Friction coefficient of the Langevin equation and drag of spherical macroscopic objects in steady flow at low Reynolds numbers are usually regarded as equivalent. We show that the microscopic friction can be different from the macroscopic drag when the mass is taken into account for particles with comparable scale to the surrounding fluid molecules. We illustrate it numerically by molecular dynamics simulation of chloride ion in water. Friction variation by the atomistic mass effect beyond the Langevin regime can be of use in the drag reduction technology as well as the electro or thermophoresis.

## I. INTRODUCTION

Brownian motion is often discussed with a linear relation between the force acting on a particle suspended in a fluid and the relative velocity between them. This is part of the framework of the Langevin equation where the friction coefficient is defined as the proportionality constant, and the friction coefficient is linked to the random force by thermal fluctuation through the fluctuation-dissipation theorem. This is the picture of mesoscopic scale between the atomistic and continuum ones. The linear relation between the force and the velocity is also observed for the macroscopic spherical object suspended in fluids where the Reynolds number is sufficiently small and the motion is in the steady state.^{1} The drag is derived from the continuum fluid mechanics without thermal fluctuation. The friction coefficient of a spherical Brownian particle is usually predicted from this Stokesian drag. When these relations are reconsidered from basics of molecular level, the questions arise as to the correspondence between the friction coefficient and the macroscopic drag, the friction coefficient of atomistically small Brownian particles, and what is the friction coefficient?

The analytical microscopic discussion on the Brownian motion has been proceeded through the expansion in powers of mass ratio of solvent to solute molecules.^{2–4} The friction coefficient of infinitely massive Brownian sphere suspended in a hard-sphere fluid has been theoretically studied.^{5} In our recent study, we have clarified the interrelated role of mass and effective size of particles in Brownian motion when they are comparable to those of the surrounding liquid water molecules.^{6} We have found that the diffusion coefficient can depend on the mass of the solute. While it has been pointed out in the studies exclusively on the systems of short-ranged interactions,^{7–12} we have elucidated the important role of effective size determined by the force field in the possible dependence of the diffusion coefficient on the mass. The Einstein relation connects the diffusion coefficient and friction coefficient where the temperature is the only parameter.^{13} This implies the existence of unknown aspects on the dynamic solute-solvent interactions when the solute mass is comparable to a solvent molecule.

The microscopic friction of particles in liquids is important in various situations including the engineering applications. Thermophoretic phenomena especially in combination with ionic contribution and charge effects have been attracting the attention with the advances of nanotechnologies.^{14–17} The engineering of fluid dynamics at nanoscale has a wide spectra of specific disciplines from the treatment of stochastic nature in material processing^{18} to transport of molecules through nanochannels by applied voltages.^{19–24} The treatment of friction coefficient is also important in the coarse-grained molecular dynamics (MD) method.^{25} Hence, time is ripe for the discussion on the possible departure of microscopic friction from the macroscopic drag. In this article, we report the departure of the microscopic friction coefficient from the macroscopic drag for atomistic solutes in liquid water. In order to reveal this difference, we propose two definitions of friction coefficient, which coincide in the macroscopic limit by construction. The significant difference is mainly attributed to the light mass of the particle, which is beyond the premise of the linear Langevin equation.

## II. METHOD

Liquid water is defined as the working fluid and Cl^{−} ion as a solute in order to show the ubiquity of the phenomena. Since atomistic discreteness of water sometimes manifests in a drastic manner as the system of interest is miniaturized to the molecular scales,^{26–28} we employ MD method in this study. The TIP3P model is employed for water. Each atomic position is defined as the mass point with partial charge, and the molecule is a electrically neutral rigid body. The interactions between the water molecules consist of Lennard-Jones (LJ) interaction between the oxygen sites and Coulomb interactions between every pair of atoms. We also examine the effect of force field by comparison with another solute particle based on the Lennard-Jones potential, where the only difference from Cl^{−} is the absence of charge. Periodic boundary conditions are applied three dimensionally in the simulation domain. The domain size was empirically determined by the examination of several sizes yielding the same friction coefficients, starting from a cubic with 5 nm on a side. The system is first equilibrated under the condition of NPT with the Lowe-Andersen thermostat and Berendsen barostat with goal temperature and pressure of 300 K and 1 atm for 100 ps, respectively. Then, the external field is applied to the system with thermostat but without barostat. The final unit domain size of the system was at least 4.8 nm on a side. It should be noted that the Lowe-Andersen thermostat is a Galilean invariant algorithm that does not accompany the problem of distinction between the flow velocity and thermal velocity. The data for friction coefficients were sampled for at least 2 × 10^{5} frames at an interval of 10 fs. Other computational details are the same as those in our previous study.^{6} Although the diffusion coefficients of dilute ions in water are dominated by the characteristics of water force field, the model used in this study reproduce diffusive property of ions at reasonable level of the standard of MD.^{29}

Since microscopic objects are subject to thermal fluctuations, we define the friction coefficient *ξ* in this study as follows:

where $ f \u0304 $ and $ v \u0304 $ are the time (i.e., frame) averages of instantaneous force *f*(*t*) and *v*(*t*) of the particle of interest. The mean forces and velocities are extracted from the nonequilibrium steady states. We consider two kinds of friction coefficients by the conditions of external field application. When the particle of interest is subject to a constant force, the particle undergoes Brownian motion in addition to the effective transport. We call it the “constant-force” condition in this article. On the other hand, it is in principle possible to theoretically consider the situation where the particle undergoes the transport at a constant velocity realized by the adaptively time-varying force on the particle. In this case, what fluctuates is not the velocity but the applied external force. We call it the “constant-velocity” condition. The adaptive force is realized by the method of our previous study.^{30} This algorithm is also applied to the unit domain of surrounding water so that the water as a whole does not acquire momentum by the interaction with the driven particle. The two distinctive conditions of constant force and velocity yield the same result in the macroscopic limit where the thermal fluctuation is negligible. Therefore, the possible difference in the friction characteristics obtained from these methods reveals the departure of microscopic friction from macroscopic drag.

## III. RESULTS AND DISCUSSION

Figure 1 shows the summary of friction coefficients *ξ* for Cl^{−} and the LJ particle obtained under the different nonequilibrium conditions. *ξ* of Cl^{−} and the LJ particle in water obtained from the constant-force condition has been found to be smaller than those under the constant-velocity conditions. Since the two definitions of *ξ* agree in the macroscopic object by construction, the difference is attributed to the microscopic origin pertaining to the thermal fluctuation. More specifically, it is likely to be caused by the fact that the mass of the particles of interest is comparable to the surrounding water molecules. In fact, the friction coefficients of Cl^{−} and the LJ particle with 10^{2} times larger mass are larger than those of the original masses, and they are close to, while not larger than, the values obtained from the constant-velocity conditions as shown in Fig. 1. The difference between the atomistic particles and the macroscopic objects in water and that between the constant force and velocity conditions is the existence/absence of Brownian motion for the particle of interest. The constant-velocity condition does not allow the fluctuation in the particle velocity. Then, it appears that this difference of friction coefficients is related to the finite-breadth properties of force and/or velocity distributions as indicated by the error bars, but symmetric breadths do not affect the friction coefficient as it is averaged out in Eq. (1).

Although *ξ* indicates the significance of mass effect, the force distribution is similar between the different mass conditions whereas the velocity distributions are substantially different as shown in Figs. 2(a) and 2(b). The relation between the mass and the thermal velocity distribution is simply understood from the correspondence between the temperature and the kinetic energy. In contrast, the difference of *ξ* between Cl^{−} and the LJ particle, caused by the existence/absence of electric charge, is reflected on the force distribution while the velocity distributions appear to be similar as shown in Figs. 2(c) and 2(d). Thus, the difference of mass does not substantially affect the force distribution, whereas the difference of the force fields does. The charge leads to more intensive interaction in a sense of the solvation as shown in Fig. 3 as the higher peak of water density profile for Cl^{−}. Figure 3 also indicates that the effect of force field on the water density profile is preserved regardless of the two types of nonequilibrium conditions. The stronger random interactions with the surrounding water lead to the larger *ξ* for Cl^{−} as shown in Fig. 1. On the other hand, the force field does not substantially affect the velocity distribution when the mass is the same. The relation between *ξ* under the constant force, constant velocity, and the larger mass is the same regardless of the charge effects.

The differences of *ξ* based on Eq. (1) are obviously caused by those in the mean value(s) of force and/or velocity, but it is nontrivial to read clear physical characteristics from them. Therefore, it is more promising to go beyond the mean and variance on the force that generates the dynamics. In theory, the friction coefficient *ξ*_{A} is analytically obtained from the force autocorrelation functions (FACFs),

where *m*, *k*_{B}, *T*, and ** f**(

*t*) are the mass, Boltzmann constant, temperature, and force at time

*t*, respectively. Although we did not use the FACFs for the derivation of the friction coefficients themselves in order to avoid the numerical subtleties,

^{31–39}the FACFs nevertheless reveal the time-dependent characteristics that is not clear in the force distributions. Therefore, we examine the normalized FACFs,

for the different conditions of the simulations. It also serves as the cross check of the deviation and correspondence in *ξ* among different conditions. Figure 4(a) summarizes the normalized FACFs of Cl^{−} obtained under different conditions. The FACF under the constant external force condition coincides with that in equilibrium. Therefore, *ξ* obtained under constant-force condition is considered to reflect the equilibrium property, including the mass-dependent dynamic aspects. The FACF under the constant-velocity condition is substantially different from that under the constant-force condition. This difference vanishes when the mass is 10^{2} times larger. They reduce to the FACF obtained under the constant-velocity condition, which suppress the thermal fluctuation of the particle position and the mass of the particle does not play any role in the friction. As shown in Fig. 4(b), the existence of charge causes the faster decay of FACF with more prominent overshoot. The existence of charge leads to larger number of interacting water molecules due to the long-ranged nature of the Coulomb interaction and the solvation effect shown in Fig. 3.

In our previous study, we have shown by the system of liquid water that the diffusion coefficient can depend on the mass when the Brownian particle of interest does not have sufficiently large mass and size compared to the surrounding water molecules.^{6} The mass dependence of *ξ* for Cl^{−} almost vanishes with 10^{2} times larger mass, but the particle still exhibits the Brownian motion with finite diffusion coefficient. Therefore, it is not the Brownian motion in general that leads to the mass dependence of the friction coefficient but the same origin that causes the mass dependence of the diffusion coefficient. This is also understood from the basic fact that the linear Langevin equation describes the Brownian motion of a particle with sufficiently large mass with a friction coefficient independent of mass. The mass dependence of *ξ* is attributed to the insufficient time scale separation of Brownian motion of the particle of interest and those among the surrounding fluid molecules, relevant to the derivation of the linear Langevin equation from Hamiltonian through projection operator formalism.^{32} The rate of decay is faster and the overshoot is prominent in the FACFs when the mass is lighter (Fig. 4(a)) and when charge exists (Fig. 4(b)), which is consistent with our previous study.^{6} The existence of charge leads to the smaller diffusion coefficient and larger friction coefficient which is interpreted as caused by the increase in the effective size. Smaller mass leads to the smaller friction coefficient. Although the mass dependence of friction coefficient is beyond the regime of the linear Langevin equation, smaller friction coefficient with larger diffusion coefficient^{6} (for smaller mass) shares the trend in common.

We note the significance of the findings to the experimental situations by addressing a few examples. The friction coefficients of microscopic particles by the constant force are directly related to the electrophoresis and electroosmosis where ionic species are driven by the electric field. The mass dependence may lead to the substantial effect when transient response against externally applied electric fields is involved, especially at the short time scales by high frequency AC signal such as in the cases of the Paul traps.^{40,41} Besides the electric fields for the charged solutes or particles in general, thermophoresis is suggested as another example.^{14–17} As it is driven by the temperature gradient, the detailed characteristics of thermal fluctuation can affect the phenomenon in a unique manner. The constant-velocity condition might seem to be a typical gedankenexperiment, but the particles of interest are not necessarily subject to free Brownian motions. Friction between a fluid and solid surfaces with complex topography, e.g., self-assembled monolayers and carbon nanotubes, may hold analogy with our discussion. Technical consideration of the surface roughness for drag reduction taking into account the dynamic aspects as well as the elasticity can be important in such situations. Furthermore, the surface charge affects the ionic distribution, and their non-bulk mobility is relevant to the electroosmosis. Therefore, it can lead to the difference in the macroscopic drag in the end, although the macroscopic measurements just indicate it as the surface property of the macroscopic object. The friction coefficients at the molecular level are determined not only by static but also by dynamic property of the objects of interest themselves as well as the surrounding fluid. In these circumstances, the dynamic interactions between the object of interest and the surrounding fluid do not allow the straightforward separation between the fluid dynamics and thermodynamics, where the energy dissipation is understood in terms of statistical mechanical viewpoint.

## IV. CONCLUSIONS

We have elucidated the possible difference of microscopic friction coefficient from macroscopic drag through the direct simulations of molecular dynamics with two types of definitions for the friction coefficient. They match at the macroscopic limit by construction, but we have found the significant difference for Cl^{−} in liquid water. The difference is caused by the atomistic mass effect in the thermal fluctuation. The fluctuation is usually regarded to vanish if only we are able to obtain sufficient number of samples by averaging. However, the difference of thermal fluctuation remains in the friction coefficient obtained under the condition of constant-force transport, which is a usual condition in the experiments, e.g., electrophoresis and electroosmosis. Since the dissipative interaction between the suspended body and fluid is one of the essential components of fluid dynamics, thermal fluctuation plays an vital role in the molecular fluid dynamics. In particular, slight yet significant difference of friction coefficients can be of use in the drag reduction whose consequence can be fully macroscopic.

## Acknowledgments

This work was partly supported by a Grant-in-Aid for Young Scientists (A), Grant No. 26709008.