We discuss the behavior of the extended sound modes of a dense binary
hard-sphere mixture. In a dense simple hard-sphere fluid the Enskog theory
predicts a gap in the sound propagation at large wave vectors. In a binary
mixture the gap is only present for low concentrations of one of the two
species. At intermediate concentrations sound modes are always propagating.
This behavior is not affected by the mass difference of the two species, but it
only depends on the packing fractions. The gap is absent when the packing
fractions are comparable and the mixture structurally resembles a metallic
glass.Comment: Published; withdrawn since ordering in archive gives misleading
  impression of new publicatio

A. A. van Well

A. Campa

E. G. D. Cohen

I. M. de Schepper

J. B. Suck

J. Bosse

J. M. Kincaid

J. P. Boon

J. R. D. Copley

M. Cristina Marchetti

P. B. Lerner

Supurna Sinha

T. R. Kirkpatrick

English

arXiv

We discuss the behavior of the extended sound modes of a dense binary hard-sphere mixture. In a dense simple hard-sphere fluid the Enskog theory predicts a gap in the sound propagation at large wave vectors. In a binary mixture the gap is only present for low concentrations of one of the two species. At intermediate concentrations sound modes are always propagating. This behavior is not affected by the mass difference of the two species, but it only depends on the packing fractions. The gap is absent when the packing fractions are comparable and the mixture structurally resembles a metallic glass

Sinha, Supurna

Marchetti, M. Cristina

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 5-3-2006 Sound-Propagation Gap in Fluid Mixtures Supurna Sinha Syracuse University M. Cristina Marchetti Syracuse University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation arXiv:cond-mat/0506350v2 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:cond-mat/0506350v2  [cond-mat.soft]  3 May 2006Sound-propagation gap in fluid mixturesSupurna Sinha and M. Cristina MarchettiPhysics Department, Syracuse University, Syracuse, New York 13244We discuss the behavior of the extended sound modes of a dense binary hard-sphere mix-ture. In a dense simple hard-sphere fluid the Enskog theory predicts a gap in the soundpropagation at large wave vectors. In a binary mixture the gap is only present for lowconcentrations of one of the two species. At intermediate concentrations sound modesare always propagating. This behavior is not affected by the mass difference of the twospecies, but it only depends on the packing fractions. The gap is absent when the packingfractions are comparable and the mixture structurally resembles a metallic glass.I. INTRODUCTIONShort-wavelength collective modes in dense simple hard-sphere fluids have been stud-ied extensively by both theoretical and experimental researchers in the past fewyears.1−4 Two interesting features of the extended hydrodynamic modes in a simplefluid are the softening of the heat mode at the wave vector k where the static struc-ture factor S(k) has its first maximum and the appearance of a gap in the soundpropagation in the same large-wave-vector region.l,2 The softening of the heat modecorresponds to the slowing down of structural relaxation in a dense fluid and hasbeen discussed extensively in the literature. The sound-propagation gap consistsin the vanishing of the imaginary part of the frequency of the sound waves, whichdescribes the propagation of the wave, over a finite region of k values.In a one-component fluid of hard spheres of size σ and density n one obtains agap in the sound propagation for all values of the packing fraction η = (pi/6)nσ3above η ≈ 0.33.5 This feature of the extended modes has been the topic of somecontroversy. de Schepper, Cohen, and Zuilhof,5 have shown that the appearance anddisappearance of the sound propagation gap can be understood as arising from thecompetition between generalized wave-vector-dependent reversible elastic restoringforces and dissipative forces in the fluid. Roughly speaking, a gap arises whenever1the dissipation exceeds the elastic forces. They also pointed out that the behav-ior of the sound modes in a one-component fluid is quite insensitive to the staticstructure factor of the fluid and the gap is present even when setting S(k) = 1.de Schepper, Van Rijs, and Cohen6 have also shown that even the long-wavelengthNavier-Stokes equations, when used outside their range of validity to describe short-wavelength phenomena, predict a gap in the sound propagation. The model basedon the Navier-Stokes equations suggests that there is no connection between localeffects leading to wave-vector-dependent transport coefficients and susceptibilitiesand the trapping of sound on molecular length scales.In a recent paper (hereafter denoted I) we used the Enskog equation as the startingpoint to study the short-wavelength collective excitations of a binary hard-spherefluid mixture.7 In a mixture the spatial ordering of the particles can be changedwhile keeping the total density of the fluid constant by changing the concentra-tion x2 = n2/(n1 + n2) of the larger species (here type-2 spheres) or the size ratioα = σ1/σ2. We find that sound propagation in the mixture depends on the con-centration x2: the trapping of sound modes and the corresponding propagation gapthat arises in dense simple fluids only occurs in mixtures with a small concentrationof small spheres in a dense fluid of large spheres (x2 ≈ 1) or a small concentrationof large spheres in a dense fluid of small spheres (x2 ≪ 1). At intermediate valuesof x2 we observe a softening of the sound propagation, but no actual gap. For eithersmall or large values of x2 the mixture closely resembles a simple fluid, as can beseen from the behavior of the partial structure factors S11(k), S22(k) and S12(k)defined in I. For instance, for x2 = 0.01 only the static structure factor S11(k) ofthe dense component is peaked [see Fig. 1(a)]. The other structure factors are es-sentially constant. Thus we expect the appearance of a sound-propagation gap asin a simple fluid. At intermediate values of the concentration x2 the dense mixtureresembles a metallic glass: there is short-range order in the spatial arrangements ofboth types of spheres, as indicated by the fact that all three static structure factorsdisplay considerable structure as functions of the wave vector [see Fig. l(b)]. Theabsence of a sound-propagation gap in this case is then consistent with the exper-imental finding that no gap occurs in the second sound of metallic glasses.8 Thisobservation seems to indicate that the local fluid structure does play a role in de-termining sound propagation and sound trapping at short wavelengths, in contrastto what was argued for a one-component fluid.In this paper we investigate in more detail the extended sound modes in binarymixtures, with the objective of clarifying their dependence on concentration, size2ratio and mass ratio, and the role of the fluid structure in determining the gap.As in our earlier paper, we keep the total packing fraction of the mixture η =(pi/6)(n1σ31+n2σ32) fixed at the value η = 0.46, which is close to the packing fractioncorresponding to freezing of a one-component hard-sphere fluid.FIG.1. The partial static structure factors S11(k), S22(k), and S12(k) for η = 0.46,α = 0.7, x2 = 0.01 (a), and x2 = 0.5(b).First we have considered the dependence of the sound modes on the ratio m1/m2 ofthe masses of the two species and the role of this ratio in determining the propaga-tion gap. By comparing the extended hydrodynamic modes displayed in I (α = 0.7and m1/m2 = 0.5) to the modes obtained for the same values of size ratio andconcentration, but equal masses, we have found that the sound propagation is es-sentially unchanged at all concentrations. This holds true for other values of thesize ratio. In general, as long as the mass ratio is not too small (m1/m2 ≥ 0.1) thedifference in mass of the two species affects only weakly the sound propagation ina mixture. This points to the fact that the size difference and the spatial orderingof the two types of spheres must play the main role in determining the absence ofsound trapping at large wave vectors.In Sec. II we discuss the dependence of the sound modes on the concentration ofone of the two species. We show that in dense mixtures, as in a one-componentfluid, the appearance of a sound propagation gap can be understood as the resultof the competition between elastic and dissipative forces in the fluid. The absenceof the sound-propagation gap in mixtures of intermediate concentrations is not de-termined by the details of the static structure factors, but it does depend on thefluid structure in the sense that it occurs when the packing fractions η1 = (pi/6)n1σ31and η2 = (pi/6)n2σ32of the two species are comparable. In this case all three staticstructure factors are peaked and the structural properties of the mixture resemblethose of a metallic glass.II. DEPENDENCE OF SOUND PROPAGATIONON CONCENTRATIONWe showed in I that at large wave vectors the extended heat and diffusion modesgovern the relaxation of the densities of the two species. These are the only long-3lived fluctuations in the fluid at these wave vectors. At large wave vectors theextended sound modes mainly describe the relaxation of temperature and longitudi-nal momentum fluctuations. Similarly, in a one-component fluid large-wave-vectordensity fluctuations are long-lived while temperature and momentum fluctuationsrelax quickly. On the basis of this observation Zuilhof and co-workerss suggestedthat extended sound modes in a one-component fluid can be described by a simplemodel obtained by neglecting the coupling to the density in the generalized hydro-dynamic equations for temperature and longitudinal momentum. The resulting twocoupled equations have eigenfrequencies that closely reproduce the extended soundmodes at large wave vectors. Clearly, at small wave vectors these modes do notreduce to the hydrodynamic sound modes.The same model can be used to describe the extended sound modes of a mixture,where the characteristic time scale for the relaxation of temperature and momentumfluctuations is well separated from that governing the relaxation of fluctuations inthe two densities. If we neglect the coupling to fluctuations in the densities in thegeneralized hydrodynamic equations for temperature and longitudinal momentum,we obtain two coupled equations that are formally identical to those obtained for aone-component fluid. The eigenvalues of these equations are obtained by solving aquadratic equation, with the resultz±(k) = − 12[Ωll(k)− ΩTT (k)]± 12{[Ωll(k)− ΩTT (k)]2 − 4[f(k)]2} 12 , (2.1)where Ωll(k) and ΩTT (k) are the damping rates in the equations for longitudinalmomentum and temperature fluctuations, respectively, and f(k) = iΩT l(k) is theelastic restoring force that couples the two equations,Ωll(k) =23ρ∑a=1,2∑b=1,22µab√nanbtEab[1 − j0(kσab)+ 2j2(kσab)] (2.2)ΩTT (k) =23ρ∑a=1,2∑b=1,22µab√nanbtEab[1− j0(kσab)] (2.3)4f(k) = k√2n/3βρ1 + ∑a=1,2∑b=1,22pinanbnσ3ab×χab j1(kσab)kσab], (2.4)where n = n1 + n2 and ρ = m1n1 + m2n2 are the total number and mass densi-ties, respectively. Also, µab = mamb/(ma + mb) is the reduced mass, χab the paircorrelation function of species a and b at contact, σab = (σa + σb)/2, and jn(x)is a spherical Bessel function of order n. Finally, tEab is the Enskog mean-freetime, tEab =√2µabβ/4√pinanbσ2abχab). At large wave vectors the two modes givenin Eq. (2.1) closely resemble the extended sound modes obtained in I by solv-ing the four coupled hydrodynamic equations. The argument of the square rootin Eq. (2.1) is generally negative (and the two modes are propagating), unlessthe dissipative damping exceeds the elastic forces. In this case Eq. (2.1) yieldstwo real roots, corresponding to diffusive modes. This is the region of the sound-propagation gap. The argument of the square root in Eq. (2.1) can be factorized as(Ωll−ΩTT +2f)(Ωll −ΩTT − 2f). The first factor is always positive. We define thefunction ∆(k) byσ122[3βρ2n]1/2(Ωll − ΩTT − 2f) = ∆(k)− kσ12,with∆(k) =2pin∑a=1,2∑b=1,2nanbσ2abσ12χab[16µabn3piρ]1/2j2(kσab)− j1(kσab) . (2.5)A gap occurs when ∆(k) > kσ12. This condition is displayed graphically in Fig. 2 atthree different concentrations. The figure shows that a gap is obtained for x2 = 0.01and x2 = 0.9, but not for x2 = 0.5. For α = 0.7 and m1/m2 = 0.5 this simplemodel predicts no sound-propagation gap for 0.05 ≤ 0.7, in agreement with whatwas obtained in I solving the full coupled hydrodynamic equations.FIG.2. The straight line is kσ12. The curves represent the function ∆(k) forη = 0.46, α = 0.7, and m1/m2 = 0.5, and x2 = 0.01 (solid line), x2 = 0.5 (dashed5line), and x2 = 0.9 (chain dashed line), as a function of kσ12.In mixtures, as in a one-component fluid, the sound-propagation gap occurs as aconsequence of the competition between dissipative and reversible forces. In mostof the wave-vector range elastic forces exceed dissipative forces. This simply reflectsthe fact that sound modes propagate in both liquids and low-density gases (in thevery-large-wave-vector limit the fluid resembles an ideal gas, since one is consideringlength scales that are too short for the interactions to be relevant). The dampingterm and the elastic forces depend on the packing fractions η1 and η2. For a fixedtotal packing fraction η and size ratio (α = 0.7), both dissipative and elastic termsdecrease with increasing x2. At all x2 there is a region of large k where the dissipa-tion rates are essentially constant (and equal the sum of the Enskog times weightedby the mass fractions) and Ωll − ΩTT ≈ 0. For x2 ≪ 1 and x2 ≃ the elastic forcesvanish even more rapidly than the dissipative term in this large-k region and a gapoccurs. For intermediate x2 the elastic forces remain finite at all k and the modesare always propagating.In a recent paper9 Campa and Cohen suggested that the static structure factorsmay play a role in determining sound-propagation gaps in a mixture. On the otherhand, the simple model described above does not contain the static structure fac-tors Sij(k). In fact, the appearance and disappearance of the sound-propagating gapare insensitive to the structure factors. This can also be seen by reconsidering theextended sound modes obtained in I from the solution of the four coupled hydrody-namic equation and arbitrarily setting S11(k) = S22(k)− 1 and S12(k) = 0 in theseequations. One finds that, while the gap itself, when it occurs, is wider if the properstatic structure factors are included, its concentration dependence, i.e., its appear-ance and disappearance, is qualitatively unchanged. The suggestion of Campa andCohen is, however, in the right direction and it would be misleading to say thatshort-wavelength sound modes in a mixture are not affected by the fluid structure.The latter does enter even in the simple model yielding Eq. (2.1) through the de-pendence on the packing fractions of the two components of the mixture. When thepacking fractions are both appreciable, there is not gap. This situation correspondsto a fluid where all three partial static structure factors are peaked (Fig. I) and thefluid structure resembles that of a metallic glass.III. CONCLUSION6We conclude with two remarks.(i) In I we neglected the coefficient of thermal diffusion in evaluating the extendedhydrodynamic modes of the mixture. This approximation was motivated bythe fact that the coefficient of thermal diffusion vanishes in a first-Sonine-polynomial approximation, and in the long-wavelength limit it is always muchsmaller than the diffusion coefficient in mixtures of spheres of not too disparatesizes and masses.1O On the other hand, in the long-wavelength limit the coef-ficient of thermal diffusion contributes to the damping of the sound modes.11Since neglecting thermal diffusion is best justified in mixtures of either verylow x2 or x2 ≃ 1, one may ask if the absence of the sound-propagation gapat intermediate x2 simply occurs because we have underestimated the sounddamping by neglecting thermal diffusion. The model used here for the de-scription of the extended sound modes based on the two coupled equations fortemperature and longitudinal momentum fluctuations neglects all couplingsto the densities and therefore does not contain the coefficient of thermal dif-fusion. This model still predicts the disappearance and reappearance of thesound propagation gap as a function of concentration, indicating that the de-pendence of the sound propagation on concentration is not an artifact of theapproximation of neglecting thermal diffusion.(ii) Previous work has focussed on unusual sound propagation in disparate-massgas mixtures and binary liquid alloys at moderate density.9,12 One interestingfeature observed in computer simulations13 is the appearance of a fast propa-gating sound mode above a certain nonzero value of the wave vector, signalingan effective separation of the dynamics of light and heavy particles. Campaand Cohen9 have shown that the fast sound is a kinetic mode rather than ahydrodynamic mode and it yields an observable shoulder in the dynamic struc-ture factor of gas mixtures. In moderately dense mixtures the fast sound modeis present only for mass ratios below 0.1 and no fast sound occurs for η ≥ 0.42.Our generalized hydrodynamic equations as derived in I cannot yield a fastsound mode since our set of independent variables only includes the conserveddensities of the fluid: we only obtain the extended hydrodynamic modes andno kinetic modes. To account for the possibility of fast sound we would needto enlarge our set of independent variables to include at least the relative mo-mentum density and the temperature difference of the two species. On theother hand, the fact that no fast sound is observed in Ref. 9 for the values ofthe total packing fraction and mass ratio considered here suggests that at such7high densities the dynamics of light and heavy particles never decouple andthe generalized hydrodynamic equations of I may indeed provide an adequatedescription of the dynamics at large wave vectors.14 A proper test of whetherour theory is relevant for real mixtures will, however, only come from a de-tailed comparison of the dynamic structure factor that can be obtained fromthe generalized hydrodynamic equations of I with neutron-scattering spectrafrom dense mixtures or with numerical simulations. Finally, the fact that theappearance and disappearance of the sound-propagation gap in our work doesnot depend on the mass ratio and takes place even for equal masses seems toindicate that the fast sound and the concentration dependence of the gap arenot related.ACKNOWLEDGMENTSOne of us (M.C.M.) thanks T. R. Kirkpatrick for stimulating discussions. This workwas supported by the National Science Foundation under Contract No. DMR-87-17337.−−−−−−−−−−−−1 I. M. de Schepper and E. G. D. Cohen, Phys. Rev. A 22, 287 (1980); J. Stat.Phys. 27, 223 (1982).2 T. R. Kirkpatrick, Phys. Rev. A 32, 3120 (1985).3 J. R. D. Copley and J. M. Rowe, Phys. Rev. Lett. 32, 49 (1973).4 A. A. van Well, P. Verkerk, L. A. De Graaf, J. B. Suck, and J. R. D. Copley,Phys. Rev. A 31, 3391 (1985).5 I. M. de Schepper, E. G. D. Cohen, and M. J. Zuilhof, Phys. Lett. 103A, 120(1984).6 I. M. de Schepper, J. C. Van Rijs, and E. G. D. Cohen, Physica 1434A, 1(1985).87 M. C. Marchetti and S. Sinha, Phys. Rev. A 41, 3214 (1990).8 J. B. Suck, H. Rudin, H. J. Guntherodt, and H. Beck, Phys. Rev. Lett. 50,49 (1983).9 A. Campa and E. G. D. Cohen, Phys. Rev. A 41, 5451 (1990).10 J. M. Kincaid, E. G. D. Cohen, and M. Lopez de Haro, J. Chern. Phys. 86,963 (1987).11 J. P. Boon and S. Yip, Molecular Hydrodynamics (McGraw-Hill, New York,1980), pp. 270 and 271.12 P. B. Lerner and I. M. Sokolov, Physica C 150, 465 (1988).13 J. Bosse, G. Jacucci, M. Ronchetti, and W. Schirmacher, Phys. Rev. Lett.57, 3277 (1986).14 1n a one-component hard-sphere fluid with η = 0.46 the generalized hydrody-namic modes, provide a good representation of the dynamic structure factorup to k ∼ 10/σ, as discussed, for instance, by E. G. D. Cohen, in Trends inApplications of Pure Mathematics to Mechanics, Vol. 249 of Lecture Notesin Physics, edited by E. Kroner and K. Krichgasser (Springer-Verlag, Berlin1986).9

Sound-propagation gap in fluid mixtures

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