reactions (Northrup et al. 1984; Allison et al. 1990). These studies were relevant in order to suggest a simple algorithm that emulates the irreversible first-order reaction on the motor’s surface. We describe the two possible scenarios for the motor as well. We calculate the driving force on the motor, and, in the case of the free motor, the osmotic velocity. BD simulations provide a means of checking and extending our theoretical analyses and allow us to design and analyze experimental systems in which different effects can be cleanly sep- arated. Theoretical predictions are compared with BD simulation and reported in Section 3.5.
In Section 3.6 we suggest how the dilute results can be scaled up to higher concentra- tions. We examine in Section 3.7 the force fluctuations resulting from random collisions between the motor and bath particles and its behavior as the speed of reaction, the bath particle concentration, and particle size are changed. In Section 3.8 we propose a formula to determine the energy conversion efficiency of chemical energy (supplied by the reactants) into motion via the dissipation of mechanical energy. Lastly, some concluding remarks are presented in Section 3.9.
F osm
z y
a
b
Figure 3.1: Model system for osmotic propulsion. A motor particle of radiusawith a first- order reaction on half its surface (located in the z-axis) surrounded by bath particles of radii binduces an osmotic force Fosm that points towards low bath particle concentration regions. Particle interactions are modeled with a hard-sphere potential.
that describes the fluid inertia is small, thus enabling the use of the Stokes equations. The suspension of bath particles generates an osmotic pressure in the system Π =nbkT, where kT is the thermal energy and nb is the number density of bath particles. Our goal is to develop a theory that models the microstructural evolution of the suspension that is driven out of equilibrium by a chemical reaction on the motor surface.
Because asymmetric deformations of the suspension microstructure are of primary im- portance to obtain a “driving” force, we must solve for the distribution of bath particles surrounding the osmotic motor. Our point of departure is the N-particle Smoluchowski equation governing the spatio-temporal evolution of theN-particle probability density func- tion PN(xN, t) for finding theN particles at position xN in the fluid:
∂PN
∂t +
N
X
i=1
∇i·ji = 0, (3.1)
where ji is the probability flux of particle i and the sum is over all the particles in the
suspension. The motion of an individual particle is governed by a balance among hydrody- namic, Brownian, and interparticle forces. Thus, the probability flux carried by particle i is given by
ji =UiPN −
N
X
j=1
DijPN·∇j(logPN+VN/kT), (3.2)
where Ui is the velocity of particle i, Dij is the relative Brownian diffusivity tensor, and VN is the N-particle interaction potential, which is assumed to be central. There is no uniform flow at infinity in the “laboratory” frame. For simplicity, we neglect hydrodynamic interactions and rotary Brownian motion. To model the interactions among the particles, it is assumed that VN is a hard-sphere potential so that the colloidal particles do not interact until their hard-sphere radii touch. The hard-sphere potential generates a force that prevents the hard-sphere radii, a and b of the motor and bath particles, respectively, from overlapping. The instantaneous thermal or Brownian force on the motor due to random collisions with solvent molecules is given by FB =−kT∇logPN. This expression for the instantaneous Brownian force is exact for all bath particle volume fractions,φb = 4πb3nb/3.
At equilibrium the absence of any external (e.g., optical tweezers) or self-induced (e.g., swimming bacteria, catalytic nanomotor) forces implies that Ui = 0 for each particle and the probability distribution is independent of time. This results in a balance between the interparticle potential and thermal forces, which is solved by the well-known Boltzmann distribution functionPNeq∼exp(−VN/kT). Application of an external force or gradient, or a reaction-induced force, will cause relative motion among the particles in the suspension, drive the system away from equilibrium, and PN away from the Boltzmann distribution PNeq.
To proceed analytically, it is necessary to restrict the analysis to the limit where bath
particles do not interact with each other and therefore behave as an ideal gas. Thus, only the motor interacts with the bath particles. We proceed by integrating Eq. (3.1) over the configurational degrees of freedom ofN −2 bath particles, neglecting interactions between bath particles. The neglect of such higher-order couplings could also be interpreted as our theory is restricted to low bath particle volume fractions, φb 1, for which only one bath particle interacts with the motor. Similarly, averaging the balance between the instantaneous Brownian and the hard-sphere force over the positions of the N −2 bath particles and integrating by parts, results in an exact formula for the “osmotic” force on the motor due to collisions with bath particles,
Fosm =−nbkT I
r=a+b
ng(r)dS. (3.3)
In (3.3) the integral is over the surface of contact between the osmotic motor and bath particles, r = a+b, and n is the normal pointing out of the motor particle. The pair- distribution function g(r) defined as n2bg(r) = ((N −2)!)−1R
PN(rN, t)dr3...drN is the probability density for finding a bath particle at r relative to the motor. The interested reader can find a detailed derivation of the above equations in Squires and Brady (2005).
Eq. (3.3) is nothing more than the osmotic pressure Π sensed by the motor times the surface area available for collisions between the motor and bath particles. In the absence of any disturbance to the surrounding medium, no structural deformation is present; thus, g(r) is isotropic and the integral in (3.3) is zero, resulting in no net osmotic force. However,
if there is a non-uniform concentration of bath particles, either caused by an externally imposed concentration gradient or by the motor itself via a chemical reaction on its surface, there will be a net osmotic force on the motor. This osmotic force must be balanced by an
externally imposed forceFext(via, e.g., optical tweezers, magnets) to hold the motor fixed, or by the hydrodynamic Stokes drag force Fhyd = −6πηaU. A similar observation was pointed out by Batchelor (1983) in the problem of multicomponent diffusion. The approach here produces precisely the result for the flux of one species due to a concentration gra- dient of another species as derived by Batchelor (1983) when hydrodynamics interactions are neglected. The surface integral ofg(r) at contact in Eq. (3.3) represents the local con- centration of bath particles. And independently of what mechanism or input is responsible for producing a variation, a nonuniform distribution can drive the motor. An externally imposed concentration gradient gives rise to what is known as diffusiophoresis (Anderson 1989), whereas a surface chemical reaction alters the local concentration of bath particles and results inautonomousmotion — namely, the osmotic motor.