Local and global properties of mixtures in one-dimensional systems. II. Exact results for the Kirkwood-Buff integral
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Local and global properties of mixtures in one-dimensional systems. II. Exact resultsfor the Kirkwood–Buff integrals
Arieh Ben-Naim ∗ The Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel
Andr´es Santos † Departamento de F´ısica, Universidad de Extremadura, E-06071 Badajoz, Spain (Dated: October 31, 2018)The Kirkwood–Buff integrals for two-component mixtures in one-dimensional systems are cal-culated directly. The results are applied to square-well particles and found to agree with thoseobtained by the inversion of the Kirkwood–Buff theory of solutions.
I. INTRODUCTION
In Part I of this series one of us has studied boththe global and the local properties of mixtures of sim-ple particles in one-dimensional system. This work hasbeen part of a more general advocacy in favor of thestudy of local properties of liquid mixtures. Instead ofthe traditional study of mixtures based on the global properties, such as excess Gibbs energy, entropy, volume,etc. we have advocated a shift in the paradigm towardsfocusing on the local properties of the same mixtures,such as affinities between two species (embodied in theKirkwood–Buff integrals), and derived quantities such aslocal composition, preferential solvation, and solvationthermodynamic quantities.The local properties, though equivalent to and deriv-able from the global properties, offer a host of new in-formation on the local environments of each molecularspecies in the mixture. This information is not conspic-uous from the global properties. Therefore, the study ofthe local quantities offer a new and more detailed andinteresting view of mixtures.In this paper we have recalculated the Kirkwood–Buffintegrals (KBI) directly for two-component mixtures ofparticles interacting via square-well (SW) potential.In the next section, we outline the derivation of thepair correlation functions for two-component systems in1D system, for arbitrary nearest-neighbor interactions.In section III we present a sample of results for mixturesof SW particles. It is shown that the results are in quan-titative agreement with those obtained in Part I, whichwere based on the partition function method and the in-version of the Kirkwood–Buff (KB) theory of solution.We have also calculated the limiting values of the KBIwhen one of the species has a vanishing mole fraction,which we could not have done from the partition func-tion methods.Another question examined both numerically and the-oretically is the deviations from symmetrical ideal solu-tions and its relation with the stability of the mixtures.It is shown that no miscibility gap can occur in such mix-tures.
II. THEORETICAL BACKGROUND
It is known that the correlation and thermodynamicproperties of any one-dimensional homogeneous systemin equilibrium can be derived exactly, provided that ev-ery particle interacts only with its nearest neighbors.
The aim of this section is to present a self-contained sum-mary of the exact solution. Although the scheme extendsto any number of components, here we focus on the two-component case. A. Correlation functions
Let us consider a binary one-dimensional fluid mix-ture at temperature T , pressure P , and number densities ρ α ( α = A, B ). The particles are assumed to interactonly between nearest neighbors via interaction potentials U αβ ( R ). Before considering the pair correlation functions g αβ ( R ), it is convenient to introduce some probabilitydistributions.Given a particle of species α at a certain position, let p ( ℓ ) αβ ( R ) dR be the conditional probability of finding as its ℓ th neighbor in some direction a particle of species β ata distance between R and R + dR . If ℓ ≥ ℓ − α in the same direction(being located at some point R ′ between 0 and R ) is alsoa first neighbor of β . Therefore, the following recurrencecondition holds p ( ℓ ) αβ ( R ) = X γ = A,B Z R dR ′ p ( ℓ − αγ ( R ′ ) p (1) γβ ( R − R ′ ) , (2.1)where p (1) αβ ( R ) is the nearest-neighbor probability dis-tribution function. On physical grounds, the ratio p (1) αA ( R ) /p (1) αB ( R ) must become the same for α = A asfor α = B in the limit of large R , i.e.,lim R →∞ p (1) AA ( R ) p (1) AB ( R ) = lim R →∞ p (1) BA ( R ) p (1) BB ( R ) . (2.2)This relation will be used later on. The total probabilitydensity of finding a particle of species β , given that aparticle of species α is at the origin, is p αβ ( R ) = ∞ X ℓ =1 p ( ℓ ) αβ ( R ) . (2.3)The convolution structure of Eq. (2.1) suggests the in-troduction of the Laplace transforms e p ( ℓ ) αβ ( s ) = Z ∞ dR e − sR p ( ℓ ) αβ ( R ) , e p αβ ( s ) = Z ∞ dR e − sR p αβ ( R ) , (2.4)so that Eq. (2.1) becomes e p ( ℓ ) αβ ( s ) = X γ = A,B e p ( ℓ − αγ ( s ) e p (1) γβ ( s ) . (2.5)Equation (2.5) allows us to express e p ( ℓ ) αβ ( s ) in terms of thenearest-neighbor distribution as e p ( ℓ ) ( s ) = he p (1) ( s ) i ℓ , (2.6)where e p ( ℓ ) ( s ) is the 2 × e p ( ℓ ) αβ ( s ). FromEqs. (2.3) and (2.6) we get e p ( s ) = ∞ X ℓ =1 he p (1) ( s ) i ℓ = e p (1) ( s ) · h I − e p (1) ( s ) i − , (2.7)where e p ( s ) is the 2 × e p αβ ( s ) and I is the 2 × g αβ ( R )and the probability density p αβ ( R ) are simply related by p αβ ( R ) = ρ β g αβ ( R ) or, equivalently in Laplace space, e p αβ ( s ) = ρ β e g αβ ( s ) , (2.8)where e g αβ ( s ) = Z ∞ dR e − sR g αβ ( R ) (2.9)is the Laplace transform of g αβ ( R ). Therefore, thanksto the one-dimensional nature of the model and the re-striction to nearest-neighbor interactions, the knowledgeof the nearest-neighbor distributions p (1) αβ ( R ) suffices toobtain the pair correlation functions g αβ ( R ). More ex-plicitly, from Eqs. (2.7) and (2.8) the Laplace transforms e g αβ ( s ) are found to be e g AA ( s ) = 1 ρ T Q AA ( s ) [1 − Q BB ( s )] + Q AB ( s ) x A D ( s ) , (2.10) e g BB ( s ) = 1 ρ T Q BB ( s ) [1 − Q AA ( s )] + Q AB ( s ) x B D ( s ) , (2.11) e g AB ( s ) = 1 ρ T Q AB ( s ) √ x A x B D ( s ) , (2.12)where ρ T = ρ A + ρ B is the total number density, x α = ρ α /ρ T is the mole fraction of species α , and we havecalled Q αβ ( s ) ≡ r x α x β e p (1) αβ ( s ) , (2.13) D ( s ) ≡ [1 − Q AA ( s )] [1 − Q BB ( s )] − Q AB ( s ) . (2.14)The KBI in the one-dimensional case are defined by G αβ = 2 Z ∞ dR [ g αβ ( R ) − . (2.15)In terms of the Laplace transform e g αβ ( s ), Eq. (2.15) canbe rewritten as G αβ = 2 lim s → (cid:20)e g αβ ( s ) − s (cid:21) . (2.16)We see that only the nearest-neighbor distribution p (1) αβ ( R ) is needed to close the problem. It can beproven that p (1) αβ ( R ) is just proportional to the Boltz-mann factor e − U αβ ( R ) /k B T times a decaying exponen-tial e − ξR , where the damping coefficient is ξ = P/k B T .Therefore, p (1) αβ ( R ) = x β K αβ e − U αβ ( R ) /k B T e − ξR , (2.17)where the proportionality constants K αβ = K βα (whichof course depend on the thermodynamic state of the mix-ture) will be determined below by applying physical con-sistency conditions. Taking Laplace transforms in Eq.(2.17) and inserting the result into Eq. (2.13) we get Q αβ ( s ) = √ x α x β K αβ Ω αβ ( s + ξ ) , (2.18)where Ω αβ ( s ) = Z ∞ dR e − sR e − U αβ ( R ) /k B T (2.19)is the Laplace transform of e − U αβ ( R ) /k B T .To recapitulate, given the interaction potentials U αβ ( R ) and given a particular thermodynamic state( P, T, x A ), the three correlation functions are obtained(in Laplace space) from Eqs. (2.10)–(2.12), supplementedby Eqs. (2.14), (2.18), and (2.19). B. Equation of state
In order to close the exact solution, it only remains todetermine the total density ρ T (equation of state) and theamplitudes K αβ as functions of P , T , and x A = 1 − x B .As said above, they can be easily obtained by applyingbasic physical conditions. First, note that Eq. (2.2) es-tablishes the following relationship K AB = K AA K BB . (2.20)Next, the physical condition lim R →∞ g αβ ( R ) = 1 impliesthat e g αβ ( s ) → /s for small s . According to Eqs. (2.10)–(2.12), this is only possible if D (0) = 0, so that D ( s ) → D ′ (0) s for small s , where D ′ ( s ) = dD ( s ) /ds . Thus, onehas [1 − Q AA (0)][1 − Q BB (0)] − Q AB (0) = 0 , (2.21) ρ T = Q AB (0) √ x A x B D ′ (0) , (2.22) ρ T = Q AA (0)[1 − Q BB (0)] + Q AB (0) x A D ′ (0) , (2.23) ρ T = Q BB (0)[1 − Q AA (0)] + Q AB (0) x B D ′ (0) . (2.24)Elimination of ρ T between Eqs. (2.22)–(2.24) yields twocoupled equations which, together with Eq. (2.21), gives K AA = 1 − x B K AB Ω AB ( ξ ) x A Ω AA ( ξ ) , (2.25) K BB = 1 − x A K AB Ω AB ( ξ ) x B Ω BB ( ξ ) . (2.26)Insertion of Eqs. (2.25) and (2.26) into Eq. (2.20) allowsone to obtain a quadratic equation for K AB whose phys-ical root is K AB = 1Ω AB ( ξ ) 1 − p − x A x B (1 − R )2 x A x B (1 − R ) , (2.27)where we have called R ≡ Ω AA ( ξ )Ω BB ( ξ )Ω AB ( ξ ) . (2.28)It is interesting to note that, since K αβ and Ω αβ are positive definite, Eq. (2.25) and (2.26) imply that x α K AB Ω AB ( ξ ) < α = A, B , i.e., K AB Ω AB ( ξ ) < min (cid:18) x A , x B (cid:19) ≤ . (2.29)Finally, the density ρ T is obtained from either of Eqs.(2.22)–(2.24). The result is ρ T ( P, T, x A ) = − x A K AA Ω ′ AA ( ξ ) + x B K BB Ω ′ BB ( ξ ) + 2 x A x B K AB Ω ′ AB ( ξ ) , (2.30)where Ω ′ αβ ( s ) is the first derivative of Ω αβ ( s ).Equations (2.25)–(2.28) and (2.30) complete the fulldetermination of e g αβ ( s ) and the equation of state for any choice of the nearest-neighbor interaction potentials U αβ ( x ) and of the thermodynamic state ( P, T, x A ). C. Kirkwood–Buff integrals
The KBI G αβ can be derived, according to Eq. (2.16),by expanding s e g αβ ( s ) in powers of s as s e g αβ ( s ) = 1 + G αβ s + · · · and identifying the linear term. After somealgebra one gets G AB = ρ T J + 2 Ω ′ AB ( ξ )Ω AB ( ξ ) , (2.31) G AA = ρ T J − x B K BB Ω ′ BB ( ξ ) x A K AB Ω AB ( ξ ) − ρ T x A , (2.32) G BB = ρ T J − x A K AA Ω ′ AA ( ξ ) x B K AB Ω AB ( ξ ) − ρ T x B , (2.33)where J ≡ x A K AA Ω ′′ AA ( ξ ) + x B K BB Ω ′′ BB ( ξ )+2 x A x B K AB Ω ′′ AB ( ξ ) − x A x B K AB Ω ′ AA ( ξ )Ω ′ BB ( ξ ) − [Ω ′ AB ( ξ )] Ω AB ( ξ ) . (2.34)The knowledge of the KBI allows us to obtain the (re-duced) isothermal compressibility χ = k B T (cid:18) ∂ρ T ∂P (cid:19) T,x A (2.35)by means of χ = 11 + ρ T x A x B ∆ AB [1 + ρ T ( x A G AA + x B G BB )+ ρ T x A x B (cid:0) G AA G BB − G AB (cid:1)(cid:3) , (2.36)where ∆ AB ≡ G AA + G BB − G AB . (2.37)It can be checked that the resulting expression of χ (which, due to its length, will be omitted here) coincideswith the one obtained as χ = ( ∂ρ T /∂ξ ) T,x A from Eq.(2.30). This confirms the exact character of the solution.Making use of Eqs. (2.30)–(2.33), it is easy to provethat 1 + ρ T x A x B ∆ AB = 2 K AB Ω AB ( ξ ) − , (2.38)which, according to Eq. (2.29), is a positive definite quan-tity. More explicitly, from Eq. (2.27) we have1 + ρ T x A x B ∆ AB = p − x A x B (1 − R ) . (2.39)Therefore, the denominator in Eq. (2.36) never vanishesand the isothermal compressibility is well defined. Thisagrees with van Hove’s classical proof that no phasetransition can exist in this class of nearest-neighbor one-dimensional models.Let us now obtain the KBI in the infinite dilution limit x A →
0. In that limit, Eqs. (2.25)–(2.27) and (2.30)become K AA = Ω BB ( ξ )Ω AB ( ξ ) , K BB = 1Ω BB ( ξ ) , K AB = 1Ω AB ( ξ ) , (2.40) ρ T = − Ω BB ( ξ )Ω ′ BB ( ξ ) . (2.41)Analogously, from Eqs. (2.31)–(2.34) one gets G AB = − Ω ′′ BB ( ξ )Ω ′ BB ( ξ ) + 2 Ω ′ AB ( ξ )Ω AB ( ξ ) , (2.42) G AA = − Ω ′′ BB ( ξ )Ω ′ BB ( ξ ) +4 Ω ′ AB ( ξ )Ω AB ( ξ ) − AA ( ξ )Ω ′ BB ( ξ )Ω AB ( ξ ) , (2.43) G BB = − Ω ′′ BB ( ξ )Ω ′ BB ( ξ ) + 2 Ω ′ BB ( ξ )Ω BB ( ξ ) , (2.44)∆ AB = 2Ω ′ BB ( ξ ) (cid:20) BB ( ξ ) − Ω AA ( ξ )Ω AB ( ξ ) (cid:21) . (2.45)Note that special care is needed to obtain K AA and G AA . D. Chemical potentials and solvation Gibbsenergies
Finally, let us get an explicit expression for the chemi-cal potential. From the KB theory of solution we have k B T (cid:18) ∂µ A ∂x A (cid:19) P,T = 1 x A − ρ T x A ∆ AB ρ T x A x B ∆ AB = 1 x A − x B p − x A x B (1 − R ) − p − x A x B (1 − R ) , (2.46) BB G AB G / AA x A G AA FIG. 1: The KBI G αβ for hard rods of different diameters σ BB /σ AA = 2 and P σ AA /k B T = 1. where in the last step we have made use of Eq. (2.39).Integration over x A yields µ A k B T = const + ln x A + ln [1 − x B (1 − R )+ p − x A x B (1 − R ) i . (2.47)For pure A ( x B = 0), we have µ PA k B T = const + ln 2 . (2.48)The solvation Gibbs energy of A in pure A may be ob-tained from (2.47) as ∆ µ ∗ A = µ A − k B T ln( ρ A Λ A ) , (2.49)where Λ A = h/ √ πm A k B T is the momentum partitionfunction of A in one-dimensional systems. Similarly,∆ µ ∗ PA = µ PA − k B T ln( ρ PA Λ A ) , (2.50)where ρ PA is the density of pure A at the same T and P as the mixture. Taking the limit x B → ρ PA = − Ω AA ( ξ )Ω ′ AA ( ξ ) . (2.51)The excess solvation Gibbs energy relative to the sol-vation Gibbs energy in pure A is defined as∆∆ µ ∗ A = ∆ µ ∗ A − ∆ µ ∗ PA (2.52)This quantity may be calculated from (2.47)–(2.52) withthe result∆∆ µ ∗ A k B T = ln (cid:20) − x B (1 − R ) + 12 p − x A x B (1 − R ) (cid:21) + ln ρ PA ρ T . (2.53) A G AA / = -1 = -6 = -8.5 = -11 G AA / = -0.75 = -0.5 = -0.25 = -0.001 = -1 FIG. 2: The KBI G AA for SW particles with parameters givenin (3.7) and k B T / | ǫ AA | = 1, P σ/k B T = 1. The lines areobtained from the exact expressions presented in Sec. II C,while the circles are the data obtained in Ref. 1. III. A SAMPLE OF RESULTS
Let us start considering a binary system composed of(additive) hard rods of different diameters (lengths) σ AA , σ BB , and σ AB = ( σ AA + σ BB ) /
2. The Laplace functionΩ αβ ( s ) defined by Eq. (2.19) isΩ αβ ( s ) = e − sσ αβ s . (3.1)In this case the parameter defined in Eq. (2.28) is R = 1and thus the limit R → K AB = 1 / Ω AB ( ξ ). The general schemeof section II can be used to obtain the KBI explicitly: G AB = − σ AA + σ BB + ξσ AA σ BB ξ ( x A σ AA + x B σ BB ) , (3.2) G AA = G AB + σ BB − σ AA , (3.3) A G BB / = -1 = -6 = -8.5 = -11 G BB / = -0.75 = -0.5 = -0.25 = -0.001 = -1 FIG. 3: The KBI G BB for SW particles with parameters givenin (3.7) and k B T / | ǫ AA | = 1, P σ/k B T = 1. The lines areobtained from the exact expressions presented in Sec. II C,while the circles are the data obtained in Ref. 1. G BB = G AB + σ AA − σ BB , (3.4)so that ∆ AB = 0. Figure 1 shows the values of G αβ for adiameter ratio σ BB /σ AA = 2 and a thermodynamic state P σ AA /k B T = 1. These results are in perfect agreementwith those calculated in Part I. Having established that the programs give the correctresults for hard rods, we next present results for a mix-ture of particles’ interaction via SW potential of the form U αβ ( R ) = ∞ , R < σ αβ ,ǫ αβ , σ αβ < R < σ αβ + δ αβ , , R < σ αβ + δ αβ . (3.5)where ǫ αβ <
0. For this SW potential the Laplace func- A G AB / = -1 = -6 = -8.5 = -11 G AB / = -0.75 = -0.5 = -0.25 = -0.001 = -1 FIG. 4: The KBI G AB for SW particles with parameters givenin (3.7) and k B T / | ǫ AA | = 1, P σ/k B T = 1. The lines areobtained from the exact expressions presented in Sec. II C,while the circles are the data obtained in Ref. 1. tion Ω αβ ( s ) isΩ αβ ( s ) = e − sσ αβ s h e − ǫ αβ /k B T − (cid:16) e − ǫ αβ /k B T − (cid:17) e − sδ αβ i (3.6)and again the general results of section II provide theKBI explicitly.We have taken the following values for the potentialparameters: σ AA = σ BB = σ AB = σ,δ AA = δ BB = δ AB = 15 σ, (3.7) ǫ BB | ǫ AA | = ǫ, ǫ AB = −√ ǫ AA ǫ BB . The thermodynamic variables are T , P , and x A . Inall the calculations we choose k B T / | ǫ AA | = 1 and A AB / = -1 = -6 = -8.5 = -11 AB / = -0.75 = -0.5 = -0.25 = -0.001 = -1 FIG. 5: Values of ∆ AB for SW particles with parametersgiven in (3.7) and k B T / | ǫ AA | = 1, P σ/k B T = 1. The linesare obtained from the exact expressions presented in Sec. II C,while the circles are the data obtained in Ref. 1. P σ/k B T = 1 to compare the present results with thoseof Part I.Figures 2–4 show the values of G AA , G BB , and G AB for these systems for various values of ǫ ranging from ǫ = − .
001 to ǫ = −
1, and from ǫ = − ǫ = − Figure 5 shows the values of ∆ AB = G AA + G BB − G AB in the entire range of composition. In all the cases theagreement with the results of Part I is quantitative.The KBI in the infinite dilution limit ( x A → − ǫ for the same system as that of Figs.2–5. We observe that both G AB and G BB are hardlysensitive to the value of ǫ . In contrast, the solute-soluteKBI, G AA , is strongly influenced by the solvent-solventpotential depth, increasing both for small and for largevalues of | ǫ | . A careful inspection of the explicit expres-sions (2.42)–(2.44) in the limit | ǫ | → ∞ shows that, while BB G AB G / G AA FIG. 6: The KBI G αβ in the infinite dilution limit ( x A →
0) for SW particles with parameters given in (3.7) and k B T / | ǫ AA | = 1, P σ/k B T = 1. G AB and G BB tend to the same constant value, G AA di-verges as G AA ∼ exp [( | ǫ BB | − | ǫ AB | ) /k B T ]. This phe-nomenon might be relevant to the study of hydrophobicinteractions, as discussed in Ref. 9. IV. DISCUSSION AND CONCLUSION
In Part I we calculated all the KBI in an indirect way. We first calculated the excess functions from the partitionfunction of the system, then we used the inversion of theKB theory to calculate the KBI. This lengthy proceduremight have introduced accumulated errors. Some readersof Part I have expressed doubts regarding the reliabilityof the results calculated along this procedure. In factsome have also claimed that there might be a miscibility gap which we might have missed by this indirect andlengthy calculations.In this paper we have repeated the calculations of theKBI directly, from the same program that was designedto calculate the pair correlation functions in mixtures oftwo components in 1D system.The agreement between the two methods was satisfy-ing, it also lent credibility to the inversion procedure andencouraged us to extend the calculations of the KBI foraqueous like mixtures. We hope to report on that in thenear future.Regarding the question of miscibility gap we haveshown that the inequality1 + ρ T x A x B ∆ AB > of solution we have the equation (cid:18) ∂ g∂x A (cid:19) P,T = 1 x B (cid:18) ∂µ A ∂x A (cid:19) P,T = k B Tx A x B (1 + ρ T x A x B ∆ AB ) , (4.2)where g = G/ ( N A + N B ) is the Gibbs energy of thesystem per mole of mixture. It follows from (4.1) and(4.2) that g is everywhere a concave (downward) functionof x A . Therefore, there exists no region of compositionswhere the system is not stable, hence no phase transitionin such a system. Acknowledgments
The research of A.S. was supported by the Minis-terio de Educaci´on y Ciencia (Spain) through GrantNo. FIS2007-60977 (partially financed by FEDER funds)and by the Junta de Extremadura through Grant No.GRU09038. ∗ Electronic address: [email protected] † Electronic address: [email protected] A. Ben-Naim, J. Chem. Phys. , 084510 (2008); Erra-tum: A. Ben-Naim, J. Chem. Phys. , 159901 (2009). A. Ben-Naim,
Molecular Theory of Solutions (Oxford Uni-versity Press, Oxford, 2006). Z. W. Salsburg, R. W. Zwanzig, and J. G. Kirkwood, J.Chem. Phys. , 1098 (1953). J. L. Lebowitz and D. Zomick, J. Chem. Phys. , 3335(1971). M. Heying and D. S. Corti, Fluid Phase Equil. , 85 (2004). A. Santos, Phys. Rev. E , 062201 (2007). L. van Hove, Physica (Amsterdam) , 137 (1950). J. G. Kirkwood and F. P. Buff, J. Chem. Phys. , 774(1951). A. Ben-Naim,
Molecular Theory of Water and Aqueous So-lutions (World Scientific, Singapore, 2009). It must be noted that the curves in Ref. 1 labeled as ǫ = − . − − .
5, and −
10 actually correspond to ǫ = − . − − .
5, and −−