cond-mat0506350.tex
\documentclass[12pt,times,endnote]{article}
\usepackage{epsfig,float,color,graphicx}
\setlength{\textwidth}{15cm}
\begin{document}
\begin{center}
{\large\bf Sound-propagation gap in fluid mixtures}\\*[0.8cm]
Supurna Sinha and M. Cristina Marchetti\\*[0.2cm]
{\it Physics Department, Syracuse University, Syracuse, New York 13244}\\*[1cm]
\end{center}


\noindent
{\small 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.} \\~\\*[0.8cm]

\noindent
{\large\bf I.~~INTRODUCTION}\\

\noindent
Short-wavelength collective modes in dense simple hard-sphere fluids have 
been studied extensively by both theoretical and experimental researchers 
in the past few years.$^{\rm 1-4}$ Two interesting features of the extended 
hydrodynamic modes in a simple fluid are the softening of the heat mode 
at the wave vector $k$ where the static structure factor $S(k)$ has its 
first maximum and the appearance of a gap in the sound propagation in the 
same large-wave-vector region.$^{\rm l,2}$ The softening of the heat mode 
corresponds to the slowing down of structural relaxation in a dense 
fluid and has been discussed extensively in the literature. The 
sound-propagation gap consists in the vanishing of the imaginary part 
of the frequency of the sound waves, which describes the propagation of 
the wave, over a finite region of $k$ values.\\

\noindent
In a one-component fluid of hard spheres of size $\sigma$ and density $n$ 
one 
obtains a gap in the sound propagation for all values of the packing 
fraction \( \eta = (\pi /6)n\sigma^{3} \) above 
\( \eta \approx 0.33.^{\rm 5} \) This feature of the extended modes has 
been the topic of some controversy. de Schepper, Cohen, and 
Zuilhof,$^{\rm 5}$ have shown that the appearance and disappearance of 
the sound propagation gap can be understood as arising from the competition 
between generalized wave-vector-dependent reversible elastic restoring 
forces and dissipative forces in the fluid. Roughly speaking, a gap 
arises whenever the dissipation exceeds the elastic forces. They also 
pointed out that the behavior of the sound modes in a one-component fluid 
is quite insensitive to the static structure factor of the fluid and the 
gap is present even when setting \( S(k) = 1.\)  de Schepper, Van Rijs, 
and Cohen$^{\rm 6}$ have also shown that even the long-wavelength 
Navier-Stokes equations, when used outside their range of validity to 
describe short-wavelength phenomena, predict a gap in the sound 
propagation. The model based on the Navier-Stokes equations suggests 
that there is no connection between local effects leading to 
wave-vector-dependent transport coefficients and susceptibilities and 
the trapping of sound on molecular length scales. \\

\noindent 
In a recent paper (hereafter denoted {\bf I}) we used the Enskog equation as 
the starting point to study the short-wavelength collective excitations 
of a binary hard-sphere fluid mixture.$^{\rm 7}$ In a mixture the spatial 
ordering of the particles can be changed while keeping the total density 
of the fluid constant by changing the concentration 
\( x_{2} = n_{2} / (n_{1} + n_{2}) \) of the larger species 
(here type-2 spheres) or the size ratio \( \alpha = \sigma_{1} / \sigma_{2}.\)
We find that sound propagation in the mixture depends on the concentration 
$x_{2}$: the trapping of sound modes and the corresponding propagation 
gap that arises in dense simple fluids only occurs in mixtures with a small 
concentration of small spheres in a dense fluid of large spheres 
$(x_{2} \approx 1)$ or a small concentration of large spheres in a dense 
fluid of small spheres $(x_{2} \ll 1)$. At intermediate values of $x_{2}$ 
we observe a softening of the sound propagation, but no actual gap. 
For either small or large values of $x_{2}$ the mixture closely resembles 
a simple fluid, as can be seen from the behavior of the partial structure 
factors $S_{11}(k),\;\;S_{22}(k)\;\;{\rm and}\;\; S_{12}(k)$ 
defined in {\bf I}. For instance, for $x_{2} = 0.01$ only the static 
structure factor $S_{11}(k)$ of the dense component is peaked 
[see Fig. 1(a)]. The other structure factors are essentially constant. 
Thus we expect the appearance of a sound-propagation gap as in a simple 
fluid. At intermediate values of the concentration $x_{2}$ the dense 
mixture resembles a metallic glass: there is short-range order in the 
spatial arrangements of both types of spheres, as indicated by the fact 
that all three static structure factors display considerable structure 
as functions of the wave vector [see Fig. l(b)]. The absence of a 
sound-propagation gap in this case is then consistent with the experimental 
finding that no gap occurs in the second sound of metallic glasses.$^{8}$ 
This observation seems to indicate that the local fluid structure does 
play a role in determining sound propagation and sound trapping at short 
wavelengths, in contrast to what was argued for a one-component fluid. \\

\noindent
In this paper we investigate in more detail the extended sound modes in 
binary mixtures, with the objective of clarifying their dependence on 
concentration, size ratio 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 
$\eta = (\pi / 6)(n_{1} \sigma^{3}_{1} + n_{2}\sigma^{3}_{2})$  
fixed at the value $\eta = 0.46$, which is close to the packing 
fraction corresponding to freezing of a one-component hard-sphere fluid. \\
%\begin{center}
%\includegraphics*[width=3.5in]{Sup-New.eps}
%\end{center}
%\vspace*{0.3cm}

\noindent
FIG.1.~~The partial static structure factors $S_{11}(k),\;\;S_{22}(k)$, 
and $S_{12}(k)$ for $\eta=0.46$, $\alpha=0.7$, $x_{2}=0.01$ (a), and
$x_{2}=0.5$(b).\\*[0.2cm]

\noindent
First we have considered the dependence of the sound modes on the ratio 
$m_{1} / m_{2}$ of the masses of the two species and the role of this 
ratio in determining the propagation gap. By comparing the extended 
hydrodynamic modes displayed in {\bf I} $(\alpha = 0.7$ and 
$m_{1} / m_{2} = 0.5)$ to the modes obtained for the same values of size 
ratio and concentration, but equal masses, we have found that the sound 
propagation is essentially unchanged at all concentrations. This holds 
true for other values of the size ratio. In general, as long as the mass 
ratio is not too small $(m_{1} / m_{2} \geq 0.1)$ the difference in mass 
of the two species affects only weakly the sound propagation in a mixture. 
This points to the fact that the size difference and the spatial ordering 
of the two types of spheres must play the main role in determining the 
absence of sound trapping at large wave vectors.\\

\noindent 
In Sec. II we discuss the dependence of the sound modes on the concentration 
of one of the two species. We show that in dense mixtures, as in a 
one-component fluid, the appearance of a sound propagation gap can be 
understood as the result of the competition between elastic and 
dissipative forces in the fluid. The absence of the sound-propagation 
gap in mixtures of intermediate concentrations is not determined by 
the details of the static structure factors, but it does depend on the 
fluid structure in the sense that it occurs when the packing fractions 
$\eta_{1} = (\pi /6)n_{1}\sigma^{3}_{1}$ and 
$\eta_{2} = (\pi /6)n_{2}\sigma^{3}_{2}$ of the two species are comparable. 
In this case all three static structure factors are peaked and the 
structural properties of the mixture resemble those of a metallic glass.\\~\\

\noindent
{\large\bf II.~~DEPENDENCE OF SOUND PROPAGATION\protect\\*[0.2cm]
\hspace*{0.9cm}ON CONCENTRATION}\\

\noindent 
We showed in {\bf I} that at large wave vectors the extended heat and 
diffusion modes govern the relaxation of the densities of the two species. 
These are the only long-lived fluctuations in the fluid at these wave 
vectors. At large wave vectors the extended sound modes mainly describe 
the relaxation of temperature and longitudinal momentum fluctuations. 
Similarly, in a one-component fluid large-wave-vector density 
fluctuations are long-lived while temperature and momentum fluctuations 
relax quickly. On the basis of this observation Zuilhof and co-workerss 
suggested that extended sound modes in a one-component fluid can be 
described by a simple model obtained by neglecting the coupling to 
the density in the generalized hydrodynamic equations for temperature 
and longitudinal momentum. The resulting two coupled equations have 
eigenfrequencies that closely reproduce the extended sound modes at 
large wave vectors. Clearly, at small wave vectors these modes do not 
reduce to the hydrodynamic sound modes.\\

\noindent
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 
momentum fluctuations is well separated from that governing the relaxation 
of fluctuations in the two densities. If we neglect the coupling to 
fluctuations in the densities in the generalized hydrodynamic equations for 
temperature and longitudinal momentum, we obtain two coupled equations 
that are formally identical to those obtained for a one-component fluid. 
The eigenvalues of these equations are obtained by solving a 
quadratic equation, with the result

\renewcommand\theequation{2.1}
\begin{eqnarray}
z_{\pm} (k) = &-& \frac{1}{2} [\Omega_{ll}(k) - \Omega_{TT}(k)] \nonumber \\
              &\pm& \frac{1}{2} \{[\Omega_{ll}(k) - 
\Omega_{TT}(k)]^{2}-4[f(k)]^{2}\}^{\frac{1}{2}}, 
\end{eqnarray}
where $\Omega_{ll}(k)$ and $\Omega_{TT}(k)$ are the damping rates in the 
equations for longitudinal momentum and temperature fluctuations, 
respectively, and $f(k) =i\Omega_{Tl}(k)$ is the elastic restoring 
force that couples the two equations,
\renewcommand\theequation{2.2}
\begin{eqnarray}
\Omega_{ll}(k) = \frac{2}{3\rho} \sum_{a=1,2}\;\;\sum_{b=1,2}\;\;
\frac{2\mu_{ab}\sqrt{n_{a}n_{b}}}{t_{E_{ab}}}\; [1 &-& j_{0}(k\sigma_{ab}) 
\nonumber \\
                                 &+& 2j_{2}(k\sigma_{ab})]
\end{eqnarray} 
\renewcommand\theequation{2.3}
\begin{equation}
\Omega_{TT}(k) = \frac{2}{3\rho} \sum_{a=1,2}\;\;\sum_{b=1,2}\;
\;\frac{2\mu_{ab}\sqrt{n_{a}n_{b}}}{t_{E_{ab}}} [1 - j_{0}(k\sigma_{ab})]
\end{equation}
\renewcommand\theequation{2.4}
\begin{eqnarray}
f(k)=k\sqrt{2n/3\beta\rho}\left[1+\sum_{a=1,2}\;\;\sum_{b=1,2}\;\;
2\pi\frac{n_{a}n_{b}}{n}\sigma^{3}_{ab}\right.\nonumber \\
\left.\times\chi_{ab}\frac{j_{1}(k\sigma_{ab})}{k\sigma_{ab}}\right],
\end{eqnarray}
where $n=n_{1}+n_{2}$ and $\rho =m_{1}n_{1} + m_{2}n_{2}$ are the total 
number and mass densities, respectively. Also, 
$\mu_{ab} = m_{a}m_{b}/(m_{a}+m_{b})$ is the reduced mass, $\chi_{ab}$ the 
pair correlation function of species $a$ and $b$ at contact, 
$\sigma_{ab} = (\sigma_{a} + \sigma_{b})/2$, and $j_{n}(x)$ is a spherical 
Bessel function of order $n$. Finally, $t_{E_{ab}}$ is the Enskog mean-free 
time, $t_{E_{ab}} = \sqrt{2\mu_{ab}\beta} / 4\sqrt{\pi n_{a}n_{b}} \sigma^{2}_{ab} \chi_{ab})$. At large wave vectors the two modes given in Eq. (2.1) 
closely resemble the extended sound modes obtained in {\bf I} by solving 
the four coupled hydrodynamic equations. The argument of the square root 
in Eq. (2.1) is generally negative (and the two modes are propagating), 
unless the dissipative damping exceeds the elastic forces. In this case 
Eq. (2.1) yields two 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 
$(\Omega_{ll} - \Omega_{TT} + 2f)(\Omega_{ll} - \Omega_{TT} - 2f)$. The 
first 
factor is always positive. We define the function $\Delta (k)$ by 
\begin{eqnarray*}
\frac{\sigma_{12}}{2} \left[ \frac{3\beta\rho}{2n} \right]^{1/2}\;
\;(\Omega_{ll}-\Omega_{TT}-2f)=\Delta(k)-k\sigma_{12},
\end{eqnarray*} 
with
\renewcommand\theequation{2.5}
\begin{equation}
\Delta (k) = \frac{2\pi}{n} \sum_{a=1,2}\;\;\sum_{b=1,2}\;\;n_{a}n_{b}
\sigma^{2}_{ab}\sigma_{12}\chi_{ab}
\left[\left[\frac{16\mu_{ab}n}{3\pi\rho}\right]^{1/2}\;\;j_{2}(k\sigma_{ab}) 
- j_{1}(k\sigma_{ab})\right].
\end{equation}
\vspace*{0cm}

\noindent
A gap occurs when $\Delta (k) > k\sigma_{12}$. This condition is displayed 
graphically in Fig. 2 at three different concentrations. The figure 
shows that a gap is obtained for $x_{2} = 0.01$ and $x_{2} = 0.9$, but
not for $x_{2} = 0.5$. For $\alpha = 0.7$ and $m_{1}/m_{2} = 0.5$ this
simple model predicts no sound-propagation gap for $0.05 \leq 0.7$, in 
agreement with what was obtained in {\bf I} solving the full coupled 
hydrodynamic equations.\\
%\begin{center}
%\includegraphics*[width=3.8in]{Sup-New2.eps}
%\end{center}
%\vspace*{0.3cm}

\noindent
FIG.2.~~The straight line is $k\sigma_{12}$. The curves represent the function
$\Delta (k)$ for $\eta=0.46$, $\alpha=0.7$, and $m_{1}/m_{2}=0.5$, and 
$x_{2}=0.01$ (solid line), $x_{2}=0.5$ (dashed line), and $x_{2}=0.9$ 
(chain dashed line), as a function of $k\sigma_{12}$.\\*[0.2cm]

\noindent 
In mixtures, as in a one-component fluid, the sound-propagation gap 
occurs as a consequence of the competition between dissipative and 
reversible forces. In most of the wave-vector range elastic forces 
exceed dissipative forces. This simply reflects the fact that sound 
modes propagate in both liquids and low-density gases (in the 
very-large-wave-vector limit the fluid resembles an ideal gas, since one 
is considering length scales that are too short for the interactions to be 
relevant). The damping term and the elastic forces depend on the packing 
fractions $\eta_{1}$ and $\eta_{2}$. For a fixed total packing fraction 
$\eta$ and size ratio $(\alpha = 0.7)$, both dissipative and elastic 
terms decrease with increasing $x_{2}$. At all $x_{2}$ there is a region 
of large $k$ where the dissipation rates are essentially constant 
(and equal the sum of the Enskog times weighted by the mass fractions) 
and $\Omega_{ll}-\Omega_{TT} \approx 0$. For $x_{2} \ll 1$ and 
$x_{2} \simeq $ the elastic forces vanish even more rapidly than the 
dissipative term in this large-$k$ region and a gap occurs. For 
intermediate $x_{2}$ the elastic forces remain finite at all $k$ 
and the modes are always propagating.\\

\noindent 
In a recent paper$^{9}$ Campa and Cohen suggested that the static structure 
factors may play a role in determining sound-propagation gaps in a mixture. 
On the other hand, the simple model described above does not contain the 
static structure factors $S_{ij}(k)$. In fact, the appearance and 
disappearance of the sound-propagating gap are insensitive to the structure 
factors. This can also be seen by reconsidering the extended sound modes 
obtained in {\bf I} from the solution of the four coupled hydrodynamic 
equation and arbitrarily setting $S_{11}(k) = S_{22}(k) - 1$ and 
$S_{12}(k) = 0$ in these equations. One finds that, while the gap itself, 
when it occurs, is wider if the proper static structure factors are 
included, its concentration dependence, i.e., its appearance and 
disappearance, is qualitatively unchanged. The suggestion of Campa 
and Cohen is, however, in the right direction and it would be 
misleading to say that short-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 dependence on 
the packing fractions of the two components of the mixture. When the 
packing fractions are both appreciable, there is not gap. This situation 
corresponds to a fluid where all three partial static structure factors 
are peaked (Fig. I) and the fluid structure resembles that of a metallic 
glass.\\~\\

\noindent
{\large\bf III.~~CONCLUSION}\\

\noindent
We conclude with two remarks. 
\begin{enumerate}
\item[(i)] In {\bf I} we neglected the coefficient of thermal diffusion 
in evaluating the extended hydrodynamic modes of the mixture. This 
approximation was motivated by the fact that the coefficient of 
thermal diffusion vanishes in a first-Sonine-polynomial approximation, 
and in the long-wavelength limit it is always much smaller than the 
diffusion coefficient in mixtures of spheres of not too disparate sizes 
and masses.$^{1O}$ On the other hand, in the long-wavelength limit the 
coefficient of thermal diffusion contributes to the damping of the sound 
modes.$^{11}$ Since neglecting thermal diffusion is best justified in 
mixtures of either very low $x_{2}$ or $x_{2}\simeq 1$, one may ask if 
the absence of the sound-propagation gap at intermediate $x_{2}$ simply 
occurs because we have underestimated the sound damping by neglecting 
thermal diffusion. The model used here for the description of the 
extended sound modes based on the two coupled equations for temperature 
and longitudinal momentum fluctuations neglects all couplings to the 
densities and therefore does not contain the coefficient of thermal 
diffusion. This model still predicts the disappearance and reappearance 
of the sound propagation gap as a function of concentration, indicating 
that the dependence of the sound propagation on concentration is not 
an artifact of the approximation of neglecting thermal diffusion.
\item [(ii)] Previous work has focussed on unusual sound propagation 
in disparate-mass gas mixtures and binary liquid alloys at moderate 
density.$^{9,12}$ One interesting feature observed in computer 
simulations$^{13}$ is the appearance of a fast propagating sound mode 
above a certain nonzero value of the wave vector, signaling an effective 
separation of the dynamics of light and heavy particles. Campa and Cohen$^{9}$ 
have shown that the fast sound is a kinetic mode rather than a 
hydrodynamic mode and it yields an observable shoulder in the dynamic 
structure factor of gas mixtures. In moderately dense mixtures the fast 
sound mode is present only for mass ratios below 0.1 and no fast sound 
occurs for $\eta \geq 0.42$. Our generalized hydrodynamic equations 
as derived in {\bf I} cannot yield a fast sound mode since our set of 
independent variables only includes the conserved densities of the fluid: 
we only obtain the extended hydrodynamic modes and no kinetic modes. 
To account for the possibility of fast sound we would need to enlarge 
our set of independent variables to include at least the relative 
momentum density and the temperature difference of the two species. 
On the other hand, the fact that no fast sound is observed in Ref. 9 
for the values of the total packing fraction and mass ratio considered 
here suggests that at such high densities the dynamics of light and 
heavy particles never decouple and the generalized hydrodynamic equations 
of {\bf I} may indeed provide an adequate description of the dynamics at 
large wave vectors.$^{14}$ A proper test of whether our theory is relevant 
for real mixtures will, however, only come from a detailed comparison 
of the dynamic structure factor that can be obtained from the generalized 
hydrodynamic equations of {\bf I} with neutron-scattering spectra from 
dense mixtures or with numerical simulations. Finally, the fact that the 
appearance and disappearance of the sound-propagation gap in our work 
does not depend on the mass ratio and takes place even for equal masses 
seems to indicate that the fast sound and the concentration dependence 
of the gap are not related.\\~\\
\end{enumerate}

\noindent
{\bf ACKNOWLEDGMENTS}\\

\noindent
One of us (M.C.M.) thanks T. R. Kirkpatrick for stimulating discussions. 
This work was supported by the National Science Foundation under 
Contract No. DMR-87-17337.\\~\\

\begin{center}
$------------$
\end{center}
\vspace*{0.5cm}

\begin{itemize}
\item[$^{1}$] I. M. de Schepper and E. G. D. Cohen, Phys. Rev. A {\bf 22}, 
287 (1980); J. Stat. Phys. {\bf 27}, 223 (1982). 

\item[$^{2}$] T. R. Kirkpatrick, Phys. Rev. A {\bf 32}, 3120 (1985). 

\item[$^{3}$] J. R. D. Copley and J. M. Rowe, Phys. Rev. Lett. {\bf 32}, 49 
(1973). 

\item[$^{4}$] A. A. van Well, P. Verkerk, L. A. De Graaf, J. B. Suck, and J. 
R. D. Copley, Phys. Rev. A {\bf 31}, 3391 (1985). 

\item[$^{5}$] I. M. de Schepper, E. G. D. Cohen, and M. J. Zuilhof, 
Phys. Lett. {\bf 103A}, 120 (1984). 

\item[$^{6}$] I. M. de Schepper, J. C. Van Rijs, and E. G. D. Cohen, 
Physica {\bf 1434A}, 1 (1985).

\item[$^{7}$] M. C. Marchetti and S. Sinha, Phys. Rev. A {\bf 41}, 
3214 (1990). 

\item[$^{8}$] J. B. Suck, H. Rudin, H. J. Guntherodt, and H. Beck, 
Phys. Rev. Lett. {\bf 50}, 49 (1983). 

\item[$^{9}$] A. Campa and E. G. D. Cohen, Phys. Rev. A {\bf 41}, 5451 (1990). 

\item[$^{10}$] J. M. Kincaid, E. G. D. Cohen, and M. Lopez de Haro, 
J. Chern. Phys. {\bf 86}, 963 (1987). 

\item[$^{11}$] J. P. Boon and S. Yip, Molecular Hydrodynamics 
(McGraw-Hill, New York, 1980), pp. 270 and 271.

\item[$^{12}$] P. B. Lerner and I. M. Sokolov, Physica C {\bf 150}, 465 
(1988). 

\item[$^{13}$] J. Bosse, G. Jacucci, M. Ronchetti, and W. Schirmacher, 
Phys. Rev. Lett. {\bf 57}, 3277 (1986).

\item[$^{14}$] 1n a one-component hard-sphere fluid with $\eta = 0.46$ the 
generalized hydrodynamic modes, provide a good representation of the 
dynamic structure factor up to $k \sim 10/\sigma$, as discussed, for 
instance, by E. G. D. Cohen, in {\it Trends in Applications of Pure 
Mathematics to Mechanics}, Vol. 249 of Lecture Notes in Physics, 
edited by E. Kroner and K. Krichgasser (Springer-Verlag, Berlin 1986). 
\end{itemize}
\end{document}
