\section{Dissipative Particle Dynamics}

Dissipative Particle Dynamics (DPD) method is a stochastic particle simulation method particularly useful for the model of soft matter systems such as polymer melt and solution. Detailed description of the implementation of the DPD functionality within \USERMESO can be found in Ref.~\cite{tang2013gpudpd}.

\subsection{The DPD formulation} \label{sec:dpd_formulation}

In DPD, the force $\textbf{f}$ acting on each particle consists of three pairwise additive parts: a conservative one, a dissipative one and a random one, \textit{i.e.} \begin{align} \textbf{f}_i = \sum_{i\neq j} \textbf{F}_{ij} = \sum_{i\neq j} ( \textbf{F}_{ij}^C + \textbf{F}_{ij}^D + \textbf{F}_{ij}^R ) \end{align} given by \begin{align} \textbf{F}_{ij}^C &= a_{ij} w_C(\textbf{r}_{ij}) \textbf{e}_{ij} \\ \textbf{F}_{ij}^D &= -\gamma_{ij} w_D(\textbf{r}_{ij}) (\textbf{e}_{ij} \cdot \textbf{v}_{ij}) \textbf{e}_{ij} \\ \textbf{F}_{ij}^R &= \sigma_{ij} w_R(\textbf{r}_{ij}) \xi_{ij} {\delta t}^{-\frac{1}{2}} \textbf{e}_{ij} \end{align} if $ |\textbf{r}_{ij}| \leq r_c $, where $r_c$ is the cutoff distance, and \begin{align} \textbf{F}_{ij}^C = \textbf{F}_{ij}^D &= \textbf{F}_{ij}^R = 0,\;\;\; |\textbf{r}_{ij}| > r_c. \end{align} Between the two weight functions $w_D$ (dissipative) and $w_R$ (random), one of them can be chosen arbitrarily, while the other is then fixed as dictated by the fluctuation-dissipation theorem \cite{espanol1995dpd}. \begin{align} w_D(\textbf{r}_{ij}) = w_R^2(\textbf{r}_{ij}) \end{align} In \USERMESO, $w_R$ (random) is assumed to be a power of $w_C$ (conservative): \begin{align} w_R(\textbf{r}_{ij}) = w_C^s(\textbf{r}_{ij}) = (1-\frac{|\textbf{r}_{ij}|}{r_c})^s \end{align} The choice of exponent $s$ is somewhat arbitrary and can be conveniently tuned for reproducing dynamical properties such as viscosity and diffusivity. The coefficients $\sigma_{ij}$ and $\gamma_{ij}$ are also related to each other by \begin{align} \sigma_{ij}^2 = 2 \gamma_{ij} k_B T \end{align} as also dictated by the fluctuation-dissipation theorem.

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