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Parameters in DPD (Dissipative P

Parameters in DPD (Dissipative P

作者: Yawei551 | 来源:发表于2020-01-15 14:30 被阅读0次

by Yawei Liu @Sydney, Australia 2020/01/16

DPD model

In the DPD model, one DPD bead represents N_m fluid (e.g. water) molecules. Here, N_m is also called coarse-graining (CG) degree. The DPD beads interact with each other via a conservative force \textbf{F}_{ij}^C, a dissipative force \textbf{F}_{ij}^D and a random force \textbf{F}_{ij}^R given by
\begin{eqnarray} \textbf{F}_{ij}^C &=& A (1-r_{ij}/r_c) \textbf{e}_{ij} && r_{ij}<r_c \\ \textbf{F}_{ij}^D &=& -\gamma(1-r_{ij}/r_c)^2 (\textbf{e}_{ij} \cdot \textbf{v}_{ij}) \textbf{e}_{ij} &&r_{ij}<r_c \\ \textbf{F}_{ij}^R &=& \xi (1-r_{ij}/r_c) \theta_{ij} (\Delta t)^{-1/2} \textbf{e}_{ij} && r_{ij}<r_c \end{eqnarray}
between bead i and j.

  • r_{ij}=|\textbf{r}_{ij}|: the centre-to-centre distance between the two beads.
  • \textbf{e}_{ij}=\textbf{r}_{ij}/r_{ij}: the direction vector pointing between the two beads.
  • \textbf{v}_{ij}: the vector difference in velocities between the two beads.
  • A: the repulsion coefficient.
  • \gamma: the dissipative coefficient.
  • \xi: the the noise strength, and \xi=\sqrt{2k_BT\gamma} with k_B the Boltzmann constant and T the temperature.
  • \theta_{ij}: a Gaussian white noise variable.
  • \Delta t: the simulation time step.
  • r_c: the cutoff distance, and also can be treated as the size (diameter) of the DPD bead.

Determine paramters

  • r_c, k_BT, m (bead mass) and \tau (time)
    In simulations, the reduced units are often used. For DPD model, all units are often scaled by the length unit r_c, the mass unit m, the energy unit k_BT, and the time unit \tau. Hence, r_c^* = m^* = (k_BT)^* =\tau^* =1 in the simulations. The superscript asterisk (^*) means the quality is in reduced units. Units of other quantities are from these four basic units. For example, the unit for the mass density is m/r_c^3, the unit for the diffusion constant is r_c^2/\tau.

  • \rho (density)
    The DPD simulations are normally carried out within a NVT ensemble, and the number of particles are often determined by setting \rho^*=3. When \rho^*>2, a simple scaling relation between the density and excess pressure exits^1. In principle the density chosen for the simulation is a free parameter, but for efficiency reasons one would thus choose the lowest possible density where the scaling relation still holds.

  • A and N_m
    In order to match the compressibility of DPD fluid with a liquid having the dimensionless compressibility of \Psi, the interparticle repulsion coefficient A is given by
    A\approx \frac{\Psi N_m -1}{0.2\rho^*}(k_BT)^*.
    For water, \Psi\approx 16 yields A^*=25 for N_m=1 and A^*=104 for N_m=4.

  • \gamma and \Delta t
    As a reasonable compromise between fast temperature equilibration, a fast simulation and a stable, physically meaningful system, simulation with \Delta t^*=0.04 and \gamma^*=4.5 is often recommended.
    However, above choice for \Delta t and \gamma yields very small Schmidt number (i.e. the ratio of viscosity to diffusion) (Sc\sim1) compared to the real fluid such as water (Sc\sim10^3). A possible solution to this problem is increasing \gamma. At the same time though, \Delta t would have to be reduced to maintain the temperature control.

  • There is no unique way to determine the parameters for DPD model. For a given physical problem, with a characteristic length scale, we may always put a given number of DPD particles and parametrize the model in order to recover some macroscopic information (e.g. compressibilities, viscosity and diffusion constant).

Units conversation

The conversation between DPD units (r_c, k_BT, m and \tau) and real units ([m], [J], [kg] and [s]) is obtained from the key macroscopic information recovered by the model. If quantities in DPD units are labeled with _{sim} and in real units are labeled with _{real}, one would have

  • m=N_m M_{real}/N_A with M_{real} the mole mass of the fluid (e.g. water) and N_A the Avogadro constant.
  • r_c=(m\rho_{sim}/\rho_{real})^{1/3} with \rho_{real} the mass density of the fluid.
  • k_BT = k_B \cdot T with k_B=1.38064853e-23 J\cdot K and T is the system temperature in K.
  • The time unit \tau can be chosen in different ways.
    • By taking the long-term self-diffusion constant into account, \tau=N_m r_c^2 D_{sim}/D_{real} with D the diffusion constant. Sometimes, this relation is also used to determine N_m.
    • If the viscous processes are the main parts, then \tau=r_c^2 \nu_{sim}/\nu_{real} with \nu the kinematic viscosity.
    • Or \tau=r_c/U_{ref} in which the reference velocity U_{ref} is chosen as either system thermal velocity (k_BT/m)^{1/2} or the characteristic velocity of the real flow.

Example

A NVT simulation with 1000 DPD beads in a cubic box is carried out by LAMMPS. The parameters are: \rho^*=3; A=104 for N_m=4; \gamma=100; \Delta t=0.005. Then, by comparing the DPD fluid with the water at T=298 K, we have:

  • Mass m=1.1969\times10^{-25} [kg] (m=N_m M_{real}/N_A)
  • Length r_c=7.1125\times10^{-10} [m] (r_c=(m\rho_{sim}/\rho_{real})^{1/3})
  • Energy k_BT = 4.1143\times10^{-21} [J] (k_BT = k_B \cdot T)
  • Time \tau = 3.8363\times10^{-12} [s] (\tau=r_c\sqrt{m/k_BT} )
    As a results, the kinematic viscosity for the DPD fluid is \nu=1.569 r_c^2/\tau = 2.069\times10^{-7} [m^2/s] (\nu=8.935\times10^{-7} [m^2/s] for water).

References

(1) Groot, R. D.; Warren, P. B. Dissipative Particle Dynamics: Bridging the Gap between Atomistic and Mesoscopic Simulation. J. Chem. Phys. 1997, 107 (11), 4423–4435.

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