Consider a collection of rigid, spherical, neutrally-buoyant colloidal reactant particles R of hydro- dynamic radius b immersed in a Newtonian, viscous, constant property solvent of density ρ and dynamic viscosity η. Among the reactant particles is situated a rigid, spherical catalytic “motor”
particle of hydrodynamic radius a. The size of the motor relative to the size of the reactant is arbitrary and is set by the ratioλ≡b/a. The crux of the catalytic nanomotor problem is to deter- mine how the motor induces a local chemical potential gradient about its surface and consequently travels through the dispersion with some mean phoretic velocity�U�of magnitudeU. The chemical potential gradients are achieved by means of an asymmetric distribution of reactive and unreactive surface area on the motor, i.e. the motor is a Janus particle with catalytic and inert faces. To systematically tune the fraction of catalytically-active surface area, the polar angleθsor the “switch angle” is introduced. For polar angles0 ≤θ≤θsas measured from the direction of mean motion, the surface of the motor catalyzes the irreversible conversion of reactantRinto productPwith some stoichiometric ratioϑ,
R→ϑP. (3.1)
For polar angles θs< θ ≤π, the surface of the osmotic motor is inert. For micron-sized colloidal particles in a solvent comparable to water at room temperature, the Reynolds number Re≡ρU a/η describing the relative magnitude of inertial to viscous forces is∼O�
10−6�
, setting our problem in the regime of Stokes flow. We should note that in the Stokes flow regime, the hydrodynamic force becomes infinite when bringing two rigid spheres into contact at their hydrodynamic radii. Thus, a
surface-mediated reaction cannot occur between colloids without attractive interparticle interactions if the particles are required to come into hydrodynamic contact. The excluded-annulus interparticle potential alleviates this issue of infinite forces preventing contact, with the additional benefit of introducing a parameter that can easily “tune” the strength of hydrodynamic interactions between the reactant particles and the catalytic motor. This potential Φ (r) is spherically symmetric and depends only on the relative separationr ≡ |r| of the catalyst and reactant particle centers. The potential is a Heaviside step function preventing the particles from approaching any closer thanrmin,
Φ (r)≡
0, r≥rmin;
∞, r < rmin.
(3.2)
We define the minimum approach distancermin≡(a+b) (1 + ∆), where∆ is the excluded-annulus parameter. The limit of fully-screened hydrodynamic interactions corresponds with∆→ ∞, while full hydrodynamic interactions are modeled by setting∆ = 0. Note that both motor and reactant can have excluded annulus character on either of their surfaces, as depicted in figure 3.1. The excluded annulus parameter for the system is related to the excluded volumes of the motor and reactant by
∆≡ ∆aa+ ∆bb
a+b . (3.3)
Reactant and product particles can have different effective hydrodynamic radii and hence different diffusivities. In order to simplify the system, we assume the following for the ratio of the diffusivity of the product,DP, to the diffusivity of the reactant, DR
DP
ϑDR → ∞. (3.4)
If this ratio were finite, one would only need to add a factor of (1−ϑDR/DP) to the resulting diffusiophoretic force as shown by Córdova-Figueroa and Brady [14]. Because of the large discrepancy in the diffusivities, product effectively disappears upon reaction, and the reaction may be thought of as a “consumption” reaction. An alternative picture would be to think of the rigid reactant particles as a collection of bound solvent monomers, and the catalytic reaction severs the bonds in the reactant particle, hence dissolving it. Thus, the product particle does not need to be accounted for in this model. The rate of consumption is given by rRhs(θ), where rR is the reactant consumption rate, and the surface catalyst geometry is described by the functionhs(θ), a Heaviside step function using the switch angleθs
r
a θ
n θ
z y
Figure 3.1: The motor-reactant geometry for the catalytic nanomotor problem. Note that the catalytic zone is symmetric about the azimuthal angleφ.
hs(θ) =
1, 0≤θ≤θs; 0, θs< θ≤π.
(3.5)
The reaction is offset from the hydrodynamic radius of the motor to occur at the minimum approach distancerminused in the definition of the interparticle potentialΦ. Unless explicitly noted, the switch angle is taken to be atθs=π/2so that the Janus particles have equal parts reactive and inert faces.
The net propulsive force on the motor is dependent on the microstructure, or configuration, of the surrounding reactant. The reactant microstructure is quantified by the pair distribution function g(r), describing the likelihood of finding a reactant particle at some spatial location r relative to the motor. The pair distribution function is defined as the ratio g(r) ≡ P1|1(r|x1)/n∞b : the conditional probability density P1|1(r|x1)of finding a reactant particle at some positionr relative to the motor center given that the motor is centered about x1 over the number density n∞b of reactant molecules in the quiescent, undisturbed suspension. Correctly solving for this steady- state pair distribution function, which is governed by the time-invariant Smoluchowski equation, allows for the prediction the diffusiophoretic velocity. The derivation for the dilute Smoluchowski equation with careful consideration of hydrodynamic interactions, becomes more complex when the
advective motion requires solute-motor collisions; the dilute treatment used in the microrheology problem derivation is insufficient. For a discussion of why three-body interactions are needed, and a derivation of the Smoluchowski equation relevant for diffusiophoretic motion, the reader is referred to recent work by Brady.
The advective term that appears in the steady Smoluchowski equation describing the motion of the catalytic nanomotor is more complex than that which governs constant-force microrheology. Only the coupling between the force monopole acting on the probe particle and the relative translational motion needed consideration for microrheology. However, in diffusiophoretic motion the motor is force free as one should expect for any type of phoretic motion. All higher moments, force dipole, force quadrupole,etc. contribute to the advective motion of reactant past the motor. In addition, the mobility functions appearing in the advection terms require averaging over three-body ensembles.
However, the leading order diffusive motion of the reactant is still described by the same relative diffusivity tensorDrel≡kT�
MU F22 −MU F21 −MU F12 +MU F11 �as for microrheology. Thus, in the limit of small Péclet number, the governing equation for the reactive bath particles is simply
∇r·Drel·∇rg= 0. (3.6)
The excluded-annulus interparticle potential prevents particles from approaching any closer than their excluded-volume radius allows. The only probability flux normal to the surface is that defined by the consumption reaction on the reactive face of the motor
−n·Drel· ∇rg=rRhs(θ) atr=rmin. (3.7) In Equation 3.7, n is the outward-facing unit normal oriented along the line of centers from the motor to the reactant particle. Far from the motor, the concentration field should not sense the presence of the motor; thus, the conditional probability density of finding a reactant particle far from the motor should approach the ambient number density of reactant
g(r)→geq as r→ ∞. (3.8)
Because of the excluded-annulus interparticle potential and the diluteness of the reactant solution, the equilibrium valuegeqof the pair distribution function is equal to unity for all radial separations r > rmin.
For our model of a surface-mediated reaction, a reaction that is first-order in the concentration of reactant is the intuitive choice, indicating that motor-reactant collisions facilitate the production
of product. We may thus define the reaction raterRin terms of the pair-distribution function
rR≡ −κn∞b g(r), (3.9)
where κ is the reaction velocity. Alternative scenarios, for example a surface-generation model, in which therapeutic molecules diffuse from a hydrogel carrier particle, may be better suited to a zero-order reaction rate.
The choice to non-dimensionalize the radial coordinate is based on the minimum approach dis- tancermin,
˜ s≡ r
rmin. (3.10)
The resulting velocity of the motor with a given distribution of reactant comprises both interparticle and Brownian contributions,
�U1�=� UP1�
+� UB1�
. (3.11)
The interparticle contribution to the motor motion is a surface integral at the excluded annulus contact distance,
�UP1�
=−n∞b Da(rmin)2L(r=rmin, λ)
˛
˜ s=1
ng(˜s) dΩ. (3.12)
The Brownian contribution to the motor motion is the divergence of the relative mobility weighted by the likelihood of the configuration averaged over all space,
�UB1�
=−n∞b Da(rmin)2 ˆ
˜ s≥1
g(˜s) ∇˜s·Mˆr d˜s. (3.13) In Equation 3.12, the function L(r=rmin, λ) is simply the value of the relative mobility hydro- dynamic functionL evaluated at the minimum approach distance between the motor and reactant molecule,rmin. The relative mobility functions depend on the reactant-to-motor particle size ratio λ≡b/a. Recall that Ldescribes the ease of relative motion along the line of centers between the motor and reactant when a force acts solely on the motor, whileM describes the facility of tangential motion. That is to say, the relative mobility tensorMˆrmay be written as,
Mˆr≡6πηa�
MU F11 −MU F12 �
=Lnn+M(I−nn), (3.14)
wherenis the outward-facing unit vector along the line of centers between the motor and reactant particles, andIis the identity tensor. The functionL(rmin, λ)increases monotonically with∆for all λ, from0, when∆ = 0, to 1, when ∆→ ∞. In the limiting case of very small reactant ,λ�1, the relative mobility functionsLandM need not consider lubrication for a certain limit: if we examine the limit where the size of the additional excluded annulus distance∆a is greater than the size of
the reactant particlesb, the gap size between the reactant and motor particles will be large enough, even at center-to-center separations of rmin, so that lubrication effects need not be considered, as depicted on the left-hand side of Figure 3.2. It is in this regime that the relative mobility functions LandM take on the following simple functional forms for all accessible motor and reactant particle configurations,
z z
z z
Figure 3.2: The possible limits of small reactant and small excluded annulus length scales. Left: When the excluded annulus is larger than the radius of the reactant particle, the lubrication regime in which the gap is the smallest length scale is inaccessible. Right: The excluded annulus is smaller than the radius of the reactant and the simple functional forms ofL andM proposed in this work do not hold for all center-to-center separations.
L(˜s, λ�1) = 1− 3
2 (1 + ∆)s˜−1+ 1
2 (1 + ∆)3s˜−3+O(λ) (3.15) M(˜s, λ�1) = 1− 3
4 (1 + ∆)˜s−1− 1
4 (1 + ∆)3s˜−3+O(λ). (3.16) Note that the limit where the bath particle radius is the smallest length scale in the problem corresponds to the ratio∆/λ >1. If the excluded annulus thickness were the smallest length scale in the problem, as seen in the right-hand side of Figure 3.2, then Equations 3.15 and 3.16 would not be valid approximations in the lubrication regime, defined bys˜−1�1. It is important to note that, for small reactant particles, the divergence of the relative mobility∇ ·Mˆrscales asO�
λ3� for all radial separations, indicating that the Brownian contribution to the motor velocity,i.e. equation 3.13, is negligible.
When also considering advection effects, we arrive at the following partial differential equation
and boundary conditions:
Pe˜ �
Uˆrel(g)�
· ∇g(˜s) +∇ ·Drel· ∇g(˜s) = 0 (3.17)
g→1 ass˜→ ∞ (3.18)
Pe˜ �
Uˆrel(g)�
·ng(˜s) +∂g
∂˜s = ˜Dahs(θ)g(˜s) ats˜= 1 (3.19) We use the same dimensionless groups to characterize the relative rates of advection, diffusion, and reaction as defined in the problem posed by Brady [10]: the Péclet�
Pe˜�
and Damköhler� Da˜ � numbers. The timescale for advection of the catalytic nanomotor to occur on the order of its excluded-volume size isτa ≡a(1 + ∆)/U, the timescale for the reaction rate to occur at the surface of the motor isτr ≡ a(1 + ∆)/κ, and the timescale for the bath particles to diffuse to fill in the trailing wake behind the motor is isτd≡(a(1 + ∆))2/DR. The Péclet number may be interpreted as the ratio of diffusive and advective timescales,
Pe˜ ≡ τd
τa =U a(1 + ∆)
DR , (3.20)
while the Damköhler number may be thought of as the ratio of diffusive and reactive timescales, Da˜ ≡ τd
τr
= κ a(1 + ∆) DR
. (3.21)
We will assume that the reactant microstructure about the motor is at a steady state. In other words, as the catalytic nanomotor constantly reacts with the particles around it, the microstructure remains fully developed. If we were to include Brownian rotation, the result would be a random, as opposed to a directed, walk. Brownian rotation would cause the long-time effective diffusivity of the motor to be
Dmotor =Da+1 6
U2
Drot, (3.22)
where Da is the diffusivity of the motor in the absence of Brownian rotation, U is the velocity of the motor, and Drot ∼ kT /8πηa3 is the rotary diffusivity. Brownian rotation can be ignored when the time scale for the concentration profile to develop, τ ∼a2/DR, is short compared to the rotary diffusion time, τrot ∼1/Drot. Since τ /τrot ∼b/a, Brownian rotation can be ignored in the small-λregime whereλ=b/a�1 as long as the time of observation of the motor is shorter than the timescale for rotation. For finite size ratios, Brownian rotation happens rapidly enough that a steady-state concentration profile may not have enough time to develop about the motor. To consider steady motion that is still governed by Equations 3.17 through 3.19 in the limit of finite
motor sizes, one can imagine a scenario where the rotational degrees of freedom are constrained.
For example a physically-tethered motor protein may be capable of translating along a surface but may not be free to change its orientation. Motor particles with a dipole magnetic moment could also align with an externally applied field while still not applying an external force.
For the remainder of this section, we examine Equations 3.17 through 3.19 in the linear-response regime where the Péclet number is much less than unity. In this limit, ensemble-averaged three body hydrodynamic interactions and the complicated dependence of the relative velocity�
Uˆrel(g)� on all moments of the microstructure function g are neglected, as only the leading-order diffusive effects described by Drel change the character of the microstructure. We examine the Brownian and interparticle velocity contributions for a half-reactive motor (θs=π/2) of various relative sizes of reactant and motor. Future directions for this colloidal approach to understanding autonomous motion, including proper consideration of advection, are highlighted.