# Interdisciplinary Applied Mathematics

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ij

include any pressure-drop terms, electrostatic or magnetic interactions, as well as    van    der    Waals    forces. This    force as    well    as    the    other two    forces

are acting    within a sphere    of    radius    rc,    which is    the    length    scale    of the

system. It corresponds to a soft interaction potential, similar to what has been proposed independently by (Forrest and Suter, 1995) in simulation studies of polymers. By averaging systematically Lennard-Jones potentials or the corresponding molecular field over the rapidly fluctuating motions of atoms during short time intervals, an effective average potential is obtained of the form shown in Figure 16.10. An approximation of this can be given as (Groot and Warren, 1997)

T7C

Fij

aij (1    rij ) Pij    for    rij    — rc 17

0    for    rij    > rc    = 1,

r

FIGURE 16.10. Lennard-Jones potential and its averaged soft potential.

where aij is a maximum repulsion between particle i and particle j. Unlike the hard Lennard-Jones potential, which is unbounded at r = 0, the soft potential employed in DPD has a finite value at r = 0 equal to aij. This reflects the fact that there is a finite probability that there will be no atoms at r = 0 for some realizations.

For water in the nondimensional units set here we have that aij = 25, while    for    other    liquids    we    have    that    aij    = 75kBT/p.    These    values    were

obtained in (Groot and Warren, 1997), by enforcing the proper compressibility of the system defined by

к

l

1 dp

kBT dn

T

where n is the number density. For water at room temperature (300 K) this dimensionless compressibility is к-1 = 15.9835. The pressure is obtained as a function of the density, e.g.,

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