Fluids confined in wedges and by edges: From cluster integrals to thermodynamic properties referred to different regions
aa r X i v : . [ c ond - m a t . s o f t ] J un Fluids confined in wedges and by edges:From cluster integrals to thermodynamic properties referred to different regions
Ignacio Urrutia †∗ † Departamento de Física de la Materia Condensada, Centro Atómico Constituyentes,CNEA, Av.Gral. Paz 1499, 1650 Pcia. de Buenos Aires, Argentina andCONICET
Recently, new insights in the relation between the geometry of the vessel that confines a fluid andits thermodynamic properties were traced through the study of cluster integrals for inhomogeneousfluids. In this work I analyze the thermodynamic properties of fluids confined in wedges or byedges, emphasizing on the question of the region to which these properties refer. In this context,the relations between the line-thermodynamic properties referred to different regions are derived asanalytic functions of the dihedral angle α , for < α < π , which enables a unified approach to bothedges and wedges. As a simple application of these results, I analyze the properties of the confinedgas in the low-density regime. Finally, using recent analytic results for the second cluster integralof the confined hard sphere fluid, the low density behavior of the line thermodynamic propertiesis analytically studied up to order two in the density for < α < π and by adopting differentreference regions. I. INTRODUCTION
The interest on confined inhomogeneous fluids coversa large length of scales of the particles size which startsat the simplest one-atom per molecule (e.g. the noblegases) and goes up to proteins, polymers (including DNAmolecules), and large colloids.[1–5] The thermodynamicproperties of these systems are influenced by the geome-try of the vessel or substrate that constrains the spatialregion where the molecules of the system are enabledto move. Important efforts are continuously devoted toreach a detailed description of the response of fluids tosome simple geometrical constraints like the confinementin pores with slit, cylindrical and spherical shapes, as wellas the case of fluids in contact with planar and curvedwalls.This work focuses on fluids confined by open dihedronsbuilt by two planar faces that meet in an edge. Previ-ous studies were dedicated to analyze the adsorption ofliquid-vapor coexisting phases on edges and wedges,[6–9]and also, to the adsorption on corrugated surfaces.[10–12] The main characteristic of the thermodynamics offluids confined by edges and wedges is the existence ofline tensions. This property also characterizes systemswith two coexisting phases adsorbed on planar substrates(sessile drops) and systems with three coexisting phasesthat meet on a common line.[13, 14]One of the particularities of the edge/wedge type ofconfinement is that it produces non-trivial spatial inho-mogeneities of the fluid. As in the case of fluids confinedby curved walls, it happens that different points of viewin the very beginning of the analysis produce dissimilarproperties.[15] Thus, it is relevant to establish the basisthat allow us to compare the thermodynamic propertiesfound by adopting these different points of view. ∗ [email protected] In this work, I analyze the statistical mechanics andthermodynamic properties of a fluid confined in anedge/wedge on the basis of the representation of its grandpotential in powers of the activity. In Sec. II differ-ent type of edge/wedge confinements are discussed andthe thermodynamics of the fluid composed by spheri-cal particles is revisited. There, I analyze the free en-ergy and related thermodynamic magnitudes of the con-fined fluid emphasizing on the explicit choice of the ref-erence region to which system properties refer. Sec. IIIdescribes the functional dependence of the cluster inte-grals with the measures of the edge/wedge spatial re-gion and the consequences that follow on system prop-erties. There, new relations between bulk- surface- andline-thermodynamic properties for different reference re-gions, are shown. They take the form of transforma-tion laws and apply to any density. Also, the behaviorof low density gases is discussed. In Sec. IV, this ap-proach is utilized to derive analytic expressions for thethermodynamic properties (pressure, surface tension, linetension, surface- and linear- adsorptions) of the confinedhard sphere fluid up to order two in density. The con-sequences of adopting different reference regions on thethermodynamics of this system are also discussed in thissection. The expressions obtained of line-tension andline-adsorption, show the dependence with the openingdihedral angle. Final remarks are presented in Sec. V.
II. DETAILED DESCRIPTION OF A FLUID INAN EDGE/WEDGE CONFINEMENT
Let us consider an open system of particles at constanttemperature T and chemical potential µ , which is con-fined by two planar walls that intersect in an edge. Forsimplicity we only refer here to spherical particles. Thewalls exert a hard potential φ ( r ) that constrains the po-sition of the center of each particle to a region A withdihedral shape (throughout this work the open-dihedron A Α<Π a A Α>Π b Figure 1. Fluid confined by a hard-wall dihedron. In theregion A in light-gray (green) particles are free to move whilethe region in white is forbidden. Note that no matter thevalue of the opening angle both light-gray (green) and whiteregions are straight-edge dihedrons. A Α<Π (cid:144) A Α>Π b Figure 2. Fluid confined by a hard wall dihedron that inducean excluded region. Painted with darker gray is the hardwall, lighter gray (green) corresponds to the fluid and theexcluded region is in white. (a) shows the case α < π while(b) corresponds to the case α > π . The wall-particle hardrepulsion distance is σ/ and dashed circles represent particlesat selected positions near the edge/wedge. The arrows showcharacteristic lengths in σ units. geometrical shape is referred to as dihedron) being α theinner angle between faces (inner to A ). I analyze two dif-ferent types of edge/wedge confinement. Fig.1 shows oneof the edge/wedge confinement considered. There, theedge/wedge available region A is defined by the Boltz-mann factor exp [ − φ ( r ) /kT ] = Θ (cid:0) | r − C| (cid:1) where k isthe Boltzmann’s constant, Θ ( x ) the Heaviside function[ Θ ( x ) = 1 if x > and zero otherwise], the dihedralregion which is the complement of A is C = R \ A ,and | r − C| is the shortest distance between r and C .Note that the faces of A meet on a straight line, thus,I call it a straight-edge dihedron. On the other hand,one has the confinement defined by the Boltzmann fac-tor exp [ − φ ( r ) /kT ] = Θ (cid:0) | r − C| − σ (cid:1) being C a soliddihedral region and σ/ the minimum distance betweenthe center of a particle and the solid dihedron. The lat-ter case, particularly relevant for colloidal particles andmacromolecules, is drawn in Fig. 2 where the forbid-den region between A and C is also indicated. Note thatfor < α < π the region A is a straight-edge dihedron(shown in Fig. 2a). On the contrary, for π < α < π (shown in Fig. 2b) A is a rounded-edge dihedron (it hasa curved end-of-fluid surface). This kind of rounded edge confinement is produced by the external hard potentialbeing the inter-particles potential arbitrary. In summary,Figs. 1a, 1b and 2a correspond to a straight-edge con-finement while Fig. 2b corresponds to a rounded-edge.Before analysing the thermodynamic properties of thisconfined fluid it is necessary to adopt a region B as thereference region (RR).[15, 16] Note that B may coincideor not with A . I wish to underline that in the study ofconfined fluids is crucial to clearly establish the adoptedRR which fix the position and shape of its boundary thesurface of tension. This question is as important as toestablish the system of reference in the study of a me-chanical system. Hence, I adopt as RR the region B which nearly follows the shape of A and has measures M B = ( V, A, L ) (being V , A and L the volume, sur-face area and length of the edge of B , respectively). Adetailed analysis of different prescriptions for B is pre-sented in Sec.III. The grand potential of the confinedfluid, relative to B , can be written as Ω = − P V + γA + T L , (1)where P is the pressure of the fluid, γ is the wall/fluid sur-face tension (or surface free-energy), and T the wall/fluidline tension (or line free-energy). The mean number ofparticles in the confined system is N = − ∂ Ω ∂µ , (2) = ρV + Γ A A + Γ L L , (3)where ρ is the mean number density, Γ A is the excessadsorption per unit area, and Γ L is the excess adsorptionper unit length. Naturally, this kind of linear decomposi-tion also applies to other magnitudes, such as the entropy S = − ∂ Ω ∂T , the energy U = Ω+ T S + µN , and higher orderderivatives like σ N ≡ (cid:10) N (cid:11) − N = kT ∂N∂µ which describefluctuations.From Eq. (1) is clear that once A is fixed, Ω be-comes independent of the adopted RR. On the contrary,since measures are relative to B , some of the magnitudes ( P, γ, T ) depend on B . Naturally, the same argumentshows that N is independent of the adopted RR althoughsome of the magnitudes ( ρ, Γ A , Γ L ) may depend on B ,and so on. In summary, even when we have an idea ofthe meaning of the magnitudes ( P, γ, T , ρ, Γ A , Γ L ) thatallow us to give a name to each one, they were not ap-propriately defined yet. Indeed, they do not describe thepure properties of the confined fluid (as it may be sug-gested by the adopted names for these magnitudes), butthey describe the properties of the fluid with regard toa given RR in a sense that will be clarified below. Somegeneral aspects of the former discussion follow the anal-ysis of macroscopic systems with coexisting phases donein Ref. [14]. III. CLUSTER INTEGRALS ANDTHERMODYNAMICS
Let us consider a system of particles interactingthrough a pair potential ψ ( r ) with finite range. Thecenter of these particles is confined by a hard externalpotential to an edge/wedge region A . It was recentlyshown that the cluster integrals of the system take theform[17, 18] τ i = i ! b i V − i ! a i A + i ! c i L . (4)Here, the i -th cluster integral τ i is linear on the extensivemeasures M A = ( V, A, L ) that geometrically character-ize the region A (its volume V , surface area A , and thelength of its edge L ). We say that M A are the mea-sures of the system relative to A . Besides, cluster inte-gral also depends on the opening dihedral angle betweenfaces, α . The volume coefficients b i are the well knownMayer’s cluster integrals for homogeneous systems andthe area coefficients a i were introduced to describe a fluidadsorbed on an infinite wall.[19, 20] b i and a i with i > depend on ψ ( r ) but are independent of α , being τ = b V with b = 1 and V = Z , the configuration integral of oneparticle. Eq. (4) was originally derived for the case ofa straight-edge dihedral region[17] A , i.e. the cases de-scribed in Figs. 1a, 1b and 2a, and was latter extendedto the rounded-edge dihedron shown in Fig.2b.[16]It is well known that in the low density regime orgaseous phase the properties of the confined fluid canbe rigorously written as power series in the activity z = Λ − exp( βµ ) (here β = 1 /kT is the inverse temper-ature and Λ the de Broglie´s thermal length).[18] TheMayer series of the grand potential for the confined fluidis given by Ω = − β − X i ≥ τ i i ! z i , (5)being its mean number of particles N = X i ≥ i τ i i ! z i . (6)By replacing Eq. (4) in Eqs. (5) and (6), one obtains β Ω = − (cid:16)X i ≥ b i z i (cid:17) V + (cid:16)X i ≥ a i z i (cid:17) A − (cid:16)X i ≥ c i z i (cid:17) L , (7) N = − (cid:16)X i ≥ ib i z i (cid:17) V − (cid:16)X i ≥ ia i z i (cid:17) A + (cid:16)X i ≥ ic i z i (cid:17) L . (8)Consistently, a similar transformation applies to otherthermodynamic magnitudes [ S , U , σ N , see Eq. (3)]. Eqs.(1) and (7) have a similar dependence with the measures.Nevertheless, Eq. (1) is in terms of measures M B thatcorrespond to the choice B as RR, while Eq. (7) is in terms of measures regarding to A as RR. Of course, thesame applies to Eqs. (3) and (8). Eqs. (1-3) show theadvantages in deriving τ i as a linear function of the mea-sures of B . If it is possible, one find the z power seriesfor the intensive thermodynamic properties P, γ, T anddensities ρ, Γ A , Γ L , referred to B .Eq. (4) transpires lineal algebra concepts. It can beseen as a vector in an abstract space with basis of co-ordinates ( V, A, L ) and components ( i ! b i , − i ! a i , i ! c i ) butalso as the inner product ( i ! b i , − i ! a i , i ! c i ) · M A betweenrow and column vectors that live in dual spaces. Theseanalogies flow to Eqs. (7, 8) and will be further investi-gated in the following Secs. III A and III B for differentranges of α .The confinement of the systems drawn in Fig. 1 ispurely characterized by the region A where the densitydistribution is non-null. This density-based choice of B will be labeled with a d subindex (d-RR). For the con-finement shown in Figs. 1a and 1b the unique simplechoice for RR is A itself, this prevents to analyze themfrom the point of view of the freedom to choose the RR.On the other hand, even when the systems shown inFig. 2 can also be analyzed under the same density-based B , other RR could be adopted. To analyze thisproblem and the relation between the thermodynamicproperties obtained under different choices of B , we studythe edges/wedges with angles < α < π and π < α < π separately. It is interesting to note that when the d-RRis adopted the system depicted in Fig. 1a and the systemdrawn in Fig. 2a are identical, and thus, their propertiesare identical too. A. Case I ( < α < π ) For the case of the wedge confinement drawn in Fig.2a, I study two different choices for the RR that are themost natural to be adopted. Under the density-basedRR that identifies B with A , the measures are M d =( V d , A d , L ) , being the i -th cluster integral τ i /i ! = b i V d − a i A d + c i ( β ) L = b d · M d , (9)where b d = ( b i , − a i , c i ) is the vector of coefficients. Thesecond simple choice for B is the empty-region (e-RR),i.e. B is taken as A joined with the white region inFig. 2a and the measures are M e = ( V e , A e , L ) . Inwhat follows we will use σ (see Fig. 2) as the unitlength. From geometrical considerations it is possible toobtain the linear relation between both sets of measures: V d = V e − A e + cot α L for < α < π , A d = A e − csc α L for < α < π , and A d = A e − cot α L for π < α < π .Thus, under the e-RR choice τ i /i ! = ˜ b i V e − ˜ a i A e + ˜ c i ( α ) L = b e · M e . (10)We introduce the matrix Y that transforms between bothsets of measures M d = Y · M e and M e = Y − · M d . (11)Its expression follows from the relations above Eq. (10) Y = −
12 14 cot α − y ( α )0 0 1 , (12)with y ( α ) = csc α if < α < π and y ( α ) = cot α if π < α < π (note that y ( α ) is a continuous non-derivablefunction at α = π ). Given that Ω remains unmodified nomatter which RR is adopted, one finds the linear relationbetween the unknown coefficients b e = (cid:16) ˜ b i , − ˜ a i , ˜ c i (cid:17) andthe known b d , through the Y matrix b e · M e = b d · Y · Y − · M d , b e = b d · Y . (13)Besides, through Eqs. (1-3) Y also transforms the ther-modynamic properties ( − P, γ, T ) e = ( − P, γ, T ) d · Y , (14) ( ρ, Γ A , Γ L ) e = ( ρ, Γ A , Γ L ) d · Y , (15)where the label outside brackets shows the adopted ref-erence region. These relations should be valid even whenthe series expansion in powers of z does not apply, andthus, they should apply to any density. Eqs. (14, 15),with Y taken from Eq. (12), are one of the main resultsof the current work. They show the transformation lawbetween the intensive properties of the confined systemwhen different RRs are adopted.Now, we turn our attention to Eqs. (1, 7) and (14).They show that P e = P d = P being P the pressure ofthe bulk fluid at the same T and µ . Furthermore, oneobtains γ e = γ d + P/ . (16)Eq. (16) found here for a wedge confinement is a knownrelation for fluids adsorbed on both planar and curvedwalls.[15, 20] The z power series representation of γ e and γ d shows that they are the surface tension of the fluidin contact with an infinite planar wall (each one for adifferent RR). The line tension transforms as T e = T d − γ d y ( α ) − P α , (17)To the best of my knowledge it is the first time that Eq.(17), which applies to any density, is derived. Turning toEqs. (3, 8) and (15), they imply that ρ e = ρ d = ρ with ρ the number density of the bulk fluid (at the same T and µ ). For the surface adsorption one finds, (Γ A ) e = (Γ A ) d − ρ/ , (18)which is known to be an exact relation for planar walls.Again, based on the z power series one finds that both (Γ A ) e and (Γ A ) d are the adsorption of the fluid on an infinite planar wall (each one for a different RR). Fur-thermore, one obtains for the excess linear adsorption (Γ L ) e = (Γ L ) d − (Γ A ) d y ( α ) + ρ α . (19)Again, this expression applies to any density and it wasnever published before. B. Case II ( π < α < π ) Now, focusing on the case shown in Fig. 2b, I will ana-lyze two different choices for the region B , which split inthree different sets of measures that are the most naturalto adopt. The first choice is the d-RR, which correspondsto identify B with A . For this d-RR one can considertwo different criteria to define the measures dependingon whether A is taken as the area of the planar part( A p d ) of the surface ∂ B or as its total area. Thus, us-ing the first criteria the measures are M d = ( V d , A p d , L ) and τ i /i ! = b i V d − a i A p d + c i ( α ) L = b d . M d , (20)with the vector of coefficients b d = ( b i , − a i , c i ) . How-ever, if one adopts the second criteria that assumes A asthe total area of ∂ B it is obtained τ i /i ! = ¯ b i V d − ¯ a i A d + ¯ c i ( α ) L = b d . M d , (21)with M d = ( V d , A d , L ) and A p d = A d − ( α − π ) L . Therelationship between both sets of measures is M d = Y · M d , M d = Y − · M d , (22)while the vectors of coefficients relate through b d = b d · Y , (23)with Y = − ( α − π )0 0 1 . (24)The relations between the equations of state ( − P, γ, T ) and also ( ρ, Γ A , Γ L ) , in d1-RR and d2-RR are given byEqs. (14) and (15) with the obvious change of labelsand with Y taken from Eq. (24). Therefore, one finds P d = P d = P (with P the bulk pressure), γ d = γ d (which are equal to the planar-wall surface tension γ d discussed for the case α < π ) and T d = T d − γ d ( α − π ) / . (25)Besides, it is obtained ρ d = ρ d = ρ , (Γ A ) d = (Γ A ) d (which are equal to the planar-wall adsorption (Γ A ) d dis-cussed for the case α < π ) and (Γ L ) d = (Γ L ) d − (Γ A ) d ( α − π ) / . (26)It seems that Eqs. (25) and (26), that apply to anydensity, are novel results.The other choice for B is the e-RR, that is, the join ofregion A and the white region in Fig. 2b. In this case,the measures are M e = ( V e , A e , L ) and τ i /i ! = ˜ b i V e − ˜ a i A e + ˜ c i ( α ) L = b e . M e , (27)with V d = V e − A e − ( α − π ) L and A p d = A e . Eqs.(22, 23) describe the transformation between both, themeasures and the coefficients, they remain valid with thechange of labels d → e and for Y = − − ( α − π )0 1 00 0 1 . (28)Yet non-surprising, following Eqs. (14) and (15) with theobvious change of labels and taking Y from Eq. (28) oneobtains P e = P and ρ e = ρ . Furthermore, both γ e and (Γ A ) e coincide with the planar-wall magnitudes foundfor the case α < π , and then Eqs. (16, 18) apply for thebroad range < α < π . Finally, one obtains T e = T d + P ( α − π ) / , (29) (Γ L ) e = (Γ L ) d − ρ ( α − π ) / . (30)Eqs. (29) and (30) were not published earlier. C. Low density
In this brief digression the confined ideal gas and thelow density regime of the confined non-ideal gas areanalyzed. I first concentrate in the d-RR (d-RR for < α < π , d1- and d2-RR for π < α < π ). To obtainthe properties of the confined ideal gas one truncates allthe series in Eqs. (6, 7, 8) at the first order in power of z . By adopting d-RR the volume V d is equal to Z andthe first cluster integral is τ = V d . Thus, βP = z , ρ = z , γ = T = 0 , and Γ A = Γ L = 0 . (31)Therefore, under d-RR the confined ideal gas is thor-oughly described by βP = ρ , i.e. the equation of stateof the bulk ideal gas . Clearly, if we turn to e-RR theedge/wedge confined ideal gas has non-null surface- andline- free energies. They can be evaluated using Eq. (31)and the transformations discussed in Secs. III B andIII A. The conclusion is that in order to obtain the sim-pler expressions for the thermodynamics of the confinedideal gas, the d-RR is better than e-RR.For the confined non-ideal gas under d-RR the firstcluster integral remains unmodified in comparison withthe ideal gas. The second and higher order τ i could becalculated by direct integration. Now, the series given inEqs. (5, 6, 7, 8) are truncated at order two in z , which gives surface and linear thermodynamic properties pro-portional to z . Using Eqs. (2, 4) and trivial series ma-nipulation one obtains the power series for z ( ρ ) and theseries representation of the thermodynamic properties inpowers of ρ . Up to order ρ it is obtained: βP = ρ − b ρ (i.e. the virial series for the bulk gas [18]) and βγ = − Γ A / , β T = − Γ L / (32)[with Γ A = − a ρ and Γ L = 2 c ρ ]. These notable rela-tions are not well known. They deal with inhomogeneousfluids and link linearly an excess free energy (times β )with the corresponding excess adsorption. It is remark-able that Eq. (32) does not include coefficients relatedto the interparticle potential. Eq. (32) resembles theequation of state of the bulk ideal gas, nevertheless, itapplies to any edge/wedge confined fluid up to order ρ .As can be easily verified, the use of e-RR provides morecomplex expressions for the surface and linear thermo-dynamic properties than Eq. (32). In summary, d-RR isappropriate to obtain a simple description for the ther-modynamics of the confined ideal gas and also of any gasat low density, but e-RR is not. IV. APPLICATION TO HARD SPHERES
Recently, through adopting the d-RR, the low densitybehavior of the hard sphere (HS) confined fluid in anedge/wedge cavity was studied using an analytic expres-sion of c ( α ) .[16] In this section we compare those prop-erties with that found by adopting the e-RR. With thispurpose the natural units for the HS system will be used(which is equivalent to set the particles diameter σ asthe unit length). In Ref. [16] was obtained the followingexact expression c ( α ) = −
115 [1 + ( π − α ) cot α ] (33)that applies for < α < π in the d-RR, while the analyticexpression for π < α < π in d1-RR is c ( α ) = 845 ( α − π ) + Q , (34)with Q = 0 . { − exp [ − .
74 ( α − π )] } . Using theknown parameters b = − π/ , a = − π/ and Eqs.(33, 34) for c ( α ) one readily finds the series expansionof { P, γ, T , Γ A , Γ L } in power of ρ up to order two, byadopting both d-RR and d1-RR. For T and Γ L it givesanalytically the angular dependence with α (up to ordertwo in ρ ).Now, we analyze the consequences of choosing a differ-ent RR on the thermodynamic properties of the confinedHS system. In particular, novel analytic expressions ofrelevant line-thermodynamic properties for e-RR and d2-RR will be derived. Through the use of Eqs. (14, 15),the matrices for RR transformation (12, 24, 28), and thedensity power series of P d , γ d , T d , T d1 , (Γ A ) d , (Γ L ) d , àà àà - - Ρ Γ (cid:144) k T a nd G A Figure 3. Surface tension and surface adsorption. Curvesfor adsorption are marked with squares. In continuous lines(blue) are plotted the magnitudes in d-RR while dot-dashedlines (red) refers to e-RR. (Γ L ) d1 one obtains the series for γ , T , Γ A and Γ L in thee-RR and d2-RR. For the wall-fluid surface tension andexcess area-adsorption, both up to terms of order O ( ρ ) ,it is obtained βγ e = ρ π ρ , (Γ A ) e = − ρ π ρ . (35)Fig. 3 displays the surface tension and surface excessarea-adsorption by adopting d-RR and e-RR. There onecan observe the effect of choosing a different RR in theproperties of the confined HS fluid. For the d-RR onefinds βγ d > and (Γ A ) d < , on the other hand forthe e-RR they yield βγ e < and (Γ A ) e > . Besides,near ρ & the null slope in βγ d and (Γ A ) d is apparentwhile βγ e and (Γ A ) e are linear with density. For T e in therange < α < π there are two branches: the first one for α < π and the second one for α > π , the correspondingexpressions are β T e = − ρ α ρ (cid:20) − (cid:18) π α (cid:19) cot α − π α (cid:21) β T e = − ρ α ρ (cid:20) π − α ) cot α − π α (cid:21) . (36)In Fig. 4 it is shown the low density behavior of theedge line tension T for d-RR and e-RR. In both cases, T is monotonous. For the M d measures T is positive andhas positive slope. On the contrary, using M e measures T is negative and has a negative slope. For both, themodulus of the slope decreases with increasing α . Theobtained expression of (Γ L ) e for < α < π is (Γ L ) e = ρ α − ρ (cid:20) π − α ) cot α + 15 π y ( α ) (cid:21) , (37) - - - - Ρ T (cid:144) k T Figure 4. Line tension of the HS system vs. density for anglesin the range < α < π . Continuous lines (blue) plot T d while dot-dashed lines (red) plot T e . The curves correspondto α = π/ , π/ , π/ while the arrow points to the directionof increasing values of α . - - Ρ G L Figure 5. Line adsorption of the HS system vs. density forangles in the range < α < π , the curves correspond to α = π/ , π/ , π/ . Continuous lines (blue) plot (Γ L ) d whiledot-dashed lines (red) plot (Γ L ) e . See Fig. 4 for details. which is non-derivable at α = π [see Eq. (12)]. Fig. 5 issimilar to Fig. 4 but for the linear adsorption Γ L . Whenthe M d measures are considered the linear adsorption ismonotonous, negative (i.e., there is local desorption) andhas negative slope which increases with increasing α . Onthe other hand, when M e measures are adopted Γ L is notmonotonous, it is positive (i.e., there is local adsorption)and decreases for larger values of α .I also present here a similar analysis for the case π < α < π . The results for the line-tension and ex-cess linear adsorption by adopting e-RR and d2-RR up - - Ρ T (cid:144) k T Figure 6. Line tension of the HS system vs. density for an-gles in the range π < α < π . The curves correspond to α = 5 π/ , π/ , π/ . In long-dashed lines (blue) it is drawn T d1 , short-dashed line (green) corresponds to T d2 and dot-dashed line (red) is for T e . The arrows point to the directionof increasing values of α . to order O ( ρ ) are β T e = α − π ρ − (cid:20)(cid:18) π
12 + 845 (cid:19) ( α − π ) + Q (cid:21) ρ , (cid:0) Γ L (cid:1) e = − α − π ρ + (cid:20) α − π ) + Q (cid:21) ρ , (38) β T d2 = (cid:20)(cid:18) π − (cid:19) ( α − π ) + Q (cid:21) ρ , (cid:0) Γ L (cid:1) d2 = (cid:20)(cid:18) − π (cid:19) ( α − π ) + Q (cid:21) ρ , (39)Fig. 6 shows the linear tension of the HS system in thecase π < α < π . The functions T d1 , T d2 and T e areshown for comparison. In all cases, T is monotonous.For the M d measures T is negative and has negativeslope which decreases with increasing α . On the otherhand, for M e and M d measures, T is positive and hasa positive slope which increases with increasing α . Even, T for M d is nearly zero in the adopted scale. From thecomparison between Fig. 6 and Fig. 4 the inversion ofthe sign of T e at α = π is evident, where the edge/wedgedisappears. Fig. 7 plots the linear adsorption of the HSsystem for the same three measure sets. In the cases of M d and M d we observe a monotonous Γ L . For the M d measures Γ L is a positive (i.e., there is local ad-sorption) increasing function and its slope increases withincreasing α . On the contrary, for M d measures Γ L isa negative (local desorption) decreasing function and itsslope decreases with increasing α . Even, Γ L for M d isnearly zero in the adopted scale. For M e measures Γ L isnot monotonous, is negative (i.e., there is local desorp-tion), attains its minimum near ρ ≃ . and decreases - - Ρ G L Figure 7. Line adsorption of the HS system vs. density forangles in the range π < α < π . In long-dashed lines (blue)it is draw (Γ L ) d1 , short-dashed line (green) corresponds to (Γ L ) d2 and dot-dashed line (red) is for (Γ L ) e . See Fig. 6 formore details. with increasing values of α . Fig. 7 and Fig. 5 show theinversion of the sign of (Γ L ) e when both edge and wedgedisappear at α = π .In the literature, both d-RR and e-RR were usedto study HS systems confined in cavities with differentgeometries.[21, 22] The behavior found with d-RR (in-cluding d1-RR and d2-RR) and e-RR, for both line ten-sion and line free energy, shows that they strongly dependon the adopted reference system. Linear thermodynamicmagnitudes that are less dependent on the adopted refer-ence region can be found by considering the mean valuesof excess density and excess free energy in a region withfinite size around the edge.[16] V. FINAL REMARKS
In this work I studied the relations between the ther-modynamic properties of fluids confined by wedges andedges when different RRs are adopted. The analysis wasbased on the activity series expansion of the grand freeenergy for inhomogenous systems, and on the propertiesof its coefficients the cluster integrals. I utilized a simpleapproach that linearly connects the geometric measures:volume, surface area and edge length, in the differentRR that are considered. From that, the law of transfor-mation of thermodynamic properties between RRs wasdeduced. A similar method was previously used to studya system of hard spheres confined by curved walls. Themethod was here refined and can be used to analyze in-homogenous fluids confined by walls with a variety ofshapes. Under this non-standard approach I have stud-ied the dependence of the linear-thermodynamic prop-erties on the adopted RR along the complete range ofdihedral angles < α < π . Analytic expressions thattransform the thermodynamic intensive properties: pres-sure, surface tension and line tension of the system whendifferent RR are adopted were derived for the first time.Surface adsorption and line adsorption were also ana-lyzed in this framework. The relevant results were givenin Eqs. (17, 19, 25-26, 29-30). Furthermore, the thermo-dynamic properties of both, the confined ideal gas andof the confined real gases at low density (up to order ρ ), were analyzed by adopting different RR. We foundthat the density-based d-RR is advantageous to obtain asimpler analytic description of the studied properties.Regarding to the confined HS fluid, which is a rele-vant reference system both for simple and colloidal fluids,the dependence of line adsorption and line tension withthe edge/wedge dihedral angle and density was analyzed. We found explicit analytic expressions truncated to ordertwo in density that describe these properties for differentRR and for the complete range < α < π . They areshown in Eqs. (36, 37, 38, 39). The new results obtainedfor HS complement those recently published.[16] Giventhat these analytic expressions are exact or quasi-exact,they constitute well defined references that should enableto validate other approximate theories like fundamen-tal measure density functional approaches to edge/wedgeconfined fluids at low density. ACKNOWLEDGMENTS
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