on the density profile, the interaction poten-tial, and is independent of the geometry of the interface. In fact, the geometry isdefined by the density profile. For instance, for a planar interface ρ ( ~r ) = ρ ( z ),and for a spherical interface ρ ( ~r ) = ρ ( | ~r | ). In general, for an arbitrary interfaceif ξ denotes the normal coordinate, the density profile is a function exclusively ofthis quantity: ρ ( ~r ) = ρ ( ξ ). By using this information, we go on to calculatingthe grand potential for an interface having an arbitrarily curved geometry, The free energy of the interfacial region, given by Eq. (7), in explicit form writesΩ S = 12 Z d~r Z d~r ′ Z dλ ˆ n α ( ~r )ˆ n β ( ~r )˜ ω ( | ~r ′ | ) r ′ α ρ ( ~r − (1 − λ ) ~r ′ ) ∇ β ρ ( ~r + λ~r ′ ) − Z d~r ˆ n α ( ~r )ˆ n β ( ~r ) ∇ ν Z d~r ′ Z dλρ ( ~r − (1 − λ ) ~r ′ )˜ ω ( | ~r ′ | ) × r ′ β [ r ′ α ∇ ν ρ ( ~r + λ~r ′ ) − r ′ ν ∇ α ρ ( ~r + λ~r ′ )] . (9)By performing some manipulations this expression may be compacted to theform [12]Ω S = 12 Z d~r Z d~r ′ Z dλρ ( ~r − (1 − λ ) ~r ′ )˜ ω ( | ~r ′ | ) ∇ α ρ ( ~r + λ~r ′ )[ r ′ β ˆ n α ( ~r )ˆ n β ( ~r ) − λr ′ β ˆ n β ( ~r ) r ′ ν ∇ α ˆ n ν ( ~r ) + λr ′ β ˆ n α ( ~r ) r ′ ν ∇ ν ˆ n β ( ~r ) + λr ′ β ˆ n β ( ~r ) r ′ ν ∇ ν ˆ n α ( ~r )] , (10)which has been previously used to describe interfaces having planar, spherical,and cylindrical geometry [12]. Direct calculation of the grand potential in eachof these geometries leads to the equationΩ S = − Z d~r Z d~r ′ Z ∞ ds ∇ ρ ( ~r ) · ∇ ′ ρ ( ~r ′ ) ˜ w ( s + ( ~r − ~r ′ ) ) , (11)the difference being contained only in the density profile.The first goal in this paper is to prove that Eq. (11) is satisfied for an arbi-trarily curved interface, and we concentrate on this for the remaining of thissection. Instead of starting from Eq. (10), we prefer to manipulate expression(9), introduce the linear change of variables ~r (1) = ~r + λ~r ′ , (12) ~r (2) = ~r ′ , (13)and use the relationship r (2) ν ∇ (1) ν (cid:20) ˆ n α ˆ n β (cid:16) ~r (1) − λ~r (2) (cid:17) (cid:21) = ∂∂λ (cid:20) ˆ n α ˆ n β (cid:16) ~r (1) − λ~r (2) (cid:17) (cid:21) , (14)5ith the consideration of the symmetry with respect to the exchange of indices α and β in this equation, and that the profile is constant at ±∞ , to obtainΩ S = − Z d~r Z d~r ′ ρ ( ~r ) ˜ w ( | ~r ′ | )ˆ n α ( ~r )ˆ n β ( ~r ) r ′ β ∇ α ρ ( ~r + ~r ′ ) − Z d~r Z d~r ′ Z dλλ ˜ w ( | ~r ′ | ) r ′ β r ′ α ˆ n α ( ~r )ˆ n β ( ~r ) × ∇ ν ρ ( ~r − (1 − λ ) ~r ′ ) ∇ ν ρ ( ~r + λ~r ′ ) − Z d~r Z d~r ′ Z dλλ ˜ w ( | ~r ′ | ) r ′ β r ′ α ˆ n α ( ~r )ˆ n β ( ~r ) × ρ ( ~r − (1 − λ ) ~r ′ ) ∇ ν ∇ ν ρ ( ~r + λ~r ′ ) . (15)The last two integrals in this expression have to be manipulated to eliminatethe arbitrary parameter λ . With this in mind we define a system of semi-orthogonal coordinates, having unit vectors (ˆ n, ˆ t , ˆ t ), and express each of thevectors ~r and ~r ′ on its own basis. That is, ~r = r n ˆ n ( ~r ) + r t ˆ t + r t ˆ t and ~r ′ = r n ′ ˆ n ′ ( ~r ′ ) + r t ′ ˆ t + r t ′ ˆ t . However, vector ~r ′ can also be expressed on thebasis of vector ~r in the following manner: ~r ′ = ( ~r ′ · ˆ n ( ~r ))ˆ n ( ~r ) + ( ~r ′ · ˆ t )ˆ t + ( ~r ′ · ˆ t )ˆ t = r ′ n ˆ n ( ~r ) + r ′ t ˆ t + r ′ t ˆ t . We now go on to manipulating expressions so asto eliminate λ . To achieve it we define an auxiliary function W ( r ′ n + r ′ t + r ′ t ),related to ˜ w ( | r ′ n + r ′ t + r ′ t | ) via ∂∂r ′ n W ( r ′ n + r ′ t + r ′ t ) = r ′ n ˜ w ( | r ′ n + r ′ t + r ′ t | ) . (16)This relationship is employed to eliminate any power of r ′ n within the integrals.After some more manipulations we get to the different expression for the grandpotentialΩ S = − Z d~r Z d~r ′ ρ ( ~r ) ˜ w ( | ~r ′ | )ˆ n α ( ~r )ˆ n β ( ~r ) r ′ β ∇ α ρ ( ~r + ~r ′ ) − Z d~r Z d~r ′ Z dλλ Z ∞ dt ˜ w ( t + r ′ n + r ′ t + r ′ t ) × ∇ ν ρ ( ~r − (1 − λ ) ~r ′ ) ∇ ν ρ ( ~r + λ~r ′ ) − Z d~r Z d~r ′ Z dλλ Z ∞ dt ˜ w ( t + r ′ n + r ′ t ) × ρ ( ~r − (1 − λ ) ~r ′ ) ∇ ν ∇ ν ρ ( ~r + λ~r ′ ) − Z d~r Z d~r ′ Z dλλ Z ∞ dt Z ∞ dt ′ ˜ w ( t + t ′ + r ′ n + r ′ t + r ′ t ) × ∂ ∂r ′ n [ ∇ ν ρ ( ~r − (1 − λ ) ~r ′ ) ∇ ν ρ ( ~r + λ~r ′ )] − Z d~r Z d~r ′ Z dλλ Z ∞ dt Z ∞ dt ′ ˜ w ( t + t ′ + r ′ n + r ′ t ) × ∂ ∂r ′ n [ ρ ( ~r − (1 − λ ) ~r ′ ) ∇ ν ∇ ν ρ ( ~r + λ~r ′ )] , (17)6hich may now be simplified by considering each of the terms separately. Asit is shown in Appendix A, the last four integrals cancel one another in pairs,yielding the result for the grand potentialΩ S = − Z d~r Z d~r ′ ρ ( ~r ) ˜ w ( | ~r ′ | )ˆ n α ( ~r )ˆ n β ( ~r ) r ′ β ∇ α ρ ( ~r + ~r ′ ) . (18)The task now is to show the equivalence between this and Eq. (11). We startby carrying out further manipulations using (16) and introducing a change ofvariables of the form (12)–(13), but now being ~r (1) = ~r + ~r ′ , (19) ~r (2) = ~r. (20)Thus, the final result obtained for the microscopic expression for the grandpotential of an arbitrarily curved interface isΩ S = − Z d~r (1) Z d~r (2) Z ∞ dt ˜ w ( t + ( ~r (1) n − ~r (2) n ) + ( ~r (1) t − ~r (2) t ) ) × ∂ (1) n ρ ( ~r (1) ) ∂ (2) n ρ ( ~r (2) n ) , (21)where each vector has been written as ~r ( i ) = r ( i ) n ˆ n ( i ) + r t ˆ t ( i )1 + r t ˆ t ( i )2 .Notice that this expression is equal to (11), which was to be proved. This is oneof the most general results for the grand potential that represents the free energyof the interfacial region. The result is simple and exact. Given the interactionpotential, it only depends on the density profile. Although the previous result is simple and exact, it is not the appropriate formto carry out comparisons with other works. In this sense, it results convenientto get an approximation as a reference value. We start by choosing a commonbasis to express all vectors. If the basis is that of vector ~r (1) , vector ~r (2) can be written as ~r (2) = ( ~r (2) · ˆ n (1) )ˆ n (1) + ( ~r (2) · ˆ t (1)1 )ˆ t (1)1 + ( ~r (2) · ˆ t (1)2 )ˆ t (1)2 = r (1) n ˆ n (1) + r (1) t ˆ t (1)2 + r (1) t ˆ t (1)2 . By introducing these into (21) we findΩ S = 14 Z ∞ dr (1) n Z dS (1) Z ∞ dr (2) n Z dS (2) Z ∞ dt × ˜ w ( t + ( r (1) n − ~r (2) · ˆ n (1) ) + ( ~r (1) t − ~r (2) t ) ) × ˆ n (1) · ˆ n (2) ∂ (1) n ρ ( ~r (1) ) ∂r (1) n ρ ( ~r (2) · ˆ n (1) ) , (22)where ∂r (1) n = ∂/∂r (1) n and each element of volume has been written as d~r ( i ) = dr ( i ) n dS ( i ) with ~r ( i ) t two-dimensional vectors on S (1) and S (2) respectively.7he expression for the grand potential that we have constructed so far containsthe description of the whole interfacial region, independently of its details. Themicroscopic expression is exact for this model; exact in the sense that the den-sity profile contains all the information of the geometry under consideration. Inorder to obtain microscopic expressions for interfacial properties it is necessaryto fix the Gibbs dividing surface. The choice in this work is through the approx-imation of the density profile as a step function. For a more general profile theprocedure becomes more complicated. We will consider that case in a futurestudy.The density profile is thus approximated as ρ ( ~r ) = ρ ( ~r ( i ) n ) = ρ l Θ( r ( i ) n − r ( i ) n ) + ρ v Θ( r ( i ) n − r ( i ) n ) , (23)where r ( i ) n is the radius of the Gibbs dividing surface. We have ∂ ( i ) n ρ ( r ( i ) n ) = − ∆ ρ δ ( r ( i ) n − r ( i ) n ) , (24)with ∆ ρ = ρ v − ρ l . Evaluation of the derivatives of the density profile isobvious. The result we obtain isΩ S = 14 (∆ ρ ) Z dS (1) Z dS (2) ˆ n (1) · ˆ n (2) Z ∞ dt ˜ w ( t +( r (1) n − r (2) n ) +( ~r (1) t − ~r (2) t ) ) , (25)To proceed we approximate the surface representing the interface as a paraboloid.This is possible as long as the radius of this surface is very large compared tothe range of the interaction potential. Under this assumption, the surface canbe approximated as a plane with corrections [17, 18].We now build a local coordinate system in the neighborhood of a point P of theGibbs dividing surface. This arbitrary point is localized on the surface by vector ~r (1) . The normal vector at that point is ˆ n (1) and we chose the z axis pointingin this direction. Thus the coordinates x and y lie on the plane tangent to thesurface at P and point along the directions of the principal radii of curvature.At point P , ~r (1) t = (0 ,
0) and ˆ n (1) = ˆ k , so that the Gibbs dividing surface islocalized at height r (1) n . Vector ~r (2) localizes another point Q on the dividingsurface, close to P but outside the local plane, at a distance z = r (1) n − r (2) n fromthe plane (see Fig. 1). If z is measured from the local system, it has the value z = 12 (cid:18) x R + y R (cid:19) + · · · (26)As vector ~r (2) is not parallel to the z axis of the coordinate system, the normalvector at point Q is given byˆ n (2) = (cid:16) − xR , − yR , (cid:17)h xR ) + ( yR ) i . (27)8igure 1: This schematic picture shows the local approximation for the surface S about a point P . Points P and Q are localized by vectors ~r (1) and ~r (2) respectively, and the normal vectors to the surface at those points are ˆ n (1) andˆ n (2) . Point P is chosen as the origin of the local coordinate system whereas Q isoutside the tangent plane. Its projection onto this tangent plane has coordinates( x, y ). The radius of localization of the Gibbs dividing surface is at r (1) n . Thedistance from Q to the local plane (projection of ˆ n (2) onto the direction ˆ n (1) ) is r (1) n − r (2) n = z , with z seen from the local system as a paraboloid.9he metric in this coordinate system is g = 1 + [ ∇ z ( x, y )] = 1 + x R + y R , where xR ≪ yR ≪
1. The surface element in the local system is dS (2) = g dxdy .The scalar product of the normals is ˆ n (1) · ˆ n (2) = g − . We incorporate the effectof the local approximation into the interaction potential to obtain the free energyof the interfacial regionΩ S = − (∆ ρ ) Z dS (1) Z ∞−∞ dx Z ∞−∞ dy Z ∞ dt × ˜ ω t + x + y + 14 (cid:18) x R + y R (cid:19) + · · · ! . (28)By expanding the interaction potential to first order about t + x + y andevaluating the integrals on the coordinates x and y , one gets to a microscopicexpression for the grand potential in terms of the principal radii of curvatureΩ S = − Z dS (cid:26) (∆ ρ ) π Z ∞ dr ˜ ω ( r ) r (cid:27) + Z dS (cid:18) R + 1 R + 23 R R (cid:19) (cid:26) ρ ) π Z ∞ dr ˜ ω ( r ) r (cid:27) + · · · (29)where we have put dS (1) = dS . On the other hand, from the definitions of themean ( H ) and Gaussian ( K ) curvatures one finds that4 H − K = 1 R + 1 R + 23 R R . (30)By introducing this into (29), we get to the most general microscopic expressionfor the grand potential within this approximation level; consistent with theHelfrich prediction [13] for the free energyΩ S = − Z dS (cid:20) π (∆ ρ ) Z ∞ drr ˜ ω ( r ) − π (∆ ρ ) (cid:18) H − K (cid:19) Z drr ˜ ω ( r ) (cid:21) . (31)which is also in agreement with previous results in the planar [5, 7, 12], spheri-cal [12], and cylindrical [12] geometries, as it is shown in Appendix B. There exist two relevant aspects of this work that are worthwhile remarking.The first one concerns a rigorous proof for a general expression, exact and sim-ple, of the grand potential within a mean field approximation. The second is afirst principles derivation of the Helfrich free energy, within this context, froma completely original approach. This, in addition, confirms that the Helfrichscheme is appropriate for the study of curved interfaces. Although the localapproximation of a surface as a plane that has been used is appropriate for the10escription of weakly curved surfaces, we observe that the result for the grandpotential representing the free energy of the system is sufficiently general. Themicroscopic expression obtained is a function of the principal curvatures of thesurface, in complete agreement with previous predictions [13, 15]. We also findcomplete consistency with previous results for the simplest geometries, whichwere obtained using a different analytic approach [12]. Finally, we point outthat within the step function approximation for the density profile, no contri-bution exists from spontaneous curvature to the free energy. We shall study theproblem of an arbitrarily curved interface for a more general density profile ina future publication.
Acknowledgments
The authors wish to thank V. Romero-Roch´ın for helpful comments and stim-ulating discussion. This work was supported partially by PFICA-UJAT undercontract No. UJAT-2009-C05-61 and by PROMEP-MEXICO under contractUJAT-CA-15. J.A.S. akcnowledges financial support from PROMEP, projectUAM-PTC-196.
A Simplifications on the Grand Potential
In this appendix we summarize the main steps to simplify Eq. (17). We startby writing it in the formΩ S = − Z d~r Z d~r ′ ρ ( ~r ) ˜ w ( | ~r ′ | )ˆ n α ( ~r )ˆ n β ( ~r ) r ′ β ∇ α ρ ( ~r + ~r ′ )+ T + T + T + T , (32)where the T i , with i = 1 , . . . ,
4, have been defined as the integrals appearing in(17) sequentially.In order to simplify and easily identify each of the terms T , T , T , and T , weintroduce explicitly the change of variables (12)–(13). Under such a coordinatetransformation the term T reads T = − Z d~r (1) Z d~r (2) Z dλλ Z ∞ dt ˜ ω ( t + | ~r (2) | ) ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1) n ρ ( ~r (1) ) , (33)where the notation ∂ (1) n = ∂/∂r (1) n has been introduced. Integrating by partswith respect to r (1) n , this now becomes T = 14 Z d~r (1) Z d~r (2) Z dλλ Z ∞ dt ˜ ω ( t + | ~r (2) | ) ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) ) . (34)11he other term with normal derivatives of the same order is T . The effect oftransformation (12)–(13) into this leads to T = − Z d~r (1) Z d~r (2) Z dλλ Z ∞ dt ˜ ω ( t + | ~r (2) | ) ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )= − T . (35)Therefore these terms cancel one another when substituted into the expressionfor the grand potential (32).To simplify the terms with normal derivatives of the second order, that is T and T , we need to take into account the following manipulations ∂∂r ′ n = ∂r (1) n ∂r ′ n ∂∂r (1) n + ∂r (2) n ∂r ′ n ∂∂r (2) n = λ ∂∂r (1) n + ∂∂r (2) n = λ∂ (1) n + ∂ (2) n , (36)from where ∂ ∂r ′ n = ∂∂r ′ n (cid:16) λ∂ (1) n + ∂ (2) n (cid:17) = λ ∂ (1)2 n + 2 λ∂ (1) n ∂ (2) n + ∂ (2)2 n . (37)By introducing this into T , we obtain T = − Z d~r (1) Z d~r (2) Z dλλ Z ∞ dt Z ∞ dt ′ ˜ ω ( t + t ′ + | ~r (2) | ) × h λ ∂ (1)2 n + 2 λ∂ (1) n ∂ (2) n + ∂ (2)2 n i h ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1) n ρ ( ~r (1) ) i . (38)We need to calculate derivatives of the product within the integrand. The firstone is ∂ (1) n [ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1) n ρ ( ~r (1) )] = ∂ (1)2 n ρ ( ~r (1) − ~r (2) ) ∂ (1) n ρ ( ~r (1) )+ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) ) , (39)and the second ∂ (1)2 n [ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1) n ρ ( ~r (1) )] = ∂ (1) n [ ∂ (1)2 n ρ ( ~r (1) − ~r (2) ) ∂ (1) n ρ ( ~r (1) )+ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )]= ∂ (1)3 n ρ ( ~r (1) − ~r (2) ) ∂ (1) n ρ ( ~r (1) )+ 2 ∂ (1)2 n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )+ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)3 n ρ ( ~r (1) ) . (40)The corresponding derivatives for ∂ (2) n are also calculated. By introducing all12hese into T one finds T = − Z d~r (1) Z d~r (2) Z dλλ Z ∞ dt Z ∞ dt ′ ˜ ω ( t + t ′ + | ~r (2) | ) × { λ [ ∂ (1)3 n ρ ( ~r (1) − ~r (2) ) ∂ (1) n ρ ( ~r (1) ) + ∂ (1)2 n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )+ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)3 n ρ ( ~r (1) )] + 2 λ [ ∂ (1)2 n ρ ( ~r (1) − ~r (2) ) ∂ (1) n ρ ( ~r (1) )+ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )] + ∂ (2)2 n [ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1) n ρ ( ~r (1) )] } . (41)or alternatively, after an integration by parts respect to r (1) n in each term, T = 18 Z d~r (1) Z d~r (2) Z dλλ Z ∞ dt Z ∞ dt ′ ˜ ω ( t + t ′ + | ~r (2) | ) × { λ [ ∂ (1)2 n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )2 ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)3 n ρ ( ~r (1) )+ ρ ( ~r (1) − ~r (2) ) ∂ (1)4 n ρ ( ~r (1) ] + 2 λ∂ (2) n [ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )+ ρ ( ~r (1) − ~r (2) ) ∂ (1)3 n ρ ( ~r (1) )] + ∂ (2)2 n [ ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )] } . (42)Now we investigate the effect of the same transformation in the expression for T . Direct substitution implies T = − Z d~r (1) Z d~r (2) Z dλλ Z ∞ dt Z ∞ dt ′ ˜ ω ( t + t ′ + | ~r (2) | ) × h λ ∂ (1)2 n + 2 λ∂ (1) n ∂ (2) n + ∂ (2)2 n i h ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) ) i . (43)Although this compact form appears simple, we need to carry out further sim-plifications to identify similar terms. We start by calculating the first derivative ∂ (1) n [ ρ ( ~r (1) − ~r (2) ) ∂ (1)2 ρ ( ~r (1) )] = ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )+ ρ ( ~r (1) − ~r (2) ) ∂ (1)3 n ρ ( ~r (1) ) . (44)and then a further derivative of this quantity in the same direction ∂ (1) n , whichyields ∂ (1)2 n [ ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )] = ∂ (1) n [ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )+ ρ ( ~r (1) − ~r (2) ) ∂ (1)3 n ρ ( ~r (1) )]= ∂ (1)2 n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )+ 2 ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)3 n ρ ( ~r (1) )+ ρ ( ~r (1) − ~r (2) ) ∂ (1)4 n ρ ( ~r (1) ) . (45)13y substituting we find T = − Z d~r (1) Z d~r (2) Z dλλ Z ∞ dt Z ∞ dt ′ ˜ ω ( t + t ′ + | ~r (2) | ) × { λ [ ∂ (1)2 n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) ) + 2 ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)3 n ρ ( ~r (1) )+ ρ ( ~r (1) − ~r (2) ) ∂ (1)4 n ρ ( ~r (1) )] + 2 λ∂ (2) n [ ∂ (1) n ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )+ ρ ( ~r (1) − ~r (2) ) ∂ (1)3 n ρ ( ~r (1) )] + ∂ (2)2 n [ ρ ( ~r (1) − ~r (2) ) ∂ (1)2 n ρ ( ~r (1) )] } = − T . (46)That is, also the terms T and T cancel one another. B Simplest Geometries
To make contact with previous results we consider here the simplest geometries.For the planar surface R = R = ∞ so that 4 H − K = 0 and thus Eq. (31)simplifies to Ω S = − Z dS (cid:20) π (∆ ρ ) Z ∞ drr ˜ ω ( r ) (cid:21) . (47)That is, the only contribution to the free energy comes from the surface tensionterm, in agreement with previous results [5, 7, 12].For a spherical interface R = R = R , H = R , and K = R , so that 4 H − K = R . From (31) one gets to the corresponding free energyΩ S = − Z dS (cid:20) π (∆ ρ ) Z ∞ drr ˜ ω ( r ) − π (∆ ρ ) R Z drr ˜ ω ( r ) (cid:21) . (48)Finally, for the cylindrical interface R = ∞ and R = R , so that 4 H − K = R . In this case the free energy isΩ S = − Z dS (cid:20) π (∆ ρ ) Z ∞ drr ˜ ω ( r ) − π (∆ ρ ) R Z drr ˜ ω ( r ) (cid:21) . (49)These last two expressions are consistent with previous results [12]. References [1] R. Evans, Adv. Phys. , 143 (1979).[2] J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity (Claren-don, Oxford, 1982).[3] R. C. Tolman, J. Chem. Phys. , 333 (1949).144] S. Dietrich and M. Napiorkowski, Physica A , 437 (1991);M. Napiorkowski and S. Dietrich, Phys. Rev. E , 1836 (1993);M. Napiorkowski and S. Dietrich, Z. Phys. B , 511 (1995);K. R. Mecke and S. Dietrich, Phys. Rev. E , 6766 (1999);K. R. Mecke and S. Dietrich, J. Chem. Phys. , 204723 (2005).[5] D. G. Triezenberg and R. Zwanzig, Phys. Rev. Lett. , 1183 (1972). Thisresult is known to have been obtained by Yvon but he did not publishedit.[6] A. J. M. Yang, P. D. Fleming, and J. H. Gibbs, J. Chem. Phys. , 3732(1976);A. J. M. Yang, P. D. Fleming, and J. H. Gibbs, J. Chem. Phys. , 74(1977).[7] E. M. Blokhuis and D. Bedeaux, Physica A , 42 (1992);E. M. Blokhuis and D. Bedeaux, Mol. Phys. , 705 (1993);A. E. van Giessen, E. M. Blokhuis, and D. J. Bukman, J. Chem. Phys. , 1148 (1998).[8] J. K. Percus, in The Liquid State of Matter: Fluids, Simple and Com-plex , E. W. Montroll and J. L. Lebowitz, eds. (North-Holland, Amsterdam,1982).[9] V. Romero-Roch´ın, C. Varea, and A. Robledo Phys. Rev. A , 8417,(1991);V. Romero-Roch´ın, C. Varea, and A. Robledo, Physica A , 367 (1992);V. Romero-Roch´ın, C. Varea, and A. Robledo, Mol. Phys. , 821 (1993);V. Romero-Roch´ın, C. Varea, and A. Robledo, Phys. Rev. E , 1600(1993).[10] J. K. Percus, J. Math. Phys. , 1259 (1996).[11] V. Romero-Roch´ın and J. K. Percus, Phys. Rev. E , 5130 (1996).[12] J´ose G. Segovia-L´opez and V´ıctor Romero-Roch´ın, Phys. Rev. E ,021601, (2006).[13] W. Helfrich, Z. Naturfosch. Teil A , 693 (1973).[14] C. Varea and A. Robledo, Physica A , 33 (1995);C. Varea and A. Robledo, Mol. Phys. , 477 (1995).[15] A. Robledo and C. Varea, Physica A , 178 (1996).[16] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon Press,Oxford, 1984).[17] R. Balian and C. Bloch, Ann. Phys. (N. Y.) , 401 (1970).[18] Bertand Duplantier, Raymond E. Goldstein, Victor Romero-Roch´ın, andAdriana I. Pesci, Phys. Rev. Lett.65