in bath particle concentration, which is utilized by the motor. Two possible scenarios for the motor are studied: one consisting of a motor held fixed and another one where the motor is free to move. We discuss the theory of osmotic propulsion in Section 4.2. In Section 4.3, we investigate the net osmotic force created by the fixed motor and its dependence on the angle θs. We compute the optimal angleθs necessary to obtain maximum osmotic force as a function of the Damk¨ohler number. For the free motor scenario, the velocity of the motor for variousθsand its implications for different bath particle concentrations and motor sizes are discussed. The behavior of the free motor is maximized by finding the optimal angleθs needed to create maximum velocity according to the Damk¨ohler number and the properties of the suspension. Theoretical results are compared to Brownian dynamics simulations (Foss and Brady 2000; Carpen and Brady 2005) modified to consider surface reactions. We offer some concluding remarks in Section 4.4.
determined by the forces of interaction among the particles; the solvent or fluid only enters in the form of a thermal bath providing kT of energy for each degree of freedom of the particles — the well-known fluctuation-dissipation theorem. A portion of the motor surface is reactive such that the bath particles undergo an irreversible first-order reaction with speed κ. For simplicity, we assume bath particles that undergo reaction are consumed1. Figure 4.1 shows a definition sketch of the osmotic motor-bath particles configuration. Note that in the figure we have chosen the reactive site to be axisymmetric with the z-axis. The interface between reactive and passive surfaces is defined by the polar angleθs, which takes values from 0 (completely inert) toπ (completely reactive), and becomes the only variable that determines the distribution of reaction on the motor. Bath particles that collide on the passive surface, or do not react at all, are reflected by a hard-sphere potential, so that the particles do not interact until their hard-sphere radii touch, whereupon a force is exerted to prevent the hard-sphere radiiaand bof the motor and bath particles, respectively, from overlapping.
Consumption of bath particles at the reactive portion of the motor’s surface creates a low bath particle concentration region (depletion zone) that drives the microstructure away from equilibrium. This creates local bath particle concentration gradients that the motor can potentially use for propulsion (or for pumping fluid). Physically, the motor experiences fewer collisions on the reactive surface than on the passive (inert) surface, thus generating an imbalance in osmotic pressure about the motor, and consequently, a net “osmotic” force.
Because rotational diffusion of particles is not included in this model (torque-free particles), the reactive site always points in the direction of this self-created net osmotic force, here
1This simple reaction could be interpreted as if reactant particles are instantaneously transformed into solvent molecules or into products that diffuse very fast away from the motor. Thus, mass is conserved and there is no need to resupply particles to the suspension. This issue was addressed in our previous work, C´ordova-Figueroa and Brady (2008), and in Chapter 3. A simple scaling argument was derived to account for other reaction stoichiometries.
F osm
z y
a
b θs
Figure 4.1: Definition sketch for the suspension: An osmotic motor of radiusawith a first- order reaction on a portion of its surface determined by the polar angle θs (located in the z-axis) surrounded by bath particles of radii b induces an osmotic force Fosm that points towards low bath particle concentration regions. Particles are modeled as hard-spheres.
The angle θs is measured from the front and takes values from 0 (completely inert) to π (completely reactive).
taken to be thez-coordinate (see Figure 4.1). To understand the resulting behavior of the osmotic motor, it is necessary to compute the local microstructural deformations caused by the nonuniform reaction, specifically, the pair-distribution function g(r) thus determining the probability of finding a bath particle at a vector separationr from the motor.
Local deformations to the microstructure are of central importance to the driving force of the motor, therefore, we must solve for the distribution of bath particles surrounding the motor. The stochastic nature of Brownian motion requires such distribution to be found statistically. We denote the probability density of finding the motor and bath par- ticles in a given configuration at time t, PN(x1,x2, ...,xN, t), where the label 1 denotes the motor and 2 → N the bath particles, and xi denotes the position of particle i. The non-equilibrium microstructure of colloidal dispersions can be directly calculated from a
Smoluchowski equation (Batchelor 1977, Felderhof 1988):
∂PN
∂t +
N
X
i=1
∇i·ji = 0 (4.1)
where ji is the probability flux carried by particle i. In the absence of hydrodynamic interactions and rotational motion, the balance of instantaneous forces acting on the motor exerted by particle j is given by
FH1 =
N
X
j=1
Fext+FPj −kT∇jlnPN
, (4.2)
whereFHis the hydrodynamic force that at low Reynolds number is linear in the velocityU; Fextis an external force (e.g., to hold particle, to pull particle through material) acting on the motor;FP is the interparticle force, which we have assumed is central and corresponding to a hard-sphere potential; and −kT∇lnPN is the entropic, or Brownian, force on the motor due to random thermal fluctuations of the solvent molecules (Batchelor 1976). Bath particles in the suspension (labels 2→ N) move under the action of hydrodynamic forces and entropic forces, given by a force balance similar to (4.2) but with zero external forces, Fext= 0.
A closed equation for the pair-distribution function g(r) is obtained by integrating the N-particle Smoluchowski equation (4.1) over the configurational degrees of freedom ofN−2 bath particles, neglecting interactions between bath particles. The neglect of such high-order couplings restricts the theory to low bath particle volume fractions, φb = 4πb3nb/3 1;
only one bath particle interacts with the motor, therefore, behaving as an ideal gas. For a detailed derivation of the pair-level Smoluchowski equation see Squires and Brady (2005) in their work on single particle motion in colloidal dispersions and microrheology in the
absence of hydrodynamic interactions.
The average force of the osmotic motor is obtained from (4.2) by averaging over the po- sitions of theN −1 bath particles. The N-particle probability distribution can be written asPN(x1,x2, ...,xN, t) =PN−1/1(r2,r3, ...,rN, t|x1, t)P1(x1, t), wherePN−1/1 is the condi- tional probability for finding the N−1 bath particles in configuration rN−1 given that the motor is at location x1. Owing the statistical homogeneity of the suspension, PN−1/1 does not depend on the position of the motor. Thus, the average force of the motor is defined
hFH1 i1= Z
FH1 PN−1/1(r2,r3, ...,rN, t)dr2...drN, (4.3)
where the subscript 1 on h i1 indicates to the conditional average relative to the motor atx1. A procedure to solve (4.3) assuming diluteness and hard-sphere interparticle forces is discussed in Squires and Brady (2005) for averaging stochastic quantities, so we shall proceed quickly. At the two-body level, the averaged forces acting on the osmotic motor becomes
FH =Fext−nbkT I
r=a+b
ng(r)dS, (4.4)
where the integral is over the surface of contact between the motor and bath particles, r=a+b, with normalndirected radially out from the motor. We have dropped the brackets and label 1 denoting the motor for clarity. The hydrodynamic Stokes drag force, defined as FH = −6πηaU (where U is the motor velocity), is balanced by a constant external force Fextand the average entropic or osmotic force, Fosm=−nbkTH
r=a+bng(r)dS, exerted on
the motor by collisions with bath particles. From Eq. (4.4) we see clearly that the thermal motion of the bath particles acts to resist the motion of the motor or, if present, the external force. In the absence of surface reaction on the motor (inert colloids), the pair-distribution
function g(r) is isotropic and the integral in (4.4) is zero. Indeed, if the particle is fully reactive over its surfaceg(r) is also isotropic. This implies that for the motor to create an anisotropic environment, or regions of low bath particle concentration necessary to drive the suspension away from equilibrium, the reaction must be on a portion of the motor to break the symmetry.
To compute the net osmotic force acting on the motor, we need to determine the pair- distribution function g(r) about the motor. In addition to studing the behavior of the osmotic motor subject to different distributions of reactive sites, we are interested in op- timizing the resulting osmotic force and providing ideas on how to design better osmotic motors in terms of the properties considered here for the simple model motor. In a particle- level description, the osmotic force increases if the motor maximizes the number of hard- sphere collisions with bath particles in the vicinity on its left hemisphere (negativez-axis in Figure 4.1). We demonstrate that this can be easily done by selecting the right distribution of reaction on the spherical motor for a given Damk¨ohler number, and also by taking into consideration the number of bath particles in the vicinity that collide with the motor. We shall see that half reactive or fast reaction is not necessarily the right design choice for spheres if maximum osmotic force is desired. Osmotic motors could take advantage of their geometry and the flux of bath particles toward the motor to minimize the reactive portion on the surface, an option often needed for designing and constructing ‘efficient’ nanodevices.
Two scenarios for the osmotic motor are considered: one in which the motor is held fixed and another one where the motor is free to translate with velocity U. A spherical coordinate system (r, θ) is adopted to model the particles.
4.2.1 Fixed motor
In the fixed motor scenario, the osmotic motor experiences no motion. The net osmotic force created autonomously by the surface reaction balances the external force that keeps the motor fixed (Fext=Fosm). Therefore, only bath particle positions evolve by Brownian motion. The osmotic force along thez-axis is given by
Fosm=−nbkBT(a+b)2 I
nzg(r= 1;Da, θs) dΩ, (4.5)
where dΩ =dS/(a+b)2 is the solid angle of integration and nz is a unit vector along the z-axis. We have made length dimensionless by the contact distance a+b. The probability density flux is simply the diffusion of bath particles j = −D∇g. Conservation of bath particles about the fixed motor satisfies the steady-state diffusion equation,
∇2g= 0, (4.6)
subject to the undisturbed microstructure condition far from the motor,
g∼1 asr→ ∞, (4.7)
and at contact, the flux to the motor is balanced by the nonuniform reaction:
n· ∇g=Da h(n)g. (4.8)
The parameter that governs the motor behavior is the ratio of the speed of reaction to that of diffusion — the so-called Damk¨ohler number Da=κ(a+b)/D, where the characteristic
length is the sum of the motor and bath particle radii andDis the relative diffusivity. Here, the relative diffusivityD is simply the diffusivity of bath particles Db. The distribution of reaction on the surface is determined by the dimensionless function h(n), which we take to be 1 on the reactive surface (0 < θ < θs) and 0 on the passive surface (θs < θ < π) (see Figure 4.1). We must compute the pair-distribution function at contact, and thus determine the net osmotic force. For simplicity, we shall refer to the integral of the pair-distribution over the surface of contact in (4.5) as F(Da, θs) = −4π3 H
r=1nzg(r)dΩ, a nondimensional function of the Damk¨ohler number and the angle θs that represents the asymmetric bath particle distribution created by the nonuniform reaction on the motor. The osmotic force can be maximized by finding the best distribution for reactive sites for each Damk¨ohler number. These values are obtained from the derivative of the nondimensional function F(Da, θs) with respect of Da and keeping the angle θs constant, which at maximum, the derivative must equal zero.
4.2.2 Free motor
The second scenario corresponds to a motor that is free to translate at velocityU due to the reaction-driven osmotic force. No external forces, Fext = 0, or imposed particle gradients act on the motor. By contrast, the osmotic motor migrates autonomously through the suspension. The resulting net osmotic force balances the hydrodynamic Stokes drag force FH, giving the motor velocity along thez-axis
U =−nbDa(a+b)2 I
nzg(r= 1;Da, P e, θs) dΩ, (4.9)
whereDa=kT /6πηa is the Stokes-Einstein-Sutherlad diffusivity of the osmotic motor. In a reference frame moving with the motor, which translates at velocityU, each bath particle is advected with velocity −U and moves diffusively. Therefore, the probability density flux is given by j = −D∇g−Ug. Continuity of bath particles requires that ∇ ·j = 0, so that, at steady-state, the pair-distribution function g(r) for the free motor satisfies the advection-diffusion equation
∇2g=−P e∂g
∂z, (4.10)
and has taken the motion to be along thez-axis. The associated boundary conditions are
g∼1 asr→ ∞, (4.11)
∂g
∂r = (Da h(n)−P eµ)gatr= 1, (4.12) so that n·(U/U) = cosθ = µ. We made all lengths dimensionless by scaling it with the contact distance a+b, and velocity with the yet unknown osmotic velocity U. The expression at contact (4.12) shows the competition between self-propulsion of the motor in driving the environment away from equilibrium and Brownian motion attempting to restore the disturbed microstructure. The P´eclet number P e is the ratio of the “self-induced”
osmotic velocity U to the relative Brownian velocityD/(a+b). The free motor undergoes Brownian motion; therefore, the relative diffusivityDis now the sum of the motor and bath particle diffusivity: D=Da+Db. From (4.9) the implicit equation for the P´eclet number is
P e= U(a+b)
D =βF(Da, P e, θs), (4.13)
whereβ =φb(1+a/b)2andF(Da, P e, θs) is now a nondimensional function of the Damk¨ohler number, the P´eclet number, and the angleθs;F(Da, P e, θs) must be solved simultaneously with the governing equations to get a solution for the motor velocityU. The P´eclet number is not an independent parameter, but rather is regulated byDaandβ. In fact, the resulting P e could behave differently depending on the portion of the reactive surface. The product β corresponds to the number of bath particles within a bath particle radius of the motor surface: β∼nb(a+b)2b.
Consumption of bath particles near the reactive surface, which is controlled by Da, creates a low bath particle concentration region in which the motor prefers to move. As it moves forward, the osmotic motion acts as a source/sink of bath particles on the motor particle surface. On the reactive surface, the osmotic motion is a source of bath particles, which eventually reduces the motor’s driving force and therefore its speed by restoring the depleted region. On the rear surface, the motor creates a wake of low bath particle concentration, again reducing the motor velocity. This has the physical effect of limiting the osmotic pressure imbalance resulting from the surface reaction. Note that at P e = 0 the free motor problem reduces to the scenario in which the motor is held fixed, giving similar microstructural deformation for active surface reaction. Here, osmotic propulsion is maximized by finding a set of values (θs,Da) where the derivative of F(Da, P e, θs) with respect toDaresults in zero. The calculations are repeated for different values ofβ.