3.5 Results
3.5.1 Fixed motor
3.5.1.2 Arbitrary Da
Having studied the two limiting cases for the fixed motor, we proceed to obtain the mi- crostructure for arbitrary values of Da, and from there we move on to the osmotic force.
Figure 3.4 shows the pair-distribution function at contact for various Damk¨ohler numbers.
As expected, the pair-distribution function at the reactive surface goes to zero as Da is increased. The concentration of bath particles jumps to higher values near π/2 (µ = 0), clearly showing the two distinctive surfaces. Note that at the passive suface, g(r) also de- creases as Dais increased, suggesting that bath particles migrate from this region to the reactive surface. This is occurring because the reactive half is a sink for bath particles. In
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
!/"
0 0.2 0.4 0.6 0.8 1
g(r=1,Da)
Da = 0.1 Da = 1 Da = 10 Da = 100
Figure 3.4: Pair-distribution function at contact for the fixed motor as a function ofθ for various Da. The reactive and passive surfaces are located from 0 to π/2 and π/2 to π, respectively.
the limit asDa→ ∞, the pair-distribution function at contact in the passive side scales as O(1), which is independent of the speed of reaction.
The major limitation of the general solution (3.39) is that asDais increased, a consider- able number of coefficientsAm in the expansion (3.39) are necessary to accurately represent the perturbed microstructure, making its implementation less practical. Even though we found that 150 terms in the series was good enough to properly represent the microstructure at high Da numbers, it is not entirely computationally feasible to calculate F(Da) for all Dausing a symbolic mathematics program. It did work quickly and accuratly in the limit as Da → ∞ because its corresponding boundary conditions are simple and independent of Da, which allows us to compute many coefficients Am in a reasonable amount of time.
Therefore, we compute the pair-distribution function with fewer terms (or coefficients) in the series (24 to be exact) and its value at contact to calculateF(Da). At first, the solution
of F(Da) agrees with the low-Dalimit, but diverges at high Danumbers as expected. A more accurate solution for F(Da) is obtained by a (9-9) Pad´e approximant used to ex- trapolate F(Da) to higherDanumbers. A (9-9) Pad´e approximant, plotted in Figure 3.5, shows that F(Da → ∞) = 0.4485, quite close to the expected value of 0.4515 from the asymptotics. We expect Figure 3.5 to represent a universal curve for the fixed osmotic force which has been made nondimensional by kT /(a+b), the fraction of bath particles in the motor volume,φ=nb(a+b)34π/3, and the stoichiometric/diffusivity factor (1−sDR/DP).
The open symbols in the figure are the results of Brownian dynamics simulations for various conditions. We have used the proposed formula forPsin order to findDaand construct this figure; it shows the formula works fine for the studied values. Clearly, the scaled osmotic force does not depend on the bath particle volume fractionφb, size ratioa/b, and from the time step ∆tused in the simulations. The transition from reaction- to diffusion-controlled regimes occurs approximately at a Damk¨ohler number of unity.
It is instructive to ask now what would be the magnitude of the force that must be exerted on the motor to keep it fixed? And what would happen to the local microstructure and the osmotic force if the motor were allowed to move? As described earlier, the maximum force occurs in the limit asDa→ ∞for large motors (ab),Fosm∼nbkBT a2, where the force saturates. (For the benefit of this illustration, we have assumed s = 0.) Consider a motor of a= 1µm with a 0.1 M bath particle concentration. The resulting osmotic force is of order 0.2µN, a respectable and easily measured force. In fact, it is rather large, as optical tweezers typically exert nano-Newton forces (Faucheux et al. 1995) and biological motors exert pico-Newton forces (Montemagno and Bachand 1999). Similar disagreement holds when compared to depletion forces that lead to depletion flocculation (Jenkins and Snowden 1996). Indeed, if the motor was released, it would start to move at a speed of
10-3 10-2 10-1 100 101 102
Da = !(a+b)/D
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
Fosm / (n R kT(a+b)2 (1-sD R/D P) 4"/3)
BD: #b=0.1, a/b=1, $t=0.001 BD: #b=0.1, a/b=1, $t=0.0001 BD: #b=0.3, a/b=1, $t=0.001 BD: #b=0.1, a/b=2, $t=0.001 dilute theory
Figure 3.5: The osmotic forceFosm scalednRkT(a+b)2(1−sDR/DP)4π/3 plotted against Da for various bath particle volume fractions, φb, size ratios a/b, and simulation time steps ∆t. The theoretical prediction (curve) is compared with Brownian dynamics (BD) simulations (symbols).
order 10 m/s. This surprising and aphysical result is resolved by noting that the motor cannot travel any faster than the bath particles can diffuse — that is, no faster than their diffusive velocity Ubath ∼D/(a+b). If the motor were to move faster than this velocity, bath particles would accumulate in front of the motor on the reactive side and a deficit would appear behind the motor, as the bath particles could not keep up. The motor would thus loose the propulsive force that caused it to move in the first place or even reverse the direction of motion. The resolution of this paradox is to recognize that in a frame of reference traveling with the free motor, there will be an advective flux of bath particles towards the motor that will alter the bath particle probability distribution about the motor and consequently, the propulsive osmotic force. In what follows, we shall see a detailed picture of the microstructure as the osmotic motor moves freely (and with directionality) and how that compares to the fixed motor problem. We also show that the velocity of the
free motor for any Damk¨ohler number is limited by the parameterβ — the number of bath particles within a bath particle radius of the motor surface.