Residual Multiparticle Entropy for a Fractal Fluid of Hard Spheres
aa r X i v : . [ c ond - m a t . s o f t ] J u l Article
Residual Multiparticle Entropy for a Fractal Fluid ofHard Spheres
Andrés Santos * ID , Franz Saija ID and Paolo V. Giaquinta ID Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad deExtremadura, E-06006 Badajoz, Spain; [email protected] CNR-IPCF, Viale F. Stagno d’Alcontres, 37-98158 Messina, Italy; [email protected] Università degli Studi di Messina, Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche eScienze della Terra, Contrada Papardo, 98166 Messina, Italy; [email protected] * Correspondence: [email protected]; Tel.: +34-924-289-651Academic Editor: nameReceived: date; Accepted: date; Published: date
Abstract:
The residual multiparticle entropy (RMPE) of a fluid is defined as the difference, ∆ s , between the excess entropy per particle (relative to an ideal gas with the same temperatureand density), s ex , and the pair-correlation contribution, s . Thus, the RMPE represents thenet contribution to s ex due to spatial correlations involving three, four, or more particles. Aheuristic “ordering” criterion identifies the vanishing of the RMPE as an underlying signature ofan impending structural or thermodynamic transition of the system from a less ordered to a morespatially organized condition (freezing is a typical example). Regardless of this, the knowledge ofthe RMPE is important to assess the impact of non-pair multiparticle correlations on the entropyof the fluid. Recently, an accurate and simple proposal for the thermodynamic and structuralproperties of a hard-sphere fluid in fractional dimension 1 < d < Phys. Rev. E , , 062126]. The aim of this work is to use this approachto evaluate the RMPE as a function of both d and the packing fraction φ . It is observed that, forany given dimensionality d , the RMPE takes negative values for small densities, reaches a negativeminimum ∆ s min at a packing fraction φ min , and then rapidly increases, becoming positive beyonda certain packing fraction φ . Interestingly, while both φ min and φ monotonically decrease asdimensionality increases, the value of ∆ s min exhibits a nonmonotonic behavior, reaching an absoluteminimum at a fractional dimensionality d ≃ ∆ s / | ∆ s min | shows aquasiuniversal behavior in the region − . φ − φ . Keywords: residual multiparticle entropy; hard spheres; fractal dimension
1. Introduction
The properties of liquids are of great interest in many science and engineering areas. Aside fromordinary three-dimensional systems, many interesting phenomena do also occur in restricted one- ortwo-dimensional geometries, under the effect of spatial confinement. Actually, there are also caseswhere the configuration space exhibits, at suitable length scales, non-integer dimensions. Indeed,many aggregation and growth processes can be described quite well by resorting to the concepts offractal geometry. This is the case, for example, of liquids confined in porous media or of assembliesof small particles forming low-density clusters and networks [1–4].Heinen et al. [5] generalized this issue by introducing fractal particles in a fractal configurationspace. In their framework the particles composing the liquid are fractal as is the configuration spacein which such objects move. Santos and López de Haro [6] have further developed reliable heuristicinterpolations for the equation of state and radial distribution function of hard-core fluids in fractaldimensions between one and three dimensions. Taking advantage of their work, we intend to study of 10 in this paper some thermostatistical properties of such fractal systems in the theoretical frameworkprovided by the multiparticle correlation expansion of the excess entropy, s ex ( ρ , β ) = s ( ρ , β ) − s id ( ρ , β ) , (1)where ρ is the number density, β = k B T is the inverse temperature, s ( ρ , β ) is the entropy per particle(in units of the Boltzmann constant k B ), and s id ( ρ , β ) = d + − ln " ρ (cid:18) h β π m (cid:19) d /2 (2)is the ideal-gas entropy; in Eq. (2), d is the spatial dimensionality of the system, h is Planck’s constant,and m is the mass of a particle.As is well known, the excess entropy can be expressed as an infinite sum of contributionsassociated with spatially integrated density correlations of increasing order [7,8]. In the absenceof external fields, the leading and quantitatively dominant term of the series is the so-called “pairentropy”, s ( ρ , β ) = − ρ Z d r [ g ( r ; ρ , β ) ln g ( r ; ρ , β ) − g ( r ; ρ , β ) + ] , (3)whose calculation solely requires the knowledge of the pair distribution function of the fluid, g ( r ; ρ , β ) .An integrated measure of the importance of more-than-two-particle density correlations in the overallentropy balance is given by the so-called “residual multiparticle entropy” (RMPE) [9]: ∆ s ( ρ , β ) = s ex ( ρ , β ) − s ( ρ , β ) . (4)It is important to note that, at variance with s ex and s , which are both negative definite quantities, ∆ s may be either negative or positive. As originally shown by Giaquinta and Giunta for hard spheres inthree dimensions [9], the sign of this latter quantity does actually depend on the thermodynamic stateof the fluid. In fact, the RMPE of a hard-sphere fluid is negative at low densities, thus contributing toa global reduction of the phase space available to the system as compared to the corresponding idealgas. However, the RMPE undergoes a sharp crossover from negative to positive values at a value ofthe packing fraction which substantially overlaps with the thermodynamic freezing threshold of thehard-sphere fluid. Such a behavior suggests that at high enough densities multiparticle correlationsplay an opposite role with respect to that exhibited in a low packing regime in that they temper thedecrease of the excess entropy that is largely driven by the pair entropy. The change of sign exhibitedby the RMPE is a background indication, intrinsic to the fluid phase, that particles, forced by more andmore demanding packing constraints, start exploring, on a local scale, a different structural condition.This process is made possible by an increasing degree of cooperativity, that is signalled by the positivevalues attained by ∆ s , which gradually leads to a more efficacious spatial organization and ultimatelytriggers the crystalline ordering of the system on a global scale.A similar indication is also present in the RMPE of hard rods in one dimension [10]. Inthis model system, notwithstanding the absence of a fluid-to-solid transition, one can actuallyobserve the emergence of a solid-like arrangement at high enough densities: tightly-packed particlesspontaneously confine themselves within equipartitioned intervals whose average length is equalto the the total length per particle, even if the onset of a proper entropy-driven phase transition isfrustrated by topological reasons. Again, even in this “pathological” case, the vanishing of the RMPEshows up as an underlying signature of a structural change which eventually leads to a more orderedarrangement.The relation between the zero-RMPE threshold and the freezing transition of hard spheresapparently weakens in four and five dimensions [11], where lower bounds of the entropy thresholdsignificantly overshoot the currently available computer estimates of the freezing density [11,12]. On of 10 the other side, a close correspondence between the sign crossover of the RMPE and structural orthermodynamical transition thresholds has been highlighted in both two and three dimensions ona variety of model systems for different macroscopic ordering phenomena other than freezing [13],including fluid demixing [14], the emergence of mesophases in liquid crystals [15], the formation of ahydrogen-bonded network in water [16], or, more recently, the onset of glassy dynamics [17].If hard-core systems in fractal geometries exhibit a sort of disorder-to-order transition, it seemsplausible that such a transition is signaled by a change of sign of ∆ s . Taking all of this into account, itis desirable to study the RMPE of hard-core fractal fluids, and this is the main goal of this paper. It isorganized as follows. The theoretical approach of Ref. [6] is described and applied to the evaluationof the RMPE in Sec. 2. The results are presented and discussed in Sec. 3. Finally, the main conclusionsof the work are recapped in Sec. 4.
2. Methods
In principle, the knowledge of the pair distribution function, g ( r ; ρ , β ) , allows one to determinethe pair entropy from Eq. (3). This is equivalent to s ( ρ , β ) = [ χ T ( ρ , β ) − ] + e s ( ρ , β ) , (5)where χ T ( ρ , β ) = + ρ Z d r [ g ( r ; ρ , β ) − ] (6)is the isothermal susceptibility and we have called e s ( ρ , β ) = − ρ Z d r g ( r ; ρ , β ) ln g ( r ; ρ , β ) . (7)Thus, Eq. (4) can be rewritten as ∆ s ( ρ , β ) = s ex ( ρ , β ) − [ χ T ( ρ , β ) − ] − e s ( ρ , β ) . (8)Equations (5)–(8) hold regardless of whether the total potential energy U ( r , r , r , . . . ) is pairwiseadditive or not. On the other hand, if U is pairwise additive, the knowledge of g ( r ; ρ , β ) yields, apartfrom s ( ρ , β ) , the thermodynamic quantities of the system via the so-called thermodynamic routes[18]. In particular, the virial route is Z ( ρ , β ) ≡ β p ( ρ , β ) ρ = − ρβ d Z d r r d u ( r ) d r g ( r ; ρ , β )= + ρ d Z d r r d e − β u ( r ) d r y ( r ; ρ , β ) , (9)where p is the pressure, Z is the compressibility factor, u ( r ) is the pair interaction potential, and y ( r ; ρ , β ) ≡ e β u ( r ) g ( r ; ρ , β ) is the so-called cavity function. Next, the excess Helmholtz free energy perparticle, a ex , and the excess entropy per particle, s ex , can be obtained by standard thermodynamicrelations as β a ex ( ρ , β ) = Z d t Z ( ρ t , β ) − t , s ex ( ρ , β ) = β ∂β a ex ( ρ , β ) ∂β − β a ex ( ρ , β ) . (10) of 10 Combining Eqs. (9) and (10), we obtain s ex ( ρ , β ) = ρ d (cid:18) β ∂∂β − (cid:19) Z d r r d e − β u ( r ) d r Z d t y ( r ; ρ t , β ) . (11)To sum up, assuming the pair distribution function g ( r ; ρ , β ) for a d -dimensional fluid of particlesinteracting via an interaction potential u ( r ) is known, it is possible to determine the excess entropy[see Eq. (1)], the pair entropy [see Eq. (3)], and hence the RMPE ∆ s . Note that, while s only requires g ( r ) at the state point ( ρ , β ) of interest, s ex requires the knowledge of g ( r ) at all densities smaller than ρ and at inverse temperatures in the neighborhood of β .A remark is now in order. The isothermal susceptibility χ T ( ρ , β ) can be obtained directly from g ( r ; ρ , β ) via Eq. (6) or indirectly via Eq. (9) and the thermodynamic relation χ − T ( ρ , β ) = ∂ρ Z ( ρ , β ) ∂ρ . (12)If the correlation function g ( r ; ρ , β ) is determined from an approximate theory, the compressibilityroute (6) and the virial route given by Eqs. (9) and (12) yield, in general, different results. Now we particularize to hard-sphere fluids in d dimensions. The interaction potential is simplygiven by u ( r ) = ( ∞ , r < σ ,0, r > σ , (13)where σ is the diameter of a sphere. In this case, the pair distribution function g ( r ; φ ) is independentof temperature and the density can be characterized by the packing fraction φ ≡ ( π /4 ) d /2 Γ ( + d /2 ) ρσ d . (14)Taking into account that dd r e − β u ( r ) = δ ( r − σ ) , Eqs. (9) and (11) become Z ( φ ) = + d − φ g c ( φ ) , (15) s ex ( φ ) = − β a ex ( φ ) = d − φ Z d t g c ( φ t ) , (16)where g c ( φ ) = g ( σ + ; φ ) = y ( σ ; φ ) is the contact value of the pair distribution function. Also, Eq. (7)can be written as e s ( φ ) = − d d − φ Z ∞ d r r d − g ( r ; φ ) ln g ( r ; φ ) . (17)In Eqs. (14)–(17) it is implicitly assumed that d is an integer dimensionality. However, in a pioneeringwork [5] Heinen et al. introduced the concept of classical liquids in fractal dimension and performedMonte Carlo (MC) simulations of fractal “spheres” in a fractal configuration space, both with the samenoninteger dimension. Such a generic model of fractal liquids can describe, for instance, microphaseseparated binary liquids in porous media and highly branched liquid droplets confined to a fractalpolymer backbone in a gel. For a discussion on the use of two-point correlation functions in fractalspaces, see Ref. [19].It seems worthwhile extending Eqs. (14)–(17) to a noninteger dimension d and studying thebehavior of the RMPE ∆ s as a function of both φ and d . To this end, an approximate theoryproviding the pair distribution function g ( r ; φ ) for noninteger d is needed. In Ref. [5], Heinen etal. solved numerically the Ornstein–Zernike relation [20] by means of the Percus–Yevick (PY) closure of 10 [21]. However, since one needs to carry out an integration in Eq. (17) over all distances, an analyticapproximation for g ( r ; φ ) seems highly desirable.In Ref. [6] a simple analytic approach was proposed for the thermodynamic and structuralproperties of the fractal hard-sphere fluid. Comparison with MC simulation results for d = g c ( φ ) = − k d φ ( − φ ) , (18)with k d = ( − d )( − d ) + ( − d )( d − ) k , k = √ π − ≃ d =
1, 2, and 3, Eq. (18) gives the exact [18], the Henderson [22], and thePY [23,24] results, respectively. Insertion into Eq. (15) gives the compressibility factor Z ( φ ) and, byapplication of Eq. (12), the isothermal susceptibility as χ T ( φ ) = (cid:20) + d − φ − k d φ ( − φ )( − φ ) (cid:21) − . (20)Analogously, Eq. (16) yields s ex ( φ ) = − d − (cid:20) ( − k d ) φ − φ − k d ln ( − φ ) (cid:21) . (21)Thus, in order to complete the determination of ∆ s from Eq. (8), only e s remains. It requires theknowledge of the full pair distribution function [see Eq. (17)]. In the approximation of Ref. [6], g ( r ; φ ) is given by the simple interpolation formula g ( r ; φ ) = α ( φ ) g (cid:16) r ; φ eff1D ( φ ) (cid:17) + [ − α ( φ )] g (cid:16) r ; φ eff3D ( φ ) (cid:17) , (22)where g ( r ; φ ) and g ( r ; φ ) are the exact and PY functions for d = φ eff1D ( φ ) = g c ( φ ) − g c ( φ ) , φ eff3D ( φ ) = + g c ( φ ) − p + g c ( φ ) g c ( φ ) (23)are effective packing fractions, and α ( φ ) = H ( φ ) − H (cid:0) φ eff3D ( φ ) (cid:1) H (cid:0) φ eff1D ( φ ) (cid:1) − H (cid:0) φ eff3D ( φ ) (cid:1) (24)is the mixing parameter. In Eq. (24), H ( φ ) = − A d φ + C d φ + ( d − ) φ [ + ( − d )( − k )( − φ ) φ ] , (25)with A d = ( − d )( − d ) + ( d − )( − d ) k , C d = ( − d )( − d ) + ( d − )( − d ) k . (26)Of course, H ( φ ) and H ( φ ) are obtained from Eq. (25) by setting d = d =
3, respectively.Summing up, the proposal of Ref. [6] for noninteger d is defined by Eqs. (22)–(24), with g c ( φ ) and H ( φ ) being given by Eqs. (18) and (25), respectively. By construction, this approximation reducesto the exact and PY results in the limits d → d →
3, respectively. Moreover, it is consistent (via of 10 both the virial and compressibility routes) with Henderson’s equation of state [22] in the limit d → ∆ s ( φ ) can be obtained from Eq. (8) by evaluating e s ( φ ) from Eq. (17) numerically. To that end,and in order to avoid finite-size effects, it is convenient to split the integration range 0 < r < ∞ into0 < r < R and R < r < ∞ , with R = σ . In the first integral the analytically known function g ( r ; φ ) is used, while in the second integral g ( r ; φ ) is replaced by its asymptotic form [6].
3. Results and Discussion ex s s e x , s d=1(a) min d=3 d=2.5d=2d=1.5d=1 s s min Figure 1. ( a ) Plot of s ex ( φ ) (solid lines) and s ( φ ) (dashed lines) for dimensions d =
1, 1.5, 2, 2.5, and3. The circles indicate the points where s ex ( φ ) and s ( φ ) cross. ( b ) Plot of ∆ s ( φ ) = s ex ( φ ) − s ( φ ) for d =
1, 1.5, 2, 2.5, and 3. The triangles indicate the location of the minima and the circles indicate thepacking fractions φ where ∆ s = Figure 1a shows s ex ( φ ) and s ( φ ) as functions of the packing fraction for a few dimensions1 ≤ d ≤
3. In all the cases, both functions become more negative as the packing fraction increases.Moreover, at a common packing fraction φ , both s ex ( φ ) and s ( φ ) decrease as the dimensionalityincreases. This is an expected property in the conventional case of integer d since, at a common φ ,all the thermodynamic quantities depart more from their ideal-gas values with increasing d . Notsurprisingly, this property is maintained in the case of noninteger d .Figure 1a also shows that the pair entropy s ( φ ) overestimates the excess entropy s ex ( φ ) forpacking fractions smaller than a certain value φ . This means that, if φ < φ , the cumulated effect ofcorrelations involving three, four, five, . . . particles produces a decrease of the entropy. The oppositesituation occurs, however, if φ > φ . At the threshold point φ = φ the cumulated effect ofmultiparticle correlations cancels and then only the pair correlations contribute to s ex .The density dependence of the RMPE ∆ s = s ex − s is shown in Fig. 1b for the same values of d as in Fig. 1a. The qualitative shape of ∆ s ( φ ) is analogous for all d : ∆ s starts with a zero value at φ = ∆ s min at a certain packing fraction φ min , after which itgrows very rapidly, crossing the zero value at the packing fraction φ . of 10 s m i n d 2.38 (a) min , m i n d - min Figure 2. ( a ) Plot of ∆ s min as a function of d . The circle and the arrow indicate the location of theminimum at d ≃ b ) Plot of φ (solid line), φ min (dashed line), and the difference φ − φ min (dotted line) as functions of d . The horizontal solid line signals the value φ − φ min = φ = d = φ = d = The dimensionality dependence of the minimum value of the RMPE, ∆ s min , is displayed in Fig.2a. Interestingly enough, as can also be observed in Fig. 1a, ∆ s min presents a nonmonotonic variationwith d , having an absolute minimum ∆ s min ≃ − d ≃ s represents the largest overestimate of the excess entropy s ex . In contrast to ∆ s min ,both φ and φ min decay monotonically with increasing d . This is clearly observed from Fig. 2b,where also the fluid-hexatic and the fluid-crystal transition points for disks and spheres, respectively,are shown. The proximity of those two points to the curve φ provide support to the zero-RMPEcriterion, especially considering the approximate character of our simple theoretical approach. Thus,if a disorder-to-order transition phase is possible for fractal hard-core liquids, we expect that it islocated near (possibly slightly above) the packing fraction φ . -0.8 -0.6 -0.4 -0.2 0.0-1.0-0.50.00.51.0 d=3d=1 s / | s m i n | - (a) -0.12 -0.08 -0.04 0.00-1.0-0.50.00.5 (b) d=1 d=1.5 d=2 d=2.5 d=3 Eq. (27) s / | s m i n | - Figure 3. ( a ) Plot of the scaled RMPE ∆ s / | ∆ s min | as a function of the difference φ − φ for dimensions d =
1, 1.5, 2, 2.5, and 3. ( b ) Magnification of the framed region of panel a . The light thick linerepresents the formula given by Eq. (27). An interesting feature of Fig. 2b is that the difference φ − φ min ≃ d . This suggests the possibility of a quasiuniversal behavior of the scaled RMPE ∆ s / | ∆ s min | in theneighborhood of φ = φ . To check this possibility, Fig. 3a shows ∆ s / | ∆ s min | as a function of φ − φ for the same dimensionalities as in Fig. 1. We can observe a relatively good collapse of the curves inthe region − . φ − φ . of 10 can be obtained as follows. Let us define X ≡ ( φ − φ ) /0.109 and Y ( X ) ≡ ∆ s ( φ ) / | ∆ s min | . Then, acubic function Y ( X ) consistent with the conditions Y ( ) = Y ( − ) = − Y ′ ( − ) = Y ′′ ( − ) > Y ( X ) = X (cid:2) + X + c ( + X ) (cid:3) with c <
1. A good agreement is found with 0.8 < c < c = ∆ s ( φ ) | ∆ s min | ≃ X h + X + c ( + X ) i , X ≡ φ − φ c = ≤ d ≤ ∆ s have been obtained from Eq. (8) by evaluating e s from Eq. (17) numerically. Since inEq. (20) we have followed the virial route, here we will refer to this method to obtain the function ∆ s as the virial route and denote the resulting quantity as ∆ s vir . On the other hand, this methodis not exactly equivalent to that obtained from Eq. (1) with s evaluated numerically from Eq. (3) byfollowing the same procedure as described above for e s . This alternative method will be referred to asthe compressibility route ( ∆ s comp ), since it is equivalent to evaluating the isothermal compressibilityfrom Eq. (6). Therefore, according to Eq. (8), ∆ s vir − ∆ s comp = − (cid:16) χ vir T − χ comp T (cid:17) . (28)We have checked that both methods (virial and compressibility) yield practically indistinguishableresults. For instance, if d = φ = φ = d = d = φ = φ = ∆ s has still a virial “relic” in the contribution comingfrom the excess free energy, Eq. (21). A pure compressibility route would require the numericalevaluation of χ T from Eq. (6) and then a double numerical integration, as evident from Eqs. (10)and (12). This procedure would complicate enormously the evaluation of s ex without any significantgain in accuracy.
4. Conclusions
In this article we have calculated the pair contribution and the cumulative contribution arisingfrom correlations involving more than two particles to the excess entropy of hard spheres in fractionaldimensions 1 < d <
3. To this end, we have resorted to the analytical approximations for the equationof state and radial distribution function of the fluid previously set up by Santos and López de Haro[6]. Over the fractional dimensionality range explored, the so-called “residual multiparticle entropy”(RMPE), obtained as the difference between the excess and pair entropies, shows a behavior utterlysimilar to that exhibited for integer 1, 2, and 3 dimensions. Hence, on a phenomenological continuitybasis, we surmise that hard spheres undergo an “ordering” transition even in a space with fractionaldimensions, which may well anticipate a proper thermodynamic fluid-to-solid phase transition.We found that the packing fraction loci of minimum and vanishing RMPE show a monotonicdecreasing behavior as a function of the dimensionality; this result is coherent with the magnificationof excluded-volume effects produced by increasing spatial dimensionalities and, correspondingly,with a gradual shift of the ordering transition threshold to lower and lower packing fractions.However, it also turns out that the minimum value of the RMPE exhibits a non-monotonic behavior,attaining a minimum at the fractional dimensionality d = d the relativeentropic weight of more-than-two-particle correlations reaches, in the “gas-like” regime, its maximumabsolute value.Finally, the quasi-universal scaling of the RMPE over its minimum value in the neighborhoodof the sign-crossover point suggests that the properties of the local ordering phenomenon should notsensitively depend on the spatial dimensionality. of 10 Author Contributions:
A.S. proposed the idea and performed the calculations; F.S. and P.V.G. participated inthe analysis and discussion of the results; the three authors worked on the revision and writing of the finalmanuscript.
Funding:
A.S. acknowledges financial support from the Ministerio de Economía y Competitividad (Spain)through Grant No. FIS2016-76359-P and the Junta de Extremadura (Spain) through Grant No. GR18079, bothpartially financed by Fondo Europeo de Desarrollo Regional funds.
Acknowledgments:
A.S. is grateful to Dr. Roberto Trasarti-Battistoni for helpul discussions and for bringing Ref.[19] to our attention.
Conflicts of Interest:
The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:RMPE Residual Multiparticle EntropyMC Monte CarloPY Percus–Yevick
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