# Microscopic Activated Dynamics Theory of Shear Rheology and the Stress
Overshoot in Ultra-Dense Glass-Forming Fluids and Colloidal Suspensions

Ashesh Ghosh  Present address: Department of Chemical Engineering, Stanford
University, CA 64305  Department of Chemistry, University of Illinois at
Urbana-Champaign, IL 61801 USA  Materials Research Laboratory, University of
Illinois at Urbana-Champaign, IL 61801 USA  Kenneth S. Schweizer
<EMAIL_ADDRESS>Department of Chemistry, University of Illinois at
Urbana-Champaign, IL 61801 USA  Materials Research Laboratory, University of
Illinois at Urbana-Champaign, IL 61801 USA  Department of Materials Science,
University of Illinois at Urbana-Champaign, IL 61801 USA  Department of
Chemical & Biomolecular Engineering, University of Illinois at Urbana-
Champaign, IL 61801 USA  Beckman Institute, University of Illinois at Urbana-
Champaign, IL 61801 USA

###### Abstract

We formulate a microscopic, force-level, activated dynamics-based theory for
the continuous startup shear rheology of ultra-dense glass-forming hard sphere
fluids and colloidal suspensions in the context of elastically collective
nonlinear Langevin equation (ECNLE) theory. This microscopic force-based
approach describes activated relaxation as a coupled local–nonlocal event
involving caging and longer-range collective elasticity. ECNLE based
theoretical predictions for the deformation-induced enhancement of mobility,
the onset of relaxation speed up at remarkably low values of stress, strain,
or shear rate, apparent power law thinning of the steady state structural
relaxation time and viscosity, a non-vanishing activation barrier in the shear
thinning regime, an apparent Herschel–Bulkley form of the rate dependence of
the steady state shear stress, exponential growth of different measures of a
dynamic yield or flow stress with the packing fraction, and reduced fragility
and dynamic heterogeneity under deformation were shown to be in excellent
agreement with experimental results. The central new question we address is
that of stress overshoot in transient shear rheology. In contrast to the
steady state flow regime, understanding the complex transient response
requires an explicit treatment of the coupled nonequilibrium evolution of
structure, elastic modulus and stress relaxation time. Theoretical predictions
for the stress overshoot are shown to be in good agreement with experimental
observations as a function of shear rate and packing fraction in all
rheological regimes, and deformation-assisted activated motion is shown to be
crucial for both the transient and steady state responses.

††preprint: AIP/123-QED

##  I Introduction

Microscopic understanding of the nonlinear mechanical response of deeply
supercooled or over-compressed liquids and glasses under strong deformation is
a fundamental problem (with high technological relevance) in nonequilibrium
statistical mechanics for diverse systems including colloids, metals and
polymers Mewis and Wagner (2012); Hebert  _et al._ (2015); Berthier and Biroli
(2011); Gratson  _et al._ (2004); Lee  _et al._ (2009a). Quiescent structural
relaxation involves thermally activated hopping processes which can be
massively sped under deformation, ultimately resulting in (plastic) flow, in a
manner that depends on thermodynamic state, interparticle interactions, and
deformation rate Lee  _et al._ (2009a); Eyring (1936); Jadhao and Robbins
(2017); Pham  _et al._ (2006); Besseling  _et al._ (2007); Lee  _et al._
(2009b); Koumakis  _et al._ (2012a). For the classic rheological protocol of
homogeneous continuous startup shear, the stress evolution exhibits multiple
regimes with increasing time or accumulated strain (see, e.g., Fig.1a): (i) an
initial linear response characterized by an elastic shear modulus (blue shaded
region in Fig. 1(a)), (ii) an ‘anelastic’ (or, nonlinear elastic) regime where
dissipative processes begin to emerge (yellow shaded region in Fig. 1(a)),
(iii) a stress maximum or “overshoot” (point iii in Fig. 1(a)), (iv) stress
softening (gray shaded region in Fig. 1(a)), and (v) steady state flow
characterized by a plateau stress which determines the nonequilibrium
viscosity (red shaded region in Fig. 1(a)) Larson (1999); Haward and Mol
(1997); Roth (2016); Chen and Schweizer (2011); Koumakis  _et al._ (2016,
2012b); Laurati  _et al._ (2012).

Of special interest is the physics of the defining feature of a non-monotonic
stress-strain curve, the overshoot. This stress maximum is often intuitively
viewed as a “dynamic yield point” that signals a mechanically driven crossover
from predominantly elastic to predominantly viscous response. For the ultra-
dense monodisperse hard sphere fluid and Brownian metastable colloidal
suspensions of present interest which have a well-defined maximum or random
close packing (RCP) fraction, the overshoot strain and stress are rich
functions of shear rate and concentration. This includes a striking non-
monotonic evolution of the overshoot amplitude with shear rate, and its
tendency to vanish at ultra-high packing fractions Koumakis  _et al._ (2016,
2012b). The microscopic physics of these observations are not understood, and
seem especially subtle since the overshoot amplitude reflects the relative
magnitudes of the transient peak versus steady state stresses, and hence can
potentially involve different nonequilibrium processes. We believe that at
least three physical effects are crucial: (i) activated structural relaxation
under quiescent and strongly deformed conditions as objectively indicated,
e.g., by the observation in experiments and simulations of intermittent
particle trajectories even far from equilibrium at high stresses Besseling
_et al._ (2007), (ii) accelerated relaxation via shear advection and/or stress
reduction of activation barriers [6,19-21], and (iii) deformation-induced
change of structure which typically weakens (in an average sense) cage scale
spatial correlations and kinetic constraints [12,16-18].

Figure 1: Schematic of (a) the stress ($\Sigma$) vs strain ($\gamma$) curve
for a fixed strain rate that shows the various regions of behavior. A stress
overshoot is observed at a strain of $\gamma=\gamma_{\text{pk}}$, which
increases weakly with shear rate. The overshoot stress and plateau (or steady
state) stress are indicated as $\Sigma_{\text{pk}}$ and $\Sigma_{\text{ss}}$
(or $\Sigma_{\text{pl}}$), respectively. For dense hard sphere colloidal
suspensions, experiments and simulations find the emergence of a non-
equilibrium steady state typically occurs at a strain 1.8-2.3 times larger
than the overshoot peak strain. Schematic of the amplitude of the stress
overshoot, defined as, $r_{O}=\Sigma_{\text{pk}}/\Sigma_{\text{pl}}-1$, as a
function of strain rate at fixed low and high packing fractions in the dense
metastable (“glassy”) regime are shown in (b) which displays a characteristic
non-monotonic behavior. (c) Monotonically decreasing overshoot amplitude as a
function of packing fraction at a fixed strain rate.

Conceptually diverse phenomenological plasticity models (e.g., soft glass
rheology (SGR), shear-transformation zone (STZ), mean-field nonlinear
deformation) have been developed for the rheology of non-polymeric systems
Falk and Langer (1998); Demetriou  _et al._ (2006); Fuchs and Cates (2002a);
Zaccone and Terentjev (2020); Zaccone  _et al._ (2014a). STZ models invoke
specific average features (such as the population density STZs) of a
relatively compact two-state shear-transformation zone with internal dynamics
to understand the large-scale deformation behavior of amorphous solids Langer
(2006). SGR models often ignore tensorial aspects of deformation for
simplicity and construct a probabilistic evolution equation of a specified
region within a material as a function of strain, strain rate, a yield stress,
and time. More recently, stress overshoots in simple yield stress fluids have
been analyzed by a “continuum fluidity” model Benzi  _et al._ (2021a, b) that
combines a fluidity equation with stress evolution per a Maxwell model.
However, all these models are phenomenological in the sense that the 3 points
enumerated above are not microscopically addressed and/or empirical fit
parameter(s) are invoked. Moreover, they generally are applied to soft matter
suspensions (e.g., microgels, pastes,) that do not exhibit a literal RCP
state, but rather a soft jamming crossover. More crucially for our interests,
the 5 zones of the stress-strain curve in Fig. 1a as a function of strain rate
and packing fraction are not predicted based on a microscopic description.

The most ambitious microscopic “first principles” approach formulated at the
level of forces and static and dynamic structural correlations is the mode
coupling theory (MCT) Fuchs and Cates (2002a); Amann  _et al._ (2013); Fuchs
and Cates (2002b, 2005); Amann  _et al._ (2015); Gotze (2008). The key physics
for nonlinear rheology is argued to emerge from local structural changes
driven by shear advection that weaken and ultimately destroy local caging
constraints Fuchs and Cates (2002a); Amann  _et al._ (2013); Fuchs and Cates
(2002b, 2005); Amann  _et al._ (2015). Though valuable progress has been made
with this approach, qualitative problems remain. These include the prediction
of a strong increase of the overshoot amplitude with packing fraction over a
tiny change of colloid concentration and a monotonic evolution of the
overshoot amplitude with shear rate Amann  _et al._ (2015), both in
disagreement with experiment and simulation Koumakis  _et al._ (2016, 2012b);
Laurati  _et al._ (2012) (see Fig. 13 in Appendix). A major conceptual issue
is that the rheological MCT is built on a hypothetical quiescent “ideal glass
transition” Gotze (2008), the existence of which reflects the neglect of
ergodicity restoring thermally activated processes. The numerical complexity
of the fully microscopic rheological MCT has also largely restricted its
applications Amann  _et al._ (2015) to further simplified semi-
phenomenological “schematic” versions which contain multiple fit parameters,
and in some cases a fit function. The version of ideal MCT for sheared
colloidal suspensions in ref. Miyazaki and Reichman (2002) utilizes a shear
distorted structure factor and a generalized hydrodynamics approach where
shearing retains the form of the force-force memory kernel in MCT with
modified time-dependent wave vector associated with advection. However, this
approach is restricted to 2D fluids due to the large numerical complexity
associated with the tensorial nature of the structure factor.

Here we take a different microscopic theoretical approach to address the
nonlinear rheology of ultra-dense metastable hard sphere systems by building
on the Elastically Collective Nonlinear Langevin Equation (ECNLE) theory
Mirigian and Schweizer (2013, 2014) of coupled local cage-nonlocal collective
elastic activated structural relaxation in equilibrium glass-forming fluids
(conceptual schematic in Fig. 2a), and its microrheology-inspired
generalization to treat deformation Ghosh and Schweizer (2020). The central
quantity is a dynamic free energy, $F_{\text{dyn}}(r)$, that is a function of
instantaneous scalar particle displacement (r) and which controls single
particle stochastic trajectories including thermal activation effects based on
a nonlinear Langevin equation (NLE) of motion. The coupling of cage scale
barrier hopping and longer range collective elasticity of the fluid is
included, with the latter becoming increasingly important in equilibrium as
the fluid packing fraction (temperature in a viscous liquid) becomes very high
(low). In the spirit of MCT, the dynamic free energy quantifies kinetic
constraints via the pair correlation function ($g(r)$ or Fourier space
structure factor $S(k)$). In prior work to date, stress enters the NLE
evolution equation solely as an external force which directly accelerates
relaxation via softening of the constraints embedded in the nonequilibrium
free energy, with stress-induced activation barrier reduction being the most
important aspect.

The formulation of ECNLE theory sketched above has previously addressed points
(i) and (ii) discussed at the start of this Introduction, but not point (iii)
since S(k) was assumed to be invariant to deformation. This zeroth order
assumption of an invariant local structure model is of course not literally
true, but has some practical support from measurements on of model silica hard
sphere suspensions Watanabe  _et al._ (1997, 1998); Maranzano and Wagner
(2002) where, (a) no significant changes of structure are observed up to a
packing fraction of 0.50, and (b) surprisingly, strong nonlinear dynamical
effects are not accompanied by significant anisotropy of structure in the
shear thinning regime Maranzano and Wagner (2002). The invariant-structure
theory Ghosh and Schweizer (2020) predicts massive deformation-induced
mobility enhancement leading to the following predictions that have been shown
to agree well with experiments and simulations: (a) nonlinearity emerges at
remarkably low values of “renormalized” Peclet number (product of shear rate
and quiescent structural or alpha relaxation time,
$\text{Pe}=\dot{\gamma}\tau_{\alpha}\ll 1$), (b) power law alpha time and
viscosity thinning in the steady state, (c) a flow curve of the Herschel-
Buckley (HB) empirical form Larson (1999), and (d) an apparent static yield
stress that grows nearly exponentially with packing fraction Ghosh and
Schweizer (2020). More broadly, we note that recent combined confocal imaging
and rheology experiments Sentjabrskaja  _et al._ (2015) provide significant
justification for our specific theoretical formulation of rheological response
built on single dynamics since a very strong link between macroscopic
nonlinear creep with microscopic single particle motion as encoded at the
elementary mean square displacement level has been observed. However, the full
transient stress-strain response, the stress overshoot, and the fundamental
problem of dynamic yielding have not been addressed. Doing so is the goal of
the present article. We formulate a minimalist microscopic model for capturing
the coupled nonequilibrium evolution of structure, relaxation time, and
rheology which is demonstrated to provide a unified understanding of the
stress overshoot, the steady state, and the role of activated dynamics.

##  II ECNLE Theory

ECNLE theory under both quiescent and driven conditions with a deformation-
invariant $S(k)$ has been thoroughly described in the literature Mirigian and
Schweizer (2013, 2014); Ghosh and Schweizer (2020); Kobelev and Schweizer
(2005). For the convenience of the reader, and as relevant background, we
briefly review without derivation the key elements germane to our new
generalization (entering solely via the deformation-dependent isotropic
structure factor, $S(k)$) in the context of the system of present interest, a
fluid or suspension of monodisperse hard spheres. The key input to the
quiescent dynamical theory is $S(k)=1+\rho h(k)=(1-\rho C(k))^{-1}$, where
$h(k)$ is the Fourier transform of the non-random part of the pair correlation
function, $h(r)=g(r)-1$, and $C(k)$ is the direct correlation function Hansen
and McDonald (2006), computed here using standard PY integral equation theory
Hansen and McDonald (2006).

Figure 2: (a) Schematic of the activated hopping of particles on the cage
scale described by NLE theory which is coupled to a longer ranged collective
elastic barrier arising from the effective cage expansion and displacement of
all particles outside the cage. Schematic of the NLE dynamic free energy is
shown with all important length scales indicated (top right), while the
collective elastic barrier arises from the small harmonic displacements of all
particles outside the cage (bottom right). (b) Evolution of the dynamic free
energy as a function of particle displacement under a constant applied stress
for a high packing fraction of 0.61. Blue, orange and black curves are for,
respectively, zero stress, an intermediate stress, and the ‘absolute yield
stress’ ($\Sigma_{y}^{\text{abs}}$) defined as the value of stress required to
first destroy the cage barrier ($\beta F_{B}=0$) Koumakis  _et al._ (2016,
2012b) and the localized form of the dynamic free energy. The inset shows the
equilibrium alpha time normalized by the short process time scale as a
function of packing fraction predicted by NLE theory (orange; no collective
elasticity) and ECNLE theory (blue). The green curve is the horizontally and
vertically shifted ideal MCT prediction based on an empirically shifted
packing fraction $\phi_{c}\sim 0.581$, which agrees with ECNLE theory over the
first few decades of relaxation indicated by a vertical black arrow. (c) Plot
of the local (blue), elastic (green) and total (red) barriers as a function of
packing fraction. Inset: the normalized changes of the 3 barriers with stress
nondimensionalized by the absolute yield stress for a packing fraction of
0.61.

###  II.1 Quiescent fluids

The dynamic starting point is the nonlinear Langevin equation (NLE) that
describes the stochastic evolution of a tagged particle trajectory controlled
by a dynamic free energy. The dynamical variable is the angularly averaged
scalar displacement of a particle, $r(t)$, which under quiescent conditions
obeys the stochastic NLE in the overdamped limit Schweizer and Saltzman
(2003); Schweizer (2005):

$-\zeta_{s}\frac{dr(t)}{dt}-\frac{\partial F_{\text{dyn}(r(t))}}{\partial
t}+\xi(t)=0$ (1)

where $\xi(t)$ is a white noise random force associated with a short time
Fickian diffusion process quantified by the friction constant $\zeta_{s}$
determined based on a binary collision plus weak caging model known to be
accurate for hard sphere colloidal suspensions when trajectories are not
strongly activated Verberg  _et al._ (1997); hydrodynamics enters only via the
single particle Stokes-Einstein diffusivity. An explicit expression for
$\zeta_{s}$ is given below. The dynamic free energy is Schweizer and Saltzman
(2003); Schweizer (2005),

$\begin{split}\beta
F_{\text{dyn}(r)}=-3\ln{\left(\frac{r}{\sigma}\right)}-\rho\int\frac{d\vec{k}}{(2\pi)^{3}}\frac{C(k)^{2}S(k)}{\left(1+S^{-1}(k)\right)}\times\\\
\exp{\left(-\frac{k^{2}r^{2}}{6}(1+S^{-1}(k))\right)}\end{split}$ (2)

where the integrand contains the spatially resolved (in Fourier space) mean
square force vertex defined as $V(k)=\rho k^{2}C(k)^{2}S(k)$ that quantifies
caging constraints on a length scale $2\pi/k$ per the MCT idea of projecting
real forces onto bilinear density variables followed by a Gaussian
factorization approximation of 4-point correlation to products of pair
correlation functions Wolynes  _et al._ (2012); Kirkpatrick (1985). The so-
called ideal MCT “transition” at the single particle level (referred to as
naïve MCT or, NMCT) is recovered from the mathematical condition of when a
local minimum of the dynamic free energy first emerges with increasing packing
fraction Schweizer and Saltzman (2003); Schweizer (2005). This connection
between NMCT and NLE theory is equivalent to stating NMCT effectively ignores
the thermal noise in the NLE evolution equation, i.e. thermal fluctuation
driven “uphill” activated motions are not allowed. The emergence of transient
localization length ($r_{L}$), defined as the local minimum of the dynamic
free energy, at $\phi=\phi_{c}=0.432$ (using PY closure Hansen and McDonald
(2006)) defines a smooth dynamic crossover to the activated regime in NLE
theory. For $\phi>\phi_{c}$ the dynamic free energy has a local cage barrier
of height $F_{B}$ at $r=r_{B}$, with a jump distance defined naturally as
$\Delta r=r_{B}-r_{L}$. Fig. 1a shows an example of the dynamic free energy in
a highly activated regime where $\phi=0.61$. The local cage barrier is shown
in Fig. 2c as a function of packing fraction.

NLE theory has been shown to capture only noncooperative hopping events and
hence is valid only at sufficiently low packing fractions (but still in the
overcompressed or metastable $\phi>0.495$ regime for monodisperse hard
spheres) Mirigian and Schweizer (2013, 2014). This motivated the extension of
NLE theory (thereby defining ECNLE theory) to include collective physics
associated with the coupling of local hopping on the cage scale and collective
elasticity associated with the facilitating displacement of all particles
beyond the cage size. The basic physical idea was inspired by the
phenomenological shoving model of Dyre Dyre (1998, 2006) (albeit with multiple
physical differences as discussed elsewhere Mirigian and Schweizer (2014))
that all particles outside the cage must elastically displace via a
spontaneous fluctuation to create the small amount of extra space required to
allow large amplitude cage scale particle hopping Mirigian and Schweizer
(2013, 2014). The corresponding elastic energy cost enters as an additional
barrier for relaxation which has been derived to be Mirigian and Schweizer
(2014),

$\begin{split}\beta F_{e}=4\pi\int_{r_{\text{cage}}}^{\infty}drr^{2}\rho
g(r)\left(\frac{1}{2}K_{0}u(r)^{2}\right)\\\ \approx 12\phi
K_{0}\left(\frac{r_{\text{cage}}}{\sigma}\right)^{3}\Delta
r_{\text{eff}}^{2}\end{split}$ (3)

where $K_{0}=3k_{B}T/r_{L}^{2}$ is the harmonic spring constant or curvature
of the dynamic free energy at its local minimum;$\Delta
r_{\text{eff}}=\frac{3}{32}\frac{\Delta r^{2}}{r_{\text{cage}}}$ is the
effective amplitude (typically small, of order $r_{L}$ or smaller) of the
elastic displacement field at the cage surface. The elastic displacement field
at larger scalar distances $r$ from the cage center is $u(r)=\Delta
r_{\text{eff}}(r_{\text{cage}}/r)^{2}$, $r_{\text{cage}}$ is the mean cage
radius defined by the first minimum of $g(r)$ ($\sim 1.5\sigma$ for hard
spheres) and the final relation in Eq(3) uses $g(r)=1$ for $r\geq
r_{\text{cage}}$ which is a technical simplification that has been shown to be
extremely accurate for hard spheres Mirigian and Schweizer (2014). In ECNLE
theory, the alpha relaxation event is a coupled local-nonlocal process
corresponding to a total barrier equal to the sum of the local cage and
collective elastic contributions, $\beta
F_{\text{{total}}}=\beta(F_{B}+F_{e})$. Importantly, all quantities needed to
compute the total barrier follow from the dynamic free energy, and hence the
structural correlations. The average hopping or alpha relaxation time is
computed using a Kramers mean first passage time analysis for barrier crossing
as Mirigian and Schweizer (2014),

$\begin{split}\frac{\tau_{\alpha}}{\tau_{s}}=\frac{2}{\sigma^{2}}\exp{(\beta
F_{e})}\int_{r_{L}}^{r_{B}}dx\exp{(\beta F_{\text{dyn}}(x))}\times\\\
\int_{r_{L}}^{x}dy\exp{(-\beta F_{\text{dyn}}(y))}\\\
\approx\frac{2\pi}{\sqrt{K_{0}K_{B}}}\exp{(\beta
F_{\text{total}})}\end{split}$ (4a)

where $\tau_{s}=\beta\zeta_{s}\sigma^{2}$ is a short relaxation timescale,
here taken as the elementary time unit, and for hard spheres is given by Cohen
_et al._ (1998)

$\frac{\tau_{s}}{\tau_{0}}=1+\frac{\sigma^{3}}{36\pi\phi}\int_{0}^{\infty}dk\left(\frac{k^{2}(S(k)-1)^{2}}{S(k)+(1-j_{0}(k\sigma)+2j_{2}(k\sigma))^{-1}}\right)$
(4b)

where $j_{n}(x)$ is spherical Bessel function of order n and the ‘bare’
relaxation time $\tau_{0}$ can be written in terms of Stokes-Einstein friction
constant ($\zeta_{\text{SE}}$) for suspensions with a binary collision
contribution as, $\tau_{0}=\beta g(\sigma)\sigma^{2}\zeta_{\text{SE}}$. The
simpler approximate Kramers result in Eq. (4a) is valid when the local barrier
is $\sim 1-2k_{B}T$ or higher.

A sample calculation of the equilibrium alpha time with and without the
elastic barrier is shown in the inset of Fig. 2b. The elastic barrier is not
important for lower packing fractions in the metastable regime where the
barrier is determined by local caging, but it becomes increasingly dominant
for $\phi\geq 0.56$. The critical power law form of ideal MCT (appropriately
shifted) is shown as the green curve in the inset of Fig. 2b. As discussed
previously Schweizer and Saltzman (2003), this critical power law form
empirically captures the initial $\sim 2-2.5$ decades of slowdown (low barrier
regime) predicted by the activated dynamics theory before it unphysically
diverges as $\tau_{\alpha}\sim(\phi_{\text{MCT}}^{\text{shift}}-\phi)^{-2.5}$.

###  II.2 Effect of external forces

Motivated by a microrheology perspective in a stress ensemble, the direct
effect of external deformation in the NLE approach enters as an additional
external force, $f$, on a tagged particle corresponding to a nonequilibrium
dynamic free energy with an extra contribution of a mechanical work form Ghosh
and Schweizer (2020); Kobelev and Schweizer (2005):

$\beta F_{\text{dyn}}(r,\Sigma)=\beta F_{\text{dyn}}(r,\Sigma=0)-f\cdot r$ (5)

where $\Sigma$ is stress. A linear connection between microscopic force and
macroscopic stress is employed: $f=A\Sigma$, where $A$ is a microscopic cross-
sectional area related to particle size. A detailed discussion of the
motivation for Eq(5) and the specific choice of $A$ is provided in refs. Ghosh
and Schweizer (2020); Kobelev and Schweizer (2005); here we briefly explain
the main ideas.

For computational tractability and conceptual simplicity reasons, the NLE
theory is formulated based on scalar displacement of particles. Hence, prior
rheological work with this approach has adopted an isotropized description of
the dynamic trajectories under deformation for which treating $S(k)$ as
isotropic is consistent. This is why only a scalar force, $f$, enters Eq(5),
which can be viewed as a pre-averaging of the anisotropy (different directions
of external force and particle trajectory) for which one obtains
$A=\pi\sigma^{2}/24$ Ghosh and Schweizer (2020); another choice is the
particle cross-sectional area of $A=\pi\sigma^{2}/4$. These 2 choices differ
only by a numerical prefactor and we have verified here and elsewhere Ghosh
and Schweizer (2020) that our results are insensitive to the precise choice
which only affects the absolute values of characteristic rheological
quantities but not the basic qualitative behaviors predicted by the theory
Ghosh and Schweizer (2020). For the rest of the article, we adopt
$A=\pi\sigma^{2}/24$.

The mean alpha time is again computed using Kramers theory but with all
required properties now stress-dependent, except for the very short time and
length scale process prefactor $\tau_{s}=\beta\zeta_{s}\sigma^{2}$ in Eq. 4a
which is fixed at its equilibrium value. With increasing applied force at a
fixed packing fraction, one physically expects (as Eq (5) predicts) that
$r_{L}$ increases indicating weaker localization. The ‘critical’ force or
stress at which the localized form of the dynamic free energy first vanishes,
the local barrier goes to zero continuously and the localization length
disappears, is termed the ‘absolute yield stress’, $\Sigma_{y}^{\text{abs}}$.
Both the local and elastic barriers decrease with applied deformation as shown
in the inset of Fig. 2(b). Crucially, the dependence of the elastic barrier on
applied deformation is much stronger since both the harmonic spring constant
and jump distance decrease with stress, and hence the elastic barrier is
destroyed much rapidly with increasing stress or force compared to its local
cage analog, as shown in the inset of Fig. 2(b). This means the theory
predicts the degree of dynamic cooperativity of the alpha relaxation process
decreases under deformation, i.e. the activated dynamics becomes more
spatially local Ghosh and Schweizer (2020).

In prior work and in the results shown in Figure 2 Ghosh and Schweizer (2020);
Kobelev and Schweizer (2005), $S(k)$ was assumed to be invariant to applied
deformation. A schematic of the evolution of the dynamic free energy with
increasing deformation (force) under this condition is shown in the main frame
of Fig. 2b. The black curve corresponds to when the stress first destroys the
barrier (where $r_{L}=r_{B}$), the ‘absolute yielding’ condition. The
dynamically relaxed elastic shear modulus at the NMCT level is calculated from
knowledge of the stress-dependent localization length using the expression
Ghosh and Schweizer (2020); Kobelev and Schweizer (2005); Nägele and
Bergenholtz (1998),

$\begin{split}G^{\prime}[\Sigma]=\frac{k_{B}T}{60\pi^{2}}\int_{0}^{\infty}~{}dk\left[k^{2}\frac{d\ln{S({k})}}{dk}\right]^{2}\exp{\left(-\frac{k^{2}r_{L}(\Sigma)^{2}}{3S(k)}\right)}\\\
\approx c\frac{\phi k_{B}T}{\sigma r_{L}(\Sigma)^{2}}\end{split}$ (6)

The last approximate equality is an analytic micro-rheology like relation
previously derived under equilibrium conditions and verified to also hold well
under stress Ghosh and Schweizer (2020); $c$ is a known numerical constant. Eq
(6) is directly relevant to real world mechanical measurements performed at
finite frequencies only beyond the ideal NMCT “transition” ($r_{L}$ finite) in
absence of ergodicity restoring activated events, and thus connects to the
concept of “absolute yielding” mentioned above.

##  III Rheological Framework, the Isostructural Limit, and Beyond

We first briefly review the adopted rheological framework in the context of
the previously developed isostructural formulation of ECNLE theory under
deformation Ghosh and Schweizer (2020). We then motivate the need to consider
deformation-induced changes of $S(k)$ to properly capture the overshoot
physics within the same theoretical description.

Our starting point is a minimalist and physically transparent generalized
Maxwell constitutive equation where the two key properties are the stress-
dependent shear elastic modulus ($G^{\prime}$) and the structural or alpha
relaxation time ($\tau_{\alpha}$) which we adopt as the characteristic mean
stress relaxation time. For a constant shear rate $\dot{\gamma}$ the time (or
strain)-dependent nonlinear stress relaxation function ($G$) obeys the first-
order evolution equation Chen and Schweizer (2011)
$\displaystyle\frac{dG[\gamma,\Sigma(\gamma)]}{d\gamma}=-\frac{G[\gamma,\Sigma[\gamma]]}{\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma)]}$,
where $\gamma=\dot{\gamma}t$ is the accumulated strain and $\Sigma$ the
stress. Formal integration through transients yields Chen and Schweizer
(2011),

$\Sigma(\gamma)=\int_{0}^{\gamma}d\gamma^{\prime}G^{\prime}[\Sigma(\gamma^{\prime})]\exp{\left(-\int_{\gamma^{\prime}}^{\gamma}d\gamma^{\prime\prime}\frac{1}{\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma^{\prime\prime})]}\right)}$
(7a)

or equivalently in differential form,

$\frac{d\Sigma(\gamma)}{d\gamma}+\frac{\Sigma(\gamma)}{\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma)]}=G^{\prime}[\Sigma(\gamma)]$
(7b)

From Eq.(7b), the steady state stress obeys the self-consistent equation,
$\displaystyle\Sigma_{\text{ss}}=\dot{\gamma}\tau_{\alpha}[\Sigma_{\text{ss}}]G^{\prime}[\Sigma_{\text{ss}}]$.

We again note that, as also true of work of others (e.g., the original
formulations of the soft glass rheology model Falk and Langer (1998), the
“isotropically-sheared” MCT approach Fuchs and Cates (2002b), NLE theory
Kobelev and Schweizer (2005)), explicit tensorial anisotropy is ignored for
tractability reasons. This simplification is consistent with the description
of colloid trajectories in terms of a scalar displacement dynamical variable
in NLE theory, and the adoption of an isotropic structure factor, $S(k)$, to
construct the dynamic free energy. Of course, neglecting explicit anisotropy
is an approximation, but the practical question is how large are the
consequences on the cage scale for the scalar observables of interest. In this
regard, simulation studies of the rheology of glassy fluids Miyazaki  _et al._
(2004); Yamamoto and Onuki (1998) found that even in the strong shear thinning
regime, for cage scale wavevectors, $k$, that define structural relaxation,
which is strongly coupled with stress relaxation, there is very little
dynamical anisotropy of the measured alpha time, mean square displacement, and
incoherent dynamic structure factor, $F_{s}(k,t)$. This supports, or at a
minimum can rationalize, why using an effectively isotropic $S(k)$ can
successfully capture many aspects of glassy colloid rheology.

Now, assuming literally no change of $S(k)$ under deformation Ghosh and
Schweizer (2020), ECNLE theory predicts the required elastic modulus and alpha
time in Eq.(7) as a function of stress. Deformation enters $F_{\text{dyn}}(r)$
solely in a mechanical work like manner (Fig. 2) and monotonically reduces
$\tau_{\alpha}$ and $G^{\prime}$ with increasing stress as calculated using
the Kramers’ formula for barrier crossing Mirigian and Schweizer (2014); Ghosh
and Schweizer (2020); Schweizer (2005) and the Green-Kubo-like elastic shear
modulus expression Kobelev and Schweizer (2005); Nägele and Bergenholtz (1998)
discussed above.

The core hypotheses in our present new work are: (1) within the generalized
Maxwell constitutive equation approach, structure changes under deformation
are necessary to obtain an overshoot in the stress- strain response, and (2)
for a steady state stress to exist, structural changes must eventually stop at
sufficiently high strain or time. We justify these hypotheses by analyzing the
generalized Maxwell constitutive equation. First, since
$\displaystyle\frac{d\Sigma(\gamma)}{d\gamma}=G^{\prime}[\Sigma(\gamma)]-\frac{\Sigma(\gamma)}{\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma)]}$,
for the invariant structure model if
$\displaystyle\frac{d\Sigma(\gamma)}{d\gamma}=0$ then strain becomes an
irrelevant variable and one has $\displaystyle
G^{\prime}[\Sigma]=\frac{\Sigma}{\dot{\gamma}\tau_{\alpha}[\Sigma]}$. With
increasing stress at a fixed shear rate, the LHS side of this equation
monotonically decreases, while the RHS monotonically increases (the alpha time
is a monotonically decreasing function of applied force), and thus only one
crossing of curves (the mathematical solution) is possible which defines the
steady state. This proves that within our formulation it is impossible to have
an overshoot if structure does not change (corresponding to an ideal plastic
response), consistent with point (1).

For a changing structure under deformation model, using again
$\displaystyle\frac{d\Sigma(\gamma)}{d\gamma}=G^{\prime}[\Sigma(\gamma)]-\frac{\Sigma(\gamma)}{\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma)]}$,
one finds the second derivative of stress with respect to strain is,

$\begin{split}\frac{d^{2}\Sigma(\gamma)}{d\gamma^{2}}=\left(\frac{\partial
G^{\prime}[\Sigma(\gamma)]}{\partial\gamma}+\frac{\Sigma(\gamma)}{\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma)]^{2}}\frac{\partial\tau_{\alpha}[\Sigma(\gamma)]}{\partial\gamma}\right)+\frac{d\Sigma(\gamma)}{d\gamma}\times\\\
\left[\frac{dG^{\prime}[\Sigma(\gamma)]}{d\Sigma}-\frac{1}{\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma)]}+\frac{\Sigma(\gamma)}{\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma)]^{2}}\frac{d\tau_{\alpha}[\Sigma(\gamma)]}{d\Sigma}\right]\end{split}$
(8)

The partial derivatives come from purely structural changes with deformation.
For both the steady state and at the stress overshoot one has
$\displaystyle\frac{d\Sigma(\gamma)}{d\gamma}=0$, and thus only the partial
derivative terms survive. Thus, for both the steady state and overshoot
maximum one can write,

$\frac{d^{2}\Sigma(\gamma)}{d\gamma^{2}}\bigg{|}_{\gamma=\gamma_{\text{pk}},\gamma_{\text{ss}}}=\left(\frac{\partial
G^{\prime}[\Sigma(\gamma)]}{\partial\gamma}+\frac{\Sigma(\gamma)}{\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma)]^{2}}\frac{\partial\tau_{\alpha}[\Sigma(\gamma)]}{\partial\gamma}\right)$
(9)

An additional criterion for the overshoot is
$\displaystyle\frac{d^{2}\Sigma(\gamma)}{d\gamma^{2}}\bigg{|}_{\gamma=\gamma_{\text{pk}}}<0$.
In general, $G^{\prime}[\Sigma(\gamma)]$ decreases with stress and strain-
induced softening of structure, so its partial derivative in Eq. 9 is always
negative. The same is true for
$\displaystyle\frac{\partial\tau_{\alpha}[\Sigma(\gamma)]}{\partial\gamma}$
(the numerical prefactor is positive), and hence
$\displaystyle\frac{d^{2}\Sigma(\gamma)}{d\gamma^{2}}<0$. This proves that
structural changes are necessary for the partial derivatives to be nonzero, a
necessary condition for a negative second derivative. This completes our
discussion of point (1) above.

A steady state requires $\displaystyle\frac{\partial
G^{\prime}[\Sigma(\gamma)]}{\partial\gamma}=0$ and
$\displaystyle\frac{\partial\tau_{\alpha}[\Sigma(\gamma)]}{\partial\gamma}=0$
for $\displaystyle\frac{d^{2}\Sigma(\gamma)}{d\gamma^{2}}=0$. This is
impossible if deformation-induced changes in $S(k)$ do not eventually vanish
at long time (large strains), which is physically obvious. To proceed, we
first assume an overshoot stress maximum exists and occurs at
$\gamma=\gamma_{m}$ where all the above-mentioned criteria are satisfied. We
explore what happens if one postulates that structure does not change
(achieves its nonequilibrium steady state) for $\displaystyle\lim_{\epsilon\to
0^{+}}\frac{d^{2}\Sigma(\gamma)}{d\gamma^{2}}\bigg{|}_{\gamma=\gamma_{m}+\epsilon}=0$
as the partial derivatives in Eq. 9 are zero, and this is the criterion for a
steady state. Thus, no overshoot in the stress strain curve is possible, and
to have an overshoot structural changes must continue beyond the overshoot. We
believe this to be a physically intuitive condition since our view is that the
overshoot qualitatively indicates a crossover from an elastic to viscous like
mechanical response. Simulation findings [16-18] are also consistent with this
perspective since they find structural changes do not saturate at the stress
overshoot, but rather require strains modestly beyond it. This completes our
discussion of point (2).

##  IV Structural Changes with Deformation

In Brownian suspensions not close to a shear jamming or thickening regime, the
cage scale packing correlations, and hence kinetic constraints, are generally
believed to on average weaken under shear deformation Larson (1999); Koumakis
_et al._ (2016, 2012b); Fuchs and Cates (2002a, 2005). As mentioned above,
simulations find that the shear rate dependences of the local structure
correlations, the anisotropy of particle diffusion (after subtracting kinetic
advection effects) and the incoherent dynamic structure factor that quantifies
cage scale structural relaxation, are very small even under strong deformation
Miyazaki  _et al._ (2004); Yamamoto and Onuki (1998).

Here we construct a simple and physically transparent treatment of this key
structural softening effect in real space which enters ECNLE theory and the
nonlinear Maxwell model via an isotropic but strain-dependent $S(k,\gamma)$
roughly in the spirit of an elastic-like analog of the shear advection Fuchs
and Cates (2002a); Amann  _et al._ (2013); Fuchs and Cates (2002a, 2005) idea.
Within this simplified approach that ignores explicit anisotropic changes of
structure, and as discussed in textbooks, one expects that under deformation
the average separation between a pair of particles (here hard spheres,
diameter $\sigma$) initially increase affinely with strain Larson (1999). The
radial distribution function (and thus $S(k)$ and the direct correlation
function, $C(k)$) becomes strain-dependent which is modeled in an effectively
isotropic manner as: $\displaystyle
g_{\gamma}(r\geq\sigma)=g_{0}\left(r\sqrt{1+\gamma^{2}/3}\right)$, where
$g_{0}(r)$ is the equilibrium pair correlation function and
$g_{\gamma}(r<\sigma)=0$. Strain reduces the contact value to $\displaystyle
g_{0}\left(\sigma\sqrt{1+\gamma^{2}/3}\right)$, and more generally weakens the
structural pair correlations. The dynamic theory requires as input the
structure factor, $\displaystyle S(k,\gamma)=1+\rho h(k,\gamma)$, and hence
one needs,

$\begin{split}h(k,\gamma)=\int
d\vec{r}\exp{(-i\vec{k}\cdot\vec{r})}h\left(\vec{r}\sqrt{1+\gamma^{2}/3}\right)\\\
=\int\frac{d\vec{R}}{(1+\gamma^{2}/3)^{3/2}}\exp{\left(-i\frac{\vec{k}}{\sqrt{1+\gamma^{2}/3}}\cdot\vec{R}\right)}\\\
=\frac{1}{(1+\gamma^{2}/3)^{3/2}}h_{0}\left(\frac{{k}}{\sqrt{1+\gamma^{2}/3}}\right)\end{split}$
(10)

This gives the final form of the deformation-dependent structure factor
$S(k,\gamma)$ we employ as,

$\begin{split}S(k,\gamma)=1+\frac{\rho}{(1+\gamma^{2}/3)^{3/2}}h_{0}\left(\frac{{k}}{\sqrt{1+\gamma^{2}/3}}\right)\\\
=S_{0}(k)+\rho\left(\frac{h_{0}\left(\frac{k}{\sqrt{1+\gamma^{2}/3}}\right)}{(1+\gamma^{2}/3)^{3/2}}-h_{0}(k)\right)\\\
\approx S_{0}(k)+\rho
h_{0}(k)\left((1+\gamma^{2}/3)^{-3/2}-1\right)\end{split}$ (11)

The last approximate relation is written only to allow one to more easily see
how strain modifies the wavevector dependence (it is mathematically exact
within this model for $k=0$). In all numerical calculations we employ the full
relation. The $k\to 0$ limit quantifies the density fluctuation amplitude (or
thermodynamic “dimensionless compressibility”) and is given as,

$S(k\to 0,\gamma)=S_{0}(0)+\rho
h_{0}(0)\left((1+\gamma^{2}/3)^{-3/2}-1\right)$ (12)

Since generically both $\displaystyle(1+\gamma^{2}/3)^{-3/2}-1<0$, and
$\displaystyle\rho h(0)<0$, one has $\displaystyle S(k\to 0,\gamma)>S_{0}(0)$.
Thus, strain weakens local packing correlations on the cage scale, and also
enhances the amplitude of long wavelength density fluctuations in qualitative
analogy with a system being at an effectively lower packing fraction. The
above model connects in spirit with prior NLE theory rheology work for polymer
glasses that adopted a coarse-grained chain model with no local liquid like
structure Chen and Schweizer (2011). In that approach, an increase of $S(k=0)$
under deformation [called a “mechanical disordering or rejuvenation effect”]
below $T_{g}$ Chen and Schweizer (2011) results in the prediction of an
overshoot in the transient response. Here for colloidal systems, aging is not
considered, and we allow deformation to change packing structure on all length
scales, with changes on the local cage scale most important.

Figure 3: (a) Static structure factor at a fixed volume fraction for various
strain values. With increasing strain ($\gamma$), an increase in $S(k\to
0)=S_{0}$ and a decrease in $S(k=k^{\star})=S_{\text{max}}$ is observed.
Implications of the changing structure model on the dynamic free energy is
shown in the inset. (b) Quantification of the softening of two local
structural metrics: normalized contact value (left axis) and normalized cage
coherence peak (right axis), for a packing fraction of 0.61. Inset shows a
schematic of the cage ‘distortion’ model that motivates the choice of
renormalized average distance between particles as a function of strain and
the isotropic scalar separation of caged particles.

Figure 3a shows the strain dependent structure factor for a fixed packing
fraction of $0.61$. Per Eq. 12, $S(k=0,\gamma)$ is an increasing function of
strain $\gamma$, while the cage coherence peak decreases indicating cage
weakening. Figure 3b shows the normalized contact value of the radial
distribution function monotonically decreases with strain. A normalized “cage
order parameter” is defined as the ratio $S(k^{\star},\gamma)/S(k^{\star},0)$
where $k^{\star}$ is the wavevector at the first maximum, and it also
decreases with strain (right legend). If packing fraction is reduced, nearly
identical curves are found in this normalized framework (not shown), and the
results in Fig. 3b are thus generic.

The deformation-dependent $S(k)$ enters the calculation of the dynamic free
energy, from which all dynamical properties relevant to our rheological
predictions follow using ECNLE theory. The new feature of the changing
structure model is deformation modifies both the force vertex (kinetic
constraints, second term in Eq. 2) of the dynamic free energy and contributes
the external force or stress dependent mechanical work term in the
nonequilibrium dynamic free energy of Eq. 5. Hence all theoretical quantities
relevant to dynamics and rheology depend directly on $\Sigma$ and $\gamma$.

##  V Effect of Deformation on the Relaxation Time and Elastic Modulus and
Onset of the Structural Steady State

Our modeling of structural softening weakens all the localizing features of
the dynamic free energy, resulting in strain-dependent decreases of
$\tau_{\alpha}$ and $G^{\prime}$, in addition to the direct effect of stress
via the mechanical work term in the nonequilibrium dynamic free energy.
Representative results using Eqs. 4 and 6 for the relaxation time and elastic
modulus are plotted as a function of stress and strain in Fig. 4 based on
treating them as independent variables. For the deformation invariant
structure model, the elastic modulus decreases relatively slowly (the
$\gamma=0$ result is the thick blue curve in Fig. 4(b)). The new rheologically
consistent approach (i.e., using the predicted stress-strain curve) with
structural changes corresponds to traversing a specific path in
$(G^{\prime},\Sigma,\gamma)$ space as shown by the dashed curve for a
dimensionless shear rate of $0.01$. The same illustrative path is shown for
the alpha relaxation time in panel (a).

Figure 4: Stress and strain dependent (a) alpha or structural relaxation time,
and (b) elastic shear modulus for a packing fraction of 0.61. A rheologically
consistent treatment follows a specific path determined from the theoretical
constitutive equation as shown above for a dimensionless shear rate of 0.01.
Thick solid curves at $\gamma=0$ represent how the relaxation time and elastic
modulus change as a function of stress for a deformation-invariant structure
model.

While a complete microscopic theory of the full stress strain response also
requires a self-consistent equation for the evolution of the static structure
factor under deformation, we proceed here by formulating a simpler and far
more computationally tractable approach to quantify when deformation no longer
changes structural correlations.

Figure 5: Evolution of the alpha relaxation time as a function of strain for a
packing fraction of (a) 0.58 and (b) 0.61 at several dimensionless shear rates
relevant to experimental studies Koumakis  _et al._ (2012b, 2016); Laurati
_et al._ (2012). The black arrows indicate the overshoot strain
($\gamma_{\text{pk}}$). Magenta and orange arrows indicate two different
strains where $\text{Pe}=\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma)]=1$ and
$\gamma=2\gamma_{\text{pk}}$ respectively. Panel (c) shows the normalized
alpha relaxation times for two different packing fractions of 0.58 and 0.61.
The downward triangles indicate overshoot location and horizontal arrows
denote orders of magnitude faster dynamics with respect to the “equilibrium”
alpha relaxation ($\gamma=0$).

Figure 5a and 5b plots the logarithm of the alpha time as a function of strain
for packing fractions of $0.58$ and $0.61$ and several shear rates. The
overshoot strains ($\gamma_{\text{pk}}(\dot{\gamma})$) are indicated by
vertical black arrows. Magenta arrows show when the renormalized Pe equals
unity in both plots, defined as at $\gamma=\gamma_{\text{Pe}=1}$. We note that
setting $S(k,\gamma\geq\gamma_{\text{Pe}=1})=S(k,\gamma_{\text{Pe}=1})$ always
destroys the overshoot, hence $\gamma_{\text{Pe}=1}>\gamma_{\text{pk}}$ is
required. This inequality is consistent with the experimental and simulation
findings for hard sphere suspensions Koumakis  _et al._ (2012b, 2016); Laurati
_et al._ (2012) that at the overshoot
$\text{Pe}(\gamma_{\text{pk}}(\dot{\gamma}))<1$. These same studies Koumakis
_et al._ (2012b, 2016); Laurati  _et al._ (2012) find the structural steady
state is effectively achieved at strains modestly beyond the overshoot of
$b\gamma_{\text{pk}}$, i.e., $S(k,\gamma\geq
b\gamma_{\text{pk}})=S(k,b\gamma_{\text{pk}})$, where $b\sim 2-3$ Jadhao and
Robbins (2017); Pham  _et al._ (2006). These considerations are what motivated
our modeling of structure change under deformation discussed in the previous
section.

With the above motivation, exploratory calculations with the new theory
including $S(k,\gamma)$ have been performed over a wide range of shear rates
and packing fractions. We find that a sensible criterion for the emergence of
a steady state structure is
$\text{Pe}(2\gamma_{\text{pk}}(\dot{\gamma}|))=a(\phi)\approx 0.15-0.30$
independent of shear rate to zeroth order, and very weakly dependent on
packing fraction for $\phi\in(0.55,0.61)$ which is a range that corresponds to
the quiescent alpha time varying by 9 decades Mirigian and Schweizer (2014);
Ghosh and Schweizer (2020). We thus take as our core hypothesis that structure
becomes effectively deformation-independent at a single fixed value of
renormalized Pe: $S(k,\gamma\geq\gamma_{c})=S(k,\gamma_{c})$ where
$\gamma_{c}$ is determined from the criterion
$\text{Pe}(\gamma_{c})=\dot{\gamma}\tau_{\alpha}[\Sigma(\gamma_{c})]=C$, with
$C$ a universal constant for a given material system. We have verified that
different choices of $C\in(0.1,0.3)$ only very weakly modify the results
discussed below, which are shown using $C=0.25$111We begin to see deviations
beyond $A=0.3$ where the magnitude of the overshoot (steady state stress)
starts to decrease (increase). Eventually for $A\geq 0.5$ the overshoot is
almost destroyed.. This fixed Pe criterion introduces just one adjustable
number of clear physical meaning. We find the above criterion is equivalent to
$\gamma_{c}/\gamma_{\text{pk}}\sim 1.8-2.3$ for all packing fractions and
shear rates studied. Nontrivially, the above perspective agrees well with
experiments and simulations on ultra-dense colloidal suspensions, and also
prior simulations Hoy and O’Hern (2010), experiments Bennin  _et al._ (2019);
Bending  _et al._ (2014); Lee and Ediger (2010); Pamvouxoglou  _et al._
(2021), and theoretical Chen and Schweizer (2011) analysis of glassy polymers
which find $Pe(\gamma_{c})\sim 0.2\pm 0.05$.

##  VI Theory Predictions

###  VI.1 Alpha Relaxation

ECNLE theory predicts Mirigian and Schweizer (2014) the quiescent alpha
relaxation slows down by 6 decades as the packing fraction grows from 0.58 to
0.61. The theory also predicts massive speed up of the alpha time under
deformation Ghosh and Schweizer (2020). Since structural correlations soften
the caging constraints, the degree of speed-up in dynamics must be stronger
than predicted based on using the invariant structure model Ghosh and
Schweizer (2020). Figure 5c presents the normalized relaxation times for two
packing fractions and several strain rates. At $\gamma\tau_{s}=1$, alpha
relaxation time at the overshoot speeds up by $\sim 2$ or $\sim 8$ orders of
magnitude for packing fractions of 0.58 or 0.61, respectively. At a lower
strain rate of 0.01, the relaxation time speeds up by $\sim 3$ to $\sim 9.5$
decades for the same packing fractions. As shown in Fig. 5b, for both the
packing fractions studied and several strain rates, the relaxation time at the
overshoot point and in the steady state
($\gamma_{\text{ss}}\sim[2-3]\gamma_{\text{pk}}$) are almost exactly the same.

Figure 6: : (a) Theoretical stress (unit of $k_{B}T/\sigma^{3}$) vs strain
curve for packing fractions 0.58 (dashed) and 0.61 (solid) and two reduced
shear rates Schweizer and Saltzman (2003). The green dash-dotted curve is for
$\phi=0.61$ and $\dot{\gamma}\tau_{s}=0.01$ using the deformation-invariant
structure model and does not exhibit an overshoot (compare to the blue solid
curve). (b) Total and local cage barriers in thermal energy units as a
function of strain. At the overshoot ($\gamma_{\text{pk}}\sim 0.22$) the total
barrier is predominantly the caging contribution. Inset: total barrier at the
overshoot as a function of shear rate for different packing fractions.

###  VI.2 Overshoot Predictions: Full Rheological Response and Role of
Activation

Figure 6a shows representative stress-strain curves for packing fractions and
shear rates 222Experiments and simulations sometimes employ a “bare” Peclet
number $\text{Pe}_{0}=\dot{\gamma}\tau_{0}$, where $\tau_{0}$ is the dilute
suspension Brownian time. Many other workers adopt a known short time process
time scale (includes local hydrodynamics or binary collisions) that obeys
$\tau_{0}<\tau_{s}\ll\tau_{\alpha}$. All our theoretical analysis uses this
short time Peclet number, $\text{Pe}_{s}=\dot{\gamma}\tau_{s}$, to display our
results. typical of ultra-dense colloidal suspension experiments Koumakis  _et
al._ (2016, 2012b); Laurati  _et al._ (2012). All exhibit stress overshoots,
which reflects a competition between the “direct” effect of stress via the
mechanical work term in the nonequilibrium dynamic free energy and the
softening of caging constraints with strain as encoded in $S(k,\gamma)$ and
$C(k,\gamma)$. Specifically, although stress and strain both accelerate
relaxation and soften the elastic modulus, they do so in different ways, and
these effects are self-consistently coupled via the nonlocal (strain history
integration) and nonlinear form of Eq.(7).

Fig.6a shows that in the low-strain elastic regime, $\Sigma$ grows with strain
in a shear-rate independent manner. A representative example of our general
finding that the deformation-invariant structure model
($S(k,\gamma)=S(k,\gamma=0)$) predicts no overshoot is also presented,
consistent with our analytic arguments in section III. The steady state stress
is smaller in the presence of structural softening, as expected.

To understand the importance of activated dynamics in general, and
specifically the role of collective elasticity on the activation barrier, we
plot the total and local cage barriers as a function of strain in Fig. 6b, and
the total barrier at the overshoot peak as a function of shear rate. The total
barrier is not negligible at the overshoot, and is larger at higher packing
fractions and lower shear rates. This existence of a nonzero barrier is
consistent with experimental and simulation observations that intermittent
trajectories persist under strong deformation Besseling  _et al._ (2007); Hoy
and O’Hern (2010). The main frame of Fig.6b also shows the total barrier is
dominated by its local cage component at and beyond the overshoot. This is a
consequence of the predicted ease of destruction of the elastic barrier under
deformation Ghosh and Schweizer (2020).

At very large shear rates (typically beyond experimental study), the inset of
Fig.5b suggests a type of “granular” or “non-Brownian” yielding may apply in
the sense that the barrier at the overshoot (dynamic yield point) is
asymptotically approaching zero, and hence no thermal activation is required
(per an ideal MCT scenario). However, as established below, the strong rate
and packing fraction dependences of the barrier do critically impact our
predictions for how key rheological signatures evolve with these control
parameters.

###  VI.3 Correlation Between Peak Curvature and Overshoot Magnitude

Figure 7: Absolute magnitude of the second derivative of the stress-strain
curve at the overshoot and the overshoot amplitude as a function of shear rate
for a packing fraction of 0.61.

Our intuition based on the analytic analysis presented in section III is that
the absolute magnitude of the second derivative of the stress-strain curve at
the overshoot and the amplitude of overshoot are highly correlated. To test
this idea numerically, we plot in Fig. 7 the rescaled absolute value of the
2nd derivative and overshoot amplitude as a function of shear rate for fixed
packing of 0.61. One sees the two curves track each other extremely well,
indicating a strong casual correlation between them as,
$\displaystyle\bigg{|}\frac{\partial^{2}\Sigma}{\partial\gamma^{2}}\bigg{|}_{\gamma=\gamma_{\text{pk}}}\approx
a+b\left(\Sigma_{\text{pk}}/\Sigma_{\text{pl}}-1\right)$ where $a$ and $b$ are
constants. This example for a packing fraction of $0.61$ is not special, and
the same level of correspondence is found for all the other high packing
fractions studied.

###  VI.4 Steady State Analysis

Figure 8: Total activation barrier (thermal energy units) in the steady state
as a function of dimensionless shear rate for a wide packing fraction range.
The dotted orange dashed straight line shows a $\beta F_{\text{total,
ss}}\sim-\log{(\dot{\gamma}\tau_{s})}$ fit of the numerical calculations.
Inset: ratio of the total barrier at the overshoot to that in the steady state
as a function of dimensionless shear rate. Colors have the same meaning as in
the main panel.

Results for the total activation barrier as a function of shear rate and
packing fraction in the steady state are shown in Fig. 8. This total barrier
is dominated by the local caging contribution since the collective elastic
effects become small beginning at low strains or stresses, as previously
discussed Ghosh and Schweizer (2020). However, a non-vanishing total barrier
is predicted even in the steady state, thereby establishing that some level of
activated dynamics remains present. The steady state total barrier roughly
follows a logarithmic decrease with dimensionless shear rate,
$\displaystyle\beta F_{\text{total,ss}}\sim-\ln{(\dot{\gamma}\tau_{s})}$, as
illustrated by the dashed blue line in Fig. 8 for a packing fraction of
$0.61$. Interestingly, the inset of Fig. 8 shows that the ratio of total
barrier at the stress overshoot to that in the steady state is predicted be
roughly constant $\sim 1.5-2$. This represents a non-trivial connection of
activated physics in the transient and steady state regimes. For dimensionless
shear rates $>10$ (not shown) the steady state barriers become very small.

Finally, we consider how deformation-induced changes of structure modifies
steady state properties compared to our prior isostructural theory results
Ghosh and Schweizer (2020) and experiment. Fig. 9 shows representative
calculations of the flow curve (steady state stress vs shear rate) and shear
thinning of the relaxation time. The prior prediction that the onset of shear
thinning begins at remarkably low Pe values is robust to including
deformation-induced structural changes, and agrees with simulations and
experiments as previously discussed Ghosh and Schweizer (2020). Only modest
increases of the apparent thinning exponent are predicted when structural
changes are included. The flow curves are also only very weakly affected by
deformation-induced structural changes. Moreover, note that the empirical
Herschel-Buckley (HB) form observed experimentally,
$\Sigma_{\text{ss}}=\Sigma_{\text{ss}}^{\text{HB}}+c\dot{\gamma}^{n}$, is
generically predicted by the microscopic theory, not assumed as in most
phenomenological models. This is significant since the HB form is generally
empirically adopted with adjustable parameters in the analysis of experimental
and simulation data. Here, the HB form emerges in a predictive way from our
approach based on analyzing our numerical results over a practical finite
range of shear rates as done in experiment and simulation. Moreover, the idea
there is a static yield stress that is nonzero in the literal zero shear rate
limit is not true in our theory where there is always finite, albeit very
slow, ergodicity restoring structural relaxation in the absence of a literal
jamming point.

Figure 9: Non-dimensionalized steady state stress as a function of
dimensionless shear rate (points) for packing fractions 0.61 (upper blue
points) and 0.58 (lower red points); results are shown using the changing
(lower stress points) and invariant with deformation structure models. The
corresponding smooth (solid, dashed) curves are HB model fits for packing
fractions of 0.61 (n=0.15 and 0.17) and 0.58 (n=0.20 and 0.21), where the HB
exponents, n, are slightly smaller for the invariant structure model. Inset:
steady state alpha time normalized to its quiescent value as a function of
renormalized Peclet number for the invariant (dashed) and changing (solid)
structure models for packing fractions of 0.61 (blue) and 0.58 (green).

Finally, we note that additional analysis reveals a nearly universal value of
roughly 10 for the ratio of local cage to collective elastic barriers at the
overshoot and in the steady-state, and hence the total barrier is dominated by
the local caging contribution. The relevant plot is not shown since the
absolute value of the collective elastic barrier at these rheological state
points is negligible ($F_{e}\ll k_{B}T$), consistent with the results in Figs.
2b, 6b and 8.

##  VII Comparison with Experiments and Simulations

###  VII.1 Stress Overshoot

Figure 10 presents predictions (not fits) for the packing fraction and shear
rate dependences of the stress overshoot amplitude,
$r_{O}=\Sigma_{\text{pk}}/\Sigma_{\text{ss}}-1$ , and its corresponding
strain, $\gamma_{\text{pk}}$; experimental results for hard sphere colloidal
suspensions from refs Chen and Schweizer (2011); Koumakis  _et al._ (2016) are
shown as discrete points. The dependence of $\gamma_{\text{pk}}$ on shear rate
for different packing fractions is presented in Fig.10a. Experiments find a
factor of $\sim 2.5$ increase (from $\sim 8$ to 20%) for a $\sim$1-1.5 decade
increase of shear rate over the range $\phi=0.54-0.6$, simulations find
qualitatively similar results with $\gamma_{\text{pk}}\sim$ 15-40% for a 2
decade increase of shear rate Chen and Schweizer (2011); Koumakis  _et al._
(2016), and the theory predicts a factor of $\sim 2-4$ growth of
$\gamma_{\text{pk}}$ depending on packing fraction. The theoretical trend of
smaller changes for higher packing fractions is consistent with simulation
Chen and Schweizer (2011); Koumakis  _et al._ (2016). We note that for
$\phi=0.61$, the peak strain follows an apparent power law
$\gamma_{\text{pk}}\sim\dot{\gamma}^{0.16}$ for all the relevant strain rates
spanning 4 decades (black dotted line in Fig. 10 (a)), with a small effective
exponent quite similar to our prediction for the HB exponent in the previous
section. The same qualitative similarity remains true for a lower packing
fraction of 0.58, where the exponent is $\sim 0.22$ for
$\dot{\gamma}\tau_{s}\in(0.001,1)$ (not shown).

Figure 10: Overshoot properties, theory and experiment. (a) Double log plot of
the overshoot strain vs short time Peclet number [59] for different packing
fractions. The hollow circles are experimental data for colloids of mean
diameter D 535nm at packing fractions of 0.595 (blue) and 0.562 (green)
Koumakis  _et al._ (2016). (b) Overshoot amplitude as a function of
dimensionless shear rate for different packing fractions (see Fig. A1 for the
qualitatively different ideal MCT prediction). Hollow black circles are
experimental results with D 366nm at $\phi=0.582$ Koumakis  _et al._ (2016).
To compare the shape of the theoretical curves, the experimental points have
been vertically shifted upwards by a factor of 3.

Fig.10b shows the shear rate dependence of the overshoot amplitude for
different packing fractions. Fully aged experimental data for weakly
polydisperse hard sphere (HS) suspensions (mean diameter $\sigma=366$nm,
$\phi=0.582$) are also shown Koumakis  _et al._ (2016). A striking non-
monotonic dependence is experimentally observed, with $r_{O}$ becoming very
small at both low and high shear rates. The theory predicts the same behavior
for at low and high packing fractions and shear rate ranges typical of
experiments [59].

Figure 11: Overshoot amplitude as a function of packing fraction for several
shear rates. The experimental data is from the same system as in Fig. 10(b)
and is shown as hollow circles for $\dot{\gamma}\tau_{s}\sim 0.1$ (red) and
$\dot{\gamma}\tau_{s}\sim 1$ (green), respectively Koumakis  _et al._ (2016).

Results for the overshoot amplitude as a function of packing fraction at
various shear rates are presented in Fig.11. For a fixed shear rate, one sees
that $r_{O}$: (a) decreases monotonically and weakly with packing fraction
with a functional form difficult to unambiguously extract, (b) is modestly
larger at lower shear rates, and (c) becomes quite small in an absolute sense
at the highest shear rates. These trends are all in good qualitative accord
with the theoretical predictions. Considering the challenge of making high
precision measurements of the overshoot features, and complexities in real
suspensions not in the theory (e.g., particle size polydispersity,
hydrodynamic lubrication forces), we find the overall agreement with theory to
be very good.

###  VII.2 Steady State Comparison

Based on HB form analysis of our numerical theoretical results like those
shown in Fig.9, the extracted theoretical “static yield stress”
($\Sigma_{\text{ss}}^{\text{HB}}$) as a function of packing fraction scaled by
its RCP value (0.64 for the monodisperse theory, 0.67 for polydisperse
experiments Koumakis  _et al._ (2016, 2012b)) is presented in Fig.12.
Structural changes under deformation again have little effect, and the
predicted nearly exponential form, and tendency to bend over at the highest
packing fractions, is in good agreement with experiments on dense hard colloid
suspensions Besseling  _et al._ (2010)333Using the deformation-dependent
structure model, a rescaling constant of 15 produces quantitative agreement
with experimental data. A detailed discussion of the possible reasons for this
factor are given in Ghosh and Schweizer (2020).. Quantitatively, the
theoretical yield stresses are roughly a decade smaller in absolute magnitude
than observed experimentally. This could have many possible origins given the
simplicity of the model and theoretical approximations adopted, including
especially how to relate force and stress via an effective cross-sectional
area, $A$, as discussed in section II.B.

Figure 12: Comparison between the experimentally extracted (with HB model fit)
static yield stress Besseling  _et al._ (2010) with its theoretical analog
based on the invariant structure model (dashed green curve shifted up by a
factor of 12) and the new theory with $S(k,\gamma)$(solid blue curve shifted
up by a factor of 15).

Recent experimental studies Cao  _et al._ (2021) of dense emulsions extracted
via a HB fit the static yield stress as a function of packing fraction. They
found $\sigma_{y}\sim e^{C\phi}$, where $C\sim 60-70$. Our previous analysis
in ref. Ghosh and Schweizer (2020) based on the invariant under deformation
structure model predicts the so-called absolute yield stress increases as
$\sim e^{60\phi}$ for packing fraction ranges of 0.58-0.61. Since
incorporating deformation-induced structural changes has nominal consequences
on the steady state predictions, our prior conclusion for the dependence of
yield stress remains unchanged and is in reasonable agreement with the
emulsion results Cao  _et al._ (2021) within the caveat they employ soft
repulsive particles and not strictly hard spheres.

Based on the above, we conclude that deformation-induced structural changes
are critical for understanding the physics of the stress overshoot, but have
only minor consequences for the packing fraction and shear rate dependences of
steady state properties. This deduction is buttressed by our analytic second-
derivative analysis of the stress-strain curve presented above. Finally, we
find that activated dynamics remains important in the nonequilibrium steady
state in the sense that activation barriers are nonzero and, most importantly,
vary in a systematic manner with packing fraction and shear rate. The latter
aspect is the origin of our ability to understand the experimental trends
discussed above. Moreover, as seen in the inset of Fig.8, we note that the
barrier height in the steady state relative to its value at the transient
overshoot is nearly a universal constant (0.5), independent of shear rate and
packing fraction.

##  VIII Comments on Other Approaches for Predicting the Transient Shear
Response

The stress-strain curves for a variety of dense (glassy) colloidal suspensions
of repulsive particles has modeled by multiple other, generally
phenomenological, approaches. In this section we briefly mention a few
prominent examples, and contrast them with our work. We note that essentially
all phenomenological models build in by hand the idea that activated processes
are important, albeit not at a fundamental microscopic level that connects in
a predictive manner interactions, structure, dynamics, and rheology as we
have.

Ideal MCT is the technically most directly related approach to our work on
glassy hard spheres in the sense that it also is a microscopic force-based
theory that relates structure to dynamics. A rheological theory has been
developed based on an integration through transients formalism where the MCT
vertex incorporating force correlations becomes a function of strain, strain
rate and time Fuchs and Cates (2002a); Amann  _et al._ (2013); Fuchs and Cates
(2002b); Amann  _et al._ (2015). However, since ideal MCT is fundamentally
built on the hypothetical idea of a literal kinetic arrest glass transition
and includes no activated processes, it can be applied in practice only over a
very narrow range of packing fractions defined by the “distance” to the
quiescent ideal glass transition (at $\phi_{c}\sim 0.516$),
$\Delta\phi\equiv\phi-\phi_{c}$. Implementation of the full MCT including
microscopic structural information for 3d dense suspensions is extremely
demanding numerically. These considerations have strongly limited
confrontation of the predictions of the full ideal MCT with experiment in a
comprehensive and no fit parameter manner. The Appendix provides a more
detailed discussion of the ideal MCT results germane to our work, including
the large qualitative differences compared to our theory.

Phenomenological “soft glass rheology” (SGR) models primarily aim to
understand the rheological response of systems such as emulsions and slurries
composed of soft particles Sollich  _et al._ (1997); Sollich (1998); Fielding
_et al._ (2000, 2009). Motivated by phenomenological trap models, deformation
and flow is built into local ‘mesoscopic’ zones that can dynamically evolve,
and the role of (effectively quenched) structural disorder in determining
macroscopic rheological response and metastability is studied. Such models can
predict a stress overshoot that increases monotonically with strain rate,
which can rationalize data on soft pastes but disagrees with experimental
observations on hard sphere suspensions Sollich (1998). Since SGR models are
not based on microscopic control parameters, nor directly relate structure,
dynamics and rheology, they cannot predict the precise packing fraction
dependences of the key rheological features we have focused on in this
article.

Phenomenological “elastoplastic models” were originally constructed to explain
the stress overshoot of bulk metallic glasses that share some qualitative
resemblances to hard-sphere systems. Based on the idea of shear transformation
zones carrying information of plastic flow, a nonaffine response is coupled to
structural rearrangements at the cage level Zaccone  _et al._ (2014b); Maestro
and Zaccone (2017). While this approach requires at least two non-trivial
fitting parameters to characterize relaxation dynamics, the predictions agree
qualitatively with some measurements on metallic glasses. However, this
approach predicts monotonic changes of the stress overshoot magnitude with
strain rate, and is unable to capture the packing fraction dependence of this
quantity for hard sphere suspensions.

A “continuum fluidity” model has been very recently implemented in detail by
combining ideas of the nucleation and growth dynamics of a ‘shear zone’ that
acts like a localized fluid state Bocquet  _et al._ (2009); Benzi  _et al._
(2019). The flow properties are controlled by a free-energy functional that
depends on a phenomenological (and non-microscopically defined) local fluidity
variable, a length-scale related to the size of a fluidized region, and a
parameter that carries information about steady-state HB rheology which is
adopted empirically. A dynamical evolution equation for the fluidity and
subsequent minimization of the free-energy gives an evolution equation for
stress. Significant progress has been made recently with this approach for the
nonlinear rheology of soft particle suspensions and pastes Langer (2006);
Benzi  _et al._ (2021a). However, since the information about the HB flow
curve is built into the model by hand, predictions are subject to parameter
fitting and a microscopic understanding of how packing fraction changes the
rheological response in hard sphere suspensions is not possible.

##  IX Conclusions and Future Directions

We have constructed a microscopic theory that relates particle interactions,
structure, slow dynamics, and rheology based on deformation-assisted thermally
activated relaxation that treats in a unified manner the transient and steady
state features of the continuous shear rheology of ultra-dense Brownian
colloidal suspensions. Important physical elements include how activated
hopping driven stress relaxation is accelerated due to barrier reduction and
the critical role of deformation-induced disordering of local structure as the
origin of the stress overshoot. The theory captures the rich experimental
behavior of dense hard sphere colloidal suspensions nearly quantitatively. We
emphasize the importance of activated dynamics which is key to our successful
predictions of the packing fraction and strain rate dependences of the stress
overshoot. As shown in the present article and ref. Ghosh and Schweizer
(2020), the collective elastic physics of central importance for the
equilibrium dynamics in highly metastable states plays an essentially
negligible role in determining both the overshoot and steady-state rheological
behavior. This is the physical reason that the prior constant structure model
version of the ECNLE theory based rheological approach predicted Ghosh and
Schweizer (2020) a suppression of dynamic heterogeneity with applied strain.
These prior conclusions remain true in the presence of deformation induced
changes of structure both for the transient and steady state responses.

The magnitude and qualitative aspects of the packing fraction and strain rate
dependences of the overshoot and steady state stresses strongly depend on the
interparticle interaction. Soft microgels, core-shell particles, star-like
micelles, and thermoresponsive particles generally show different and
sometimes stronger overshoot effects Koumakis  _et al._ (2012c); Carrier and
Petekidis (2009); Pham  _et al._ (2008). The theory presented here can
potentially be extended to such systems by employing an appropriate
interparticle interaction pair potential and/or intraparticle degrees of
freedom. For example, for classic microgels, a Hertzian contact potential is
often adopted Ghosh  _et al._ (2019). We believe our same basic theoretical
framework can also be generalized to address the rheology of other hard
nanoparticle and colloidal dense gels and attractive glasses Ghosh and
Schweizer (2019), including potentially those that display a “double yielding”
behavior Pham  _et al._ (2008) associated with caging and physical bonding.

Finally, our theoretical approach has invoked simplifications that would be
valuable to address in the future. For example, use of an effectively
isotropic description of structure and particle trajectories. Addressing this
aspect in a microscopic force-level theory will be very complex and
computationally difficult, not only because of the explicit vectorial
dependence of $S(k)$, particle displacements and dynamic properties, but also
because tractable and accurate theories for $S(\vec{k},\gamma)$ largely do not
exist. A second challenge is to formulate coupled self-consistent evolution
equations for $S(k,\gamma)$ and $\Sigma(\gamma)$. We note that this problem
was addressed in prior work on polymer glasses in the presence of physical
aging and mechanical rejuvenation based on NLE theory and the generalized
Maxwell description for a highly coarse grained description of polymer chain
structure and $S(k)$ Chen and Schweizer (2011). Ideas in that work may be
relevant for constructing in the future a fully microscopic theory of
nonequilibrium structure and stress evolution in dense particle suspensions.
Other real-world complications such as (many body) hydrodynamic interactions
Nabizadeh and Jamali (2021); Nakayama and Yamamoto (2005); Whittle and
Dickinson (1997), colloids with rough surfaces and dissipative aspects like
sliding and rolling friction Singh  _et al._ (2020); Fuchs  _et al._ (2014),
and the shear thickening and/or jamming regimes Fernandez  _et al._ (2013);
Wyart and Cates (2014) remain outstanding challenges for microscopic
nonequilibrium statistical mechanical approaches.

## Acknowledgements

The early stages of this research was supported by the U.S. Department of
Energy, Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering under Award No. DE-FG02-07ER46471 through the Materials Research
Laboratory at the University of Illinois at Urbana-Champaign. The latter
stages of the research was supported by the Army Research Office via a MURI
grant with contract W911NF-21-0146.

## Appendix: Ideal Mode Coupling Theory Predictions

Figure 13: Ideal MCT theory predictions for the overshoot amplitude for
monodisperse 3D hard spheres as a function of bare Peclet number for two
different packing fractions straddling the quiescent ideal glass transition
[30].

Ideal Mode Coupling Theory (MCT) [31] based approaches for the nonlinear
rheology of dense colloidal suspensions [21, 27-30] have largely considered
only deformation-induced changes of the static structure factor, $S(k)$, via a
simple shear advection mechanism (per a time-dependent wavevector, $k(t)$).
This is argued to be the crucial process that weakens caging constraints,
eventually leading to complete cage ‘melting’ of an “ideal glass” and
deformation driven flow as $t\to\infty$. Ideal MCT is built on a hypothetical
‘ideal glass transition’ that for hard spheres under quiescent conditions
occurs at a critical packing fraction $\phi_{c}\approx 0.516$ [31] based on
structural input from PY integral equation theory. These defining features
have restricted applications of ideal MCT to an exceptionally small range of
packing fractions $\phi\in(0.51-0.52)$, significantly below the experimentally
studied range of $\phi\sim 0.56-0.61$ or higher where activated dynamics is
important in equilibrium. Computational complexity has rendered MCT difficult
to implement with the full microscopic structural information retained. Thus,
most prior MCT-based analyses of experimental stress-strain curves and the
overshoot have been carried out very close to $\phi_{c}=0.516$, and are often
performed in the context of the further simplified “schematic”
$F_{12}^{(\dot{\gamma})}$-model [21] and related semi-microscopic models [21,
28-30] which typically have multiple fit parameters.

Schematic MCT calculations predict that over a very large $\sim 8$ decade
change in shear rate the overshoot strain changes only slightly from
$\gamma_{\text{pk}}\sim 0.25$ to 0.30[28-30]. This is a much weaker variation
than observed in experiments and simulations [16-18], and much weaker than
predicted by our theory. MCT also predicts more than a factor of $\sim 2$
growth of the overshoot amplitude over a tiny change of packing fraction (and
a roughly linear growth law with packing fraction) at all fixed shear rates
[28-30]. This is much too strong and does not agree with simulations or
experiments [16-18] nor our theory which find: (i) much weaker changes of the
overshoot amplitude, and (ii) an opposite trend with increasing packing at a
fixed shear rate, i.e., the overshoot becomes smaller for ultra-dense systems.
One could in principle calculate the peak strain and magnitude of the stress
overshoot from the full microscopic MCT (e.g., using the simplified
isotropically sheared hard sphere model [30]) that avoids the schematic
aspects and most, but not all, fitting parameters. However, results based on
this more computationally intensive implementation also lead to a much too
large overshoot strain and amplitude with a similar highly limited range of
applicability tightly straddling the hypothetical ideal glass transition
packing fraction [26-30].

Figure 13 shows a specific example for the overshoot amplitude predicted by
the full numerical version of MCT for 3D monodisperse hard spheres [30]. The
results are dramatically different compared to our theory and the
experimental/simulation findings discussed in the main text. Ideal MCT results
for the overshoot amplitude are shown as a function of the bare
$\text{Pe}_{0}$ for two packing fractions that differ by $\sim 0.001$ on the
two sides of the quiescent ideal glass transition at a packing fraction of
0.516. As noted above, non-monotonic trends are not predicted, and as the
shear rate increases, MCT continues to predict larger values of the overshoot
magnitude.

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