#  Memory switching due to thermal noise in amorphous solids subject to cyclic
shear

Debjyoti Majumdar  Environmental Physics, Jacob Blaustein Institutes for
Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus 84990,
Israel  Ido Regev<EMAIL_ADDRESS>Environmental Physics, Jacob Blaustein
Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer
Campus 84990, Israel

###### Abstract

The discovery that memory of particle configurations and plastic events can be
stored in amorphous solids subject to oscillatory shear has spurred research
into methods for storing and retrieving information from these materials.
However, it is unclear to what extent the ability to store memory is affected
by thermal fluctuations and other environmental noises, which are expected to
be relevant in realistic situations. Here, we show that while memory has a
long lifetime at low temperatures, thermal fluctuations eventually lead to a
catastrophic loss of memory, resulting in the erasure of most or all of the
stored information within a few forcing cycles. We observe that an escape from
the memory-retaining state (limit cycle) is triggered by a change in the
switching of plastic events, leading to a cascade of new plastic events that
were not present in the original limit cycle. The displacements from the new
plastic events change the particle configuration which leads to the loss of
memory. We further show that the rate of escaping from a limit cycle increases
in a non-Arrhenius manner as a function of temperature, and the probability of
staying in a limit cycle decays exponentially with an increase in the shearing
frequency. These results have important implications for memory storage since
increasing the temperature offers a means of effectively erasing existing
memories and allowing for the imprinting of new ones that can then be stored
for a long time at low temperatures.

##  I Introduction

The study of mechanical memory in materials is an emerging field Keim  _et
al._ (2019); Bense and van Hecke (2021); Pashine  _et al._ (2019); Shohat  _et
al._ (2022); Fiocco  _et al._ (2014); Keim and Paulsen (2021); Merrigan  _et
al._ (2022); Royer and Chaikin (2015); Khirallah  _et al._ (2021); Jules  _et
al._ (2022); Liu  _et al._ (2021); Pine  _et al._ (2005) that is expected to
play an important role in applications ranging from soft robotics Laschi  _et
al._ (2016) to materials with tunable properties Chen  _et al._ (2021) and
mechanical computation Kwakernaak and van Hecke (2023). One important type of
material that was shown to exhibit mechanical memory is amorphous solids Regev
_et al._ (2013); Priezjev (2013); Schreck  _et al._ (2013); Keim and Arratia
(2013); Kawasaki and Berthier (2016). Amorphous solids subject to low
amplitude periodic straining at athermal, quasistatic conditions can be
trained to remember past configurations and different training protocols Keim
_et al._ (2019); Adhikari and Sastry (2018); Fiocco  _et al._ (2014); Mukherji
_et al._ (2019). However, under most conditions and in most environments,
amorphous solids are not sheared in athermal quasistatic conditions and are
subject to random vibrations of thermal, mechanical, or seismic origin. It is,
therefore, desirable to understand how such vibrations affect the formation of
new memory and the longevity of stored memory.

In amorphous solids subject to athermal quasistatic shear (AQS) periodic
straining has the form of increasing the strain in small straining steps and
minimizing the strain after each deformation step. This repeats until a
maximal (minimal) strain of $\gamma_{max}$ ($-\gamma_{max}$) is obtained. The
strain is then changed in an alternating manner of the form
$0\rightarrow\gamma_{max}\rightarrow-\gamma_{max}\rightarrow 0$ which
comprises one forcing cycle. If the amplitude $\gamma_{max}$ is smaller than
the yielding amplitude, the system eventually reaches a state (usually
referred to as a “limit cycle”), where the particle trajectories repeat after
one or more forcing cycles Szulc  _et al._ (2022); Keim and Paulsen (2021);
Ness and Cates (2020); Lindeman and Nagel (2021); Szulc  _et al._ (2020).
Inside such a limit cycle, plastic deformation is mediated by a finite set of
hysteretic plastic elements called “soft spots” (sometimes referred to as
“shear transformation zones”) Manning and Liu (2011); Falk and Langer (1998);
Bouchbinder  _et al._ (2007). Each soft spot switches at a given strain, and
some can switch back at a different strain. When a soft spot switches, it
creates a strain field that affects the switching fields of other soft spots.
When the system reaches a limit cycle, soft spots switch back and forth, and
at the end of the limit cycle, they all go back to their zero-strain states
Mungan  _et al._ (2019). Both particle positions and the switching order of
soft spots contain memory that can be retrieved using suitable methods. At
athermal conditions, the order of soft spot switching is entirely
deterministic, and leaving a limit cycle is impossible. Intuitively, we expect
that thermal fluctuations will allow the change of switching ordering and/or
the switching of soft spots that do not participate in the limit cycle, which
will cause the system to exit the limit cycle. However, it is unclear if such
thermally activated switching causes a large or small change in the limit
cycle and if such activation involves an Arrhenius-type process or a more
complicated thermal activation.

The current knowledge regarding the formation and retention of mechanical
memory at finite temperatures is rather limited as most of the work on
amorphous solids under oscillatory shear at finite frequencies and
temperatures, focused on post-yield failure fatigue Sha  _et al._ (2015);
Cochran  _et al._ (2022); Priezjev (2020); Lindström  _et al._ (2012); Bhowmik
_et al._ (2022) and not on the response of memory to such conditions. It is
known that amorphous solids subject to oscillatory shear can reach a limit
cycle in the presence of small thermal fluctuations Priezjev (2016);
Nagamanasa  _et al._ (2014), but it was found that they exhibit a reduced
degree of mechanical reversibility when subjected to a single forcing cycle,
compared to the athermal case Farhadi and Arratia (2017). In another example,
it was shown that thermal fluctuations can cause systems to leave a limit
cycle that was prepared by athermal shearing and to reach a new, stable, limit
cycle Regev  _et al._ (2013). However, to the best of our knowledge, there are
no detailed numerical studies of the behavior of limit cycles under a range of
temperatures and frequencies and only one theoretical study where it was
assumed that such systems leave a cycle due to standard Arrhenius activation
Mungan and Sastry (2021).

In this manuscript, we present quantitative results from non-equilibrium
molecular dynamics (NEMD) simulations of shearing of amorphous solids at
different temperatures and frequencies that clearly show that limit cycles
containing mechanical memory retain configurational memory to a very good
degree even at finite temperatures and strain rates. However, we find that
after a certain time, thermal fluctuations induce a change in the switching of
soft spots, which causes new soft spots to appear that in turn leads to a
rapid erasure of memory. We then show that the escape rate as a function of
temperature exhibits a sub-Arrhenius behavior and that the probability of
leaving a limit cycle as a function of frequency exhibits an exponential
decay. We further show that the time to leave a limit cycle exhibits a power-
law distribution. We then discuss the implications for understanding the
physics of plastic deformation in amorphous solids and storing memory in such
systems.

##  II Memory in the presence of thermal noise

To study the effect of different temperatures and frequencies on memory
stability in amorphous solids subject to oscillatory shear, we simulated two-
dimensional binary mixtures of particles interacting via a Lenard-Jones
potential. Each system comprises $N$ particles enclosed in a square simulation
box of size $L$ with periodic boundary conditions. We used two different
number densities: $\rho=0.8$ with linear size $L=32$, and $\rho=0.7$ with
$L=35$. Each realization was prepared by quenching a liquid from high
temperature to $T=10^{-4}$ and applying oscillatory strain of the form:

$\gamma=A\sin(\omega t)\,,$ (1)

using the LAMMPS Plimpton (1995) implementation of the SLLOD algorithm Evans
and Morriss (1984) together with Lees-Edwards periodic boundary conditions
Lees and Edwards (1972). Here $\gamma$ is the applied strain, $t$ is time, $A$
is the strain amplitude and $\omega$ the shearing frequency.

In Fig. 1, we show examples of the zero-strain (stroboscopic) potential energy
per particle ($U/N$) of an amorphous solid sheared at three different
temperatures, each differing from the other by an order of magnitude, though
still well below the liquefaction point. In the top-most sub-figure, the
temperature is the lowest ($T=10^{-4}$ in simulation units) and the system
reaches a periodic state (highlighted background) after a transient of about
$100$ training cycles.

Figure 1: Zero-strain (stroboscopic) energy per particle in amorphous solids
subjected to oscillatory shear at a frequency $\omega=0.001$ and amplitude
$A=2.2$, for three different temperatures $T=10^{-4}$, $10^{-3}$ and $10^{-2}$
in a system of $N=841$ particles at a density of $\rho=0.7$. Periodic states
(highlighted) are manifested by a constant zero-strain potential energy. Of
importance is the fact that at intermediate temperatures the system is able to
jump in and out of the limit cycle for multiple times as shown for $T=0.001$.

For this temperature, the limit cycle reached is stable and the solid stays in
the same state, indicated by the fact that the potential energy remains
unchanged after each cycle for an extended period of time. This observation
suggests that the memory obtained is stable against the thermal fluctuations
at that temperature and strain rate. In the second case (middle sub-fig of
Fig. 1), where the temperature is higher ($T=10^{-3}$), the system also
reaches a periodic state after about 150 forcing cycles and stays in the same
limit cycle for an extended period of time. However, in this case, after about
$200$ more forcing cycles, the system leaves the periodic state, starts a new
random transient, and after $50$ more forcing cycles, reaches a new limit
cycle. Nevertheless, the new limit cycle is also unstable, and the system
leaves it after a relatively small number of cycles. The dynamics involving
jumping in and out of limit cycles keep repeating, seemingly forever. In this
case, it is easy to observe that the memory has a limited, random expiration
time. Lastly, at an even higher temperature ($T=10^{-2}$), shown in the bottom
sub-fig of Fig. 1, the system spends most of the time in non-repetitive
dynamics but still manages to find a short-lived limit cycle. A question
arises as to the extent to which the particle configuration changes when
jumping between limit cycles, and this will be discussed below.

Figure 2:  The hysteresis curves of two limit cycles that were reached during
the same simulation at $T=0.003$. Note that the transient connecting them is
not shown.

##  III Memory loss due to thermal noise

In Fig. 2, we show two limit cycles connected by a random transient (not
shown). The plastic events and thermal noise are visible in both cycles, and
it seems that most plastic events in the two cycles are different. However,
one may wonder to what extent the particle configurations reached after the
system escapes from a limit cycle differ from those inside the limit cycle. To
check this, we calculated the self-part of the intermediate scattering
function, given by:

$\mathcal{F}_{s}({\bf k},t)=\frac{1}{N}\sum_{i=1}^{N}\cos({\bf k}\cdot[{\bf
r}_{i}(t)-{\bf r}_{i}(0)]),$ (2)

where ${\bf r}_{i}$ is the coordinate of the $i$th particle and ${\bf k}$ is a
wave-vector. When ${\bf k}$ is chosen to have a size comparable to the average
distance between particles, the decay of the function indicates diffusion away
from the first shell of neighboring particles (this function was previously
used to study self-diffusion close to the glass transition) Donati  _et al._
(1998); Berthier and Biroli (2011).

Figure 3: Self-part of the intermediate scattering function for a single
sample at $T=0.006$ exhibiting a step-like change from $1$ inside a limit
cycle to $0$ when the system escapes the limit cycle. (Inset) The
corresponding displacement between the zero-strained configurations across the
point where $F_{s}(k,t)$ decays to zero. Figure 4: The displacements of
particles between zero-strain configurations before and after a thermal escape
from a limit cycle.

In Fig 3, we show a typical case, where we use
$k_{x}=k_{y}=2\pi/\sigma_{\alpha}$ with $\sigma_{A}=1$ for the small particles
and $\sigma_{B}=1.4$ for the large particles. Intriguingly,
$\mathcal{F}_{s}(k,t)$ changes sharply from $1$ to $0$ as the system jumps out
of the limit cycle (Fig. 3) in a manner resembling a Heaviside step function.
This indicates an abrupt change in the configuration, which implies that the
memory stored in the limit cycle is lost in a small number of forcing cycles
following the point from which the system leaves the limit cycle. To
understand the origin of this behavior, we looked at the displacements between
the zero strain configurations before and after an escape from a limit cycle
(see Fig. 3 inset and 4). We observed that the displacements due to the new
plastic events cover a significant part of the material, which explains the
abrupt change in $\mathcal{F}_{s}(k,t)$.

To identify the point of departure from the limit cycle, we plot the
hysteresis curve for two consecutive cycles, one inside and another
immediately outside the limit cycle in Fig 5. During forcing cycle 187
(counting from the onset of shearing), the system is still inside a limit
cycle, but at forcing cycle 188 leaves the limit cycle. We can see that the
deviation of cycle 188 from the stress-strain curve of 187 starts when one
plastic event (marked by an arrow) does not occur. This, next, triggers the
switching of new plastic events that were not present in 187 and prevents
plastic events that were present in 187 from appearing. Looking at the
displacement fields of the individual plastic events (not shown), we found
that the displacements due to these new events accumulate during the cycle
such that they cover a large part of the simulation box. We, therefore,
conclude that the sharp structural change observed in Fig 3 is due to the
combination of two effects - a plastic event that was not switched due to
thermal fluctuations allows the switching of a new plastic event that was not
switched as part of the limit cycle. This leads other plastic events that did
not take part in the cycle to occur, which causes the particle configuration
to change such that the memory is erased.

Figure 5: Hysteresis curves of two consecutive cycles where the system leaves
the limit cycle during cycle 188. In both cycles, the temperature was
$T=7\times 10^{-4}$ and the amplitude was $A=2$ and the frequency was
$\omega=0.001$. The transition out of the limit cycle occurs due to one
plastic event (marked with an arrow) being “missed” which causes plastic
events that were not present in cycle 187 to occur.

In Fig 6 we show $\mathcal{F}_{s}(k,t)$ averaged over 30 realizations for two
different temperatures $T=0.003$ and $0.006$. The configuration corresponding
to the first limit cycle for each individual sample is taken as the reference
configuration for that sample i.e. ${\bf r}(0)$. In all of the realizations,
we observed the same step-like switching observed in Fig 3 where
$\mathcal{F}_{s}(k,t)$ switches abruptly from $1$ to $0$ at the vicinity of a
specific forcing cycle. However, in different realizations, the switching
occurs at different times, which causes the more gradual decay observed in Fig
6. In fact, for the temperatures shown, the distribution of switching times
was found to be a power-law, as shown in Fig 7. We verified that when the
number of samples is small, summing a set of step-functions (realizations)
that switch at a random time (cycle number) reproduces behavior similar to the
one shown in Fig 6, while when the number of step-functions increases,
$\mathcal{F}_{s}(k,t)$ exhibits a power-law decay, as expected. From Fig 6 it
seems that the temperature has an effect on the average decay rate. A
quantitative analysis of this difference is discussed in the next section.

Figure 6: Self part of the intermediate scattering function averaged over $30$
samples for two different temperatures $T=0.003$ and $0.006$. Here $t$ is the
number of cycles.  Figure 7: The distribution of limit cycle longevity $\tau$
exhibiting a power-law decay with exponent $2$.

##  IV Memory stability at different temperatures

As demonstrated in Fig. 1, temperature strongly affects the dynamics, which is
reflected in the limit cycle escape times. To quantify the effect of
temperature on the escape time, we measured the average escape rate using 100
realizations. The initial realizations were prepared by applying 1000 forcing
cycles at an amplitude $A=2.1$, frequency $\omega_{i}=10^{-3}$ and temperature
$T_{i}=10^{-4}$. We verified that all the realizations reached a limit cycle
which was used as the initial state. The prepared samples were next subjected
to shearing with the same amplitude and frequency but using a temperature
$T\geq T_{i}$. We then recorded the fraction of realizations that stayed in
the initial limit cycle after 10 forcing cycles $P_{lc}$ as a function of
temperature. $P_{lc}(T)$ is shown in Fig. 8 for the two densities, $\rho=0.8$,
and $\rho=0.7$. We found that $P_{lc}$ exhibits a stretched exponential decay
$P_{lc}\sim\exp(-(T/T_{0})^{\beta})$ with an exponent $\beta=0.59$ and a
temperature scale $T_{0}=0.007$. In the inset, we show an Arrhenius plot of
the escape rate $1-P_{lc}$ as a function of $1/T$. We can see that at low
temperatures, the escape rate is Arrhenius-like, but at temperatures above
$T\sim 10^{-3}$, we see a strong deviation from Arrhenius behavior for both
densities. In Fig. 9, we show the same quantity but at a constant temperature
$T_{i}=10^{-4}$ and a varying frequency $\omega$. In this case,
$P_{lc}(\omega)$ exhibits an exponential decay. This indicates that the
stability of limit cycles is very sensitive to temperature increases, while it
is somewhat less sensitive to an increase in the shearing frequency.

Figure 8: The probability $P_{lc}$ to stay in the initial limit cycle as a
function of temperature $T$ of a system consisting of $N=841$ particles at a
density of $\rho=0.8$, and $N=900$ particles at a density of $\rho=0.7$. The
initial samples were prepared at $T_{i}=10^{-4}$, $\omega_{i}=0.001$, and
amplitude $A=2.1$. A fit (continuous red line) indicates a decaying exponent
behavior with an exponent of $\beta=0.59$ and $T_{0}=0.007$. Inset: Arrhenius
plot of the escape rate $1-P_{lc}$ as a function of $1/T$ showing a deviation
from Arrhenius behavior at low temperatures.  Figure 9: $P_{lc}$ as a function
of frequency ($\omega$) exhibiting an exponential decay. The initial
configurations are prepared at $T_{i}=10^{-4}$ and $\omega_{i}=0.001$,
amplitude $A=2.1$. (Inset) Same as the outer plot but in the semi-log axis.
The continuous line is a fit with $\omega_{0}=0.067$.

##  V Discussion

In this study, we investigated the impact of finite temperatures and strain
rates on the formation and stability of memory in amorphous solids subject to
oscillatory shear. Our findings indicate that memory of particle positions and
plastic events can be stored accurately for a large range of temperatures and
frequencies and that the main difference between high and low temperatures is
the amount of time for which memory is stored rather than the quality of the
memory stored. Our simulations indicate that when the system is inside a limit
cycle, the main effect of thermal fluctuations is to shift the strain values
at which different plastic events are switched during the cycle. However,
after some time, these fluctuations become large enough to cause a
“scrambling” of the switching fields of the soft spots, an effect that was
attributed to soft spot interactions in previous work Bense and van Hecke
(2021); Szulc  _et al._ (2022). This then causes the original soft spots to
become disabled and a completely new set of soft spots to switch. It is not
clear how the switching of one new soft spot can cause such an extensive loss
of memory but it is most likely a result of the elastic interactions between
soft spots, interactions that were shown to cause the shifting of the
switching strains Mungan  _et al._ (2019). Another intriguing result is the
sub-Arrhenius decay of the rate at which the system escapes a limit cycle.
Since the set of energy minima visited by the system during a limit cycle
cannot change, as long as the system is inside the same limit cycle, this
cannot be a result of a change in the energy landscape region visited
(contrary to what is found in supercooled liquids Debenedetti and Stillinger
(2001); Sastry  _et al._ (1998)) but is most likely a result of the combined
effects of inertia and thermal fluctuations. We believe that most of the
computer experiments that we performed should be accessible to actual
experiments in colloidal glasses and it will be interesting to see how
experiments and simulations compare. Lastly, we believe that the abrupt memory
loss may also be useful as a means of erasing an existing memory and preparing
the system for the imprinting of a new memory by ramping the temperature up
and down. Future work will focus on understanding the relationship between
temperature, frequency, and amplitude on memory lifetime, the origin of the
power-law behavior, $T_{0}$ and $\omega_{0}$, and the mechanism behind the
abrupt memory loss.

Acknowledgements: DM and IR were supported by the Israel Science Foundation
(ISF) through Grant No. 1301/17. DM and IR would like to thank Itamar
Procaccia for useful comments.

##  VI Materials and methods

###  VI.1 Model and simulation details

We simulated a 50:50 binary mixture of two-dimensional particles interacting
via a Lenard-Jones (LJ) pair-wise potential using LAMMPS Plimpton (1995):

$V_{\alpha\beta}(r)=4\epsilon_{\alpha\beta}\left[\left(\frac{\sigma_{\alpha\beta}}{r}\right)^{12}-\left(\frac{\sigma_{\alpha\beta}}{r}\right)^{6}\right]~{}~{}~{}(r<r_{c}),$
(3)

where $\alpha,\beta\in\\{A,B\\}$,
$\epsilon_{AA}=\epsilon_{AB}=\epsilon_{BB}=1$, $\sigma_{AA}=1$,
$\sigma_{AB}=1.2$, and $\sigma_{BB}=1.4$, whereas the masses of the two
particle sizes are equal $m_{A}=m_{B}=1$. The cutoff radius is taken to be
$r_{c,\alpha\beta}=2.5\sigma_{\alpha\beta}$. In all simulations we predefined
the linear system size $L$ and the number density $\rho$ whereas the number of
particles $N$ was determined from the relation $\rho=N/L^{2}$. We used two
densities, $\rho=0.8$ with $L=32$, and $\rho=0.7$ with $L=35$ which keeps the
particle number roughly equal. Note that the density is well above the jamming
density $\rho_{J}=0.64$ in LJ systems. We choose half of the particles to be
$1.4$ times larger than the other half which inhibits crystallization Perera
and Harrowell (1999). All simulation results are reported in LJ units, with
the time - being measured in units of $\tau=\sqrt{m\sigma^{2}/\epsilon}$.

Each amorphous solid realization is first equilibrated for $10^{6}$ time steps
of value $\delta t=10^{-4}$ at a constant volume $V$ and a high temperature
$T=2$. Next, the temperature is ramped down to the target temperature of
$T=10^{-4}$ with - additional equilibration steps at two intermediate
temperatures, $T=1$ and $T=0.1$ for $10^{6}$ steps each. Finally, the system
is again equilibrated for another $10^{6}$ time steps at the target
temperature. In order to simulate time-dependent shearing, we used the Lees-
Edwards boundary conditions Lees and Edwards (1972) coupled with the SLLOD
equations of motion Evans and Morriss (1984) where the temperature was
maintained using a Nosé-Hover thermostat Nosé (1984); Allen and Tildesley
(1989) . Time-periodic shear is imposed along the x direction as
$\gamma(t)=A\sin(\omega t)$, where $A$ is the strain amplitude and $\omega$ is
the frequency, using the same time step as $\delta t=10^{-4}$.

###  VI.2 Limit cycle detection

To detect the onset of a limit cycle, we looked at the potential energy per
particle ($U/N$) of the zero-strain configuration after each shearing cycle,
and calculated the variance of the zero-strain configurational energy for
three consecutive cycles, using the expression:

$\mbox{Var}(u)=\frac{1}{3}\left[(u_{n}-\mu)^{2}+(u_{n-1}-\mu)^{2}+(u_{n-2}-\mu)^{2}\right],$
(4)

where $\mu=\frac{1}{3}(u_{n}+u_{n-1}+u_{n-2})$ is the mean zero-strain energy
of the last three cycles $n$, $n-1$ and $n-2$. At low temperatures, the
variance inside a limit cycle could be as low as Var$(u)\sim 10^{-11}$ while
outside a limit cycle it is Var$(u)\sim 10^{-5}$.

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