pair_style edpd command

pair_style mdpd command

pair_style mdpd/rhosum command

pair_style tdpd command

Syntax

pair_style style args

style = edpd or mdpd or mdpd/rhosum or tdpd

args = list of arguments for a particular style

edpd args = cutoff seed
  cutoff = global cutoff for eDPD interactions (distance units)
  seed = random # seed (integer) (if <= 0, eDPD will use current time as the seed)
mdpd args = T cutoff seed
  T = temperature (temperature units)
  cutoff = global cutoff for mDPD interactions (distance units)
  seed = random # seed (integer) (if <= 0, mDPD will use current time as the seed)
mdpd/rhosum args =
tdpd args = T cutoff seed
  T = temperature (temperature units)
  cutoff = global cutoff for tDPD interactions (distance units)
  seed = random # seed (integer) (if <= 0, tDPD will use current time as the seed)

Examples

pair_style edpd 1.58 9872598
pair_coeff * * 18.75 4.5 0.41 1.58 1.42E-5 2.0 1.58
pair_coeff 1 1 18.75 4.5 0.41 1.58 1.42E-5 2.0 1.58 power 10.54 -3.66 3.44 -4.10
pair_coeff 1 1 18.75 4.5 0.41 1.58 1.42E-5 2.0 1.58 power 10.54 -3.66 3.44 -4.10 kappa -0.44 -3.21 5.04 0.00

pair_style hybrid/overlay mdpd/rhosum mdpd 1.0 1.0 65689
pair_coeff 1 1 mdpd/rhosum  0.75
pair_coeff 1 1 mdpd -40.0 25.0 18.0 1.0 0.75

pair_style tdpd 1.0 1.58 935662
pair_coeff * * 18.75 4.5 0.41 1.58 1.58 1.0 1.0E-5 2.0
pair_coeff 1 1 18.75 4.5 0.41 1.58 1.58 1.0 1.0E-5 2.0 3.0 1.0E-5 2.0

Description

The edpd style computes the pairwise interactions and heat fluxes for eDPD particles following the formulations in (Li2014_JCP) and Li2015_CC. The time evolution of an eDPD particle is governed by the conservation of momentum and energy given by

\begin{aligned} \frac{d^{2} r_{i}}{d t^{2}} = \frac{d v_{i}}{d t} = F_{i} = \sum_{i \neq j} (F_{i j}^{C} + F_{i j}^{D} + F_{i j}^{R}) \\ C_{v} \frac{d T_{i}}{d t} = q_{i} = \sum_{i \neq j} (q_{i j}^{C} + q_{i j}^{V} + q_{i j}^{R}), \end{aligned}

where the three components of $F_{i}$ including the conservative force $F_{i j}^{C}$ , dissipative force $F_{i j}^{D}$ and random force $F_{i j}^{R}$ are expressed as

\begin{aligned} F_{i j}^{C} & = α_{i j} ω_{C} (r_{i j}) e_{i j} \\ F_{i j}^{D} & = - γ ω_{D} (r_{i j}) (e_{i j} \cdot v_{i j}) e_{i j} \\ F_{i j}^{R} & = σ ω_{R} (r_{i j}) ξ_{i j} Δ t^{- 1 / 2} e_{i j} \\ ω_{C} (r) & = 1 - r / r_{c} \\ α_{i j} & = A \cdot k_{B} (T_{i} + T_{j}) / 2 \\ ω_{D} (r) & = ω_{R}^{2} (r) = (1 - r / r_{c})^{s} \\ σ_{i j}^{2} & = 4 γ k_{B} T_{i} T_{j} / (T_{i} + T_{j}) \end{aligned}

in which the exponent of the weighting function s can be defined as a temperature-dependent variable. The heat flux between particles accounting for the collisional heat flux $q^{C}$ , viscous heat flux $q^{V}$ , and random heat flux $q^{R}$ are given by

\begin{aligned} q_{i}^{C} & = \sum_{j \neq i} k_{i j} ω_{C T} (r_{i j}) (\frac{1}{T_{i}} - \frac{1}{T_{j}}) \\ q_{i}^{V} & = \frac{1}{2 C_{v}} \sum_{j \neq i} {ω_{D} (r_{i j}) [γ_{i j} {(e_{i j} \cdot v_{i j})}^{2} - \frac{{(σ_{i j})}^{2}}{m}] - σ_{i j} ω_{R} (r_{i j}) (e_{i j} \cdot v_{i j}) ξ_{i j}} \\ q_{i}^{R} & = \sum_{j \neq i} β_{i j} ω_{R T} (r_{i j}) d t^{- 1 / 2} ξ_{i j}^{e} \\ ω_{C T} (r) & = ω_{R T}^{2} (r) = {(1 - r / r_{c t})}^{s_{T}} \\ k_{i j} & = C_{v}^{2} κ (T_{i} + T_{j})^{2} / 4 k_{B} \\ β_{i j}^{2} & = 2 k_{B} k_{i j} \end{aligned}

where the mesoscopic heat friction $κ$ is given by

κ = \frac{315 k_{B} υ}{2 π ρ C_{v} r_{c t}^{5}} \frac{1}{P r},

with $υ$ being the kinematic viscosity. For more details, see Eq.(15) in (Li2014_JCP).

The following coefficients must be defined in eDPD system for each pair of atom types via the pair_coeff command as in the examples above.

A (force units)
$γ$ (force/velocity units)
power_f (positive real)
cutoff (distance units)
kappa (thermal conductivity units)
power_T (positive real)
cutoff_T (distance units)
optional keyword = power or kappa

The keyword power or kappa is optional. Both “power” and “kappa” require 4 parameters $c_{1}, c_{2}, c_{3}, c_{4}$ showing the temperature dependence of the exponent $s (T) = {p o w e r}_{f} (1 + c_{1} (T - 1) + c_{2} (T - 1)^{2} + c_{3} (T - 1)^{3} + c_{4} (T - 1)^{4})$ and of the mesoscopic heat friction $s_{T} (T) = κ (1 + c_{1} (T - 1) + c_{2} (T - 1)^{2} + c_{3} (T - 1)^{3} + c_{4} (T - 1)^{4})$ . If the keyword power or kappa is not specified, the eDPD system will use constant power_f and $κ$ , which is independent to temperature changes.

The mdpd/rhosum style computes the local particle mass density $ρ$ for mDPD particles by kernel function interpolation.

The following coefficients must be defined for each pair of atom types via the pair_coeff command as in the examples above.

cutoff (distance units)

The mdpd style computes the many-body interactions between mDPD particles following the formulations in (Li2013_POF). The dissipative and random forces are in the form same as the classical DPD, but the conservative force is local density dependent, which are given by

\begin{aligned} F_{i j}^{C} & = A w_{c} (r_{i j}) e_{i j} + B (ρ_{i} + ρ_{j}) w_{d} (r_{i j}) e_{i j} \\ F_{i j}^{D} & = - γ ω_{D} (r_{i j}) (e_{i j} \cdot v_{i j}) e_{i j} \\ F_{i j}^{R} & = σ ω_{R} (r_{i j}) ξ_{i j} Δ t^{- 1 / 2} e_{i j} \end{aligned}

where the first term in $F_{C}$ with a negative coefficient $A < 0$ stands for an attractive force within an interaction range $r_{c}$ , and the second term with $B > 0$ is the density-dependent repulsive force within an interaction range $r_{d}$ .

The following coefficients must be defined for each pair of atom types via the pair_coeff command as in the examples above.

A (force units)
B (force units)
$γ$ (force/velocity units)
cutoff_c (distance units)
cutoff_d (distance units)

The tdpd style computes the pairwise interactions and chemical concentration fluxes for tDPD particles following the formulations in (Li2015_JCP). The time evolution of a tDPD particle is governed by the conservation of momentum and concentration given by

\begin{aligned} \frac{d^{2} r_{i}}{d t^{2}} & = \frac{d v_{i}}{d t} = F_{i} = \sum_{i \neq j} (F_{i j}^{C} + F_{i j}^{D} + F_{i j}^{R}) \\ \frac{d C_{i}}{d t} & = Q_{i} = \sum_{i \neq j} (Q_{i j}^{D} + Q_{i j}^{R}) + Q_{i}^{S} \end{aligned}

where the three components of $F_{i}$ including the conservative force $F_{i j}^{C}$ , dissipative force $F_{i j}^{C}$ and random force $F_{i j}^{C}$ are expressed as

\begin{aligned} F_{i j}^{C} & = A ω_{C} (r_{i j}) e_{i j} \\ F_{i j}^{D} & = - γ ω_{D} (r_{i j}) (e_{i j} \cdot v_{i j}) e_{i j} \\ F_{i j}^{R} & = σ ω_{R} (r_{i j}) ξ_{i j} Δ t^{- 1 / 2} e_{i j} \\ ω_{C} (r) & = 1 - r / r_{c} \\ ω_{D} (r) & = ω_{R}^{2} (r) = (1 - r / r_{c})^{p o w e r_f} \\ σ^{2} = 2 γ k_{B} T \end{aligned}

The concentration flux between two tDPD particles includes the Fickian flux $Q_{i j}^{D}$ and random flux $Q_{i j}^{R}$ , which are given by

\begin{aligned} Q_{i j}^{D} & = - κ_{i j} w_{D C} (r_{i j}) (C_{i} - C_{j}) \\ Q_{i j}^{R} & = ϵ_{i j} (C_{i} + C_{j}) w_{R C} (r_{i j}) ξ_{i j} \\ w_{D C} (r_{i j}) & = w_{R C}^{2} (r_{i j}) = (1 - r / r_{c c})^{p o w e r_c c} \\ ϵ_{i j}^{2} & = m_{s}^{2} κ_{i j} ρ \end{aligned}

where the parameters kappa and epsilon determine the strength of the Fickian and random fluxes. $m_{s}$ is the mass of a single solute molecule. In general, $m_{s}$ is much smaller than the mass of a tDPD particle m. For more details, see (Li2015_JCP).

The following coefficients must be defined for each pair of atom types via the pair_coeff command as in the examples above.

A (force units)
$γ$ (force/velocity units)
power_f (positive real)
cutoff (distance units)
cutoff_CC (distance units)
$κ_{i}$ (diffusivity units)
$ϵ_{i}$ (diffusivity units)
power_cc_i (positive real)

The last 3 values must be repeated Nspecies times, so that values for each of the Nspecies chemical species are specified, as indicated by the “I” suffix. In the first pair_coeff example above for pair_style tdpd, Nspecies = 1. In the second example, Nspecies = 2, so 3 additional coeffs are specified (for species 2).

Example scripts

There are example scripts for using all these pair styles in examples/USER/meso. The example for an eDPD simulation models heat conduction with source terms analog of periodic Poiseuille flow problem. The setup follows Fig.12 in (Li2014_JCP). The output of the short eDPD simulation (about 2 minutes on a single core) gives a temperature and density profiles as

The example for a mDPD simulation models the oscillations of a liquid droplet started from a liquid film. The mDPD parameters are adopted from (Li2013_POF). The short mDPD run (about 2 minutes on a single core) generates a particle trajectory which can be visualized as follows.

The first image is the initial state of the simulation. If you click it a GIF movie should play in your browser. The second image is the final state of the simulation.

The example for a tDPD simulation computes the effective diffusion coefficient of a tDPD system using a method analogous to the periodic Poiseuille flow. The tDPD system is specified with two chemical species, and the setup follows Fig.1 in (Li2015_JCP). The output of the short tDPD simulation (about one and a half minutes on a single core) gives the concentration profiles of the two chemical species as

Mixing, shift, table, tail correction, restart, rRESPA info:

The styles edpd, mdpd, mdpd/rhosum and tdpd do not support mixing. Thus, coefficients for all I,J pairs must be specified explicitly.

The styles edpd, mdpd, mdpd/rhosum and tdpd do not support the pair_modify shift, table, and tail options.

The styles edpd, mdpd, mdpd/rhosum and tdpd do not write information to binary restart files. Thus, you need to re-specify the pair_style and pair_coeff commands in an input script that reads a restart file.

Restrictions

The pair styles edpd, mdpd, mdpd/rhosum and tdpd are part of the USER-MESO package. It is only enabled if LAMMPS was built with that package. See the Build package doc page for more info.

pair_style edpd command

pair_style mdpd command

pair_style mdpd/rhosum command

pair_style tdpd command

Syntax

Examples

Description

Restrictions

Related commands