(Eq. 12)

where, the superscript 0 denotes the value of the variable at the beginning of the time step, r_i is the position of the i^th nanoparticle, D_ij is the diffusion tensor, and F_j refers to the force acting on the j^th particle. The displacement R_i is the unconstrained Brownian displacement with a white noise having an average value of zero and a covariance of 2D_ij⁰ 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 r_i(t) are updated in the streaming steps, and their velocities U_i(t) are obtained through multi-particle collisions in the collision steps: r_i(t+Δt)=r_i(t)+ ΔtU_i(t), and U_i(t+Δt)=U(t)+ R • U_i(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., ΔU_i(t)=U_i-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].