
\documentclass[showpacs,a4paper,preprint,titlepage,secnumarabic,groupedaddress,prl]{revtex4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage[final]{graphicx}
\usepackage{dcolumn}
\usepackage{bm}
\usepackage{verbatim}

\setcounter{MaxMatrixCols}{10}
%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=5.50.0.2953}
%TCIDATA{<META NAME="SaveForMode" CONTENT="2">}
%TCIDATA{BibliographyScheme=BibTeX}
%TCIDATA{LastRevised=Saturday, March 16, 2013 20:59:38}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=revtex4.cst}
%TCIDATA{PageSetup=72,72,72,72,0}
%TCIDATA{Counters=arabic,1}
%TCIDATA{<META NAME="PrintViewPercent" CONTENT="100">}
%TCIDATA{AllPages=
%H=36
%F=36
%}


\bibliographystyle{apsrev}

\begin{document}

\title{Supplementary Material for Complex oscillatory yielding of model hard
sphere glasses}
\author{N. Koumakis$^{1}$, J. F. Brady$^{2}$ and G. Petekidis$^{1}$}
\affiliation{$^{1}$FORTH/IESL and Department of \ Materials Science \& Technology,
University of \ Crete, 71110, Heraklion, Greece}
\affiliation{$^{2}$Division of Chemistry and Chemical Engineering, Caltech, Pasadena, CA
91125, USA}
\maketitle

\section{Computer Simulations}

Brownian dynamics (BD) simulations were conducted according to \cite%
{FossBrady}, with the addition of an oscillatory shear field and the
incorporation of a moderate polydispersity in the particles in order to
avoid shear induced crystallization.

In general since colloidal particles are much larger than solvent molecules
the clear separation of time and length scales allow us to treat the fluid
as a continuum; however colloids are still small enough to be affected by
collisions with the fluid molecules and are thus still Brownian. BD can be
described as a simplification of SD, where hydrodynamic interactions between
particles are neglected. For $N$ rigid particles of radius $R$ and density $%
\rho $ in a medium of viscosity $\eta $ moving with velocity $U$, we examine
states where the Reynolds number (the dimensionless ratio of inertial forces 
$\rho U^{2}/R$ to viscous forces $\eta U/R^{2}$) is $Re<<1$. The motion of
the particles is described by the N-body Langevin equation:

\begin{center}
$\mathbf{m}(d\mathbf{U}/dt)=\mathbf{F}^{H}+\mathbf{F}^{B}+\mathbf{F}^{P}$
\end{center}

where $\mathbf{m}$ is the generalized mass/moment tensor, $\mathbf{U}$ is
the particle translational/rotational velocity vector, $\mathbf{F}^{H}$ is
the hydrodynamic force vector, $\mathbf{F}^{B}$ is the stochastic force
vector that gives rise to Brownian motion, and $\mathbf{F}^{P}$ is the
deterministic non-hydrodynamic force vector. Since inertia is not important
in colloidal dispersions $(Re<<1)$ the equation reduces to $\mathbf{F}^{H}+%
\mathbf{F}^{B}+\mathbf{F}^{P}=0$. The Brownian forces due to thermal fluid
fluctuations are random so their ensemble average is $\overline{\mathbf{F}%
^{B}}=0$, and follow the fluctuation-dissipation theorem so that $\overline{%
\mathbf{F}^{B}(0)\mathbf{F}^{B}(t)}=2k_{B}T(6\pi \eta R)\mathbf{I}\delta (t)$
with $\delta (t)$ the delta function and $\mathbf{I}$ the unit tensor. Here
we choose to use BD simulations were Hydrodynamic Interactions (HI) between
particles are ignored \cite{FossBrady} instead of the much more
computational demanding Stokesian Dynamics (SD) method where the full HIs
between particles are explicitly computed. In this case the hydrodynamic
force reduces to Stokes drag $\mathbf{F}^{H}=-6\pi \eta R\mathbf{U}$,
allowing the study of larger and more concentrated systems, in a reasonable
computation time.

The non-hydrodynamic force vector for a simple hard sphere system becomes
the hard sphere interaction occurring at contact $\mathbf{F}^{P}=\mathbf{F}%
^{HS}$. This is calculated through the \textit{potential-free} algorithm of 
\cite{HeysMelrose,FossBrady} in which the overlap between pairs of particles
is corrected by moving the particles with equal force along the line of
centres, back to contact. In order to calculate the stress, the algorithm
directly calculates the pairwise interparticle forces that would have
resulted in the hard sphere displacements during the course of a time step 
\cite{FossBrady}. Therefore

\begin{center}
$\mathbf{F}^{HS}=-6\pi \eta R(\Delta x^{HS}/\Delta t)$
\end{center}

i.e. the average Stokes drag on the particle during the course of the
hard-sphere displacement. Once the interparticle forces from each collision
are known, they can be used to calculate the stress matrix\cite{FossBrady}
according to:

\begin{center}
$\left\langle \mathbf{\ \Sigma }\right\rangle =-N\left\langle \mathbf{xF}%
^{P}\right\rangle $
\end{center}

We should note that at rest and for relatively low $Pe$, Brownian Dynamics
simulations qualitatively capture experimental stresses and particle
motions, even with the absence of hydrodynamics \cite{FossBrady}, although
stresses cannot be quantitatively compared.  We can further note that at at
high volume fractions long range hydrodynamics can be partly screened and
that lubrication effects leading to shear thickening may become important.
However, the fact that shear thickening is absent in our experiments (in the
regime studied), further supports our observation that BD simulations
describe well the experiments.

In order to have greater clarity of the structural information and avoid
crystallization under shear \cite{KoumakisPhD}, polydispersity was added to
the simulations, which is represented by a Gaussian distribution of radii
with a rms value of $10\%$. Runs with $N=1005$ and $4860$ particles were
examined by averaging data sets over multiple oscillations (30-100) after
reaching a steady state (typically 10 oscillations). We use a time step of $%
10^{-4}\tau _{B}$ for $Pe<1$ (and rest), which scales to $10^{-4}$ per unit
strain for $Pe>1$ \cite{FossBrady}. Initial configurations were constructed
by quenching from a dilute liquid state by appropriately increasing the
particle size. The system was then properly equilibrated at rest allowing
for a steady state to be reached by following the osmotic pressure and
particle mean square displacements until the both remained stable (typically
after about $50\tau _{B}$). However, steady states under non-linear shear
were found to be generally independent of the initial conditions.

\section{Microscopic Analysis}

Using the positions of the particles from simulations, we are able to
calculate structural and dynamic properties. We use the radial pair
distribution function, $g(r)$, for the description of the microscopic
structure. In the main text, we quantify the structural anisotropy under
shear by taking the projection of the radial distribution function, $g(r)$,
in the velocity-gradient (xy) plane of the applied shear. In order to
highlight the distortions, the projection is restricted to a distance of $%
0.7 $ radii from the plane.

The Mean Squared Displacement (MSD) describes the single particle dynamics.
For example in the x axis:

\begin{center}
$\langle \Delta x^{2}(\tau )\rangle _{N,t}=\frac{1}{N}\left\langle
\sum\limits_{i=1}^{N}\left[ x_{i}(t+\tau )-x_{i}(t)\right]
^{2}\right\rangle _{t}$
\end{center}

with $x_{i}$ is the position of a particle $i$ in the $x$ axis and $N$ the
total number of particles. In our case, $x_{i}$ is the position the particle
calculated after subtraction of affine motion due to shear. In the text we
show and discuss the displacements only the vorticity ($z$) direction,
however the velocity ($x$) and velocity gradient ($y$) give similar results.
An effective diffusion coefficient defined within the period as $%
D_{eff}=\left\langle \Delta z^{2}(T)\right\rangle _{N,t}/T$ (see fig 4) may
convey the changes in microscopic motions during the period of a single
oscillation. Note that since during an oscillation period shear rate varies,
the displacements vary as well and therefore $D_{eff}$ cannot be described
as a simple diffusivity.

\section{Time-scales}

We have used a combination of experimental oscillatory shear rheometry and
Brownian Dynamics (BD) simulations in a wide range of frequencies, $\omega $%
, non-dimensionalized in the dilute regime by $Pe_{\omega }^{0}$ $=\omega
\tau _{B}$, with $\tau _{B}=R^{2}/D_{0}$ and $D_{0}=k_{B}T/6\pi \eta R$ the
free Stokes-Einstein-Sutherland diffusion coefficient. However, the short
time diffusion slows down with volume fraction due to HIs and thus we use a $%
\varphi $ dependent short-time self diffusion coefficient, $D_{s}(\varphi )$%
, \cite{SierouBrady04} in a $Pe_{\omega }=\omega R^{2}/D_{s}(\varphi )$.
Since in the BD simulations HI are excluded, the oscillation frequency is
non-dimesionalized by $Pe_{\omega }$.

\bibliographystyle{apsrev}
\bibliography{Oscillatory_Nick_lett}

\end{document}
