GPU-Accelerated Large-Scale Molecular Simulation Toolkit
Station: Research

Dissipative particle dynamics (DPD) method is widely used to investigate the self-assembly of soft matters and diffusion behaviors of fluids. DPD simulations were used by us in the study on how to design nanopatterns with block copolymers[1-2], the fabrication of diblock copolymer patchy particles[3], the formation of phenylboronic acid-functionalised nanoparticles for emodin delivery[4], the self-assembly of nanoparticles based on rigid body model[5-7], and the effect of side chains on the self-assembly of 'rod-coil' copolymers[8]. The DPD method has been proven to be an effective mesoscopic simulation tool to study fluid events occurring on millisecond time scales and micrometer length scales via tracking the motion of coarse-grained particles (composed of a group of atoms or molecules). The fundamental equation in the DPD method is Newton's equation of motion. For a particle $i$, each DPD bead is subjected to three types of forces: conservative, dissipative, and random. Newton's equation of motion for the particle $i$ is given by
\begin{equation}
{{m}_{i}}\frac{d{{v}_{i}}}{dt}={{\overrightarrow{F}}_{i}}=\sum\limits_{j}{{{\overrightarrow{F}}_{ij}}^{C}+{{\overrightarrow{F}}_{ij}}^{D}+{{\overrightarrow{F}}_{ij}}^{R}},
\end{equation}
where $F^C$ is the conservative force, $F^R$  is the pairwise random force, and $F^D$  is the dissipative force. The force acting on a particle is summed over all the inter-bead forces between particles $i$ and $j$. The conservative force is weakly repulsive and is given by
\begin{equation}
{{\overrightarrow{F}}_{ij}}^{C}=-\nabla V({{r}_{ij}})=\left\{\begin{array}{ll}
{{\alpha }_{ij}}(1-\frac{{{r}_{ij}}}{{{r}_{c}}}){{\overrightarrow{e}}_{ij}}&,{{r}_{ij}}\le {{r}_{c}}\\
0&,{{r}_{ij}}>{{r}_{c}}
\end{array} \right.\,,
\end{equation}
where $r_{ij}$ is the distance between particles $i$ and $j$, ${{r}_{ij}}=|{{\overrightarrow{r}}_{ij}}|=|{{\overrightarrow{r}}_{i}}-{{\overrightarrow{r}}_{j}}|,{{\overrightarrow{e}}_{ij}}={{\overrightarrow{r}}_{ij}}/{{r}_{ij}}$. Here, $\alpha_{ij}$ is a parameter to determine the magnitude of the repulsive force between particles $i$ and $j$, and $r_c$ is the cutoff distance. Random force ($F_{ij}^R$) and dissipative force ($F_{ij}^D$) are given by
\begin{eqnarray}
{{\overrightarrow{F}}_{ij}}^{D}&=&-\gamma {{\omega }^{D}}({{r}_{ij}})({{\overrightarrow{e}}_{ij}}\cdot {{\overrightarrow{v}}_{ij}}){{\overrightarrow{e}}_{ij}},{{r}_{ij}}\le {{r}_{c}} \\
{{\overrightarrow{F}}_{ij}}^{R}&=&\sigma {{\omega }^{R}}({{r}_{ij}}){{\xi}_{ij}}{{\overrightarrow{e}}_{ij}},{{r}_{ij}}\le {{r}_{c}}
\end{eqnarray}
where $\overrightarrow{v}_{ij}$ = $\overrightarrow{v}_j$ - $\overrightarrow{v}_i$, $\sigma$ is the noise parameter, $\gamma$ is the friction parameter, and ${\xi}_{ij}$ is the random number based on the Gaussian distribution. Here ${\omega }^{R}$ and ${\omega }^{D}$ are r-dependent weight functions, which are given by
\begin{eqnarray}
{{\omega }^{D}}({{r}_{ij}})={{\left[ {{\omega }^{R}}({{r}_{ij}}) \right]}^{2}}=\left\{\begin{array}{ll}
{{(1-\frac{{{r}_{ij}}}{{{r}_{c}}})}^{2}}&,{{r}_{ij}}\le {{r}_{c}}\\
0&,{{r}_{ij}}>{{r}_{c}}
\end{array} \right.\,,
\end{eqnarray}
The temperature is controlled by a combination of dissipative and random forces. The noise parameter $\sigma$ and friction parameter $\gamma$ are connected to each other by the fluctuation-dissipation theorem in the following equation:
\begin{eqnarray}
{{\sigma }^{2}}=2\gamma {{k}_{B}}T
\end{eqnarray}
where $k_B$ is the Boltzmann constant and $T$ is the temperature.
The polymer chain has beads connected with harmonic springs. The spring force($F_{ij}^S$) is given by
\begin{eqnarray}
{{\overrightarrow{F}}_{ij}}^{S}=-k({{r}_{ij}}-{{r}_{0}}){{\overrightarrow{e}}_{ij}}
\end{eqnarray}
where $k$ is the spring constant and $r_0$ is the equilibrium bond distance.

References

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