Computing finite--temperature elastic constants with noise cancellation

Abstract

Elastic constants are central material properties, frequently reported in experimental and theoretical studies. While their computation is straightforward in the absence of thermal fluctuations, finite--temperature methods often suffer from poor signal--to--noise ratios or the presence of strong anharmonic effects. Here, we show how to compute elastic constants in thermal ordered and disordered systems by generalizing a noise--cancellation method originally developed for piezoelectric coupling coefficients. A slight strain is applied to an equilibrated solid. Simulations of both the strained and unstrained (or oppositely strained) reference systems are performed using identical thermostatting schemes. As demonstrated theoretically and with generic one--dimensional models, this allows stress differences to be evaluated and elastic constants to be determined with much reduced thermal noise. We then apply this approach across a diverse set of systems, spanning crystalline argon, ordered silicon as well as amorphous silicon, poly(methyl methacrylate), and cellulose derivatives.

Figures (8)

• Figure 1: Relaxation of the modulus $E(t)$ in a linear, ideally harmonic chain composed of three diatomic molecules. Three cases are shown: a heterogeneous (hetero) chain with alternating stiff, short bonds and soft, long bonds at finite temperature (black circles) and at zero temperature (blue stars), and a homogeneous (homog) chain with only one bond type. In all cases, the static modulus is $E_\infty = 1$. The inset displays the relative error of the estimate. Solid lines indicate fits using a single-exponential relaxation model, $E(t) = E_\infty + \Delta E \exp(-t/\tau)$.
• Figure 2: Relaxation of the elastic--tensor estimator for a linear Lennard--Jones chain using the noise--cancellation technique with $\varepsilon = 10^{-3}$, shown for two relaxation times: $\tau = 0.109$ (large damping) and $\tau = 0.436$ (small damping). Averages were taken over 200,000 samples. Instantaneous error bars (std) have been reduced by a factor of 100 for clarity.
• Figure 3: Instantaneous standard deviation for data like that shown in Figure \ref{['fig:LJ_relax']}. In the calculation using uncorrelated random numbers for the follow--up runs, a strain of $\varepsilon = 10^{-4}$ was imposed and large damping $\tau = 0.109$ was used. The dashed line shows a $\sqrt{t}$ power law.
• Figure 4: Monomer structures of poly(methyl methacrylate) (part a), cellulose (part b), and cellulose acetate (part c) ivestigated in this study.
• Figure 5: Time evolution of the tetragonal shear--modulus estimator defined in Equation \ref{['eq:shear_modulus_estimator_1']} for crystalline argon at a temperature $T = 10$ K. Different integration schemes are used: microcanonical (blue, dotted line), canonical--sampling by velocity rescaling (CSVR, red, dashed lines), and Langevin (full, black line) using the Grønbech--Jensen algorithm.