^{1,a)}and Raymond Kapral

^{1,b)}

### Abstract

The chemically powered self-propelled directed motions of nanodimer motors confined in a rectangular channel and subject to an applied external conservative force are investigated using hybrid molecular dynamics/multiparticle collision dynamics. The influence of factors, such as dimer sizes, chemical reaction type, and the nature of the interaction potentials between dimer monomers and solvent molecules, on the propulsion force and friction constant are examined. The stall force, for which the nanodimer has zero net velocity, and the thermodynamic efficiency of the motor are calculated. Both irreversible and reversible chemical reactions are considered. The simulation results are compared to theoretical predictions which are able to capture the major features of the self-propelled motion.

Research supported, in part, by a grant from the Natural Sciences and Engineering Research Council of Canada.

I. INTRODUCTION

II. NANODIMER MODEL AND DYNAMICS

III. PROPULSION AND STALL FORCES

A. Stall force

IV. EFFICIENCY

V. CONCLUSION

## Figures

Instantaneous configuration of (dots) molecules in the vicinity of the nanodimer. The rectangular simulation box has dimensions and contains 512 000 solvent molecules in total. The molecules, which constitute the majority of the solvent particles, are not shown. The diameters of the catalytic (small) and the noncatalytic (large) spheres are and , respectively. Attractive LJ interactions between the solvent molecules and the noncatalytic sphere with an energy parameter are used. The arrow shows the direction along which a conservative external force is applied to the nanodimer center-of-mass. The self-propelled directed motion of the nanodimer is opposite to that of the applied force.

Instantaneous configuration of (dots) molecules in the vicinity of the nanodimer. The rectangular simulation box has dimensions and contains 512 000 solvent molecules in total. The molecules, which constitute the majority of the solvent particles, are not shown. The diameters of the catalytic (small) and the noncatalytic (large) spheres are and , respectively. Attractive LJ interactions between the solvent molecules and the noncatalytic sphere with an energy parameter are used. The arrow shows the direction along which a conservative external force is applied to the nanodimer center-of-mass. The self-propelled directed motion of the nanodimer is opposite to that of the applied force.

The solvent molecule velocity field in the vicinity of the nanodimer. (a) No external force is applied. (b) The external force is , for which the velocity of the nanodimer is nearly zero. In these simulations, the monomer diameters are and , and the wall separations are . The noncatalytic monomer interacts with solvent particles through attractive LJ potentials with energy parameter .

The solvent molecule velocity field in the vicinity of the nanodimer. (a) No external force is applied. (b) The external force is , for which the velocity of the nanodimer is nearly zero. In these simulations, the monomer diameters are and , and the wall separations are . The noncatalytic monomer interacts with solvent particles through attractive LJ potentials with energy parameter .

The average velocity of the nanodimer as a function of the applied external force . The monomer sizes are and in these simulations. For each value of the applied external force, the self-propulsion velocity of the nanodimer along its internuclear axis was determined from an average over ten independent realizations. Circles denote results where no chemical reaction occurs at the catalytic monomer; thus, the simulation box contains only solvent molecules. Triangles and diamonds denote results where the interactions between the noncatalytic sphere and solvent particles are through either repulsive or attractive LJ potentials, respectively. In these two cases, the reaction probability is and the reaction is irreversible. The squares denote results for reversible reactions with for attractive interactions between the sphere and particles. All simulations were carried out in rectangular boxes with wall separations of .

The average velocity of the nanodimer as a function of the applied external force . The monomer sizes are and in these simulations. For each value of the applied external force, the self-propulsion velocity of the nanodimer along its internuclear axis was determined from an average over ten independent realizations. Circles denote results where no chemical reaction occurs at the catalytic monomer; thus, the simulation box contains only solvent molecules. Triangles and diamonds denote results where the interactions between the noncatalytic sphere and solvent particles are through either repulsive or attractive LJ potentials, respectively. In these two cases, the reaction probability is and the reaction is irreversible. The squares denote results for reversible reactions with for attractive interactions between the sphere and particles. All simulations were carried out in rectangular boxes with wall separations of .

The stall force as a function of (a) the energy parameter and (b) the sphere diameter . The interaction potentials between the sphere and particles are repulsive and attractive in panels (a) and (b), respectively. Panel (c) plots as a function of the sphere diameter in the absence of an external force. The reaction probability in both cases is . The sphere diameters are and in (a). The sphere diameter is fixed at in (b) and (c), while the potential parameter .

The stall force as a function of (a) the energy parameter and (b) the sphere diameter . The interaction potentials between the sphere and particles are repulsive and attractive in panels (a) and (b), respectively. Panel (c) plots as a function of the sphere diameter in the absence of an external force. The reaction probability in both cases is . The sphere diameters are and in (a). The sphere diameter is fixed at in (b) and (c), while the potential parameter .

Thermodynamic efficiency of the nanodimer as a function of the applied external force. Curves are fits using Eq. (15). In (a), the interaction forces between the monomer and particles are through repulsive LJ potentials, while in (b) they are attractive. The energy parameter is in both cases. The reaction probability of the reversible reaction varies from to 0.99. The diameters of the and monomers are and , respectively. Each value of the thermodynamic efficiency is the result of an average over ten independent realizations.

Thermodynamic efficiency of the nanodimer as a function of the applied external force. Curves are fits using Eq. (15). In (a), the interaction forces between the monomer and particles are through repulsive LJ potentials, while in (b) they are attractive. The energy parameter is in both cases. The reaction probability of the reversible reaction varies from to 0.99. The diameters of the and monomers are and , respectively. Each value of the thermodynamic efficiency is the result of an average over ten independent realizations.

## Tables

Average velocities of the center of mass of the nanodimer along its internuclear axis for an irreversible reaction. The applied constant external force varies from to −9.0. The diameters of the and spheres are and , respectively. The internuclear separation is . There are attractive interactions between sphere and molecules with . The theoretical values are determined from Eq. (9) and the theoretical estimate for the friction in Eq. (11), while the hybrid results are calculated using the friction coefficient determined from simulations instead of the Oseen approximation. The lower portion of the table presents the results for a reversible reaction with with external forces ranging from to −4.0. Other parameters are the same as those in the top and middle parts of the table.

Average velocities of the center of mass of the nanodimer along its internuclear axis for an irreversible reaction. The applied constant external force varies from to −9.0. The diameters of the and spheres are and , respectively. The internuclear separation is . There are attractive interactions between sphere and molecules with . The theoretical values are determined from Eq. (9) and the theoretical estimate for the friction in Eq. (11), while the hybrid results are calculated using the friction coefficient determined from simulations instead of the Oseen approximation. The lower portion of the table presents the results for a reversible reaction with with external forces ranging from to −4.0. Other parameters are the same as those in the top and middle parts of the table.

Comparison of various quantities for systems with either attractive or repulsive LJ potentials between the sphere and solvent particles. The simulations are for a system in a box with size . The reaction probability is . The monomer diameters are and , respectively. The internuclear separation is .

Comparison of various quantities for systems with either attractive or repulsive LJ potentials between the sphere and solvent particles. The simulations are for a system in a box with size . The reaction probability is . The monomer diameters are and , respectively. The internuclear separation is .

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