(Eq. 12)
where, the superscript 0 denotes the value of the variable at the
beginning of the time step, ri is the position of the
ith nanoparticle, Dij is the diffusion
tensor, and Fj refers to the force acting on the
jth particle. The displacement Ri is
the unconstrained Brownian displacement with a white noise having an
average value of zero and a covariance of
2Dij0 δ(t). The Rotne-Prager-Yamakawa
hydrodynamic mobility tensor [33, 34] is a commonly employed
diffusion tensor to approximate the hydrodynamic interactions mediated
by the fluid. The trajectories and interactions between the
coarse-grained molecules are calculated using the stochastic
differential equation (Eq. 12), which is integrated forward in time,
allowing for the study of the temporal evolution and the dynamics of
complex fluids. Stokesian dynamics also represent a class of methods
under this paradigm [35].
3.4.2 Multi-particle collision dynamics (MPCD): Multi-particle
collision dynamics (MPCD) is an algorithm that can model both
hydrodynamic interactions and Brownian motion with relatively low
computational costs [36]. The algorithm consists of discrete
streaming and collision steps at fixed discrete time intervals that have
been shown to yield the correct long-time hydrodynamics. The effects of
Brownian motion and hydrodynamic interactions are incorporated into the
simulation through the collision step. The solvent is characterized by a
large number N of point-like particles with a given mass m that move in
space with a continuous distribution of velocities. The positions of the
solvent particles ri(t) are updated in the streaming
steps, and their velocities Ui(t) are obtained through
multi-particle collisions in the collision steps:
ri(t+Δt)=ri(t)+ ΔtUi(t),
and Ui(t+Δt)=U(t)+ R • Ui(t). The
stochastic rotational dynamics (SRD) is one of the most widely used MPCD
algorithms in which the collision step consists of a random rotation R
of the relative velocities of the particles, i.e.,
ΔUi(t)=Ui-U, in a collision cell, where
U is the mean velocity of all particles in a cell. Gompper et al.
provided a review of several widely used MPC algorithms and recent
applications of MPCD algorithm to study colloid and polymer dynamics as
well as the behavior of vesicles and cells in hydrodynamic flow
environments [36].
3.4.3 Dissipative particle dynamics (DPD): To reach larger time-
and lengthscales, the dissipative particle dynamics (DPD) method uses a
much coarser mapping, in which one site may represent many molecules in
a small fluid volume [37-39]. There are three types of forces
present in DPD models: a conserved soft repulsion force, pairwise
dissipation forces, and pairwise random forces. The balance of
dissipation and random forces provides the thermostat for the DPD model,
and since this thermostat preserves the momentum of individual
particles, these models provide correct hydrodynamic behavior. In
addition to using a coarser mapping, DPD simulations use a longer
timestep due to the use of soft repulsion forces. It is necessary to
match the observed compressibility in a DPD simulation to the target
fluid in order to study the phase behavior and interfacial tension of
the model fluid. The DPD method has been applied to biological lipid
bilayers, membrane fusion processes, and bilayers with proteins, and its
connections to the mesoscale have been reviewed extensively [40,
41].