Study of arbitrarily low shear rate rheology using dissipative particle dynamics

TL;DR

This work extends the transient-time correlation function (TTCF) approach to dissipative particle dynamics (DPD) to achieve high-precision viscosity measurements at arbitrarily low shear rates, addressing the limitations of direct averaging at high \\dot{\\gamma}. It shows that mappings, which enforce zero initial dissipation for LJ, do not reliably translate to DPD due to the dissipative force, requiring a corrected TTCF formulation and, in some cases, unmapped trajectories with bootstrap error estimation. The results demonstrate that TTCF delivers lower and stable errors across shear rates for DPD compared with direct averaging, revealing Newtonian behavior for the simple DPD fluid and enabling more realistic rheological studies of complex fluids. The methods pave the way for recovering conversion factors from system characteristics and applying low-shear TTCF to structured fluids, with practical implications for linking mesoscopic simulations to experimental rheology.

Abstract

The use of dissipative particle dynamics (DPD) simulation to study the rheology of fluids under shear has always been of great interest to the research community. Despite being a powerful tool, a limitation of DPD is the need to use high shear rates to obtain viscosity results with a sufficiently high signal-to-noise ratio (SNR). This often leads to simulations with unrealistically large deformations that do not reflect typical stress conditions on the fluid. In this work, the transient time correlation function (TTCF) technique is used for a simple Newtonian DPD fluid to achieve high SNR results even at arbitrarily low shear rates. The applicability of the TTCF on DPD systems is assessed, and the modifications required by the nature of the DPD force field are discussed. The results showed that the standard error (SE) of viscosity values obtained with TTCF is consistently lower than that of the classic averaging procedure across all tested shear rates. Moreover, the SE resulted proportional to the shear rate, leading to a constant SNR that does not decrease at lower shear rates. Additionally, the effect of trajectory mapping on DPD is studied, and a TTCF approach that does not require mappings is consolidated. Remarkably, the absence of mappings has not reduced the precision of the method compared with the more common mapped approach.

Figures (9)

• Figure 1: Values of $\lvert \langle P_{yx}(0)\rangle\rvert$ for different models using mappings. The yellow circles refer to the LJ-WCA fluid, the purple triangles to a standard DPD model, the light blue square to a DPD model with only conservative force, the green pentagons to a DPD model without dissipative force and with a constant number $\xi$ for the random force, and the red diamonds to a DPD model with the three standard forces but a constant random number $\xi$. $\langle P_{yx}(0) \rangle$ can assume negative values, so its absolute value is plotted to use a logarithmic scale.
• Figure 2: Time response of $P_{yx}$ for a LJ-WCA fluid with an applied shear rate of $\dot{\gamma} = 10^{-6}\ (\text{--})$. Four mappings are used, the error bars are equal to the standard error.
• Figure 3: Time response of $P_{yx}$ for a LJ-WCA fluid with an applied shear rate of $\dot{\gamma} = 10^{-6}$ (--). Mappings are not used, and the correction in \ref{['eq:pyx_sllod_ttcf_modification']} is adopted. a) The value of $\langle P_{yx}(0) \rangle$ is calculated as an ensemble average. b) The value of $\langle P_{yx}(0) \rangle$ is imposed equal to zero for the TTCF formula.
• Figure 4: Comparison of the standard error of $P_{yx}$ obtained with and without the mappings using the TTCF on $4\times 10^4$ daughter trajectories. The shear rate applied on the LJ-WCA fluid is $\dot{\gamma} = 10^{-6}$ (--).
• Figure 5: Results obtained with the TTCF method on a DPD simple fluid: a) time evolution of the shear pressure $P_{yx}$ divided by the shear rate $\dot{\gamma}$ for different shear rates; b) value for different shear rates at the last timestep of the simulation. The error bars represent the 95% confidence interval for the mean value. All values are expressed in reduced DPD units.