Measuring spatial distribution of the local elastic modulus in glasses

Glasses exhibit spatially inhomogeneous elastic properties, which can be investigated by measuring their elastic moduli at a local scale. Various methods to evaluate the local elastic modulus have been proposed in the literature. A first possibility is to measure the local stress-local strain curve and to obtain the local elastic modulus from the slope of the curve or, equivalently, to use a local fluctuation formula. Another possible route is to assume an affine strain and to use the applied global strain instead of the local strain for the calculation of the local modulus. Most recently, a third technique has been introduced, which is easy to be implemented and has the advantage of low computational cost. In this contribution, we compare these three approaches by using the same model glass and reveal the differences among them caused by the nonaffine deformations.

Figure 1

Schematic illustration of the simple shear deformation: (a) affine strain approach and (b) frozen matrix approach. The cubic box drawn by the thick (yellow) lines indicates the local cube $m$. In the affine strain approach (a), all the particles (red particles) are allowed to move nonaffinely. In the frozen matrix approach (b), only the particles in the local cube $m$ (red particles) can move freely, whereas the particles in the other frozen part (blue particles) are restricted to only move affinely.

Figure 2

(a) Pressure and (b) potential energy per particle versus strain $3{\epsilon }_b$, under the quasistatic bulk deformation ${\mathit{\epsilon }}_b$. We show two deformation curves: affine deformation with relaxation (solid curve) and affine deformation without relaxation (dashed curve). The macroscopic bulk modulus obtained from the slope of the stress-strain curve is $K=59.7$ (affine+relaxation) and $K^B=60.2$ (affine). We also obtained the same values of $K$ and $K^B$ from a quadratic fit of the potential energy.

Figure 3

(a) Shear stress and (b) potential energy per particle versus strain $2{\epsilon }_s$ under the quasistatic shear deformation ${\mathit{\epsilon }}_{s3}$. We show two deformation curves: affine deformation with relaxation (solid curve) and affine deformation without relaxation (dashed curve). The macroscopic shear modulus obtained from the slope of the stress-strain curve is $G=14.9$ (affine $+$ relaxation) and $G^B=36.5$ (affine). We also obtained the same values of $G$ and $G^B$ from the quadratic fitting of the potential energy. Note that we obtained five values of $G\simeq 15$ and $G^B\simeq 36$ from the five shear deformations ${\mathit{\epsilon }}_{s1}$, ${\mathit{\epsilon }}_{s2}$, ${\mathit{\epsilon }}_{s3}$, ${\mathit{\epsilon }}_{s4}$, and ${\mathit{\epsilon }}_{s5}$. The values of $G=14.9$ and $G^B=36.5$ are the averages over these values.

Figure 4

Distributions of bulk modulus and shear modulus for different cube sizes $W=3.16$, $5.27$, and $7.90$ calculated by the three approaches described in the text: fully local approach [(a) and (b)], affine strain approach [(c) and (d)], and frozen matrix approach [(e) and (f)]. The shear modulus distribution $P\left(G^m\right)$ is obtained by averaging the five distributions $P\left(G_1^m\right)$, $P\left(G_2^m\right)$, $P\left(G_3^m\right)$, $P\left(G_4^m\right)$, and $P\left(G_5^m\right)$. We also show the Gaussian distribution functions fitted to each distribution (solid lines). The vertical solid lines indicate the values of the macroscopic moduli obtained from Figs. 2 and 3, i.e., $K=59.7$ and $G=14.9$. In (a)–(e), the average value is independent of the cube size $W$ and coincides with the macroscopic value, whereas in (f), the average value varies with $W$ and seems to tend to the macroscopic value with increasing $W$.

Figure 5

Comparisons of the probability distributions for (a) bulk modulus and (b) shear modulus for the three approaches: fully local approach (open circle), affine strain approach (square), and frozen matrix approach (triangle). The local cube size is $W=3.16$. The distribution of the Born term, calculated by the fluctuation formula Eq. (8), is also plotted for comparison (closed circle). The vertical lines indicate the macroscopic values, $K=59.7$ (solid line) and $K^B=60.2$ (dashed line) in (a), and $G=14.9$ (solid line) and $G^B=36.5$ (dashed line) in (b).

Figure 6

Comparison of average and standard deviation of the distributions calculated by the three approaches, and of the Born term: fully local approach (open circle), affine strain approach (square), frozen matrix approach (triangle), and the Born term (closed circle). The averages are shown in (a) and (b), and the standard deviations are presented in (c) and (d). The horizontal lines indicate the macroscopic values, $K=59.7$ (solid line), and $K^B=60.2$ (dashed line) in (a), and $G=14.9$ (solid line) and $G^B=36.5$ (dashed line) in (b).

Figure 7

$xy$ layers taken from the spatial distributions of the bulk modulus ${\widehat{K}}^m$ and the shear moduli ${\widehat{G}}_1^m$, ${\widehat{G}}_3^m$ obtained by three approaches: fully local approach [(a)–(c)], affine strain approach [(d)–(f)], and frozen matrix approach [(g)–(i)]. The value of the local modulus is normalized by its average and standard deviation, i.e., ${\widehat{X}}^m=(X^m-X_{\text{ave}})/X_{\text{std}}(X\in K,G_1,G_3)$ [see Eq. (13)]. We also show the spatial maps of the Born terms [(j)–(l)], for comparison. $X$ and $Y$ coordinates are presented in units of the box length $L=15.8\sigma$. The $Z$ coordinates of all the layers correspond to $L/2$. Note that no average over the configurations ensemble is involved, data refer to one arbitrary system's instance only.

Figure 8

Comparison of the distributions of (a) bulk modulus and (b) shear modulus for two systems with $N=4000$ (open circles) and $N=32000$ (closed circles) particles. The distributions were calculated by the fluctuation formula Eq. (8) (i.e., fully local approach). The local cube size is $W=3.16$.