Molecular dynamics simulations of the rotational and translational diffusion of a Janus rod-shaped nanoparticle

TL;DR

This study uses molecular dynamics simulations to analyze how Janus rod-shaped nanoparticles diffuse in a fluid, revealing that wettability contrast influences their rotational and translational diffusion behaviors.

Contribution

It introduces a detailed MD simulation analysis of Janus rod-shaped nanoparticles, highlighting the effects of wettability contrast on their diffusion properties and orientation dynamics.

Findings

Higher wettability contrast increases diffusion coefficients.

Diffusion coefficients align with analytical predictions for uniform particles.

Janus particles tend to rotate to minimize drag by orienting nonwetting sides along displacement.

Abstract

The diffusion of a Janus rod-shaped nanoparticle in a dense Lennard-Jones fluid is studied using molecular dynamics (MD) simulations. The Janus particle is modeled as a rigid cylinder whose atoms on each half-side have different interaction energies with fluid molecules, thus comprising wetting and nonwetting surfaces. We found that both rotational and translational diffusion coefficients are larger for Janus particles with higher wettability contrast, and these values are bound between the two limiting cases of uniformly wetting and nonwetting particles. It was also shown that values of the diffusion coefficients for displacements parallel and perpendicular to the major axis of a uniformly wetting particle agree well with analytical predictions despite a finite slip at the particle surface present in MD simulations. It was further demonstrated that diffusion of Janus particles is markedly different from that of uniform particles; namely, Janus particles preferentially rotate and orient their nonwetting sides along the displacement vector to reduce drag. This correlation between translation and rotation is consistent with the previous results on diffusive dynamics of a spherical Janus particle with two hemispheres of different wettability.

Figures

Tables

Table 1. Table 1: The variation of the slip length L s / σ subscript 𝐿 𝑠 𝜎 L_{s}/\sigma as a function of the wall-fluid interaction energy computed at flat interfaces between a fluid phase with the density ρ = 0.749 ​ σ − 3 𝜌 0.749 superscript 𝜎 3 \rho=0.749\,\sigma^{-3} and crystalline surfaces with the lattice constants 0.707 ​ σ × 0.577 ​ σ 0.707 𝜎 0.577 𝜎 0.707\,\sigma\times 0.577\,\sigma (see text for details). The typical error bars for the slip length are about ± 1.0 ​ σ plus-or-minus 1.0 𝜎 \pm 1.0\,\sigma .

${\epsilon }_{\mathrm{pf}}/\epsilon$	$0.1$	$0.3$	$0.5$	$0.7$	$1.0$
$L_s/\sigma$	$36.5$	$20.5$	$12.0$	$8.0$	$4.5$

Table 2. Table 2: The diffusion coefficients for uniformly wetting and nonwetting particles as well as Janus particles with the indicated wettability contrast. The following parameters were evaluated from the fit to the MD data: D ⊥ subscript 𝐷 bottom D_{\bot} using Eq. ( 9 ), D | | D_{||} using Eq. ( 8 ), D t = ( 2 ​ D ⊥ + D | | ) / 3 D_{t}=(2D_{\bot}+D_{||})/3 , D r subscript 𝐷 𝑟 D_{r} using Eq. ( 6 ), τ r subscript 𝜏 𝑟 \tau_{r} using Eq. ( 5 ). The data in the last column were computed using Eqs. ( 2 )-( 4 ) from Ref.  Torre84 . Typical error bars are 6 × 10 − 4 ​ τ ​ σ − 2 6 superscript 10 4 𝜏 superscript 𝜎 2 6\times 10^{-4}\,\tau\sigma^{-2} .

${\epsilon }_{\mathrm{pf}}/\epsilon$	$(1.0,1.0)$	$(0.1,0.1)$	$(1.0,0.7)$	$(1.0,0.5)$	$(1.0,0.3)$	$(1.0,0.1)$	$\text{Ref.}\left[40\right]$
$D_{\perp }\tau {\sigma }^{-2}$	$0.0149$	$0.0242$	$0.0159$	$0.0165$	$0.0171$	$0.0179$	$0.0146$
$D_{\left|\right|}\tau {\sigma }^{-2}$	$0.0177$	$0.0291$	$0.0186$	$0.0197$	$0.0203$	$0.0209$	$0.0174$
$D_t\tau {\sigma }^{-2}$	$0.0158$	$0.0258$	$0.0168$	$0.0176$	$0.0182$	$0.0189$	$0.0155$
$D_r\tau$	$0.00176$	$0.00312$	$0.00185$	$0.00197$	$0.00202$	$0.00229$	$0.00122$
${\tau }_r/\tau$	$296.3$	$161.9$	$274.3$	$260.6$	$245.1$	$216.8$	$÷$
$\tau /2{\tau }_r$	$0.00169$	$0.00309$	$0.00182$	$0.00192$	$0.00204$	$0.00231$	$÷$

Equations

$$V_{LJ}(r)=4\,\varepsilon\,\Big{[}\Big{(}\frac{\sigma}{r}\Big{)}^{12}\!-\Big{(}\frac{\sigma}{r}\Big{)}^{6}\,\Big{]},$$

$$D_{\perp }\approx \frac{k_{B}T}{8\pi \eta b}(\text{ln}\omega +0.839+0.185/\omega +0.233/{\omega }^2),$$

$$D_{||}\approx \frac{k_{B}T}{4\pi \eta b}(\text{ln}\omega -0.207+0.980/\omega -0.133/{\omega }^2),$$

$$D_r\approx \frac{3k_{B}T}{8\pi \eta b^3}(\text{ln}\omega -0.662+0.917/\omega -0.050/{\omega }^2),$$

$$⟨{\text{e}}_1(0)\cdot {\text{e}}_1(t)⟩={\text{e}}^{-t/{\tau }_r},$$

$$⟨\Delta {\vec{\phi }}^2(t)⟩=\frac{1}{N}\sum _{i=1}^N|{\vec{\phi }}_i(t_0+t)-{\vec{\phi }}_i(t_0){|}^2=4D_{r}t,$$

$$\vec{\phi }(t)={\int }_0^t\Delta \vec{\phi }(t^{\prime })d{t}^{\prime },$$

$$\Delta r_{||}^2(t)=⟨((\text{r}(t_0+t)-\text{r}(t_0))\cdot {\text{e}}_1(t_0){)}^2⟩=2D_{||}t,$$

$$\Delta r_{\perp }^2(t)=⟨|(\text{r}(t_0+t)-\text{r}(t_0))\times {\text{e}}_1(t_0){|}^2⟩=4D_{\perp }t,$$

Full text

I Introduction

The design of functional nanomaterials, with diverse applications in biomedicine, optoelectronics and microfiltration, requires a detailed understanding of the diffusion process of anisotropic particles during their self-assembly in bulk fluid and at interfaces Busseron13 . The synthesis of Janus nanoparticles often involves a masking step where particles are temporarily trapped at the interface between two phases and only one side can be treated chemically leading to asymmetric functionalization Lattuada11 . It was shown numerically that structural evolution of polymer blends can be significantly influenced by orientation of Janus nanorods relative to the phase interface, and, thus, it allows fabrication of polymer nanocomposites with robust photovoltaic and mechanical properties Xu12 ; Li13 . Common mechanisms of colloidal self-assembly include depletion-assisted structure formation, where effective forces between neighboring colloidal particles arise due to decreasing volume available to the depletant, and shape-complementary colloidal suspensions where only particles with matching building blocks bind together Pine13 . Interestingly, a variety of prescribed crystal or gel phases can be obtained via ‘programmable’ self-assembly of DNA-grafted particles due to formation of bridges between neighboring particles Rogers16 . Regardless of the specific interaction between particles, a complete picture of the diffusive motion even of isolated anisotropic particles is still missing.

The results of equilibrium molecular dynamics simulations have shown that the diffusion dynamics of a single spherical particle depends on its wetting properties, local boundary conditions, mass and size relative to the solvent molecules, as well as viscosity and temperature of the solvent Levesque00 ; Schmidt03 ; Kapral04 ; Schmidt04 ; Levesque07 ; Li09 ; Shin10 ; Chakraborty11 ; Ohtori16 . Originally, it was discovered by Alder and Wainwright Alder70 that the velocity autocorrelation function exhibits a characteristic power-law decay at long times due to hydrodynamic coupling to the solvent. More recently, it was demonstrated that in the presence of a liquid-solid interface, the power-law exponent is changed from the bulk value $-3/2$ to $-5/2$, while the amplitude of the velocity autocorrelation function increases for diffusive motion near slipping boundary Izabela15 ; Felderhof05 . It was also shown that as the particle size decreases down to a few molecular diameters, the Stokes-Einstein relation breaks down, and the effective radius of the particle might increase due to formation of an adjacent fluid layer if the surface energy is sufficiently large Li09 . Despite significant computational efforts, the exact relationship between the local slip at confining boundaries and the position-dependent diffusion coefficient has yet to be established.

In recent years, various aspects of diffusive motion of a single Janus particle in the bulk and at liquid interfaces were studied using continuum Swan08 ; Willmott08 ; Willmott09 ; Chan13 and molecular dynamics Drazer15 ; Kharazmi15 ; Zadeh16 ; Archereau16 simulations. In addition, a high-speed experimental technique to track translation and rotation of colloidal particles in three dimensions was recently developed and used to determine accurately the diffusion coefficients of silica rods and spherical Janus particles in water Manoharan14 . Depending on the geometry of Janus particles (spherical, rod- or disk-like) and the degree of amphiphilicity, it was observed that particle rotational dynamics at sheared liquid-liquid interfaces involves either a smooth tilt or a tumbling motion Zadeh16 . In the previous study, the translational and rotational diffusion of a spherical Janus particle in a dense fluid was investigated using molecular dynamics simulations Kharazmi15 . In particular, it was shown that Janus particles with lower surface energy diffuse faster, and their nonwetting hemispheres tend to orient along the displacement vector of the center of mass during the rotational relaxation time Kharazmi15 . However, the combined effect of particle shape anisotropy and wettability contrast on diffusion remains not fully understood.

In this paper, we investigate the diffusion of a rod-shaped Janus particle in the limit of infinite dilution using molecular dynamics simulations. The particle consists of atoms rigidly fixed at the lattice sites that form a cylinder, which undergoes a diffusive motion under random forces from fluid molecules. In our model, the wall-fluid interaction energy at each half-side can be adjusted to control local wetting properties at the particle surface. We show that with increasing wettability contrast, the translational and angular displacements become larger and the effective center of rotation is displaced toward the wetting end. We also demonstrate that a Janus particle on average is rotated by its nonwetting side along the displacement vector of the center of mass to reduce drag.

The paper is organized as follows. The details of the simulation procedure and the particle model are described in the next section. In Sec. III, we report the fluid density profiles around particles, determine the local slip boundary conditions, and estimate translational and rotational diffusion coefficients from the particle trajectories and make a comparison with theoretical predictions. Brief conclusions are given in the last section.

II Simulation Method

We use molecular dynamics simulations to study the translational and rotational diffusion of a single particle in an explicit solvent Lammps . The model system consists of a rod-shaped Janus particle immersed in a monatomic fluid at equilibrium. A snapshot of the system is presented in Fig. 1. In this model, any two fluid atoms interact via the truncated Lennard-Jones (LJ) potential as follows:

[TABLE]

where the parameters $\varepsilon$ and $\sigma$ denote the energy and length scales of the fluid phase. For computational efficiency, the cutoff radius was set to $r_{c}=2.5\,\sigma$ for fluid-fluid and fluid-solid interactions. The fluid phase consists of $46\,536$ monomers of mass $m$ confined in a three-dimensional periodic cell with the linear side of $39.62\,\sigma$. When the finite size of a Janus particle is taken into account, the uniform fluid density away from the particle is $\rho=0.749\,\sigma^{-3}$. Periodic boundary conditions were applied in the $\hat{x}$, $\hat{y}$, and $\hat{z}$ directions. The MD simulations were carried out in the $NVT$ ensemble, where the temperature, $T=1.1\,\varepsilon/k_{B}$, was regulated by the Nosé-Hoover thermostat with the damping time of $1.0\,\tau$. Here, $k_{B}$ is the Boltzmann constant. The equations of motion for fluid monomers and the Janus particle were solved using the Verlet integration algorithm Allen87 ; Lammps with a time step $\triangle t_{MD}=0.005\,\tau$, where $\tau=\sigma\sqrt{m/\varepsilon}$ is the characteristic LJ time.

The Janus rod-shaped particle was constructed by arranging $72$ atoms at vertices of 12 hexagons that are stacked together and by adding two atoms at both ends, as shown in Fig. 2. Hence, the particle consists of total $74$ atoms that are fixed relative to each other and form a symmetric rod, which moves as a rigid body in the surrounding fluid. In this configuration, the distance between the outer hexagons along the $\textbf{{e}}_{1}$ axis is $6.35\,\sigma$ and the hexagon side is $0.71\,\sigma$, which is the same as the radius of a cylinder that contains all vertices. The size of all particle’s atoms is the same as the size of fluid monomers. The interaction between solid atoms of the Janus particle and fluid monomers is also described by the LJ potential but with different energies. On the wetting side, the interaction energy is fixed to $\varepsilon_{\text{pf}}=1.0\,\varepsilon$, while on the nonwetting side $\varepsilon_{\text{pf}}$ was varied from $0.1\,\varepsilon$ to $1.0\,\varepsilon$. For reference, the cases of uniformly wetting and nonwetting particles were also considered, where the interaction energy with fluid monomers was set to $\varepsilon_{\text{pf}}=1.0\,\varepsilon$ and $\varepsilon_{\text{pf}}=0.1\,\varepsilon$, respectively. Finally, the total mass of the rod-shaped particle is fixed to $M=50\,m$ in all simulations. The particle mass $M$ was chosen to be much larger that the mass of a fluid monomer $m$ in order to reduce backscattering effects at short times Levesque00 ; Shin10 , but, on the other hand, this mass is small enough so that the particle can undergo large displacements in a dense fluid, leading to an accurate determination of diffusion coefficients without the need of excessive computational resources.

The large-scale molecular dynamics simulations were performed using the LAMMPS parallel code Lammps . First, the system with the Janus particle and the fluid was equilibrated for $2\times 10^{7}$ MD steps (or $10^{5}\,\tau$), followed by a production run of about $10^{8}$ MD steps (or $0.5\times 10^{6}\,\tau$). The data were gathered in $50$ independent samples for uniform and Janus particles. For each sample, the position of the center of mass of the particle, its orientation vectors, as well as velocities and positions of all particle atoms were saved every $20$ MD steps, and these data were used later for post-processing.

III Results

We begin with a discussion of the fluid structure and local slip boundary conditions at the particle surface. The fluid density profiles around the nonwetting and wetting sides of Janus particles as well as around uniformly wetting and nonwetting particles are presented in Fig. 3. The density profiles were averaged in thin cylindrical shells of radius $r/\sigma$ around either wetting or nonwetting sides of Janus particles (see Fig. 2). It can be seen in Fig. 3 that in all cases the density profiles level off to the bulk value $\rho=0.749\,\sigma^{-3}$ at $r\gtrsim 6\,\sigma$; however, a pronounced density layering is present at smaller distances, and the amplitude of density oscillations increases at larger surface energies. Notice that the height of the first peak appears to be slightly larger near the nonwetting side of the Janus particle with $\varepsilon_{\rm pf}=0.1\,\varepsilon$ than in the case of the uniformly nonwetting particle with the same surface energy because fluid monomers near the center of the Janus particle interact with atoms of the wetting side. In other words, the contact density of the adjacent fluid monomers varies gradually along the main axis of Janus particles due to the finite cutoff radius of the LJ potential and difference in surface energy at both sides. Correspondingly, the first peak in the density profiles around the wetting side of Janus particles is slightly smaller for larger wettability contrast, as shown in the inset of Fig. 3. A similar effect of the contact density variation was observed in the previous MD studies of a spherical Janus particle in an explicit solvent Kharazmi15 and liquid films confined by surfaces of patterned wettability Priezjev05 ; Priezjev11 .

In order to determine the local flow boundary conditions at the particle surface, we carried out a set of MD simulations on a different system that consists of a monatomic fluid confined by smooth crystalline walls (but without Janus particles). The density of the fluid phase and solid walls were chosen to be the same as in the particle-fluid system described in the previous section. Special care was taken to match the nearest-neighbor distances between adjacent atoms at the particle surface (see Fig. 2) and the lattice constants of the crystalline walls ($0.707\,\sigma\times 0.577\,\sigma$) that consist of a single plane each. The steady Poiseuille flow was induced by applying a small force, $f\!=\!0.0005\,\varepsilon/\sigma$, to each fluid monomer in a direction parallel to the walls, while both walls remained at rest. As usual, the slip length was extracted from a parabolic fit to the velocity profile and then averaged over both interfaces Priezjev07 ; PriezjevJCP ; Priezjev10 . The results are presented in Table 1. It can be seen that the slip length increases for less wetting surfaces and it becomes larger than the particle size. We comment that this trend is expected to hold for atomically smooth interfaces when the wall-fluid interaction energy is sufficiently large Priezjev17 . We also remind that in the presence of curved surfaces, the slip boundary condition is modified by the local radius of curvature Priezjev06 ; Niavarani10 ; Koplik14 ; Robbins16 . Finally, the fluid viscosity was measured, $\eta=1.66\pm 0.03\,\varepsilon\tau\sigma^{-3}$, in steady flow at density $\rho=0.749\,\sigma^{-3}$ and temperature $T=1.1\,\varepsilon/k_{B}$.

It was previously shown that translational and rotational diffusion coefficients for a flat-end, rigid cylinder at short lag times can be approximated as follows:

[TABLE]

[TABLE]

[TABLE]

where $\omega=b/a$ is the aspect ratio and parameters $a$ and $b$ are the semi-minor and semi-major axes of the cylinder, and $\eta$ is the viscosity of the solvent Torre84 ; Manoharan14 . These predictions are expected to hold at lag times smaller than the typical rotational relaxation time scale, since at larger lag times the orientation of the major axis of the cylinder is decorrelated from its initial direction and the translational diffusion becomes isotropic. The interpolation equations, Eqs. (2)–(4), were derived for relatively short rigid cylinders with the aspect ratio in the range $2\lesssim\omega\lesssim 20$ Torre84 . In our study, the parameters $a=0.71\,\sigma$ and $b=0.32\,\sigma$ were augmented by $0.5\,\sigma$ to take into account the finite size of LJ atoms, which gives the aspect ratio $\omega\approx 3.04$ that was used for the comparative analysis described below.

We first plot the rotational autocorrelation function of the unit vector $\textbf{{e}}_{1}$ along the major axis of the rod-shaped particle $\langle\textbf{{e}}_{1}(0)\cdot\textbf{{e}}_{1}(t)\rangle$ in Fig. 4. As expected, the rotational diffusion is enhanced for the uniformly nonwetting particle and for Janus particles with lower surface energy at the nonwetting side. It can be observed from Fig. 4 that the data for different surface energies can be well described by the exponential decay

[TABLE]

where $\tau_{r}$ is the rotational relaxation time scale, and the corresponding rotational diffusion coefficient is given by $1/(2\tau_{r})$ Manoharan14 . The inset in Fig. 4 shows the variation of $\tau_{r}$ as a function of the surface energy at the nonwetting side of the particle. It can be seen that the rotational relaxation time gradually varies between bounds determined by the limiting cases of uniformly wetting and nonwetting particles. The same data for $\tau_{r}$ are listed in Table 2. We comment that the effect of the particle shape, rod-like versus spherical, on rotational diffusion is evident from the results reported in Fig. 4 in the present study and Fig. 4 in Ref. Kharazmi15 . Namely, in the case of rod-shaped particles, the relaxation time associated with rotation of the unit vector $\textbf{{e}}_{1}$ is larger by a factor $\approx 1.5\div 3$ for the same particle mass, surface wettability, fluid density and temperature Kharazmi15 .

Another way to quantify the rotational diffusion is to compute the average mean square angular displacement during the time interval $t$ as follows:

[TABLE]

where $D_{r}$ is the rotational diffusion coefficient and $\vec{\varphi}(t)$ is the total angular displacement vector defined by

[TABLE]

with the magnitude of $\Delta\vec{\varphi}(t^{\prime})$ given by $\textrm{cos}^{-1}(\textbf{{e}}_{1}(t)\times\textbf{{e}}_{1}(t+t^{\prime}))$, which is the rotation angle of the vector $\textbf{{e}}_{1}$ during the time interval $t^{\prime}$ Kob97 ; Weeks12 . In this definition, the mean square angular displacement $\langle\Delta\vec{\varphi}^{2}(t)\rangle$ is unbounded and the diffusion coefficient is evaluated in the linear regime at large times Kob97 .

The results for the mean square angular displacement obtained from MD simulations are displayed in Fig. 5 for uniform and Janus particles. It is clearly seen that Janus particles with larger wettability contrast diffuse faster, and their angular displacements are greater (smaller) than that of uniformly wetting (nonwetting) particles. The values of the rotational diffusion coefficient obtained from the linear fit to Eq. (6) are listed in Table 2 along with the prediction of Eq. (4). Thus, we conclude that both methods of evaluation of the diffusion coefficient, i.e., $D_{r}$ from Eq. (6) and $1/(2\tau_{r})$ from Eq. (5) give consistent results. However, we find that the value of the diffusion coefficient for a uniformly wetting particle is significantly larger than the analytical prediction of Eq. (4), possibly due to finite slip at the particle surface in MD simulations. It should be noted that a similar discrepancy between the prediction of Eq. (4) and MD results for the rotational diffusion of a carbon nanotube in a LJ fluid was recently reported by Cao and Dong Dong14 .

The mean square displacement in the directions parallel and perpendicular to the main axis of the uniformly wetting particle ($\varepsilon_{\rm pf}=1.0\,\varepsilon$) as well as its total mean square displacement are presented in Fig. 6. As is evident, all curves have a unit slope at large times but with different proportionality coefficients. By definition, the translational diffusion coefficient is a combination of parallel and perpendicular diffusion coefficients, i.e., $D_{t}=(2D_{\bot}+D_{||})/3$. To facilitate comparison, the data for the parallel, perpendicular, and total mean square displacements were multiplied by factors $6$, $3$, and $2$, respectively, and replotted in the inset of Fig. 6. It can be observed that diffusion in the direction parallel to the major axis is faster than in the perpendicular direction at $t\lesssim 500\,\tau$, while at larger times diffusion becomes isotropic as the orientation of the vector $\textbf{{e}}_{1}$ is decorrelated (see Table 2). As shown in Fig. 7, the same trends were observed for uniformly nonwetting and Janus particles. We also note that the gradual crossover from short-time anisotropic to long-time isotropic diffusion was observed for isolated ellipsoidal particles in water Lubensky06 .

In our analysis, the translational diffusion coefficients in the directions parallel and perpendicular to the major axis of a particle (along the vector $\textbf{{e}}_{1}$) were evaluated using the following equations:

[TABLE]

[TABLE]

where $t_{0}$ is the reference time and the brackets $\langle..\rangle$ indicate averaging over all $t_{0}$. The Eqs. (8)-(9) were used to fit the data shown in Figs. 6 and 7 in the range $20\,\tau\lesssim t\lesssim\tau_{r}$. The results from numerical simulations and theoretical predictions of Eqs. (2)-(4) are reported in Table 2. It can be observed that in the case of uniformly wetting particle, the values of the diffusion coefficients obtained from the particle trajectory agree well with predictions of Ref. Torre84 . We comment that this agreement might be coincidental because of the no-slip boundary condition assumed in derivation of Eqs. (2)-(4), while in MD simulations the local slip length at the particle’s surface is relatively large even for the uniformly wetting case (see Table 1). Generally, the data in Table 2 follow the same trend; namely, the diffusion coefficients are largest (smallest) for uniformly nonwetting (wetting) particles, and the diffusion is enhanced for Janus particles with less wetting surfaces. The same behavior was reported for spherical Janus and uniform particles in the previous MD study Kharazmi15 .

We next discuss more subtle aspects of the rotational dynamics arising due to the wettability contrast and asymmetrical shape of particles. The time dependence of the correlation function $\langle\textbf{{e}}_{1}(0)\cdot\textbf{{e}}_{1}(t)\rangle$, shown in Fig. 4, provides an estimate of the rotational relaxation time scale $\tau_{r}$, but it does not describe the relative motion of the wetting and nonwetting sides. From the analysis of particle trajectories, we computed the total length of paths traveled by the centers of the wetting and nonwetting sides separately. As expected, the lengths of such trajectories will be on average the same for uniformly wetting and nonwetting particles (data not shown). We also comment that the mean square displacements of the centers of wetting and nonwetting sides will also approach the same values at long times because they are determined by the displacement of the particle’s center of mass. In contrast, as shown in Fig. 8, the center of the nonwetting side follows longer trajectory than the center of the wetting side of a Janus particle. The maximum difference in length of the trajectories is about $2.2\times 10^{3}\,\sigma$ at $t=3\times 10^{5}\,\tau$ for the Janus particle with the maximum wettability contrast of ($1.0\,\varepsilon$, $0.1\,\varepsilon$). These results demonstrate that during diffusive motion of Janus particles, the nonwetting side ‘effectively’ rotates around the wetting side. In other words, the center of rotation is displaced along the main axis toward the wetting side of the Janus particle.

Another peculiar feature of the diffusive motion of Janus particles is the correlation between translation and rotation due to asymmetric wetting Kharazmi15 . When, due to thermal fluctuations, a Janus particle acquires a translational velocity in a certain direction, its nonwetting side will tend to rotate toward the displacement to reduce drag. This effect can be quantified via the average rotation angle of the major axis along the displacement vector of the center of mass. We first consider the displacement vector of the center of mass $\triangle\textbf{r}$ during a time interval $t$ and then estimate the difference between the angles that vectors $\textbf{{e}}_{1}(0)$ and $\textbf{{e}}_{1}(t)$ make with respect to $\triangle\textbf{r}$. In Fig. 9, the rotation angle averaged over particle trajectories is plotted as a function of time for Janus and uniformly wetting particles. It can be seen that Janus particles indeed preferentially rotate along the displacement, and the maximum rotation angle is attained at intermediate times that roughly correspond to the rotational relaxation time scales $\tau_{r}$ (listed in Table 2). Interestingly, the maximum values of the rotation angle shown in Fig. 9 are very close to the values reported in Fig. 7 in Ref. Kharazmi15 for spherical Janus particles with the same wettability contrast. However, the maximum rotation of rod-shaped Janus particles occurs at larger times than for spherical Janus particles due to larger moment of inertia in the former case. Finally, as shown in Fig. 9, the effect of correlated rotation is absent for uniformly wetting and nonwetting particles.

IV Conclusions

In this paper, molecular dynamics simulations were performed to investigate the diffusive dynamics of Janus rod-shaped particles in an explicit solvent. We considered the limit of low dilution where interaction between particles can be neglected. The Janus particle was modeled as a rigid body where atoms are fixed at the vertices of adjacent hexagons and form a rod with the aspect ratio of about 3. The interaction energy between fluid monomers and particle’s atoms was set to different values on both half-sides of a Janus particle. Two limiting cases of uniformly wetting and nonwetting particles were also considered for reference.

It was shown that both rotational and translational diffusion are enhanced for Janus particles with higher wettability contrast, while the largest (smallest) values of the corresponding diffusion coefficients were obtained for uniformly nonwetting (wetting) particles. Moreover, the estimate of the diffusion coefficients for displacement of the center of mass in the direction perpendicular and parallel to the major axis agree well with theoretical predictions in the case of uniformly wetting particle. The numerical analysis of the particle trajectories revealed that the effective center of rotation of Janus particles is displaced along the major axis toward the wetting side. Finally, the results of our MD simulations indicate an unusual feature of the diffusive motion of Janus particles; namely, the nonwetting side of the Janus particle is rotated on average along the displacement vector of the center of mass in order to reduce the friction force from the surrounding fluid. Interestingly, the maximum value of the rotation angle for a given wettability contrast of rod-shaped particles is very close to the values reported in our previous study on diffusion of spherical Janus particles Kharazmi15 .

Acknowledgments

Financial support from the National Science Foundation (CBET-1033662 and CNS-1531923) is gratefully acknowledged. Computational work in support of this research was performed at Michigan State University’s High Performance Computing Facility and the Ohio Supercomputer Center. The molecular dynamics simulations were conducted using the LAMMPS numerical code developed at Sandia National Laboratories Lammps .