# pair_style edpd command # pair_style mdpd command # pair_style mdpd/rhosum command # pair_style tdpd command ## Syntax ``` LAMMPS 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 ``` LAMMPS 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)](Li2014_JCP) and [Li2015_CC](Li2015_CC). The time evolution of an eDPD particle is governed by the conservation of momentum and energy given by $$\begin{aligned} \frac{\mathrm{d}^2 \mathbf{r}_i}{\mathrm{d} t^2}= \frac{\mathrm{d} \mathbf{v}_i}{\mathrm{d} t} =\mathbf{F}_{i}=\sum_{i\neq j}(\mathbf{F}_{ij}^{C}+\mathbf{F}_{ij}^{D}+\mathbf{F}_{ij}^{R}) \\ C_v\frac{\mathrm{d} T_i}{\mathrm{d} t}= q_{i} = \sum_{i\neq j}(q_{ij}^{C}+q_{ij}^{V}+q_{ij}^{R}), \end{aligned}$$ where the three components of $F_{i}$ including the conservative force $F_{ij}^C$, dissipative force $F_{ij}^D$ and random force $F_{ij}^R$ are expressed as $$\begin{aligned} \mathbf{F}_{ij}^{C} & = \alpha_{ij}{\omega_{C}}(r_{ij})\mathbf{e}_{ij} \\ \mathbf{F}_{ij}^{D} & = -\gamma {\omega_{D}}(r_{ij})(\mathbf{e}_{ij} \cdot \mathbf{v}_{ij})\mathbf{e}_{ij} \\ \mathbf{F}_{ij}^{R} & = \sigma {\omega_{R}}(r_{ij}){\xi_{ij}}\Delta t^{-1/2} \mathbf{e}_{ij} \\ \omega_{C}(r) & = 1 - r/r_c \\ \alpha_{ij} & = A\cdot k_B(T_i + T_j)/2 \\ \omega_{D}(r) & = \omega^2_{R}(r) = (1-r/r_c)^s \\ \sigma_{ij}^2 & = 4\gamma k_B T_i T_j/(T_i + T_j) \end{aligned}$$ 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 $$\begin{aligned} q_i^C & = \sum_{j \ne i} k_{ij} \omega_{CT}(r_{ij}) \left( \frac{1}{T_i} - \frac{1}{T_j} \right) \\ q_i^V & = \frac{1}{2 C_v}\sum_{j \ne i}{ \left\{ \omega_D(r_{ij})\left[\gamma_{ij} \left( \mathbf{e}_{ij} \cdot \mathbf{v}_{ij} \right)^2 - \frac{\left( \sigma _{ij} \right)^2}{m}\right] - \sigma _{ij} \omega_R(r_{ij})\left( \mathbf{e}_{ij} \cdot \mathbf{v}_{ij} \right){\xi_{ij}} \right\} } \\ q_i^R & = \sum_{j \ne i} \beta _{ij} \omega_{RT}(r_{ij}) d {t^{ - 1/2}} \xi_{ij}^e \\ \omega_{CT}(r) & =\omega_{RT}^2(r)=\left(1-r/r_{ct}\right)^{s_T} \\ k_{ij} & =C_v^2\kappa(T_i + T_j)^2/4k_B \\ \beta_{ij}^2 & = 2k_Bk_{ij} \end{aligned}$$ where the mesoscopic heat friction $\kappa$ is given by $$\kappa = \frac{315k_B\upsilon }{2\pi \rho C_v r_{ct}^5}\frac{1}{Pr},$$ with $\upsilon$ being the kinematic viscosity. For more details, see Eq.(15) in [(Li2014_JCP)](Li2014_JCP). The following coefficients must be defined in eDPD system for each pair of atom types via the [pair_coeff](pair_coeff) command as in the examples above. - A (force units) - $\gamma$ (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 $s(T) = \mathrm{power}_f ( 1+c_1 (T-1) + c_2 (T-1)^2 + c_3 (T-1)^3 + c_4 (T-1)^4 )$ and of the mesoscopic heat friction $s_T(T) = \kappa (1 + c_1 (T-1) + c_2 (T-1)^2 + c_3 (T-1)^3 + c_4 (T-1)^4)$. If the keyword *power* or *kappa* is not specified, the eDPD system will use constant power_f and $\kappa$, which is independent to temperature changes. ------------------------------------------------------------------------ The *mdpd/rhosum* style computes the local particle mass density $\rho$ for mDPD particles by kernel function interpolation. The following coefficients must be defined for each pair of atom types via the [pair_coeff](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)](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 $$\begin{aligned} \mathbf{F}_{ij}^C & = Aw_c(r_{ij})\mathbf{e}_{ij} + B(\rho_i+\rho_j)w_d(r_{ij})\mathbf{e}_{ij} \\ \mathbf{F}_{ij}^{D} & = -\gamma {\omega_{D}}(r_{ij})(\mathbf{e}_{ij} \cdot \mathbf{v}_{ij})\mathbf{e}_{ij} \\ \mathbf{F}_{ij}^{R} & = \sigma {\omega_{R}}(r_{ij}){\xi_{ij}}\Delta t^{-1/2} \mathbf{e}_{ij} \end{aligned}$$ 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](pair_coeff) command as in the examples above. - A (force units) - B (force units) - $\gamma$ (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)](Li2015_JCP). The time evolution of a tDPD particle is governed by the conservation of momentum and concentration given by $$\begin{aligned} \frac{\mathrm{d}^2 \mathbf{r}_i}{\mathrm{d} t^2} & = \frac{\mathrm{d} \mathbf{v}_i}{\mathrm{d} t}=\mathbf{F}_{i}=\sum_{i\neq j}(\mathbf{F}_{ij}^{C}+\mathbf{F}_{ij}^{D}+\mathbf{F}_{ij}^{R}) \\ \frac{\mathrm{d} C_{i}}{\mathrm{d} t} & = Q_{i} = \sum_{i\neq j}(Q_{ij}^{D}+Q_{ij}^{R}) + Q_{i}^{S} \end{aligned}$$ where the three components of $F_{i}$ including the conservative force $F_{ij}^C$, dissipative force $F_{ij}^C$ and random force $F_{ij}^C$ are expressed as $$\begin{aligned} \mathbf{F}_{ij}^{C} & = A{\omega_{C}}(r_{ij})\mathbf{e}_{ij} \\ \mathbf{F}_{ij}^{D} & = -\gamma {\omega_{D}}(r_{ij})(\mathbf{e}_{ij} \cdot \mathbf{v}_{ij})\mathbf{e}_{ij} \\ \mathbf{F}_{ij}^{R} & = \sigma {\omega_{R}}(r_{ij}){\xi_{ij}}\Delta t^{-1/2} \mathbf{e}_{ij} \\ \omega_{C}(r) & = 1 - r/r_c \\ \omega_{D}(r) & = \omega^2_{R}(r) = (1-r/r_c)^{\rm power_f} \\ \sigma^2 = 2\gamma k_B T \end{aligned}$$ The concentration flux between two tDPD particles includes the Fickian flux $Q_{ij}^D$ and random flux $Q_{ij}^R$, which are given by $$\begin{aligned} Q_{ij}^D & = -\kappa_{ij} w_{DC}(r_{ij}) \left( C_i - C_j \right) \\ Q_{ij}^R & = \epsilon_{ij}\left( C_i + C_j \right) w_{RC}(r_{ij}) \xi_{ij} \\ w_{DC}(r_{ij}) & =w^2_{RC}(r_{ij}) = (1 - r/r_{cc})^{\rm power_{cc}} \\ \epsilon_{ij}^2 & = m_s^2\kappa_{ij}\rho \end{aligned}$$ 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)](Li2015_JCP). The following coefficients must be defined for each pair of atom types via the [pair_coeff](pair_coeff) command as in the examples above. - A (force units) - $\gamma$ (force/velocity units) - power_f (positive real) - cutoff (distance units) - cutoff_CC (distance units) - $\kappa_i$ (diffusivity units) - $\epsilon_i$ (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/PACKAGES/mesodpd. 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)](Li2014_JCP). The output of the short eDPD simulation (about 2 minutes on a single core) gives a temperature and density profiles as ![image](JPG/examples_edpd.jpg){.align-center} 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)](Li2013_POF). The short mDPD run (about 2 minutes on a single core) generates a particle trajectory which can be visualized as follows. ::: only html ![image](JPG/examples_mdpd.gif){.align-center} ::: ![image](JPG/examples_mdpd_first.jpg){.align-center} ![image](JPG/examples_mdpd_last.jpg){.align-center} 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)](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 ![image](JPG/examples_tdpd.jpg){.align-center} ------------------------------------------------------------------------ ## 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](pair_modify) shift, table, and tail options. The styles *edpd*, *mdpd*, *mdpd/rhosum* and *tdpd* do not write information to [binary restart files](restart). 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 DPD-MESO package. They are only enabled if LAMMPS was built with that package. See the [Build package](Build_package) page for more info. ## Related commands [pair_coeff](pair_coeff), [fix mvv/dpd](fix_mvv_dpd), [fix mvv/edpd](fix_mvv_dpd), [fix mvv/tdpd](fix_mvv_dpd), [fix edpd/source](fix_dpd_source), [fix tdpd/source](fix_dpd_source), [compute edpd/temp/atom](compute_edpd_temp_atom), [compute tdpd/cc/atom](compute_tdpd_cc_atom) ## Default none ------------------------------------------------------------------------ ::: {#Li2014_JCP} **(Li2014_JCP)** Li, Tang, Lei, Caswell, Karniadakis, J Comput Phys, 265: 113-127 (2014). DOI: 10.1016/j.jcp.2014.02.003. ::: ::: {#Li2015_CC} **(Li2015_CC)** Li, Tang, Li, Karniadakis, Chem Commun, 51: 11038-11040 (2015). DOI: 10.1039/C5CC01684C. ::: ::: {#Li2013_POF} **(Li2013_POF)** Li, Hu, Wang, Ma, Zhou, Phys Fluids, 25: 072103 (2013). DOI: 10.1063/1.4812366. ::: ::: {#Li2015_JCP} **(Li2015_JCP)** Li, Yazdani, Tartakovsky, Karniadakis, J Chem Phys, 143: 014101 (2015). DOI: 10.1063/1.4923254. :::