A Hybrid Approach for the Simulation of the Thermal Motion of a Nearly Neutrally Buoyant Nanoparticle in an Incompressible Newtonian Fluid Medium

The chief motivation for the present study is the simulation of the thermal motion of a nanoparticle in a fluid flow that occurs in targeted drug delivery [1–6] where the drug carrying nanocarriers are intravascularly introduced into the vasculature. Nanoparticles allow more precise and successful infiltration of drugs to target cells. In general, nanoparticle drug-delivery systems have been shown to enhance the solubility of compounds, and to reduce the impact of drugs on nontarget tissue, thereby eliminating unwanted and dangerous side effects. In order to more broadly integrate this technology into medicine, a precise understanding of how to guide the nanoparticle to the target site is necessary. To achieve this goal, as a first step, it is necessary to determine the motion of a nanocarrier (due to thermal and hydrodynamic effects) in a fluid medium. The nanocarrier with a loaded cargo may be studied subsequently. A nanoparticle suspended in a fluid undergoes random motion due to the thermal fluctuations in the fluid. As a consequence, translational and rotational degrees of freedom become important. In determining the translational and rotational motions of the nanoparticle in an incompressible Newtonian medium, there exist two methods that couple the thermal fluctuations with the hydrodynamic interactions. These are the generalized Langevin method [7] and the fluctuating hydrodynamics method [8]. Either procedure would require numerical simulations for covering extensive parameter space. In the fluctuating hydrodynamics approach, the nanoparticle motion incorporates both the Brownian motion and the effect of hydrodynamic force acting on its surface imparted from the surrounding fluid. This method essentially consists of adding stochastic stresses (random stress) to the stress tensor in the momentum equation of the fluid and stochastic fluxes to the heat flux where the energy equation is present [9]. The stochastic stress tensor depends on the temperature and the transport coefficients of the fluid medium [10,11]. Numerical simulations of the fluctuating hydrodynamics approach have been carried out employing the finite volume method [12–14], Lattice-Boltzmann method [15–21], finite element method [8,22,23] and stochastic immersed boundary method [24]. In the Langevin dynamics method, the effect of thermal fluctuations are incorporated as random forces and torques in the particle equation of motion [7,25–30]. The properties of these forces depend on the grand resistance tensor. The tensor in turn depends on the fluid properties, particle shape, and its instantaneous location such as its proximity to a wall or a boundary. This is a robust thermostat, which preserves equilibrium distributions at constant temperatures (i.e., adheres to the equipartition theorem). Clearly, coupling to a thermostat will alter the hydrodyamics of the nanoparticle system. The characterizations of the performance of the thermostat as well as how it alters the associated hydrodynamic correlations are important. Numerical schemes for studying the nanoparticle motion in a fluid must simultaneously consider the momentum (Langevin) equation for the particle and the Navier–Stokes equation for the fluid. The random force/torque in the particle equation can then be related to the frictional force/torque via the generalized fluctuation–dissipation theorem [31,32]. The implementation can occur in two ways: (i) directly adjust the variance of the random force term in the classical Langevin equation to play the role of a thermostat and (ii) a second, more direct approach that preserves the structure of the generalized Langevin equation, is to consider the power spectrum for the variance of the random force term using a correlated or colored noise with a well defined characteristic memory time. Such a formalism simultaneously preserves the equipartition theorem and the nature of the long time hydrodynamic correlations, and proves to be a versatile thermostat [7]. The fluctuating hydrodynamics approach in an incompressible fluid [8] captures the correct hydrodynamic correlations and conserves thermal equipartition only after adding the mass correction [10]. On the other hand, the generalized Langevin dynamics yields the correct thermal equipartition (without any mass correction), but modifies the nature of the hydrodynamics correlations (due to the coupling of the fluid equations with the thermostat degrees of freedom) [7]. Recently, we have formulated a novel hybrid approach combining Markovian fluctuating hydrodynamics of the fluid and the non-Markovian Langevin dynamics with the Ornstein–Uhlenbeck noise perturbing the translational and rotational equations of motion of the nanocarrier [33]. For this hybrid approach, we have verified the conservation of thermal equipartition and the nature of hydrodynamic correlations by comparisons with well-known analytical results [10]. This approach effectively produces a thermostat that also simultaneously preserves the true hydrodynamic correlations [33]. With this procedure, we have also evaluated adhesive interactions between a receptor on the nanocarrier surface and a ligand (tethered by a spring force) on the surface of an endothelial cell lining the cylindrical vessel wall at a specified finite temperature [34]. In this paper, we demonstrate the thermal equilibrium and hydrodynamic correlations of a nearly neutrally buoyant nanoparticle using our recently developed hybrid approach. A direct numerical simulation procedure adopting an arbitrary Lagrangian–Eulerian based finite element method is employed to simulate the Brownian motion of a nearly neutrally buoyant particle in an incompressible Newtonian fluid contained in a horizontal micron sized cylindrical vessel. The results for the attainment of thermal equilibrium between the particle and the surrounding medium, diffusivity for the particle in the medium, effect of the presence of the confining vessel wall on particle displacement and diffusivity are evaluated and discussed in detail. A major advantage of our novel hybrid approach employed in this paper as compared to either the fluctuating hydrodynamics approach or the generalized Langevin approach by itself is that only the hybrid method has been shown to simultaneously preserve both hydrodynamic correlations and equilibrium statistics in the incompressible limit.