Measurable geometric indicators of local plasticity in glasses

TL;DR

This work tackles the challenge of quantifying local plasticity in glasses without a crystalline reference by introducing two geometry-based indicators derived from local distortions: the continuous Burgers vector $\mathbf{b}$ and the quadrupolar strain $Q_{ij}$. Using both simulations of a bead-spring glass under simple shear and diffraction-based measurements in a colloidal glass, the authors show that $\mathbf{b}$ robustly marks coordinated slips and concentrates at plastic events, while $Q_{ij}$ reveals deviatoric distortions with more limited, context-dependent usefulness. A key insight is that $\mathbf{b}$ can be inferred from observable distortions even when the initial configuration is unknown, enabling diffraction-based strain tomography in disordered materials and linking microscopic geometry to plasticity. The framework provides a pathway to engineer more stable amorphous materials by connecting local structural symmetry, non-affine motion, and defect-like distortions to macroscopic mechanical response.

Abstract

The notion of defects in crystalline phases of matter has been extremely powerful for understanding crystal growth, deformation and melting. Many of these discontinuities in the periodic order of crystals are well described by the Burgers vector, derived from the particle displacements, which encapsulates the direction and magnitude of slip relative to the undeformed state. Since the reference structure of the crystal is known a priori, the Burgers vector can be determined experimentally using both imaging and diffraction methods to measure the final lattice distortion, and thus infer the particle displacements. Glasses have structures that lack the periodicity of crystals, and thus a well-defined reference state. Yet, measurable structural parameters can still be obtained from diffraction from a glass. Here we examine the usefulness of these parameters to probe deformation in glasses. We find that co-ordinated transformations in the centrosymmetry of local particle arrangements are a strong marker of plastic events. For a glass, determining the local distortions corresponding to these plastic events requires measurements before and after deformation. We investigate two geometric indicators that can be derived from these distortions, namely the continuous Burgers vector and the quadrupolar strain. We find that the Burgers vector again emerges as a robust and sensitive metric for understanding local structural transformations due to mechanical deformation, even in disordered glasses.

Figures (13)

• Figure 1: Scanning micro-beam x-ray scattering (A) A thin colloidal glass is scanned under a micro-x-ray beam defined by a near-field aperture. Many local structural parameters can be measured from the diffraction patterns such as (B) degree of centrosymmetry in the plane of the specimen from examining angular correlations at angular separations of $\Delta$ (C) strain or distortion (D) stability from cross-correlations at different times, separated by $\Delta t$.
• Figure 2: Model glass undergoing simple shear. (A) Stress-strain curve (pink line) for the range of strain values studied $0.00 \leq \gamma \leq 0.25$. Green arrows show the values of strain in the elastic and plastic regimes for the configurations studied in Figure \ref{['fig2']}. Green and purple lines overlaid show the average magnitudes of the displacements ($\sqrt{u_{x}^2+u_{z}^2}$) and centrosymmetry ($F_{IS}$) from slices in the x-z plane, respectively. (B) Left - Particle positions in a slice of the model before (green) and after (pink) a strain increment is applied that induce distortion of nearest-neighbor configurations illustrated - Right.
• Figure 3: In-plane ($\sqrt{u_{x}^2+u_{z}^2}$) and out-of-plane ($\sqrt{u_{y}^2}$) non-affine displacement magnitudes, Burgers vector magnitude ($\sqrt{b_{x}^2+b_{z}^2}$), quadrupolar magnitude ($\sqrt{Q_{xx}^2+Q_{xz}^2}$) and change in local structural centrosymmetry ($\Delta F_{IS}$) for a x-z slice of the glass simulation at (A) $\gamma = 0.025$ (B) $\gamma = 0.093$ (C) $\gamma = 0.108$ (see green arrows overlaid on the stress-strain curve of Figure \ref{['fig1']} (A)). Histograms of the mapped parameters are shown below each map. Note the order of magnitude increase in all the parameter values at the system-spanning plastic event shown in (C).
• Figure 4: (A) Average values and standard error of localized parameters from simulation (centrosymmetry $F_{IS}$, displacement magnitude $\sqrt{u_{x}^2+u_{z}^2}$, Burgers magnitude $\sqrt{b_{x}^2+b_{z}^2}$ and quadrupole magnitude $\sqrt{Q_{xx}^2+Q_{xz}^2}$) that correspond to the different quartiles in the value of the centrosymmetry in the configuration prior to the application of a strain increment. These values were averaged over the whole simulation volume (sliced into twenty x-z slices) and all strain increments. (B) Average values and standard error of localized parameters from experiment (centrosymmetry $\sum c^{2n+2}\slash \sum c^{2n+1}$, stability $C(\Delta t)$ , Burgers magnitude $\sqrt{b_{x}^2+b_{z}^2}$ and quadrupole magnitude $\sqrt{Q_{xx}^2+Q_{xz}^2}$) that correspond to the different quartiles in the value of the centrosymmetry prior to the application of strain. Values were determined from raw experimental maps of each parameter (Methods \ref{['expt']}). The magnitude of the local centrosymmetry in different quartiles (shown to the left of the black lines in (A) and (B)) predicts the values of the other parameters.
• Figure 5: Visualizations of the local structural centrosymmetry (green - as an isosurface with a value of $F_{IS} = 0.75$) at $\gamma - \delta \gamma$ and the non-affine in-plane displacement field (pink) at $\gamma$ for (A) $\gamma = 0.025$ (B) $\gamma = 0.093$ (C) $\gamma = 0.108$ (see green arrows overlaid on the stress-strain curve of Figure 1 from the main text). These parameters are displayed as a volume and also as a slice from the middle of the volume. The local structural centrosymmetry forms a stabilizing network that is complementary to the magnitude of the non-affine displacements. Note the order of magnitude increase in the non-affine displacements in (C). Settings for the visualizations are detailed in Methods \ref{['calcparam']}.