Dynamic effective mass of granular media
Chaur-Jian Hsu$^1$
David L. Johnson$^1$
Rohit A. Ingale$^2$
John J. Valenza$^1$
Nicolas Gland$^2$
and Hern\'an A. Makse$^2$
%are a sharp resonance %and a broad background, which %A fundamental property of granular matter is the very efficient %dissipation of energy though the network of contacts or the medium. %allow us to understand some % of the microscopic origins of We develop the concept of frequency dependent effective mass, $\tilde{M}(\omega)$, of jammed granular materials which occupy a rigid cavity to a filling fraction of 48\%, the remaining volume being air of normal room condition or controlled humidity. The dominant features of $\tilde{M}(\omega)$ provide signatures of the dissipation of acoustic modes, elasticity and aging effects in the granular medium. We perform humidity controlled experiments and interpret the data in terms of a continuum model and a ``trap'' model of thermally activated capillary bridges at the contact points. The results suggest that attenuation in the granular materials studied here can be influenced significantly by the kinetics of capillary condensation between the asperities at the contacts. % leading to logarithmic aging in $\tilde{M}$. \linebreak\linebreak{PACS: 45.70.-n, 46.40.-f, 81.05.Rm}
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\title{
Dynamic effective mass of granular media}


\author{ Chaur-Jian Hsu$^1$, David L. Johnson$^1$, Rohit
A. Ingale$^2$, John J. Valenza$^1$, Nicolas Gland$^2$, and Hern\'an
A. Makse$^2$}

\affiliation{
$^1$Schlumberger-Doll Research, One Hampshire Street,
Cambridge, MA 02139\\
$^2$Levich Institute and Physics Department, City College of New York, New
York, NY 10031}
\date{\today}


\begin{abstract}
%are a sharp resonance
%and a broad background, which
%A fundamental property of granular matter is the very efficient
%dissipation of energy though the network of contacts or the medium.
%allow us to understand some
%  of the microscopic origins of
  We develop the concept of frequency dependent effective mass,
  $\tilde{M}(\omega)$, of jammed granular materials which occupy a
  rigid cavity to a filling fraction of 48\%, the remaining volume
  being air of normal room condition or controlled humidity.  The
  dominant features of $\tilde{M}(\omega)$ provide signatures of the
  dissipation of acoustic modes, elasticity and aging effects in the
  granular medium. We perform humidity controlled experiments and
  interpret the data in terms of a continuum model and a ``trap''
  model of thermally activated capillary bridges at the contact
  points.  The results suggest that attenuation in the granular
  materials studied here can be influenced significantly by the
  kinetics of capillary condensation between the asperities at the
  contacts.
  % leading to logarithmic aging in $\tilde{M}$.
  \linebreak\linebreak{PACS: 45.70.-n, 46.40.-f, 81.05.Rm}
\end{abstract}

% that at the grain-grain contact network.  The humidity controlled
% experiments reveal that dissipation and aging are determined

% We report an experimental and theoretical investigation of the
% frequency-dependent effective mass, $\tilde{M}(\omega)$, of loose
% granular particles which occupy a rigid cavity to a filling fraction
% of 48\%, {\bf the remaining volume being air of normal room
%   condition or controlled humidity}. The dominant features of
% $\tilde{M}(\omega)$ are a sharp resonance and a broad background,
% {\bf which allows to understand the microscopic origins of
%   dissipation and concomitant aging effect occurring in granular
%   assemblies}. We analyze these results within the context of {\bf a
%   simple model} as well as with molecular dynamic simulations and
% conclude that, on a fundamental level dissipation is dominated by
% that at grain-grain contacts, not by global damping through the
% medium. {\bf The humidity controlled experiments further support
%   this hypothesis and reveals the origin of aging phenomena as
%   resulting from microscopic changes of the contacts over time.}



\maketitle


%\clearpage


%\begin{multicols}{2}

A distinct feature of jammed or loosely packed granular materials
made of a variety of different materials such as sand, steel,
polymer or glass is the ability to dissipate acoustic energy
through the network of interparticle contacts or viscous
dissipation through the surrounding medium. Indeed, loose grains
damp acoustic modes very efficiently
\cite{cremer,duran,liu,bourinet,bourinet2} and they are routinely
used as an effective method to optimize the damping of unwanted
structure-borne acoustic signals \cite{cremer}. Despite its
fundamental importance and practical applications, the microscopic
origins of the mechanisms of dissipation in jammed granular
materials are still unknown.

In this Letter, we pursue the concept of a frequency dependent
effective mass, $\tilde{M}(\omega)$, of a loose granular aggregate
contained within a rigid cavity \cite{bourinet2}. The effective
mass $\tilde{M}(\omega)$ is complex valued; its real part reflects
the inertial and elastic properties while its imaginary part
reflects the dissipative properties of the granular medium. We
demonstrate how the features of $\tilde{M}(\omega)$ allow the
study of some of the mechanisms of damping of acoustic modes,
aging and elasticity in granular matter.

%The
%sound speed estimate is based on a {\bf continuum model which
%interprets the main peak in $\tilde{M}(\omega)$ as a 1/4
%wavelength resonance of the compressional sound speed.
%We further
%validate this behavior by effective mass measurements with simple
%liquids. We have developed molecular dynamic (MD) simulations to
%analyze the expected behavior of $\tilde{M}(\omega)$. The MD
%simulations reproduce the experimental features and indicate that
%the dominant mechanism for damping is at the grain-grain contact
%network and not due to viscous damping.
%Experiments under controlled humidity demonstrates
%contact damping being dominant.

Generally speaking, the real part of $\tilde{M}(\omega)$ exhibits
a sharp resonance which we interpret in terms of an effective
sound speed. The imaginary part of $\tilde{M}(\omega)$ shows a
broad resonance peak which quantifies the attenuation of acoustic
waves in the system. We observe significant changes in the
stiffness and attenuation as a function of humidity. By monitoring
the effective mass in time, we find a logarithmic aging effect in
the resonance frequency as well as an increase of the damping upon
humidification.
%and show an interesting aging
%phenomena \cite{bocquet,restango,Connell,damour} as manifested in
%$\tilde{M}(\omega)$ measurements. We show that aging observed at
%the microscopic level during humid-dry cycles is due to aging
%occurring at the microscopic asperities.}
These effects can be modeled as capillary condensation occurring
between the asperities at the contact surfaces between the grains
during humidity-dry cycles. We interpret this phenomenon in the
context of a ``trap model'' of thermally activated liquid bridges.
% and
%show that this mechanism underlies the kinetic of aging in the
%effective mass.
Our results suggest that, in the granular materials investigated
in the present study, dissipation of acoustic energy is dominated
by the asperities at the interparticle contact surfaces. In
addition, humidity drastically affects the attenuation of the
material through the capillary condensation of liquid bridges.

\begin{figure}[b]
\centerline{ \hbox{ \resizebox{8cm}{!} {
\includegraphics{fig1.eps}} } }
\caption {Effective mass of tungsten particles in a rigid cavity
as a function of driving frequency. Inset: Scanning Electron
Microscope image of the tungsten particles used in our
experiments. The scale at the bottom of the image corresponds to
$300 \mu$m.} \label{experiments}
\end{figure}



{\it Experiments.---} A cylindrical cavity (of diameter $D$=25.4 mm)
excavated in a rigid aluminum cup is filled with tungsten particles up
to a height of 30.7 mm.  We use tungsten particles due to their large
density, which maximizes the effects we are studying.  Each of these
particles consists of four or five equal-axis particles, of nominal
size 100 $\mu$m, fused together (see inset of
Fig. \ref{experiments}). Such irregular arrangement produces low
volume fractions (in this case $48\%$, in comparison with random close
packing of spherical particles, $64\%$) and allows to study the loose
packings of interest in the present work.  To investigate the
applicability of our results to other materials we perform experiments
with spherical glass beads as well as lead beads, see below.
% In the terminology of the jamming transition, we are interested in
% disordered packings just above jamming.  The cup is mounted on a
% shaker ($B\&K$ mini-shaker) and subjected to a vertical sinusoidal
% vibration at frequency $f$. The resulting acceleration is in the
% order of $0.1 ms^{-2}$ to $1 ms^{-2}$, greater at high frequencies,
% and is much less than $1g$.
The cup is mounted on a B\&K shaker and subjected to a vertical
sinusoidal vibration at frequency $\omega=2 \pi f$, with the frequency
scan from sub kHz to 10 kHz. Since the bulk behavior depends on the
contacts between the grains, the way of filling the granular medium in
the container is important. Prior to the measurement, we pour the
grains into the cup and then shake the cup near its resonance
frequency (see below) at large amplitudes up to 3g for 30 minutes to
allow the compaction of the material.  During the measurement, the
granular medium is driven with cup acceleration of approximately 2 to
3 $ms^{-2}$. The amplitude of vibration is approximately 1 $\mu$m for
the lower frequencies and decreases to below 1 nm for the higher
frequencies.

Force $F(\omega)$ and acceleration $a(\omega)$ are independently
measured with a force gauge and an accelerometer attached to the
bottom of the cup. The signals are analogically filtered with a
lock-in amplifier, which is synchronized with the driving signal.
These filtered signals are then digitally recorded.  The force
gauge is sandwiched between the shaker and the cup container. At
each frequency, the force and acceleration are averaged over a 500
ms or longer time window. Prior to this time window, there is a 3
sec waiting time following the onset of the driving frequency in
order to allow the system to reach a steady state, and achieve
accurate measurements. Since $F(\omega)$ and $a(\omega)$ are the
net force exerted on and the response of the cup-granules
assembly, their ratio, the effective mass, is independent of the
shakers frequency response. Taking into account the mass of the
empty cup, $M_c$, we have $ \tilde{M}(\omega) + M_c =
\frac{F(\omega)}{a(\omega)},$ where, $\tilde{M}(\omega) \equiv
M^{'}(\omega)+iM^{''}(\omega)$, reflects the partially in-phase,
$M^{'}$, partially out-of-phase, $M^{''}$, motion of the
individual grains, relative to the driving force.  Thus $M^{'}$
carries information about the inertia and elastic response while
$M^{''}$ relates to attenuation of particle response.

\begin{figure}[b]
\centerline{
%\vbox{ (a) \resizebox{5.cm}{!}
%    {\includegraphics{w_pb.eps}}
%\vspace{1cm}
\resizebox{8.5cm}{!}
    {\includegraphics{fig2.eps}}
%(c) \resizebox{4.5cm}{!}
%    {\includegraphics{sound_speed.eps}}
}
 \caption{(a) Real ($M^{'}$) and (b) Imaginary part ($M^{''}$) of effective
  mass for spherical lead
%($180-250\mu m$)
and
  glass
%($150\mu m$)
  beads.  (c) Effective mass of fluorocarbon fluid. (d) Crossplot of
  measured values and book values of densities and sound speeds for
  various liquids: Methanol (o and $\square$, first symbol density,
  second sound speed), Ethanol (+ and $\times$), Fluorocarbon fluid
  ($\diamond$ and $\triangledown$), Silicon
  oil, (viscosity 10 mPa-s, $\vartriangle$ and $\ast$). (e) Effective
  speed of sound in tungsten granules as a function of filling depth,
  $L$, for constant cavity diameter, $D=25.4$mm.
(f) Continuum model prediction of $
  \tilde{M}(\omega)$.}
%The model interprets the resonance peak as a 1/4
%  wavelength resonance of the compressional sound speed. The values of
%  the bulk modulus $K_0$ and $\xi$ in the model are chosen to mimic
%  the observed frequency position and resonance width, respectively,
%  in the experiments. }
\label{fluid_grains}
\end{figure}

Fig. \ref{experiments} shows the result for a frequency sweep of
$\tilde M(\omega)$. We note that in the low frequency limit,
$M^{'}(\omega\to 0)\to 149$g, the static mass of the grains.  More
interestingly, there is a relatively sharp resonance peak around
$f_0\equiv \omega_0/2\pi = $1.8 kHz and a relatively broad tail
that diminishes with increasing frequency. $M^{''}$ exhibits a
broad peak centered at $w_0$. We have conducted the experiments
with other granular materials. Fig. \ref{fluid_grains}(a)-(b)
shows the results for spherical lead beads of the size 180-250
$\mu$m and glass beads of 150 $\mu$m. These results are
qualitatively similar to those of tungsten particles, in both the
real and imaginary parts; the tungsten particles exhibit larger
effects.

%A wide variety of real liquids satisfy the assumptions of the above
%model. Accordingly,
We also applied the effective mass technique to liquid samples.
Fig. \ref{fluid_grains}c shows a typical plot for the effective
mass of one such liquid (fluorocarbon fluid, density 1800
kg/m$^3$) as measured by the effective mass technique.
$M^{'}(\omega)$ shows a sharp resonance while the $M^{''}(\omega)$
curve is almost zero over the whole frequency range (except near
resonance) indicating negligible dissipation.  We can extract a
value for the effective compressional sound speed from the
resonance frequency as $v_s(L) = 4 L f_0$, where L is the depth of
the medium. The values of density and sound speed for a variety of
liquids, as measured with the effective-mass technique, are
cross-plotted against those determined by more conventional means
\cite{crc} in Fig. \ref{fluid_grains}d. The density of the liquids
is measured using Archimedes principle and buoyancy (accuracy of
0.1\%). The sound speed is measured with ultrasonic (centered at
500 kHz) through-transmission method (accuracy of 1\%).  The grain
density of the tungsten sample is measured with a Micromeritics
helium pycnometer (accuracy 1\%) and the bulk density is
calculated from the measured weight of tungsten grains filling a
cup of a known volume. Volume fraction is then estimated from the
grain and bulk density. The results in Fig. \ref{fluid_grains}d
suggests that our technique for measuring $\tilde M(\omega)$ is an
accurate one.

% We have measured the values of density and sound speed the resonance
% frequency provides an estimate of an effective compressional sound
% speed as $v_s(L) = 8 \pi L f_0$.  Similarly, from the measured
% low-frequency asymptote of $M^'(\omega)$ we can extract a value for
% the density.  for a variety of liquids, as measured with the
% effective-mass technique, and cross-plotted against those determined
% by more conventional means \cite{crc} and found good agreement.  in
% Fig. \ref{fluid_grains}c.

We investigate how the effective sound speed changes with the
filling depth $L$, based on the peak frequency $f_0$.  The sound
speed of the granular media is estimated based on the hypothesis
of 1/4 wavelength resonance frequency. Fig. \ref{fluid_grains}e
shows the trend of greater speed with greater depth.  The values
we are reporting are of the same order of magnitude as those
reported in the literature \cite{cremer,bourinet2,liu,shield}.
%We may expect deviations from the linearity $v_s(L) \sim L$ when $L$
%becomes increasingly larger, so that larger filling fractions are
%needed to assess the limit of validity of the approach.
In the range studied here, the linear relation $v_s(L) \sim L$
seems to hold up to $L=61$ mm as seen in Fig. \ref{fluid_grains}e.
% For larger $L$, our data are limited by the load limit of the
% shaker.


%{\it Continuum Models.---}
We may model the general features of Fig. \ref{experiments} in terms
of a simplified continuum model. The granular medium is considered to
be a lossy fluid, with negligible viscous effects at the walls.  The
acoustic pressure in the fluid is described by a one-dimensional
model: $P(z)= A\sin(qz) + B\cos(qz)$. The boundary conditions are: (a)
The displacement of the fluid at $z=0$ matches that of the cup,
$u(z=0) = u_{\rm cup}$. (b) The acoustic pressure at the top of the
fluid vanishes, $P(z=L) = 0$.  Thus, $P(z) = u_{\rm cup} \rho \omega^2
[\sin(qz) - \tan(qL) \cos(qz) ]/q$. The force the cup exerts on the
fluid is $F=\pi a^2 P(z=0)$ and the acceleration of the cup is
$-\omega^2 u_{\rm cup}$ from which we obtain the effective mass:
% Eq. (1) follows by definition of the effective mass.  that is, the
% viscous skin depth $\delta = \sqrt{2 \eta/(\rho \omega)}$ is small
% compared to the radius of the cup, $a$.  Here, $\eta$ is the
% viscosity and $\rho$ is the density of the medium.  The
\begin{equation}
\tilde{M}(\omega) = \pi a^2  \sqrt{\rho K}/\omega \tan(qL),
\label{M}
\end{equation}
where $K= K_0[1-i \omega \xi]$ is the (lossy) bulk modulus of the
medium, and $q = \omega \sqrt{\rho/K}$ is the complex wave vector with
$\rho$ the density of the medium. This model
%assumes that
% 100\% of $\tilde{M}(\omega)$ is on the bottom surface of the cup and
shows that resonance peaks, as seen in Fig. \ref{fluid_grains}f,
occur when $L$ equals odd multiples of 1/4 wavelength of the
acoustic wave in the medium, with the main resonance peak at
$f_0=1/(4 L) \sqrt{K_0/\rho}$.  Encouraged by these results, we
interpret the main resonance in Fig.  \ref{experiments} as being a
1/4 wavelength resonance of the compressional sound speed.




%this resonance
%shifts to higher frequencies as the filling depth, $L$, is
%changed.
% Throughout the volume of grains in the cup, the sound
%speed must be depth dependent; the gravity-attributed stiffness is
%small at the surface and maximum at the bottom.  Nevertheless, w
%and on the filling depth of the
%tungsten particles:
% {\bf where $f_0(L)$ is
%the depth dependent resonance frequency.}



% Next, we use the features in $\tilde M(\omega)$ to investigate the
% mechanisms of dissipation in the granular medium.


Next, we use the features in $\tilde M(\omega)$ to investigate the
dissipation in the granular medium. The dissipation mechanisms
can be grouped into two categories: 1) Local damping throughout the
contact network that includes asperity deformation and bulk
attenuation \cite{poschel2}, and wetting dynamics in the liquid
bridges caused by relative motion of neighboring particles
\cite{1}. Wetting dynamics also results in greater contact stiffness
\cite{damour}.  If large interparticle tangential displacements occur
then, there exist dissipation through Coulomb friction as well. 2)
Global damping due to drag caused by particle motion in a surrounding
viscous medium \cite{wolf-md}.

% Generally speaking dissipation can be classified in two broad
% categories: (a) Contact damping through the contact network, that
% includes bulk attenuation as particle deformation
% \cite{poschel2,wolf-md} in which the damping force is proportional
% to the relative velocity between two contacting grains and also
% viscous dissipation through the liquid bridges caused by moisture
% condensing from the surrounding gas at the point of contacts
% \cite{adams} Liquid bridges also provide a cohesive force at the
% contact that contributes to the stiffness of the material
% \cite{rietema}.  The conclusions we draw from our numerical
% simulations are relatively insensitive to the assumed values for the
% damping parameters.  (b) Global damping, in which there is a
% rotational and translational drag due to the surrounding viscous
% fluid, such as air (see \cite{thornton}).
%
% modeled as a force term proportional to the velocity of the
% particles (not the relative velocity).

% The continuum model is illuminating as far as it goes but t

%To test the  different mechanisms of  dissipation,
Next, we perform further experimentation by changing the ambient
conditions.  Ambient humidity can modify the dissipation and the
stiffness at the grain to grain surface contacts \cite{damour}.
For instance, small amounts of liquid greatly affects the maximum
angle of stability in sandpiles \cite{barabasi}, while capillary
action of the interstitial liquid results in increased cohesion
\cite{damour}, modified friction \cite{riedo} and aging in the
angle of repose of granular materials \cite{bocquet}.  We place
the effective mass apparatus in a chamber where humidity is
controlled by first using an aqueous saturated salt solution
\cite{restango} and subsequently a dessicant. We measure the
relative humidity as the ratio of the partial pressure of water to
the saturating vapor pressure $H_{\rm
  R}=p_{\rm vap}/p_{\rm sat}$.


We follow a humid-dry cycling process as follows: we start from
room humidity $H_{\rm R}= 26\%$ at $t<0$, and then we humidify at
$t=0$ to $H_{\rm R}= 75\%$ until $t=126$ hr, and then we dry to
$H_{\rm R} = 0\%$ until $t=174$ hr. Fig. \ref{humidity} shows the
effective mass measurements under the humidity controlled
environment carried out at various time intervals. We observe a
very slow shift in resonant frequency during the humidification
cycle, followed by a further positive shift upon drying. The
effect is more important when we repeat the same cycling but to
$95\%$ humidity.

During humidification, we observe an aging phenomenon as evidenced by
the approximate logarithmic behavior of the shift in resonant
frequency $\Delta f_0(t)$ (top inset of Fig. \ref{humidity}), $\Delta
f_0(t) = C(H_{\rm R}) \log(t/t_i),$
%\label{aging}
%\end{equation}
where $ C(H_{\rm R})$ is a constant that increases with humidity,
$ C(75\%)=292$ Hz and $C(95\%)=494$ Hz and $t_i\approx 2$ hr for
both measures.

Furthermore, the humid system shows a large increase in the
attenuation as illustrated by the increase in damping rate
$\Gamma$ (bottom inset of Fig. \ref{humidity}). Based on the
experimental effective mass data,
% for real valued $\omega$,
we search for poles in $\tilde{M}(\tilde\omega)$ for $\tilde\omega$ in
the complex plane. Once the complex resonance frequency
($\tilde{\omega}_{0}=\omega_{0}^{'}+i\omega_{0}^{''}$) is found, the
damping rate is given by $\Gamma=-\omega_{0}^{''}$.
% The damping rate and quality factor are related as follows:
% $\Gamma=\omega_{0}^{'}/2Q$.
The damping rate at ambient humidity is 300 $s^{-1}$, which rises to
1200 $s^{-1}$ under prolonged humidification
% at 75$\%$ relative humidity
(see bottom inset of Fig. \ref{humidity}). The damping data suggests
logarithmic increase with time.
% however the trend is not conclusive.

\begin{figure}[t]
\centerline { \hbox{ \resizebox{9.5cm}{!} {
\includegraphics{fig3.eps}}}}
\caption{Imaginary part of $\tilde M(f)$ measured under humidity
  controlled conditions (the real part shows similar behavior). The
  solid lines represent measurements under $H_{\rm R}=75\%$ and the
  dashed lines are for the dry state, $H_{\rm R}=0\%$. Top inset:
  Frequency shift vs time for $H_{\rm R}$=75\% (o) and 95\%
  ($\square$). Bottom inset: Damping rate ($\Gamma$) vs time for
  $H_{\rm R}$=75\%. The dashed lines in the insets are logarithmic
  fits. The up and down arrows indicates the large variations in
  stiffness and damping upon drying, respectively.}
\label{humidity}
\end{figure}

A possible interpretation of the observed behavior suggests that
contact forces through capillary liquid bridges are important for
damping, stiffness and aging. As demonstrated in \cite{bocquet},
the humidification process leads to the formation of capillary
bridges around the roughness asperities at the interparticle
contact surfaces. The aging effect observed in the logarithmic
variation of the frequency shift with time can be interpreted as
originating from capillary condensation. The key quantity to
calculate is the time dependence of the number of wetted
asperities $n_a(t)$.  Capillary condensation at the surface
asperities can be seen as a thermally activated process following
a model proposed in \cite{bocquet}.  The energy barrier to form a
liquid bridge of cross sectional area $A$ between two asperities
separated by a gap $h$ is $\Delta E = \Delta \mu \rho_{l} Ah$,
where $\rho_{l}$ is the molecular density of the condensing
liquid, $\Delta \mu = k_{\rm B} T \ln(1/H_{\rm R})$ is the
undersaturation of the chemical potential, and $T$ the room
temperature. The activation process follows an Arrhenius dynamics
such that the time to create a bridge is $\tau = t_0 \exp(\Delta
E/ k_{\rm B} T)$, with $t_0$ the microscopic condensation time.

Due to the roughness of the particles we expect a distribution of gaps
$P(h)$, which leads to a distribution of energy barriers, $P(\Delta
E)$. The calculation of the probability distribution $P(\tau)$ is then
analogous to obtaining the probability of the time to escape from a
trap with energy barrier $\Delta E$ for a thermally activated particle
in a random energy potential. Such a process corresponds to the ``trap
model'' used to describe the phenomenology of glasses
\cite{bouchaud}. The probability $P(\tau)$ satisfies $P(\tau) d\tau=
% P(\Delta E) dE =
P(h) dh$.  While the probability $P(h)$ is unknown, we expect it to be
a bounded distribution determined by the surface roughness of the
particles. The two most common cases lead to the same result: for an
exponential or Gaussian distribution of gaps, we find $P(\tau) \sim
\tau^{-1}$.
%with average $h_0$: $P(h) = (1/h_0)
%\exp(-h/h_0)$ leads to $P(\tau) = xt_0^x \tau ^{-(1+x)}$ where $x=1/
%(A \rho_l h_0 \ln (1/H_{\rm R}))$. Clearly $x\ll1$, then
%[Notice that in the terminology of the trap model, the
%system is deep into the glassy phase. A small but nonzero value of $x$
%assures that the probability is normalizable]. A Gaussian model of
%gaps leads to a distribution which can be approximated by a
%$P(\tau)\sim \tau^{-1}$ scaling.
%a Gaussian distribution  $P(h) = 1/(\sqrt{\pi}\sigma)
%\exp[(h-h_0)^2/\sigma^2]$,
%where $h_0$ is the typical surface
%roughness and $sigma$ its fluctuation.
%In the former case we find a power-law $P(\tau) \sim \tau ^{1+x}$
%where $x=1/ (A \rho_l h_0 \ln H_{\rm R})$,
%while in the Gaussian model
%a log-normal distribution is obtained.
%We argue that both models leads to
%since the
%and the log-normal distribution can
%be approximated by $1/\tau$
%as well.
%Therefore both distributions lead to
%$P(\tau)\sim \tau^{-1}$.

The number of activated wetted asperities at time $t$ is $n_a(t)
\propto \int_{t_0}^{t} P(\tau) d\tau$.  Then $n_a(t) \approx
% [n_f/\log(t_f/t_0)]
\log (t/t_0)$.
% where $n_f$ is the final number of wetted asperities at
%time $t_f$.
Liquid bridges induce a significant cohesion between the particles
which for loose grains under gravity and large humidity outweigh the
weight of the grains. Therefore, the logarithmic increase in $n_a$
leads to an additional stiffness with the concomitant increase in the
compressional sound speed. Since the resonance frequency is
proportional to the stiffness of the contact,
% as seen from Eq.  (\ref{fluid1}),
the logarithmic aging can be interpreted as arising from the wetted
asperities.
% results as observed experimentally.  The total number of wetted
% asperities increases with $H_{\rm R}$ suggesting the behavior of
% $C(H_{\rm R})$.


Upon drying the particles to $H_{\rm R} = 0\%$ after humidification,
%both the real and imaginary part $\tilde{M}(\omega)$
%show dramatic changes.
%During drying,
the physics is very different from the humid process.  As shown by
the experiments of \cite{damour}, the capillary rings dry out and
the asperities are sucked down by the increasing capillary
pressure in the evaporating film leading to an even larger
increase in stiffness of the contacts. Thus, in the dried out
state there is much more solid on solid contact and the sound
speed is increased.  This increase in stiffness upon drying
results in even larger positive shifts in resonant frequency as
seen in the top inset of Fig. \ref{humidity}.

We also find important changes in the dissipation in the dry state.
% also changes
%drastically.
Upon drying, the decrease in overall attenuation is evident from
the much sharper resonance in $M^{''}$ (we find a very large mass
at resonance $M^{''}(f_0)\approx 3500$ g). Accordingly, the
damping rate decreases to $\Gamma=50$ $s^{-1}$ in the dry case
from $\Gamma=1200$ $s^{-1}$ at $H_{\rm R}=75\%$ (see bottom inset
of Fig. \ref{humidity}). In the driest state, no liquids bridges
are left and the dissipation is purely through asperities
deformation with the concomitant lower damping strength. These
results suggest
% lead us to conclude
that, for the granular media studied here
%grains under gravity and
when humid conditions prevail (including ambient), liquid bridges
provide larger attenuation of acoustic waves than the viscoelastic
dissipation in the bulk of the particles or viscous global damping.

%The broader resonance peak during the humidification process indicates
%higher dissipation.
%During this process, the $Q$ factor
%changes from a $Q=100$ at the ambient humidity to
%$Q=10$ as seen in Fig. \ref{humidity}.


%{\bf OUT


%molecular
%dynamic simulations of $\tilde M(\omega)$.  Here, we consider the
%much simpler system of spherical beads confined in a rectangular
%box.

%In our simulations, a static packing at a pre-determined pressure
%is first achieved by methods previously described \cite{mgjs}, and
%then we incorporate walls, friction, and the force of gravity. The
%simulations consider the typical Hertz-Mindlin contact forces for
%the normal and tangential components, respectively, and the
%presence of Coulomb friction between the particle characterized by
%a friction coefficient $\mu$.  See Ref. \cite{mgjs} and references
%therein for a discussion of the underlying physics of these
%contact forces.

%\begin{figure}[t]
%\centerline{ \hbox{ \resizebox{4cm}{!} {
%\includegraphics{viscosity_v02.eps}} }}
%\caption {
%Log-log plot of the imaginary part
%of effective mass of loose granular media.
%using molecular dynamics
%simulations for a system with only contact damping or only global
%damping and comparison with experiments.
%The experimental results can be better fitted with a
%model  based on
%contact damping is suggestive of a macroscopic viscosity $M^{"}
%\propto f^{-1/2}$ whereas that based on global damping $M^{"}
%\propto f^{-1}$ is not.
%The simulation results have been shifted
%vertically. (b) Computed real part of effective mass assuming only
%local contact damping ($\mu=0.5$). $M^{'}$ clearly reproduces the
%resonance peak.}
%}
%\label{viscosity}
%\end{figure}




%Results for the effective mass versus frequency are displayed in
%Fig. \ref{viscosity}(a) for a system in which there is assumed to
%be only local contact damping {\bf and another with only viscous
%damping}. {\bf We show separately, $M^{'}$ for a system with only
%contact damping and $\mu=0.5$ (see Fig. \ref{viscosity}(b)). The
%fundamental resonance and the broad high-frequency tail are
%clearly evident.} The simulation results can be summarized as
%follows:






%Simulations allow us to differentiate between {\bf the two}
%different {\bf mechanisms} of dissipation.
%We find that either
%{\bf viscous} or {\bf contact} damping can capture the main
%features of the experiments: the main resonance peak and a broad
%background. However,



%The analysis of the high-frequency asymptotic behavior of $\tilde M$
%further confirms our experimental findings.
%Viscous damping predicts
%$\tilde{M}(f) \sim i f^{-1}$,
%while, as long as the viscous skin depth
%%$\sqrt{2 \eta_{\rm
%%eff}/\rho f}$
%is large compared to the grain size but small
%compared to the cup radius, then contact damping predicts $\tilde
%M(\omega) \sim (i/ f)^{1/2}$. Figure \ref{viscosity}
%%as can be seen, more or
%%less, in the result plotted in Fig. \ref{viscosity}(a). On the
%%other hand, as long as the viscous skin depth $\sqrt{2 \eta_{\rm
%%eff}/\rho f}$ is large compared to the grain size but small
%%compared to the cup radius, then contact damping predicts $\tilde
%%M(\omega) \sim (i/ f)^{1/2}$. {\bf Fig. \ref{viscosity}(a)
%shows the comparison between the asymptotic behavior of
%$M^{"}(f)$ between experiments and both predictions.
%%simulations in an attempt
%%to distinguish between both mechanisms. We recognize that this is
%While we do not attempt to fit the data since
%the large $f$  behavior is clearly masked by
%the noisy high frequency resonance peaks, the comparison suggests
%that, asymptotically, $f^{-1/2}$ follows better the trend of the
%experimental data than the $f^{-1}$ dependence of viscous
%damping. Although not conclusive, this result agrees with the
%dominance of the contact damping mechanism as found
%above.


%Thus, our $\tilde{M}(\omega)$ measurements clearly indicate that
%contact damping is the main mechanism for dissipation.

In conclusion, we have demonstrated the usefulness of the effective
mass technique to study the dissipation of acoustic waves, stiffness
and aging in granular media. Experiments and theoretical models
suggest that capillary condensation at the interparticle contacts is
an important mechanism for dissipation and aging.
%The humidity controlled experiments clearly show
%aging phenomena due to microscopic changes occurring at the
%grain-grain contacts.
The kinetics of aging can be modeled in terms of a trap model for the
formation of liquid bridges at the asperities.  The large variations
in dissipation found in this study demonstrate the conditions for
effective particle damping and are relevant to a variety of
applications for optimizing attenuation of structure-borne acoustic
waves.
%Our results have also important implications to
%understand acoustic propagation and elasticity of jammed matter near
%the jamming transition.  While many studies assume Hertz-Mindlin
%interactions between grains near jamming, here we show that the main
%interaction is through the capillary liquid bridges which provides a
%totally different outlook on the problem of jamming; Our results
%suggest a different scaling than that predicted by Hertz-Mindlin
%theory of the elastic moduli of the material near the jamming
%transition under humid or dry conditions.


%\begin{acknowledgments}
{\it Acknowledgments. }
We are grateful to L.  McGowan for technical assistance.  We
acknowledge financial support from DOE, Geosciences Division.
%\end{acknowledgments}



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L. Cremer and M. Heckl, {\it Structure Borne Sound}  (Springer, Berlin, 1973).

\bibitem{duran}
J. Duran. {\it Sands, powders and grains. An introduction
to the physics of granular materials} (Springer-Verlag,
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\bibitem{liu}
C.-h. Liu and S. Nagel, {\it Phys. Rev. Lett.} {\bf 68}, 2301 (1992).

\bibitem{bourinet}
J. M. Bourinet and D. Le Houedec, {\it Computers and Structures} {\bf 73},
395 (1999).

\bibitem{bourinet2}
J. M. Bourinet and D. Le Houedec, {\it Powders \& Grains 97} eds.
R. Behringer and J. Jenkins (Balkema, Rotterdam, 1997).

%\bibitem{liu2}
%C.-h. Liu and S. Nagel, {\it Phys. Rev. B} {\bf 48}, 15646 (1993).


%\bibitem{hsu}
%R. Ingale, C.-J. Hsu, and D. L. Johnson, (unpublished).

%\bibitem{selfridge}
%A. Selfridge, {\it IEEE Trans. Sonics and Ultrasonics} {\bf 32} (1985).


\bibitem{crc}
Handbook of Chemistry and Physics, 81st Edition (CRC Press, Cleveland, 2000).


\bibitem{shield}
F. D. Shields, J. M. Sabatier and M. Wang, {\it J. Acoust. Soc. Am.} {\bf 108}, 1998 (2000).

%\bibitem{mgjs}
%H. A. Makse, N. Gland, D. L. Johnson, and L. M. Schwartz, {\it Phys. Rev. E} {\bf 70}, 061302 (2004).

\bibitem{poschel2} N. V. Brilliantov, F. Spahn, J.-M. Hertzsch, and
T. P\"oschel, {\it
Phys. Rev E} {\bf 53}, 5382 (1996).


\bibitem{1} J. Crassous, E. Charlaix, H. Gayvallet, and J.-L. Loubet,
  {\it Langmuir} {\bf 9}, 1995 (1993); J. Crassous, E. Charlaix, and J.-L. Loubet, {\it
    Phys. Rev. Lett} {\bf 78}, 2425 (1997); M. J. Adams and V. Perchard, {\it Inst. Chem. Engr. Symp.} {\bf 91},
147 (1985).

%\bibitem{rietema}
%K. Rietema, {\it The Dynamics of Fine Powders} (Elsevier Applied
%Science, London, 1991).

\bibitem{damour}
J. N. D'Amour, {\it et al.}
%J. J. R. St\aa lgren, K. K. Kanazawa, C. W. Frank,
%M. Rodahl, and D. Johannsmann,
{\it Phys. Rev. Lett.} {\bf 96},
058301 (2006).


\bibitem{wolf-md} J. Sch\"afer, S. Dippel, and D. E. Wolf,
{\it J.
Phys. I} (France) {\bf 6}, 5 (1996).



%\bibitem{rayleigh} Lord Rayleigh, {\it Theory
%of Sound}, (Dover Publications, New York, 1945 edition, 1877).

%\bibitem{thornton} C. Thornton, and D. J. Barnes, {\it Acta Mechanica}
%  {\bf 64}, 45 (1986).




\bibitem{barabasi}
D. J. Hornbaker, R. Albert, I. Albert, A.-L. Barab\'asi, P. Schiffer,
{\it  Nature}
 {\bf 387}, 765 (1997).


\bibitem{riedo} E. Riedo, F. L\'evy, and H. Brune, {\it
    Phys. Rev. Lett.} {\bf 88}, 185505 (2002).


\bibitem{bocquet} L. Bocquet, E. Charlaix, S. Ciliberto and
  J. Crassous, {\it Nature } {\bf 396}, 735 (1998).

\bibitem{restango}
F. Restagno, C. Ursini, H. Gayvallet, and E. Charlaix, {\it Phys.
Rev. E} {\bf 66}, 021304 (2002).

%\bibitem{damour} R. J. O'Connell and B. Budiansky,{\it J. Geophys.
%Res.} {\bf 79}, 5412 (1974);  J. N. D'Amour, J. J. R. St\aa lgren,
%K. K. Kanazawa, C. W. Frank, M. Rodahl, and D. Johannsmann, {\it
%Phys. Rev. Lett.} {\bf 96}, 058301 (2006).

%\bibitem{Connell} R. J. O'Connell and B. Budiansky, {\it J. Geophys.
%Res.} {\bf 79}, 5412 (1974).



\bibitem{bouchaud}
C. Monthus and J.-P. Bouchaud, {\it J. Phys. A} {\bf 29}, 3847
(1996).

%\bibitem{denny} R. A. Denny, D. R. Reichman, and J.-P. Bouchaud,
%{\it Phys. Rev. Lett.} {\bf 90}, 025503 (2003).

\end{thebibliography}





\end{document}

cond-mat0605716.extracted_bib
\begin{thebibliography}{99}

\bibitem{cremer}
L. Cremer and M. Heckl, {\it Structure Borne Sound}  (Springer, Berlin, 1973).

\bibitem{duran}
J. Duran. {\it Sands, powders and grains. An introduction
to the physics of granular materials} (Springer-Verlag,
2000).

\bibitem{liu}
C.-h. Liu and S. Nagel, {\it Phys. Rev. Lett.} {\bf 68}, 2301 (1992).

\bibitem{bourinet}
J. M. Bourinet and D. Le Houedec, {\it Computers and Structures} {\bf 73},
395 (1999).

\bibitem{bourinet2}
J. M. Bourinet and D. Le Houedec, {\it Powders \& Grains 97} eds.
R. Behringer and J. Jenkins (Balkema, Rotterdam, 1997).

%\bibitem{liu2}
%C.-h. Liu and S. Nagel, {\it Phys. Rev. B} {\bf 48}, 15646 (1993).


%\bibitem{hsu}
%R. Ingale, C.-J. Hsu, and D. L. Johnson, (unpublished).

%\bibitem{selfridge}
%A. Selfridge, {\it IEEE Trans. Sonics and Ultrasonics} {\bf 32} (1985).


\bibitem{crc}
Handbook of Chemistry and Physics, 81st Edition (CRC Press, Cleveland, 2000).


\bibitem{shield}
F. D. Shields, J. M. Sabatier and M. Wang, {\it J. Acoust. Soc. Am.} {\bf 108}, 1998 (2000).

%\bibitem{mgjs}
%H. A. Makse, N. Gland, D. L. Johnson, and L. M. Schwartz, {\it Phys. Rev. E} {\bf 70}, 061302 (2004).

\bibitem{poschel2} N. V. Brilliantov, F. Spahn, J.-M. Hertzsch, and
T. P\"oschel, {\it
Phys. Rev E} {\bf 53}, 5382 (1996).


\bibitem{1} J. Crassous, E. Charlaix, H. Gayvallet, and J.-L. Loubet,
  {\it Langmuir} {\bf 9}, 1995 (1993); J. Crassous, E. Charlaix, and J.-L. Loubet, {\it
    Phys. Rev. Lett} {\bf 78}, 2425 (1997); M. J. Adams and V. Perchard, {\it Inst. Chem. Engr. Symp.} {\bf 91},
147 (1985).

%\bibitem{rietema}
%K. Rietema, {\it The Dynamics of Fine Powders} (Elsevier Applied
%Science, London, 1991).

\bibitem{damour}
J. N. D'Amour, {\it et al.}
%J. J. R. St\aa lgren, K. K. Kanazawa, C. W. Frank,
%M. Rodahl, and D. Johannsmann,
{\it Phys. Rev. Lett.} {\bf 96},
058301 (2006).


\bibitem{wolf-md} J. Sch\"afer, S. Dippel, and D. E. Wolf,
{\it J.
Phys. I} (France) {\bf 6}, 5 (1996).



%\bibitem{rayleigh} Lord Rayleigh, {\it Theory
%of Sound}, (Dover Publications, New York, 1945 edition, 1877).

%\bibitem{thornton} C. Thornton, and D. J. Barnes, {\it Acta Mechanica}
%  {\bf 64}, 45 (1986).




\bibitem{barabasi}
D. J. Hornbaker, R. Albert, I. Albert, A.-L. Barab\'asi, P. Schiffer,
{\it  Nature}
 {\bf 387}, 765 (1997).


\bibitem{riedo} E. Riedo, F. L\'evy, and H. Brune, {\it
    Phys. Rev. Lett.} {\bf 88}, 185505 (2002).


\bibitem{bocquet} L. Bocquet, E. Charlaix, S. Ciliberto and
  J. Crassous, {\it Nature } {\bf 396}, 735 (1998).

\bibitem{restango}
F. Restagno, C. Ursini, H. Gayvallet, and E. Charlaix, {\it Phys.
Rev. E} {\bf 66}, 021304 (2002).

%\bibitem{damour} R. J. O'Connell and B. Budiansky,{\it J. Geophys.
%Res.} {\bf 79}, 5412 (1974);  J. N. D'Amour, J. J. R. St\aa lgren,
%K. K. Kanazawa, C. W. Frank, M. Rodahl, and D. Johannsmann, {\it
%Phys. Rev. Lett.} {\bf 96}, 058301 (2006).

%\bibitem{Connell} R. J. O'Connell and B. Budiansky, {\it J. Geophys.
%Res.} {\bf 79}, 5412 (1974).



\bibitem{bouchaud}
C. Monthus and J.-P. Bouchaud, {\it J. Phys. A} {\bf 29}, 3847
(1996).

%\bibitem{denny} R. A. Denny, D. R. Reichman, and J.-P. Bouchaud,
%{\it Phys. Rev. Lett.} {\bf 90}, 025503 (2003).

\end{thebibliography}