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Shear-stress fluctuations in self-assembled transient elastic networks

Focusing on shear-stress fluctuations we investigate numerically a simple generic model for self-assembled transient networks formed by repulsive beads reversibly bridged by ideal springs. With $Δdt$ being the sampling time and $t_*(f) \sim 1/f$ the Maxwell relaxation time (set by the spring recombination frequency $f$) the dimensionless parameter $Δx = dt/t_*(f)$ is systematically scanned from the liquid limit ($Δdx \gg 1)$ to the solid limit ($Δx \ll 1$) where the network topology is quenched and an ensemble average over $m$ independent configurations is required. Generalizing previous work on permanent networks it is shown that the shear-stress relaxation modulus $G(t)$ may be efficiently determined for all $Δx$ using the simple-average expression $G(t) = μ_A - h(t)$ with $μ_A = G(0)$ characterizing the canonical-affine shear transformation of the system at $t=0$ and $h(t)$ the (rescaled) mean-square displacement of the instantaneous shear stress as a function of time $t$. This relation is compared to the standard expression $G(t) = C(t)$ using the (rescaled) shear-stress autocorrelation function $C(t)$. Lower bounds for the $m$ configurations required by both relations are given.