Free motor

3.3 Microstructural deformation by chemical reaction

3.3.2 Free motor

fusivity as the reactant, the net osmotic force would be zero. This would correspond, if you will, to simply changing the “color” of the reactant which cannot produce any net force. However, if the reactants and products have different diffusivities, say because the reaction changes the “shape” of the particle, or perhaps its interaction with the solvent (hydrophilic/hydrophobic), then even if only one product is produced for each reactant there will still be a net osmotic force on the motor. The sign of the force will depend on which diffuses faster. As another example, suppose that s = 2, but the products and reactants have the same diffusivity. There will be now a net increase of bath particles on the reactive side and the osmotic force will be to the left in Figure 3.1 rather than to the right. If we restrict ourselves to strictly hard-sphere particles of the same density, then D_R/D_P =b_P/b_R=s^−1/3 from conservation of mass, and thus the stoichiometric/diffusivity factor becomes (1−sD_R/D_P) = (1−s^2/3). Since one needs only to solve (3.14) in any situation, all cases can be treated at the same time and the precise details of stoichiometry and diffusivity, an be addressed later in the final scaling factor, (1−sD_R/D_P), for the net osmotic force. And, of course, Eq. (3.16) correctly gives zero osmotic force in the absence of any reactive particles: nR= 0.

flux has diffusive and “advective” terms,

j_R=−Ug_R−D_R∇g_R, (3.17)

where DR is now the sum of the osmotic motor diffusivity and the reactant particles.

Then, conservation of reactant particles at steady-state requires that ∇ ·j_R = 0. The microstructural deformation functionf_R(r) satisfies a pair-level Smoluchowski or advection- diffusion equation made dimensionless by scaling lengths with the contact distance a+bR

and the velocity with the yet unknown motor velocity U, which we must finally determine.

Thus,

∇²fR=−P e_Rzˆ· ∇f_R, (3.18)

and we have taken the motion to be along thez-axis (ˆz is a unit vector in the z-direction).

Far from the motor, the reactant microstructure is undisturbed, givingf_R∼0. At contact, the boundary condition becomes

n· ∇f_R= (Da h(n)−P eRn·z) (1 +ˆ fR). (3.19)

Again, the products satisfy a similar equation with the subscript R replaced by P. The above expression reflects the competition between motion due to self-propulsion of the motor in driving the environment away from equilibrium and Brownian motion attempting to restore the disturbed microstructure. The P´eclet numbers P e_R and P e_P, which appear from the scaling, may be considered as a ratio of the motor velocity U to the relative Brownian velocity D_R,P/(a+b_R). The scaled total microstructural deformation function

f(r) now satisfies

∇²f =− P e_R−P e_P 1−s^D_D^R

P

! ˆ

z· ∇f_R−P e_P ˆz· ∇f. (3.20)

The nondimensional versions of the boundary conditions become: f ∼0 asr → ∞, and

n· ∇f+P e_P n·zˆ



f + 1

nR

nT

1−s^D_D^R

P



= Dah(n)− P eR−P eP

1−s^D_D^R

P

! n·zˆ

!

(1 +fR) at r= 1. (3.21)

Note that equations (4.10) and (4.12) diverges in the special case sD_R/D_P ≡1, for which there is no osmotic force. The osmotic force is still scaled as before, however, and the unknown velocity is found from balancing the hydrodynamic Stokes drag forceF^hydon the motor with the osmotic force. The osmotic velocity of the motor is

U =− kT

6πηanR(a+b)²

1−sDR

DP

I

r=1

nf(r;Da, P eR, P eP) dΩ, (3.22)

where the microstructural deformation function f now depends on the Damk¨ohler and P´eclet numbers. Note that the motor velocity, and thus the P´eclet numbers, are unknown and must be found self-consistently along with the microstructural deformation functions f and f_R from the advection-diffusion equations.

Since now the reactant and the total microstructural deformations are coupled, two simultaneous equations need to be solved to compute the osmotic force and thus the motor velocity. Therefore, the detailed stoichiometry/diffusivity is not just a scale factor as in the osmotic force for the fixed motor case. This is somewhat involved and is taken up in the

proposed work, so we discuss below the case in which the reactants are “consumed” upon reaction. This corresponds to sD_R/D_P →0, which would occur if the products are much more diffusive than the reactants or when the reactant is indeed consumed (s= 0) by the motor. Thus, the product distribution drops out and the total microstructural deformation function is the same as the reactants. Actual consumption of reactants may indeed occur if the reactant particles irreversibly adsorb on the motor’s surface or are absorbed into the interior of the motor. In either case, the motor’s size would change over time and this effect would need to be included in the analysis. We have not done so here. The reader should note that this is just the language we choose to discuss the basic physics of the osmotic propulsion. The reader may wish to think instead of two specific cases: i) 2R → P, in which two reactant particles are joined to form a single product particle, corresponding to s = 1/2 and scale factor (1−sD_R/D_P) = (1−(1/2)^1/3) > 0. There will now be a net depletion of bath particles on the reactive side and a propulsive force to the right as sketched in Figure 3.1. ii) R → 2P, in which one reactant is split into two product particles, with corresponding scale factor (1−sDR/DP) = (1−(2)^1/3) < 0. Now there are more bath particles on the reactive side, leading to a propulsive force to the left in Figure 3.1. With this understanding in mind, we shall discuss the physics of the problem as if the reactant was consumed, s= 0.

From (4.9) the implicit equation for the P´eclet number is

P e=− kT 6πηa

(a+b)³ D n_b

I

r=1

n_zf(r;Da, P e)dΩ =φ_b 1 +a

b 2

F(Da, P e), (3.23)

where F(Da, P e) = −_4π³ R

r=1nzf(r)dΩ is a nondimensional function of the Damk¨ohler and P´eclet numbers, and must be solved in order to get a final expression for the motor

velocity. Note that in this and what follows we have dropped the subscript R for the reactant and will simply refer to the reactant as a bath particle. The P´eclet number is not an independent parameter, but rather is set by the Damk¨ohler number, the bath particle volume fractionφb, and the size ratioa/b. The term φb(a+b)² in Eq. (3.23) is defined to be a new parameter β, such that β and Da determine the motor behavior. The product β corresponds to the number of bath particles within a bath particle radius of the motor surface: β ∼nb(a+b)²b. Clearly, atP e= 0 (orβ →0) the free motor problem becomes the fixed motor, giving identical microstructural deformation for non-equilibrium conditions.

Therefore, for the fixed motor F(Da) is a function of the Damk´ohler number only. The fixed motor corresponds to an infinitely dilute suspension of bath particles. Whether the motor is fixed or free to move is just a change of reference frame. For a fixed motor there will be an advective flux at infinity to supply reactive particles to the motor.

The problem of osmotic propulsion consists of determining the microstructural defor- mation function (f = g−1) and then the dimensionless function F, which can also be interpreted as the dimensionless self-generated concentration gradient, for different physical parameters relevant to the motor and the suspension characteristics. In addition to the dis- cussed theory, we extend this study by using Brownian dynamics simulations, which allow us to build a different method to check our proposed theoretical framework and expand it to more complex systems.