# Mechanical Tuning of Residual Stress, Memory, and Aging in Soft Glassy Materials

Paolo Edera, Minaspi Bantawa, Stefano Aime, Roger T Bonnecaze, Michel Cloitre

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Paolo Edera, Minaspi Bantawa, Stefano Aime, Roger T Bonnecaze, Michel Cloitre. Mechanical Tuning of Residual Stress, Memory, and Aging in Soft Glassy Materials. Physical Review X, 2025, 15 (1), pp.011043. 10.1103/physrevx.15.011043. hal-04975607

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Distributed under a Creative Commons Attribution 4.0 International License
# Mechanical Tuning of Residual Stress, Memory, and Aging in Soft Glassy Materials

Paolo Edera,¹,*, Minaspi Bantawa,² Stefano Aime,¹ Roger T. Bonnecaze,² and Michel Cloitre¹,†

¹Molecular, Macromolecular Chemistry and Materials, ESPCI Paris, CNRS,
PSL Research University, 75005 Paris, France

²McKetta Department of Chemical Engineering and Texas Materials Institute,
University of Texas at Austin, Austin, Texas 78712, USA

<figure><img src="image_2.png" /></figure>

(Received 11 July 2024; accepted 21 January 2025; published 25 February 2025)

Glassy materials rapidly quenched from a liquid to a solid state upon flow cessation or cooling solidify in an out-of-equilibrium configuration, trapping residual stresses and retaining the memory of the processing conditions for very long times, which compromises their physical characterization and can adversely affect processing operations. Erasing the mechanical history encoded in disordered materials constitutes a great challenge. Here, we address this problem using experiments and particle dynamic simulations for the case of colloidal glasses made of soft particles densely packed at high volume fractions. We propose a conceptual framework that connects residual stresses, directional memory, and aging of colloidal glasses to the distribution of local shear stresses in the shearing plane. The mean value of the distribution corresponds to the macroscopic stress, the skewness carries information about directional memory, and the standard deviation is related to mechanical aging. Periodically training soft particle glasses near the yield point with a sequence of stress-controlled oscillations provides a fine-tuning of the particle stress distribution. Asymmetric shear stress distributions resulting from previous flow are transformed into symmetric distributions, thereby successfully erasing residual stresses and directional memory. The same methodology is successfully applied to colloidal and polymer gels with thixotropic properties, suggesting that it is general and may be extended to other classes of disordered materials.

DOI: 10.1103/PhysRevX.15.011043

Subject Areas: Fluid Dynamics,
Interdisciplinary Physics, Soft Matter

## I. INTRODUCTION

Disordered materials such as glasses, gels, and granular materials remember their mechanical history: They trap residual stresses, store directional memories, and age. Whereas significant efforts have been devoted to encode information in glasses, erasing their memory is not yet possible. We address this challenge using particle dynamic simulations and well-controlled experiments. We propose a theoretical framework that connects residual stresses, directional memory, and aging of colloidal glasses to the microscopic stress statistics. Then, we implement training methods that erase residual stresses and memory, and gain control over aging. Promising results are presented for other amorphous materials with various microstructures.

Many disordered materials of practical interest are processed in the liquid state and rapidly quenched to the

solid state upon flow cessation or cooling. Upon solidification, the microstructure becomes frozen in an out-of-equilibrium configuration, and mechanical stresses remain trapped in the materials. Residual stresses have been described and analyzed in a variety of systems, including metallic glasses [1,2], Gorilla glass [3], polymer melts [4], hard sphere suspensions [5,6], jammed suspensions [7–11], colloidal gels [12–15], and biological materials [16]. The relaxation of residual stresses in glassy materials generally involves two distinct processes that are decoupled in time [1,4,5,7,8,10,11,17]: a fast relaxation associated with ballistic particle motion corresponding to the elastic recoil of the microstructure and a much slower relaxation involving longer length scales. During the slow relaxation process, the metastable microstructure undergoes rearrangements through a process known as aging, which progressively lowers the free energy and increases the stiffness [18]. Since aging involves the existence of very long relaxation processes associated with frustration and cooperativeness, most materials do not reach mechanical equilibrium over accessible timescales, resulting in persistent residual stresses.

Residual stresses and aging are often associated with memory effects. Shearing the materials along a direction encodes directional memory: The material remembers the direction along which it has been sheared. An important

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

<footnote>*Contact author: paolo.edera@espci.psl.eu</footnote>

<footnote>†Contact author: michel.cloitre@espci.psl.eu</footnote>
consequence is that the response to an external drive is not symmetric upon flow reversal. For instance, a material initially sheared along one given direction appears stiffer in this direction than in the opposite direction because of the action of accumulated restoring stresses. Directional memory has been described in many disordered materials, including jammed suspensions [19–22], metallic glasses [23], granular materials [24,25], nanocomposites [26], and colloidal gels [27,28].

Residual stresses, aging, and memory effects are of great importance. While the manipulation of residual stresses has been exploited to fabricate materials with improved mechanical properties [29], in general, uncontrolled residual stresses are detrimental to the integrity and usage of glassy materials [30]. One manifestation of directional memory, known as the Bauschinger effect in the field of crystal and metal plasticity, leads to a weaker elastic modulus and yield stress under reverse loading, as compared with reloading after an initial plastic deformation [31]. The Bauschinger effect thus plays an important role in the processing and usage of metallic glasses. It adversely affects the physical characterization of materials [27] and can be a major issue in processing design and operation [32]. Directional memory is also at the origin of the spontaneous motion occurring without forced flow in colloidal glasses and biological materials [16].

Recently, the topic of residual stresses and memory formation in disordered materials has attracted a lot of attention [33]. Significant effort has been devoted to enhance memory effects and encode information in glasses using well-designed thermal [34] or mechanical procedures [35–37]. For instance, cyclically training a glassy material using an oscillatory shear deformation creates a memory of the shearing amplitude, which can be subsequently read by another sequence of oscillations with increasing amplitude. The possibility to encode multiple memories in a system and the comparative merit of various write-and-read protocols have also been demonstrated [38–40].

The opposite problem, which is to erase residual stresses and directional memory in athermal glassy materials, has received less attention. Whereas residual stresses in conventional glasses can be manipulated with temperature, annealing athermal soft glasses must be achieved mechanically. This process is an outstanding challenge because, memory being of elastic origin, annealing necessarily involves some irreversible rearrangements in order to relax trapped stresses, which, in turn, may induce nondesirable mechanical history. It is common to preshear soft materials prior to rheological measurements, an operation sometimes called rejuvenation. Preshearing has the merit to prepare materials in a reproducible state, but it leaves them unrelaxed [19,28]. In the opposite case where glassy suspensions are allowed to recover under zero stress, residual stresses vanish, and symmetry upon flow reversal is eventually restored. However, this procedure generally

takes prohibitively long times and is impracticable for a broad range of materials [20]. In this context, a recent approach that decomposes strains into recoverable and unrecoverable components and connects directional memory with recoverable strain is particularly appealing [41]. Applying a step strain of amplitude equal to the recoverable strain in the direction opposite to the initial flow, in combination with the zero-stress condition, restores reversibility upon flow inversion [28]. In this method, the recoverable strain is not known *a priori*, and it has to be estimated using a trial-and-error process prior to measurements. Important questions left open by this approach are whether the material has reached equilibrium after the macroscopic strain has been reset and how to deal with samples that still slowly evolves, by aging for instance, after macroscopic relaxation. Finally, this procedure, based on resetting the reversible strain component, is essentially mean-field type and does not provide information about the microscopic mechanisms that store mechanical information. In conclusion, there is a clear need to address the task of preparing glassy materials free of residual stresses and not affected by memory effects along two complementary directions: (i) build a robust annealing protocol potentially applicable to a wide range of materials; (ii) establish a generic framework that connects the macroscopic response associated with residual stresses to the underlying microscopic mechanisms.

We address this problem using soft particle glasses (SPG), which can be considered as an prototype of athermal glassy materials. SPGs are dense assemblies of soft and deformable particles packed in an amorphous state at high volume fraction. An experimental realization of SPG is a suspension of polyelectrolyte microgels swollen in water at concentrations above the jamming transition [42]. The particles have a well-defined microstructure and interact through purely repulsive contact forces of elastic origin. First, we show that training SPGs periodically with a sequence of well-chosen stress-controlled oscillations erases any directional memory originating from the mechanical history of the materials. Second, we use particle dynamic simulations to analyze the microscopic origin of directional memory in these materials. Simulations quantitatively reproduce the macroscopic behavior found in experiments and give access to the per-particle shear stress distribution, which we express in terms of the probability density function $P(\sigma_{yx}^{(p)})$, where $\sigma_{yx}^{(p)}$ is the shear stress carried by particles and its evolution over time. We demonstrate that the directional memory originates from the asymmetry of $P(\sigma_{yx}^{(p)})$ parametrized by the skewness parameter, which progressively decreases during training as the particles contributing to the tails of the distribution yield. The other components of the stress tensor do not contribute to memory effects relevant for the shear rheology experiments considered in this paper. The width of $P(\sigma_{yx}^{(p)})$
parametrized by the standard deviation becomes narrower as aging takes place. We are thus able to disentangle directional memory and aging within this framework. We show that our protocol to anneal the macroscopic residual stress and the directional memory applies to other types of amorphous materials, like colloidal and polymer gels.

## II. MATERIALS AND METHODS

### A. Microgel glasses

The microgels used in this study are synthesized by Arkema. The particles are well-defined spherical networks, made of ethyl acrylate, methacrylic acid, and diallyl phthalate as a cross-linker. Suspensions are prepared by diluting the stock solution (31 wt%) with ultrapure water and adding sodium hydroxide that ionizes the methacrylic acid units and provokes the swelling of the particles ($R_h \cong 165$ nm). The suspensions undergo a jamming transition at $C \cong 1$ wt% above which they become SPGs that behave like solids at rest but flow when the stress is large enough [42]. The concentration of the suspensions used in this study is $C \cong 2$ wt%. At this concentration, the particles are packed in an amorphous state and interact through contact elastic forces, thermal effects being negligible [42,43]. The elastic properties are characterized by the plateau value of the storage modulus $G_0 = 435$ Pa, which is determined from the viscoelastic spectrum shown in the Supplemental Material [44].

### B. Colloidal and polymer gels

A colloidal gel is prepared from fumed silica in an organic solvent. Fumed silica was donated by EVONIK. The gel is prepared as described in Ref. [28]. The powder is dispersed at a concentration of 6.7 wt% in a mixture of paraffin oil (70 wt%) and polybutylene (30 wt%). The dispersion is homogenized by mixing overnight with an overhead stirrer (IKA RW16) at a rate of 10 s⁻¹ at a temperature of 80 °C.

A polymer gel is obtained by dissolving xanthan gum powder (donated by Farmalabor) in deionized water at a concentration of 4 wt% by hand mixing at room temperature for a few minutes.

### C. Rheological measurements

We use an Anton Paar MCR 502 equipped with firmware FM4.92, working either in controlled rate mode or controlled stress mode. The geometry is a 50-mm-diameter cone and plate (angle: 2°; truncation: 49 µm), with rough surfaces (5 µm) to avoid the occurrence of wall slip. To minimize solvent evaporation from the microgel glasses and the xanthan gel, the geometry is protected from the exterior by a solvent trap, available from Anton Paar, and the atmosphere of the trap is saturated using a small quantity of water. Three main rheological tests are performed: (a) stress sweep experiments that consist in

subjecting the material to an oscillatory stress at constant frequency ($\omega = 1$ Hz) and amplitude increasing or decreasing in the range 0.5 Pa to 100 Pa; (b) stress-growth experiments, also called start-up flow experiments, that consist in applying a constant shear rate $\dot{\gamma}_0$ at rest; and (c) flow curves that are built by performing a series of start-up flow experiments where the applied shear rates vary between $10^3$ s⁻¹ and $10^{-3}$ s⁻¹ and recording the stress value when a steady state is reached, typically when the accumulated strain is larger than 1 [see Figs. 2(c)–2(c')]. All experiments are performed at 20 °C.

### D. Simulation method

We perform particle dynamics simulations of a model SPG, made of slightly polydisperse deformable spheres with Hertzian contacts, dispersed above the jamming transition in a solvent with a viscosity $\eta_s$. The SPG consists of $N = 10^5$ particles, with an average radius $R$, confined in a cubic box of volume $V$, where they are jammed at a volume fraction $\phi = 0.84$. The particles interact elastically through Hertzian contacts characterized by the contact modulus $E^* = E/(1 - \nu^2)$, where $E$ is the Young modulus and $\nu$ the Poisson ratio [43]. The contact modulus for microgels has been estimated to be of the order of $E^* = 10^4$ Pa [17,42]. Initially, the glass is equilibrated in a state where the force and torque on each particle are zero. Oscillatory shear flow, start-up flow, and continuous flow are generated using Lees-Edwards boundary conditions in the LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) package [45]. Two particles denoted by $\alpha$ and $\beta$ interact through repulsive contact forces of elastic origin ($f_{\alpha\beta}^e$) and elastohydrodynamic lubrication forces ($f_{\alpha\beta}^{EHD}$). The equations of motion are made dimensionless by scaling time, stress, and length by $\eta_s/E^*$, $E^*$, and $R$, respectively. The dynamics is thus solely characterized by the dimensionless shear rate $\tilde{\gamma} = \dot{\gamma}\eta_s/E^*$ (steady flow) and dimensionless frequency $\tilde{\omega} = \omega\eta_s/E^*$ (oscillatory flow). A description of the mathematical formulation, the derivation of the forces acting on the particles, and the simulation technique in steady, oscillatory, and transient solicitations can be found in previous publications [17,43,46].

From the interparticle forces and particle positions ($\mathbf{x}_\alpha$ and $\mathbf{x}_\beta$), we implement the Kirkwood formula [47] to determine both the average stress and per-particle stress. The average stress of the suspension is computed as

$$ \sigma = \frac{1}{V} \sum_{\beta}^{N} \sum_{\alpha>\beta}^{N} (\mathbf{f}_{\alpha\beta}^{e} + \mathbf{f}_{\alpha\beta}^{EHD}) (\mathbf{x}_{\alpha} - \mathbf{x}_{\beta}), \quad (1) $$

where $V$ is the volume of the simulation box. The per-particle stress is calculated by individually accounting for the contribution of each particle to the total stress as follows:
$$ \boldsymbol{\sigma}_p = \frac{N}{V} \sum_{\beta}^{N} (\mathbf{f}_{\alpha\beta}^{e} + \mathbf{f}_{\alpha\beta}^{\text{EHD}})(\mathbf{x}_{\alpha} - \mathbf{x}_{\beta}). \quad (2) $$

We can thus compute the stress components along the flow (x), gradient (y), and vorticity directions (z), from which we obtain the per-particle shear stress $\sigma_{yx}^{(p)}/E^*$, and the first and second normal stress differences, $N_1^{(p)}/E^* = (\sigma_{yy}^{(p)} - \sigma_{zz}^{(p)})/E^*$ and $N_2^{(p)}/E^* = (\sigma_{xx}^{(p)} - \sigma_{yy}^{(p)})/E^*$. Probability distribution functions are obtained by binning the values obtained from the calculations and creating histograms of $\sigma_{yx}^{(p)}/E^*$, $N_1^{(p)}/E^*$, and $N_2^{(p)}/E^*$. The probability distribution functions are analyzed in terms of their moments $\mu_i$, which give access to the mean value ($\mu_1$), standard deviation ($\mu_2^{1/2}$), and skewness ($S = \mu_3/\mu_2^{3/2}$).

The dynamic pair distribution function $g(\mathbf{r})$ in the flow gradient plane is used to connect the microstructure to the macroscopic properties. Note that $g(\mathbf{r})$ can be expressed as a series of spherical harmonics [48,49]:

$$ g(\mathbf{r}) = g_0(r) + \sum_{l=1}^{\infty} \sum_{-l}^{l} g_{l,m}(r) Y_{l,m}(\theta, \phi), \quad (3) $$

where $g_0(r)$ is the static pair distribution function; the functions $Y_{l,m}(\theta, \phi)$ are a set of orthogonal functions obtained from the solutions of the Laplace equation in spherical coordinates; and $g_{l,m}(r)$ are coefficients that can be calculated from $g(\mathbf{r})$ and the functions $Y_{l,m}(\theta, \phi)$. The coefficient $g_{2,-2}(r)$ gives access to the shear component of the stress tensor using the elastic force $f^e(r)$ between two particles:

$$ \sigma = -\rho^2 \sqrt{\pi/15} \int_0^{2R} r^3 f^e(r) g_{2,-2}(r) dr, \quad (4) $$

where $\rho = N/V$ is the number density of particles. It is computed as

$$ g_{2,-2}(r) = \frac{\int g(\mathbf{r}) Y_{2,-2}(\theta, \phi) \sin \phi d\theta d\phi}{\int Y_{2,-2}(\theta, \phi) Y_{2,-2}(\theta, \phi) \sin \phi d\theta d\phi}, \\ Y_{2,-2}(\theta, \phi) = \sqrt{\frac{15}{16\pi}} \sin 2\theta \sin^2 \phi. \quad (5) $$

### III. RESULTS

#### A. Oscillatory yielding of SPGs

Both the microgel glass studied experimentally and the model glass used in the simulations exhibit a yielding transition from solidlike to fluidlike behaviors under shear. To characterize this transition, we perform large amplitude oscillatory shear experiments (LAOS), keeping the frequency constant ($\omega = 1$ Hz for experiments; $\tilde{\omega} = 10^{-6}$ for simulations) and increasing the stress amplitude. Figure 1

<figure><img src="image_3.png" /><figcaption>FIG. 1. Yielding of the microgel glass (experiments; filled diamonds) and the model SPG (simulations; filled circles). The stress amplitude $\sigma_0$ and the strain amplitude $\gamma_0$ are scaled by the yield stress amplitude $\sigma_y$ and the yield strain amplitude $\gamma_y$, respectively. The inset shows the variations with the relative strain amplitude $\gamma_0/\gamma_y$ of the nondimensional storage (solid symbols) and loss modulus (open symbols) that are obtained experimentally (filled and empty diamonds) and numerically (filled and empty circles).</figcaption></figure>

shows the strain amplitude-stress amplitude curves measured experimentally and computed in the simulations when the stress amplitude is scaled by the yield stress and the strain amplitude by the yield strain. The variations of the viscoelastic moduli $G'$ and $G''$ are also represented in the inset. The raw data are shown in the Supplemental Material [44]. The observed behavior is characteristic of that found for soft glassy materials [42]. The stress amplitude $\sigma_0$ first increases linearly with the strain amplitude $\gamma_0$, signaling the linear response regime. The value of the storage modulus at low strain amplitude provides the low-frequency plateau modulus ($G_0$). Linearity breaks down at some point, above which $\sigma_0$ varies as a power law of $\gamma_0$ with a weak exponent, which characterizes the flow regime. The maximum in the loss modulus is associated with the transition from elastically dominated behavior to steady flow, revealing that yielding is gradual and takes place over an extended range of strain amplitudes [41,50]. For convenience, we use the intersection between the linear variation in the elastic response regime and the power-law variation in the flow regime as a practical indicator of yielding. In the following, we call it the yield point, which is characterized by the yield stress amplitude ($\sigma_y$) and yield strain amplitude ($\gamma_y$). These quantities are only weakly dependent on the oscillatory frequency (see Supplemental Material [44]). Note that $\sigma_y$ and $\gamma_y$ can also be determined from the steady-state flow curves, giving the variations of the stress with the applied shear rate in continuous shear flow (see Supplemental Material [44]).
<table><caption>TABLE I. Material properties extracted from small and large amplitude shear rheology (LAOS), and the flow curves data presented in the Supplemental Material [44]. Note that $E^*$ is the contact modulus of the elastic spheres of the model SPG used in simulations. The experimental value of $E^*$ for the experimental microgel glass has been estimated independently to be of the order of $10^4$ Pa [17,42].</caption><thead><tr><th></th><th>Experiments</th><th>Simulations</th></tr></thead><tbody><tr><td>Low frequency modulus</td><td>$G_0 = 435$ Pa</td><td>$G_0/E^* = 4.95 \times 10^{-2}$</td></tr><tr><td>Yield stress (LAOS)</td><td>$\sigma_y^{LAOS} = 31 \pm 2$ Pa</td><td>$\sigma_y^{LAOS}/E^* = (1.8 \pm 0.2) \times 10^{-3}$</td></tr><tr><td>Yield strain (LAOS)</td><td>$\gamma_y^{LAOS} = 0.071 \pm 0.005$</td><td>$\gamma_y^{LAOS} = 0.036 \pm 0.005$</td></tr><tr><td>Yield stress (flow)</td><td>$\sigma_y^{flow} = 32 \pm 2$ Pa</td><td>$\sigma_y^{flow}/E^* = (1.5 \pm 0.2) \times 10^{-3}$</td></tr><tr><td>Yield strain (flow)</td><td>$\gamma_y^{flow} = 0.075 \pm 0.005$</td><td>$\gamma_y^{flow} = 0.030 \pm 0.005$</td></tr><tr><td>Shear thinning exponent</td><td>$n = 0.42$</td><td>$n = 0.52$</td></tr></tbody></table>

The values of $G_0$, $\sigma_y$, and $\gamma_y(=\sigma_y/G_0)$ measured in the experiments and computed in the simulations are listed in Table I.

## B. Residual stress

Figures 2(a)–2(b) show that preshearing the microgel suspension prior to rheological measurements encodes a persisting mechanical memory that manifests either as a

time-dependent strain recovery at zero imposed stress or as a stress increase at constant imposed deformation. In the experiment, the microgel suspension is presheared at $\dot{\gamma} = 20 \text{ s}^{-1}$ for a time interval of 50 s. At $t = 0$, the preshear flow is stopped, the strain origin is set to zero by convention, and the applied stress is set to zero. We observe that the material is slowly pulled back on itself in the reverse direction, which indicates that it has developed a directional memory during the preshear. The effect is

<figure><img src="image_4.png" /><figcaption>FIG. 2. Strain recovery [(a),(a')], stress relaxation [(b),(b')], and start-up flow experiments [(c),(c')] after preshear flow (top) and optimal oscillatory annealing (bottom). Preshear is a continuous shear flow ($\dot{\gamma} = 20 \text{ s}^{-1}$) of duration 50 s (not shown). Annealing uses two sequences of stress-controlled oscillations ($\omega = 1$ Hz) applied for 30 s each, at stress amplitudes $\sigma = 100$ Pa (conditioning) and $\sigma_y = 31$ Pa (training). In panels (a) and (b), the stress is set to zero at the end of preshear, which marks the time origin; the strain is maintained constant after $t_1 = 80$ s. In panels (a') and (b'), the time origin is at the end of annealing; the stress is first set to 0, and at $t_2 = 20$ s, the strain is maintained constant. Note that $t_1 = 80$ s has been chosen as the sum of $t_2 = 20$ s and the duration of the annealing interval (60 s). The start-up flow experiments are performed by applying two opposite shear rates $\dot{\gamma}_0 = +1 \text{ s}^{-1}$ (blue curve) and $\dot{\gamma}_0 = -1 \text{ s}^{-1}$ (red curve) at $t_1$ and $t_2$, respectively. The material is a microgel suspension at concentration $C = 2$ wt% (see Table I).</figcaption></figure>
quantified in Fig. 2(a), where we use blue points to plot the time evolution of the strain during the recovery: The accumulated strain slowly decreases but does not reach a constant value over a period of time as long as $10^4$ s, as depicted by the thin line in Fig. 2(a). The recovery is always in the direction opposite to that of the applied preshear.

Since the total recoverable strain, which controls directional memory [28], cannot be reached in this experiment, we interrupt the recovery at $t_1 = 80$ s by stopping the mobile geometry of the rheometer. For $t > t_1$, the current deflection angle of the geometry is maintained constant, as indicated by the black data points in Fig. 2(a). The stress, which was set to zero during the recovery, starts increasing approximately as a logarithmic function of time, as shown by the black data points in Fig. 2(b). This experiment proves that setting the macroscopic stress to zero is not sufficient to erase directional memory. While in the equilibrated state the net forces and torques acting on each particle are negligible, the preshear flow creates long-lived structural distortions that provoke a directional imbalance of the local strains (when $\sigma = 0$ Pa) or stresses (when $\dot{\gamma} = 0$ s$^{-1}$).

### C. Memory upon shear flow inversion

The long-time directional memory encoded during preshear makes the rheological responses measured upon flow inversion nonsymmetric. This finding is probed by performing two start-up flow experiments, which consist in applying a constant shear rate of magnitude $|\dot{\gamma}_0| = 1$ s$^{-1}$ at time $t = t_1$, after the recovery that follows preshear, either in the direction of preshear (hereafter named the forward direction) or in the opposite direction (backward direction) [6,26,28,31]. Figure 2(c) shows that, in both cases, the absolute value of the stress increases from zero, passes through a maximum called the static yield stress, and slowly decreases to its steady-state values. The static yield stress represents the stress barrier that the material overcomes before reaching a steady flow. However, the stress growth functions for the forward and backward directions do not superimpose. In particular, the static yield stress is systematically larger for the forward direction. This observation recalls the Bauschinger effect that takes place during the loading and unloading of crystalline and glassy metals [31]. In the following, to quantify the asymmetry between the forward and backward start-up flow experiments, we introduce the asymmetry parameter $A$ as the area between the two curves over a fixed strain interval encompassing the entire transient responses ($10^{-2} \le \gamma \le 10^0$):

$$A = \int_{-2}^{0} [\sigma^{+}(\gamma, \dot{\gamma}_{0}) - |\sigma^{+}(\gamma, -\dot{\gamma}_{0})|]d(\log_{10}(\gamma)). \quad (6)$$

Experimentally, $A$ decreases very slowly with increasing waiting time $t_1$, yet it does not vanish for $t_1$ up to $10^4$ s, in line with previous reports [19]. This result motivates the

search for an efficient way to relax macroscopic residual stresses and erase the directional memory imprinted by the preshear flow.

### D. Annealing protocol

To anneal the residual stress and erase the directional memory, we have designed the mechanical protocol schematized in Fig. 3(a). It comprises three successive steps, namely, conditioning, training, and reading. Conditioning is intended to fully fluidize the material and put it in a controlled and reproducible initial state irrespective of its

<figure><img src="image_5.png" /><figcaption>FIG. 3. (a) Controlled-stress protocol used to anneal residual stresses. (b) Asymmetry parameters measured for the jammed microgel suspension (experiments) as a function of the amplitude of the training oscillation. (c) Asymmetry parameters (filled circles) and skewness parameters (empty circles) of the local stress distributions for the model glass (simulations).</figcaption></figure>
mechanical history. The role of training is to reorganize the microstructure in order to erase directional memory and restore symmetry upon flow reversal. The conditioning and training steps consist of two series of stress-controlled oscillations, of stress amplitude $\sigma_{0(C)}$ and $\sigma_{0(T)}$, respectively, at the same frequency $\omega$. The choice of controlling the stress instead of the strain is motivated by the lack of well-defined reference strain, with the recoverable strain not known a priori and the strain endlessly changing with time. The training sequence is designed such that the last oscillation ends when the instantaneous stress is zero; the stress is then maintained equal to zero for 20 s before the reading step. The reading step is used to assess the performance of the training operation. As above, it is achieved by performing two start-up flow experiments in the forward ($\dot{\gamma}_0$) and backward directions ($-\dot{\gamma}_0$) from which the asymmetry parameter A is computed.

The annealing protocol is now tested experimentally. The microgel glass is presheared at $\dot{\gamma} = 20 \text{ s}^{-1}$ as above, to imprint the mechanical history. Conditioning and training consist of 30 stress-controlled oscillations at constant frequency $\omega = 1 \text{ Hz}$. The conditioning stress amplitude is $\sigma_{0(T)} = 100 \text{ Pa}$. The training stress amplitude is varied between 100 Pa and 0.5 Pa. In Fig. 3(b), we plot the asymmetry parameter A, computed using expression (6), as a function of the reduced variable $\gamma_0/\gamma_y$, where $\gamma_0$ is the strain amplitude measured for the applied stress amplitude $\sigma_0$. We use this representation for clarity and for the sake of comparison with Fig. 1, despite the fact that the protocol is stress controlled. The asymmetry exhibits a deep and fairly broad minimum for training amplitudes around the yield value, i.e., $\gamma_{0(T)}/\gamma_y \approx 1$ or, equivalently, $\sigma_{0(T)}/\sigma_y \approx 1$. The value of the parameter A at the minimum is of the order of experimental uncertainty estimated from two identical experiments ($A \approx 0.4 \text{ Pa}$).

It is interesting to perform the same experiments as in Figs. 2(a) and 2(b) after the microgel suspension has been annealed using the training stress that minimizes the asymmetry parameter. Setting the stress to zero after the annealing sequence, we observe that the strain plateaus to a constant value as small as 0.01 after a short period of time, as shown in Fig. 2(a'). At time $t_2$, the strain is maintained constant and equal to its plateau value. In Fig. 2(b'), we find that the stress remains close to zero, showing that there is no net residual stress trapped inside the glass. In Fig. 2(c'), the start-up flow responses measured in the forward and backward directions are exactly superimposed, confirming that the minimum of the asymmetry parameter grants the best stress reversibility upon flow inversion. This result, obtained with a protocol that lasts about one minute, exceeds what was previously achieved by keeping the materials at rest for a long time [20]. We conclude that minimal asymmetry is the signature of the optimal training conditions.

To test the robustness of the protocol, we have repeated similar experiments where the frequency of the training

oscillation and the value of the shear rate used for reading are systematically varied. The variations of the asymmetry parameters are shown in the Supplemental Material [44]. A clear minimum is always observed when the stress amplitudes or the strain amplitudes are close to their respective yield values. We have also checked that the quality of annealing is independent of the mechanical history imprinted by the initial preshear (see Supplemental Material [44]).

To finish with this discussion, we have experimentally evaluated the quality of some variants of the annealing protocol, for instance, applying strain-controlled oscillatory deformations and omitting the conditioning step. The results, reported in the Supplemental Material [44], demonstrate that the asymmetry parameter consistently shows a well-pronounced minimum near the yield point in all cases. The minimum is slightly deeper and broader, i.e., annealing is better, when conditioning and training are combined in stress-controlled oscillation mode.

## IV. DISCUSSION

### A. Connection with local stress distributions

We now implement the simulation of the model SPG. The preshear flow is characterized by the nondimensional shear rate $\tilde{\gamma} = 10^{-6}$. For simplicity, we omit the conditioning step and train the SPG with a sequence of 20 oscillations at a constant frequency $\tilde{\omega} = 10^{-6}$ applied in strain-controlled mode (see Supplemental Material [44]). As above, reading consists of two start-up flow experiments in the forward and backward directions ($\tilde{\gamma} = 10^{-6}$), from which the asymmetry parameter is computed. The latter is plotted in its dimensionless form $A/E^*$ as a function of the nondimensional strain amplitude $\gamma_{0(T)}/\gamma_y$ in Fig. 3(c). The simulated curve has the same characteristic shape as the experimental one, with the asymmetry parameter exhibiting a clear minimum in the vicinity of the yield point, which indicates that the directional memory is minimized at this point. It is also interesting to observe that the values of the asymmetry parameter measured from experiments and simulations are in good agreement ($E^* \approx 10^4 \text{ Pa} [17,42]$).

The experimental results reported in Figs. 2(a) and 2(b) suggest the existence of a persisting strong asymmetry of residual stresses after preshearing. We now use the simulations to compute the shear stress at the particle scale and obtain a deeper understanding of the microscopic origin of directional memory in SPGs in relation to the stress distributions. Figure 4(a) shows that the particles just after the preshear flow carry local shear stresses $\sigma_{yx}^{(p)}$, which can be positive or negative and are widely distributed. This observation is quantified in Fig. 4(b) in terms of the probability density functions (PDF) $P(\sigma_{yx}^{(p)}/E^*)$ of the per-particle stress distribution. It shows that the PDF obtained after the glass has been presheared by a steady flow exhibits a strong asymmetry with an excess of particles carrying a large and positive stress. Such
<figure><img src="image_6.png" /><figcaption>FIG. 4. (a) Three-dimensional visualization of the simulation box showing local shear stress values on individual particles just after flow cessation ($\tilde{\gamma} = 10^{-6}$), with particles colored in red (blue) bearing positive (negative) stress; x, y, and z denote the flow, gradient, and vorticity directions, respectively. (b)–(d) Probability density functions of particle shear stresses after preshear flow ($\tilde{\gamma} = 10^{-6}$), training at $\gamma_{0(T)}/\gamma_y \approx 1$ and $\tilde{\omega} = 10^{-6}$, and initial equilibration following preparation. The blue lines indicate the mean values of the PDF; the red dotted lines represent the Gaussian functions built using the mean and standard deviation of the PDFs, which are indicated in the graphs.</figcaption></figure>

asymmetric distributions have been predicted by a mean-field elastoplastic model [51]. Excess positive stresses are the signature of microscopic distortions that store mechanical energy, which result in a nonzero residual stress during relaxation ($\tilde{\gamma} = 0$) or a nonzero residual strain during recovery ($\sigma = 0$). To be quantitative, we analyze the PDF in terms of its statistical moments. The mean value represents the macroscopic residual stress after preshear, and the skewness, which is nonzero, characterizes the excess of particles carrying a positive stress. Figure 4(c) represents the PDF after optimal annealing at $\gamma_{0(T)}/\gamma_y \approx 1$. The particles contributing to the positive (and to the smaller negative) tails of the distribution yield under the action of the oscillatory deformation; the distribution is now symmetric with negligible skewness and is well approximated by a Gaussian function. It is interesting to compare the PDF after training to that computed in the fully equilibrated state following the initial preparation of the glass. The latter is also symmetric and close to a Gaussian function, but it is broader, suggesting that another effect of training is to sharpen particle stress distributions.

In Fig. 3(c), we track the value of the skewness at the end of the annealing protocol as a function of the strain amplitude applied during the oscillatory training. Remarkably, the variations of the asymmetry parameter $A/E^*$ and of the skewness $S$ of the distribution of particle stresses, which are computed from simulations, can be mapped one on the other. Moreover, Figs. 3(b) and 3(c) are similar, demonstrating that the skewness of the per-particle shear stress distribution is well associated with directional memory at the microscopic level. After optimal annealing, the probability density function acquires a symmetric Gaussian shape centered around zero, expressing the absence of net residual stress and directional memory.

To investigate the tensorial nature of the stress tensor, we calculate the probability distribution functions of the

per-particle first and second normal stress differences, $N_1^{(p)}/E^*$ and $N_2^{(p)}/E^*$. They are reported in the Supplemental Material [44]. The mean values of the probability distributions, $N_1/E^*$ and $N_2/E^*$, indicate that the first normal stress differences are of opposite sign and at least 1 order of magnitude less than the shear stress with $N_2/E^* \approx -2N_1/E^*$, $N_1/E^* > 0$ [43]. Most importantly, skewness immediately after preshear is negligible, which shows that the initial preshear used to create the memory effect does not introduce any loss of isotropy in the planes perpendicular to the shear plane, which would be otherwise reflected in the normal stress differences. In conclusion, memory is essentially encoded in the probability distribution function of the shear components $\sigma_{yx}^{(p)}$.

## B. Aging

Upon annealing, the particle stress distribution becomes narrower with a smaller standard deviation [compare Figs. 4(c) and 4(d)], suggesting that prolonged training progressively reduces the amplitude of the stress fluctuations inside the SPG. We investigate this effect by training the glass with an increasing number of cycles, keeping constant the frequency ($\tilde{\omega} = 10^{-6}$) and the strain amplitude that corresponds to optimal annealing ($\gamma_{0(T)}/\gamma_y \approx 1$). The results are shown in Fig. 5(a). The standard deviation, plotted in the inset, decreases in a logarithmic way with the number of cycles, indicating that the stress distribution keeps changing very slowly after directional memory has been erased. The logarithmic form of the decay suggests that the slow evolution that is found corresponds to an aging process taking place under deformation [52]. In the main graph, we plot the variations of the transient stress computed during reading. The stress overshoot increases steadily with the number of cycles applied during training. This finding establishes a correlation between the extent of
<figure><img src="image_7.png" /><figcaption>FIG. 5. (a) Start-up flow simulations for the model SPG as the number of cycles applied during training, $N_c$, increases ($\bar{\omega} = 10^{-6}$ and $\gamma_{0(T)}/\gamma_y \simeq 1$). The inset shows the decrease in the standard deviation of the probability density function with increasing $N_c$. (b) Start-up flow experiments for the microgel glass with increasing $N_c$.</figcaption></figure>

the stress fluctuations and the barrier to overcome before macroscopic flow. In Fig. 5(b), the simulations are well supported by start-up flow experiments performed on the microgel glass after optimal annealing using an increasing number of cycles in the training step, $N_c$. The experimental results are in agreement with the simulations (with $E^* = 10^4$ Pa), with the stress overshoot increasing with the number of cycles applied during training.

### C. Significance of optimal training

We have shown that it is possible to prepare materials free of macroscopic residual stress and directional memory using periodic shear deformations. Optimal annealing is signaled by minima of both the asymmetry parameter and

the stress distribution skewness when the amplitude of the oscillatory training sequence is around the yield point. Note that the yield point is an experimental indicator of yielding, although the process is gradual, which may explain that the minima of the asymmetry and skewness span a range of amplitudes [41,50]. This condition for optimal training echoes the peculiar properties of the dynamics near the yield point. It has been established numerically, experimentally, and theoretically that the yield point in soft glasses is associated with a dynamical transition at the microscopic scale: At small strain or stress amplitudes, the displacements of the particles are localized, whereas around the yield point, irreversible displacements leading to shear-induced diffusion take place [17,35,53–57]. The vicinity of the yield point is also where cooperative relaxation leading to large-scale rearrangements is maximal [37,41,58,59]. When a material with an asymmetric local stress distribution is exposed to cyclic shear deformation near the yield point, particle configurations contributing to the tails of the stress distribution are driven beyond the local yield stress and relax their stress. If training is prolonged after directional memory has been erased, the tails of the distribution are slowly eroded in a symmetric way, which decreases the elastic energy stored in the material and slows down further rearrangements. This finding is in line with previous observations that the range of stress near the yield stress is connected to overaging [60]. By contrast, in the flow regime ($\gamma_{0(T)}/\gamma_y \gg 1$), oscillations induce irreversible rearrangements that enrich the tails of the local stress distribution and lead to large asymmetry parameters, revealing directional memory. In the linear response regime $\gamma_{0(T)}/\gamma_y \ll 1$, the asymmetry persists because the amplitude of the oscillations is too small to make particles contributing to the tails yield, leaving the local stress distributions nearly unchanged. In conclusion, applying periodic oscillations close to the yield point is an efficient tool to manipulate the local stress distributions.

The annealing protocol described in Fig. 3(a) involves a second condition for optimal annealing: Since yielding is a continuous intracycle process, the training oscillation must stop when the instantaneous stress is zero. This case is well demonstrated by considering the variations of the skewness over one cycle (see Supplemental Material [44]). For the optimal training condition, i.e., when the material is periodically sheared at the yield point, the skewness is zero—i.e., the stress distribution is symmetric, and there is no memory—only if the training stops when the instantaneous stress is zero. Otherwise, the material is left with significant residual stress and memory. At the macroscopic level, our protocol brings the system to minimal recoverable strain [41]. When the amplitude of the training oscillation is large, the skewness describes a loop, with the value at zero stress depending on the direction of the last oscillation, indicating that some memory is encoded.
### D. Consequence of mechanical annealing on the microstructure

Another interesting question concerns the effect of training on the microstructure of SPGs. In Fig. 6, we examine the pair distribution functions $g(r)$ of the model SPG computed in the flow-gradient plane before and after training. Figure 6(a) shows that the microstructure is distorted by the preshear flow, with an accumulation of particles in the compressive quadrant of the shear flow and a depletion of particles in the extension region. Note that the microstructure in the gradient-vorticity and flow-vorticity planes remains isotropic, in line with the properties of the first and second normal stress differences (see Supplemental Material [44]). The anisotropy of the pair distribution function is further reflected in the variations of the coefficient $g_{2,-2}(r)$ introduced earlier in Sec. II D. The variations of $g_{2,-2}(r)$ with the radial distance are shown in Fig. 6(b): It is nonzero and exhibits two well-defined extrema. The negative minimum at $r_{min}/R \approx 1.79$ expresses the accumulation of particles in the compression quadrant; the positive maximum at $r_{max}/R \approx 1.85$ is due to the depletion of particles in the extension quadrant. The radial distances at which the pair distribution function reaches its extrema, $r_{min}$ and $r_{max}$, are lower than the particle diameter as a consequence of the deformability and compression of the particles. Note that $r_{min}$ and $r_{max}$ are different due to anisotropic cage deformation. The accumulation-depletion phenomenon has been shown to be a

<figure><img src="image_8.png" /><figcaption>FIG. 6. Dynamic pair distribution functions and coefficients $g_{2,-2}(r)$ computed numerically just after preshearing at $\tilde{\gamma} = 10^{-6}$ [(a),(b)], training at $\gamma_{0(T)}/\gamma_{\gamma} \approx 1$ and $\tilde{\omega} = 10^{-6}$ [(c); black line in (b')], and equilibration following the initial preparation [(d); red line in (b')]. The dashed lines in panel (a) indicate the extension ($\theta = \pi/4$) and compression ($\theta = 3\pi/4$) directions.</figcaption></figure>

generic property of concentrated suspensions under shear [7,8,20,21,43,46,49,61,62]. However, after optimal training, there is no longer accumulation depletion, and the pair distribution function is isotropic with an equal probability of finding neighboring particles in all directions [Fig. 6(c)] and $g_{2,-2}(r) \approx 0$ within the statistical fluctuations [Fig. 6(b')]. Interestingly, the trained microstructure is similar to the fully equilibrated microstructure built after preparation [Figs. 6(d) and 6(b)'.]. In conclusion, we have shown that the training procedure, when it is optimal, restores the isotropy of the microstructure expected for a mechanically equilibrated glass.

### E. Extension to other materials

An important question concerns the generality of our findings. In Fig. 7, we focus on a colloidal and a polymer gel, which are both thixotropic. Their rheological behavior

<figure><img src="image_9.png" /><figcaption>FIG. 7. Start-up flow experiments in the forward direction (blue) and backward direction (red) for the fumed silica gel (a) and xanthan gel (b). The main graphs show the results for optimal annealing; the results for the untrained presheared materials are presented in the insets.</figcaption></figure>
is particularly complex [27], and a complete investigation of their flow and yielding properties is beyond the scope of this work. However, in the following, we generalize the protocol proposed for SPGs and show that it can be successfully applied to erase the residual stresses accumulated in gels during shearing and overcome the strong memory effects that are associated with them. Figure 7(a) shows the results obtained for the colloidal silica gel. Figure 7(b) shows those obtained for the xanthan polymer gel. When the samples are prepared using a traditional preshear protocol at a constant rate, we observe tremendous directional memory as revealed by stress-growth experiments performed in the forward and backward directions after flow cessation, as shown in the insets of Figs. 7(a) and 7(b). In fact, the silica and xanthan gels have even more tendency to develop long-lived residual stresses. Since the yield point is not well defined and time dependent, training is performed using a reverse amplitude sweep in stress-controlled mode. After annealing, the symmetry upon flow reversal is perfectly restored, as shown in the main panel of Figs. 7(a) and 7(b). This result supports the general framework proposed in this paper, based on the control of the distribution of local stresses, irrespective of the microscopic details of the sample, and opens the way to further investigations aimed at assessing the minimal requirements for the successful applicability of the proposed experimental protocol.

## V. CONCLUSION

Our results demonstrate that residual stress, directional memory, and aging in athermal soft glasses take their origin in the stress distributions imprinted at the particle scale during the mechanical history of the material. Subjecting a soft glass to a macroscopic shear deformation results in a distribution of local stresses, with nonzero mean, large variance due to large stress fluctuations, and nonzero skewness. When the shear deformation is stopped, the microscopic stress distribution remains trapped because the relaxation of particle configurations involves co-operativeness, frustration, and metastability. The memory of the material after flow cessation is embedded in the properties of the local stress distributions: The mean particle stress represents the residual stress stored inside the material, the asymmetry that can be quantified by the skewness characterizes directional memory, and the standard deviation is associated with aging. This conceptual framework complements and extends mean-field interpretations based on the distortions of the average microstructure.

We show that oscillatory training at a prescribed stress amplitude induces a rapid evolution of this stress distribution, dissipating residual stresses, efficiently erasing directional memory, and accelerating aging dynamics. The optimal training point corresponds to the reversibility-irreversibility transition that is associated with yielding. In our experiments, it is signaled by a minimum of the

asymmetry parameter around the yield point. However, the shape of the minimum and the quality of annealing depend on experimental parameters like the oscillatory mode. Training with stress-controlled cyclic deformations after a conditioning step constitutes the fastest and most efficient annealing conditions with little sensitivity to the training stress. Detailed investigations have been carried out and will be reported elsewhere.

Our protocol efficiently erases residual stresses and memory, corresponding to the full relaxation of the recoverable strain and stress [28]. However, the material keeps evolving, by undergoing aging, indicating the interest in a description beyond average quantities such as stress, recoverable, and unrecoverable strain. This is possible if we consider the local stress distributions and their time evolution after equilibration of the macroscopic residual stresses and strains. This interpretation prompts the microscopic investigations of shear-induced nonaffine rearrangements at the particle scale [58]. The evolution of probability distribution functions under flow is also a promising tool to predict the rheological response to steady and transient flows [63]. Finally, our protocol can be applied to many different amorphous materials with various microstructures, which have the capacity to store local stresses and strains, opening the route to the preparation and processing of materials free of directional memory.

## ACKNOWLEDGMENTS

We greatly acknowledge Jean-Marc Suau from Arkema for the synthesis of the microgels used in this study. P. E. is grateful to Evonik and Farmalabor for donating the fumed silica and xanthan polymer used to prepare the colloidal gel and polymer gel. P. E. and M. C. thank Anton Paar for technical support and advice in the development of the experimental protocols.

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