Test case: a dimer solvent¶

Here we describe a physical system in which the software modules dealing with charge dipole moments in DPD simulations can be tested. It is a polarizable fluid made of harmonically bonded dimers $(+q,-q)$ , pictorially represented on the left (not in scale). Fixing appropriately the partial charge $q$ and the Bjerrum length $l_B$ , this system mimics water in an oil background, as long as the dielectric properties are concerned.

We recall that the electric permittivity is $\epsilon_0$ for vacuum, and $\epsilon<\epsilon_0$ for a medium. The medium effect can be split into a background and a relative term $\epsilon/\epsilon_0=\epsilon_b\epsilon_r$ . The background is constant and uniform, whereas the explicit term is due to dynamic microscopic objects (dimers in this case) which carry a charge dipole moment. The strength of electrostatic interactions in a background is set by the bare Bjerrum length $l_B=e^2/(4 \pi\epsilon_0 \epsilon_b k_BT)$ . On the other hand, from linear response theory, the bulk value of the relative permittivity is $\epsilon_r = 1 + \frac{\langle\vec{P}^2\rangle_{\vec{E}=\vec{0}}}{3\epsilon_0\epsilon_{b}\, k_BT\,V}$ , where tin-foil boundary conditions are assumed.

Two types of beads are present in the simulation, solp and solm, the solvent positive and negative partial charges, respectively. We fix the bare Bjerrum length $l_B=42$ (appropriate for oil [1]), the repulsion parameter $A=25$ , the harmonic spring constant $k=5$ , the bead density $\rho=3$ , the partial charges $|q|=0.46$ and the Gaussian smearing length $\sigma=0.5$ . All quantities are given in DPD units, where $k_BT=1$ , $r_c=1$ and $m=1$ . This fluid has a relative permittivity $\epsilon_r\simeq 40$ , as can be checked using the gen_dipole.f90 utility. This value is compatible with the ratio of water and oil permittivities $\epsilon^{water}/\epsilon^{oil}\simeq 40$ .

The FIELD file defining the composition and interactions for a system of volume $V=64$ is

DL_MESO charged harmonic dimers with dpd repulsion

SPECIES 2
solp  1.0   0.46  0
solm  1.0  -0.46  0

MOLECULES 1
DIMER
nummols 96
beads 2
solp  0.0 0.0 0.0
solm  0.1 0.0 0.0
bonds 1
harm 1 2 5.0 0.0

finish

INTERACTIONS 3
solp  solp  dpd 25.0 1.0 4.5
solm  solm  dpd 25.0 1.0 4.5
solp  solm  dpd 25.0 1.0 4.5

CLOSE

[1]

Notice that the physical length scale is set choosing $r_c$ : if we choose $r_c=0.646nm$ (appropriate to match water density at room temperature if $N_m=3$ , i.e., one bead represents three water molecules), the Bjerrum length of oil in DPD units is $l_B=27 nm\simeq 42 r_c$ , hence the value given above for the oil background.