highDa, the force saturates and simply scales asFosm∼nRkBT(a+b)2(1−sDR/DP). We found this force to be large compared to typical colloidal forces indicating that this mecha- nism could be useful for self-propulsion (or for pumping fluids). It was shown that when the motor is set free, it moves rapidly toward the self-created low bath particle concentration region (g 1) located near the reactive surface. Eventually (almost instantaneously) the motor catches up with bath particles reducing the gradient in bath particle concentration.
Thus, the osmotic force is balanced by the viscous force acting on the moving motor.
For the free motor case, we assumed that the stoichiometry value s is zero, which simplifies greatly the governing equations. This assumption is unnecessary in the limit of small P´eclet numbers, i.e., the fixed motor, and in the limit of large P´eclet numbers where the effects of advection are the same for the reactants and products. The effect of nonzerosDR/DP apart from being a scalar factor in the motor velocity will be quantitative, not qualitative. Thus, the osmotic velocity was calculated as a function of Da for various β =φb(1+a/b)2. Also we demonstrated again that BD simulations are in agreement with the theoretical results. In general, we showed that the physical properties of the microstructure, a/bandφb (or the productβ), directly contribute to the speed of the motor, an observation not shared by the fixed motor problem. For small Damk¨ohler numbers, we showed the osmotic velocity scales as U ∼κ. In the limit of highDa, the free motor velocity saturates and scales as U ∼ nbDa(a+b)2. The advective flux of bath particles toward the motor alters the bath particle probability distribution relative to the motor and consequently, the propulsive force.
We also examined the influence of the productβ on the osmotic velocity. The results in Figure 3.10 showed that the fixed motor is the limit as β → 0 (P e= 0) of the free motor;
the fixed motor corresponds to an infinitely dilute suspension of bath particles. This is,
as it should be, because whether the motor is fixed or free it is just a change of reference frame. For a fixed motor there will be an advective flux at infinity to supply reactive bath particles to the motor. This also implies that the motor will induce a fluid flow to supply the bath particles and can be used as a pump — a novel microfluidic pump (and mixer).
In the limit asβ → ∞ (very large motors) and for finiteDa, the P´eclet number also scales asP e ∼O(1), giving the motor velocity U ∼D/(a+b). But at highDa and β, we found that the P´eclet number goes to infinity asP e∼(βDa)3/5. In general, we observed that at βDa1 the P´eclet number is small (slow propulsion). In the limit ofβDa1, advection dominates over diffusion in the local microstructure, thus P e is large (fast propulsion).
The reader should not be confused that these results are specific to the assumed reaction rate (heterogenous irreversible first-order reaction) and the size of the reactive area (half- reactive motor). Other scaling conditions and interesting features could arise for motors with different distributions of reactive area.
For a 1-µm-sized half-reactive motor and nanometer-sized bath particles (large β) the motor velocity U ∼D/(a+b) is now of order 20 µm/s, a much more reasonable velocity, and one that is in fair agreement with the motivating experiments of Paxton et al. (2004).
Although the mechanism suggested for the catalytic nanomotor is different from that for osmotic propulsion (Paxton et al. 2006), this result shows the significance of the speed magnitudes created by the osmotic motor for processes at nanoscale that require directed motion.
We have proposed methods to modify this theory to account for less dilute systems.
Comparisons with measurements and (more) simulations will be necessary to determine whether these ideas are applicable. This could be helpful to understand the disagreement between the simulation results and the dilute theory at highDaand to extend this ideas to
cases when the osmotic motor and bath particles are different in size. Also, we computed the fluctuations resulting from the collisions between the motor and bath particles as a function of the Damk¨ohler number and the product β. It was found that the scaled fluctuations are O(Da2) at small Da, and are independent of Da in the limit as Da→ ∞. For small and highβ, we found that the scaled velocity fluctuations are independent ofβ. In addition, we calculated the ratio of parallel to perpendicular fluctuations for variousDaand β, giving a better picture of how the fluctuations relate to the microstructural deformation relative to the motor.
We have derived an expression for the efficiency of conversion of free energy into mechan- ical energy, which measures the ability of the motor in harnessing its environment to create useful work. For an irreversible first-order reaction rate (consumption of bath particles), the motor efficiency ξ goes to zero asDa→ ∞. We found the efficiency of the fixed motor to be independent of Da forDa 1. For fast reactions, the fixed motor efficiency scales asξ ∼β/(lnDa). On the other hand, the efficiency of the free motor also scales as ξ ∼β for slow reactions, but scales as ξ ∼1/(βlnDa) for highDa and β. Many questions arise from investigating the motor efficiency. Can we define other types of efficiency? How does the motor efficiency compare to other reaction-driven transport mechanisms? Is it possible to design an osmotic motor as efficient as biological machines? There are many variables that could be manipulated to improve the efficiency that require further analysis, such as different portions of reactive surface on the motor and other types of reaction rate.
This work opens up many questions and future extensions for the theory and the sim- ulations. Clearly, neither the motor nor the bath particles need to be spherical. A variety of behaviors are possible depending on the nature of the chemical reaction at the motor surface. We have considered only the simplest of chemical reactions, the irreversible first-
order reaction of product particles. And simplified to consumption of reactant particles for analytical application. How is the osmotic force modified for other chemistries? A reversible reaction? Production of bath particles rather than consumption at the free motor surface?
What if there are enthalpic effects — specific interactions between the motor and the reac- tive species — in addition to entropic? One obvious question to ask is what is the optimal distribution of the reactive site on the motor surface? We considered one half of the motor surface to be reactive. Is this the best? Or is there an optimum for a different fraction?
How does that optimum vary with the Damk¨ohler number and the nature of the chemical reaction? What fraction of the limiting bath particle diffusion velocity can be obtained by a motor? Just how fast can it move?
In the analysis we have ignored the fact that a small motor will also be subject to its own Brownian motion, and in particular its rotary Brownian motion. As the motor rotates in response to Brownian torques the reactive side will no longer be in the same direction and this may limit the extent of its directed motion. The time scale for the establishment of the concentration profile about the motor is the diffusive time of the bath particles τb ∼a2/D. The time scale for rotary Brownian motion of the motor is its rotary diffusivity Dr = kT /8πηa3. Thus, rotary motion of the motor will not be important as long as τbDr ∼ b/a 1, which is the case when the motor is much larger than the bath particles. Thus, the work described above is restricted to this limit. If this restriction is relaxed a large motor could travel at its osmotic velocity U for a time 1/Dr after which it could establish a new bath particle concentration profile and travel again at U but in a new (random) direction. Thus, for long times compared to 1/Dr the motor will undergo a random walk with a step length U/Dr.
In addition, we neglected hydrodynamic interactions between particles which would
affect the motor speed. Hydrodynamics would be expected to slow the motor’s motion, but to what extent? At the pair level (one motor, one bath particle) hydrodynamics can be included analytically (following the work on microrheology, Khair and Brady (2006)), while for more concentrated systems, Stokesian dynamics (Brady and Bossis 1988; Banchio and Brady 2003) can be adapted to simulate reacting bath and motor particles.
The analysis can be generalized to have more than one solute (bath particle) species, and more than one motor. How will two or more motors act when they compete for the same reactant? Will a group of motors swarm together? Can this have relevance for the swarming of biological organisms?
Our investigation has demonstrated that autonomous motion can be generated quite simply by exploiting the ever-present thermal fluctuations via a chemical reaction at the motor surface. Osmotic propulsion provides a simple means to convert chemical energy into mechanical motion and work, and can impact the design and operation of nanodevices, with applications in directed self-assembly of materials, thermal management of micro- and nanoprocessors, and the operation of chemical and biological sensors. This opens up many possibilities for exploiting autonomous motion to ether propel particles and/or pump fluid, some of which are outlined in this work. Studies of autonomous motors may also help to understand chemomechanical transduction observed in biological systems (Theriot 2000) and to create novel artificial motors that mimic living organisms and which can be harnessed to perform desired tasks.