Article pubs.acs.org/accounts
Cite This: Acc. Chem. Res. XXXX, XXX, XXX−XXX
Synthetic Nanomotors: Working Together through Chemistry Published as part of the Accounts of Chemical Research special issue “Fundamental Aspects of Self-Powered Nano- and Micromotors”. Bryan Robertson,† Mu-Jie Huang,† Jiang-Xing Chen,*,‡ and Raymond Kapral*,† †
Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada Department of Physics, Hangzhou Dianzi University, Hangzhou 310018, China
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‡
CONSPECTUS: Active matter, some of whose constituent elements are active agents that can move autonomously, behaves very differently from matter without such agents. The active agents can self-assemble into structures with a variety of forms and dynamical properties. Swarming, where groups of living agents move cooperatively, is commonly observed in the biological realm, but it is also seen in the physical realm in systems containing small synthetic motors. The existence of diverse forms of selfassembled structures has stimulated the search for new applications that involve active matter. We consider active systems where the agents are synthetic chemically powered motors with various shapes and sizes that operate by phoretic mechanisms, especially self-diffusiophoresis. These motors are able to move autonomously in solution by consuming fuel from their environment. Chemical reactions take place on catalytic portions of the motor surface and give rise to concentration gradients that lead to directed motion. They can operate in this way only if the chemical composition of the system is maintained in a nonequilibrium state since no net fluxes are possible in a system at equilibrium. In contrast to many other active systems, chemistry plays an essential part in determining the properties of the collective dynamics and self-assembly of these chemically powered motor systems. The inhomogeneous concentration fields that result from asymmetric motor reactions are felt by other motors in the system and strongly influence how they move. This chemical coupling effect often dominates other interactions due to fluid flow fields and direct interactions among motors and determines the form that the collective dynamics takes. Since we consider small motors with micrometer and nanometer sizes, thermal fluctuations are strong and cannot be neglected. The media in which the motors operate may not be simple and may contain crowding agents or molecular filaments that influence how the motors assemble and move. The collective motion is also influenced by the chemical gradients that arise from reactions in the surrounding medium. By adopting a microscopic perspective, where the motors, fluid environment, and crowding elements are treated at the coarse-grained molecular level, all of the many-body interactions that give rise to the collective behavior naturally emerge from the molecular dynamics. Through simulations and theory, this Account describes how active matter made from chemically powered nanomotors moving in simple and more complicated media can form different dynamical structures that are strongly influenced by interactions arising from cooperative chemical reactions on the motor surfaces.
1. INTRODUCTION Small-scale synthetic molecular machines and motors that move autonomously are interesting objects with many unusual properties. Molecular machines in the cell often use chemical fuel to induce conformational changes that allow them to walk on biofilaments in order to carry out various biological functions.1 The fuel molecules themselves are synthesized in the cell through a series of biochemical reactions that take place under nonequilibrium conditions. Synthetic molecular motors that do not rely on conformational changes also use chemical fuel to move autonomously in solution.2 One class of such motors comprises synthetic motors that are propelled by phoretic mechanisms, and these motors will be considered here. In a phoretic mechanism, a self-generated gradient of some variable, such as concentration or temperature, gives rise © XXXX American Chemical Society
to a force that acts on the body of the motor. Since no external forces (or torques) are applied to the system, the momentum of the entire system is conserved, and this force must be balanced by an equal and opposite force on the surrounding fluid. Fluid flows are then generated in the vicinity of the motor that are responsible for its propulsion. Thus, both chemistry and hydrodynamics are essential for the operation of these motors. The steady-state product concentration (cB) and fluid flow (v) fields in the vicinity of a Janus motor involved in propulsion by self-diffusiophoresis are shown in Figure 1a,b, respectively. A spherical Janus motor comprises a noncatalytic Received: May 29, 2018
A
DOI: 10.1021/acs.accounts.8b00239 Acc. Chem. Res. XXXX, XXX, XXX−XXX
Article
Accounts of Chemical Research
Figure 1. (a) Product concentration and (b) fluid velocity fields in the laboratory frame with the origin at the center of the Janus particle with a small catalytic cap (red) and a large noncatalytic cap (blue). The color bar shows the values of the product concentration (cB) and fluid velocity (v) fields normalized by the total concentration of fuel and product particles (c0) and motor velocity (Vu), respectively. (c) A collection of Janus motors with hemispherical caps together with the inhomogeneous distribution of product particles (small spheres) produced by motor chemical reactions.
Since chemical gradients and hydrodynamic flows are integral parts of propulsion by self-diffusiophoresis, these effects also enter into the consideration of the collective motions of many motors. Motors respond not only to their self-generated concentration gradients but also to those from other motors in the system. Similarly, the flow fields produced by other motors can influence the dynamics of a given motor. Figure 1c shows an instantaneous configuration of a collection of Janus motors, along with the particles of product B that they produce in catalytic reactions. One can see that the global concentration field is complicated and that interactions among motors mediated by this field, along with the accompanying fluid flow fields and direct intermolecular interactions among motors, may lead to different types of collective behavior. Modification of the flow and chemical fields associated with motor motion can also arise from the presence of confining boundaries or walls.15 While this Account focuses on diffusiophoretic motors, the collective behavior of many active particles is not unique to motors propelled by this mechanism. There are many examples of collective motion in the biological realm on both macroscopic and microscopic scales, and in addition, there is a large body of literature on synthetic motors of various kinds that has been summarized in reviews.2,4,10,16−28
surface (N, blue) and a catalytic cap (C, red). While various sizes of catalytic cap have been considered,3 in the figure we have shown a Janus motor with a small catalytic cap to illustrate the hydrodynamic and chemical effects discussed in section 3. The inhomogeneous distribution of product (B) particles in both the radial and tangential directions can be seen in Figure 1a. Figure 1b shows the corresponding fluid velocity field in the vicinity of the motor. Fluid flow is directed toward the motor at its front and back and away from the motor at its sides, a flow characteristic of a puller force-dipole swimmer.4 The flow pattern changes with the area of the catalytic cap and other parameters such as the interaction potentials.3 The continuum theory for the propulsion velocity of such a motor is well-known,5−7 and for the A → B reaction its u component is given by Vu =
kBT Λ⟨û ·∇θ c B⟩S, η
where û is a
unit vector directed from the center of the noncatalytic cap to the catalytic cap, kBT and η are the thermal energy and fluid viscosity, respectively, ∇θ is the tangential gradient, ⟨···⟩S denotes the surface average taken at the outer edge of the boundary layer where the motor−fluid potentials vanish, and ∞ Λ = ∫ dr r(e−UB(r)/ kBT − e−UA(r)/ kBT ), where UA and UB are 0
the A and B species−motor intermolecular potentials. Some of the chemically powered nanomotors that were first synthesized and studied operated by such phoretic mechanisms. For instance, the gold−platinum bimetallic rod motors were propelled by self-electrophoresis.8,9 Although many different motors have been made with different shapes and sizes that catalyze different chemical reactions to produce motion,10 this Account of our work on many-motor systems will focus on Janus, sphere-dimer, and oligomeric motors that operate by self-diffusiophoresis. Like spherical Janus motors with catalytic and noncatalytic faces, sphere-dimer motors, which have been studied experimentally and investigated through simulation and theory,11−13 also have catalytic and noncatalytic components, but these are spheres linked by a bond. Oligomeric motors, which have not yet been realized experimentally, consist of three or more linked spheres (catalytic, noncatalytic but diffusiophoretically active, and neutral linker).14 The nonspherical geometry of the spheredimer and oligomeric motors allows one to consider the effects of shape on the motor dynamics.
2. SIMULATION OF DYNAMICS The research on motor dynamics described here adopts a microscopic perspective that includes a coarse-grained particlelevel description of the fluid particles. The dynamics satisfies the basic conservation laws of mass, momentum, and energy and preserves phase-space volumes; thus, coupling between the motor motion and the fluid concentration and velocity fields will be described properly. The hydrodynamic and reaction− diffusion equations will follow from the microscopic dynamics on long distance and time scales, and links can be made between the microscopic and macroscopic descriptions.29 Specifically, we consider systems containing many motors with various geometries in either binary fluids containing reactive A and B species or ternary fluids that have chemically inert species S in addition to the reactive species.30 The fluid particles are taken to be mechanically identical in their interactions with each other, so the diffusiophoretic properties are the same for both binary and ternary fluids3,31 apart from B
DOI: 10.1021/acs.accounts.8b00239 Acc. Chem. Res. XXXX, XXX, XXX−XXX
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Accounts of Chemical Research an overall reduction in motor catalytic activity due to the lower concentrations of reactive species in ternary systems. The motors interact with both the fluid particles and each other through intermolecular potentials that can be either attractive or repulsive; here we consider only repulsive potentials and motor−motor interactions that are sufficiently strong to avoid solvent depletion forces. The interactions among the fluid particles do not involve intermolecular potentials and are carried out using multiparticle collision dynamics.29,32−34 The full dynamics then consists of two steps: molecular dynamics evolution governed by the above intermolecular potentials and multiparticle collisions among the fluid particles at discrete time intervals, which capture the effects of many real collisions. The input parameters for such dynamical simulations are the particle masses, the intermolecular potential functions, and the parameters that describe the multiparticle collisions and determine the transport properties of the fluid. In order to properly take into account self-diffusiophoresis and sustain motor motion, we incorporate both reactions at the catalytic regions of the motors and reactions in the bulk fluid phase. The reaction at the motor, A + C → B + C, converts the fuel A to product B whenever a fuel species is close to the catalytic site C. The bulk-phase reactions are used to maintain the system in a nonequilibrium state; for instance, the simple bulk-phase reaction B → A may be used to remove product and add fuel to the system. In our simulations, these reactions occur through probabilistic birth−death processes that accompany each multiparticle collision event (reactive multiparticle collision dynamics35). Bulk-phase reactions can also be used to establish more complex, spatially inhomogeneous states (see section 4). These particle-based simulations provide a way to explore collective behavior that incorporates all of the relevant manybody coupling effects. The examples discussed below will show the important role that chemistry plays in the motor collective dynamics. Particle-based methods can be extended to incorporate other driving forces such as temperature gradients.36,37 Stochastic models have also provided a considerable amount of information on the collective dynamics of active particles.18,38−41 These models often neglect hydrodynamic effects and approximate chemical interactions.
g (r ) = (4πr 2nM)−1
NM
∑ j UBN, Λ > 0) act as pullers for short bond lengths and as pushers for long bond lengths.13,51 Figure 3 (left) shows an instantaneous
differences between forward- and backward-moving motors can be understood since forward-moving motors are attracted to high product concentrations and thus move toward other motors while backward-moving motors tend to avoid other motors that produce product. Thus, chemical interactions play an essential role in determining these structures.
4. ACTIVE CHEMICAL MEDIA The surrounding medium need not have such a limited part in determining the nature of the collective motor dynamics. For instance, the medium may support nonlinear chemical reactions under far-from-equilibrium conditions where symmetry-breaking bifurcations produce stable chemical patterns with sharp concentration gradients or chemical oscillations.52−54 To investigate the motor dynamics in systems with chemical patterns, we consider idealized chemical domains with various shapes constructed in the following way. In a system containing fuel A, product B, and a species S that does not participate in motor reactions, a domain with a specific shape is chosen, and any fluid particles that diffuse into the domain are converted to A particles. Motor-catalyzed reactions in the interior of the domain can produce B particles. When A or B particles diffuse out of the domain, they undergo the bulk kcat
⎯ S with a large rate constant kcat. phase reactions A, B ⎯→ Domains constructed in this way do not arise from symmetrybreaking bifurcations; nonetheless, these chemical patterns consist of domains with different concentrations separated by sharp chemical gradients and can be used to study the collective phenomena of interest here.55 An example of a stationary annular chemical pattern is shown in Figure 4. The annular domain has a high concentration of A particles, while other regions are occupied by S particles. We again consider sphere-dimer motors in the quasi-twodimensional geometry described earlier. Figure 4a shows a small number of motors initially distributed randomly in the medium, where they execute Brownian motion in the fuel-poor solution. Motors whose catalytic C spheres reach the edge of the annular pattern trigger reactions A + C → B + C that lead to self-propulsion in the +û direction. The motors respond to the A and B gradients and move into the annular domain, forming dynamic clusters, as shown in Figure 4b. Most of the motors are oriented with their catalytic heads pointed outward while the noncatalytic monomers lie in the inner edge, as confirmed by the two peaks in Figure 4c for the radial distribution functions g(r) for the C and N monomers, where r denotes the distance from the center of the annulus. The C−C spheres of neighboring motors tend to align in these dynamic clusters. The local orientational order is characterized by
Figure 3. Upper panels: forward-moving (left) and backward-moving (right) sphere-dimer motors (C sphere, red; N sphere, gold) with B concentration fields (blue). Lower panels: dynamic cluster formation for forward-moving motors (left) and strong density fluctuations for backward-moving motors (right). Colors are used to distinguish different cluster sizes.
configuration of 1000 forward-moving motors in a simulation that started from an approximately homogeneous distribution of motors.49 The fluid and motors are contained in a slabshaped three-dimensional volume with confining walls on the top and bottom and periodic boundary conditions in directions parallel to the walls. Lennard-Jones 9−3 sphere−wall potentials are employed so that the motor spheres are largely confined to the midplane and the sphere-dimer dynamics is quasi-two-dimensional; any hydrodynamic screening effects due to the presence of walls are taken into account in the simulations. The nonequilibrium steady state is again established by bulk-phase reactions. The forward-moving motors form large dynamical clusters, whose interiors are solidlike crystalline structures with defects, coexisting with a very dilute gas phase. One may construct backward-moving motors (−û direction, UAN < UBN, Λ < 0) by interchanging the UAN and UBN potentials so that Λ changes sign. As shown in the right panel of Figure 3, no such large cluster states are observed. The
N
ψu(r ) =
n(r )−1
∑ j UBN, Λ > 0) when unpinned. These rotors align with one another to form highly correlated domains, with defects occurring where rotors are antialigned. The rotors fluctuate as a result of Brownian motion, and a flip in orientation in one rotor influences the alignment of adjacent rotors. This can be seen in Figure 6b, which shows a space−time plot of ûx(x, t) = x̂· û(x, t), where x̂ is a unit vector along the +x direction. In order to evaluate these two influences, the modified simulation algorithm in which the chemical interactions are excluded may be used. The orientational order parameter is characterized by
motors in such media. Here we consider a simpler situation of an oscillatory supply of fuel. Figure 5 shows two instantaneous configurations of 40 sphere-dimer motors drawn from the evolution of a system with an oscillatory flux of fuel at the upper and lower confining boundaries. When the fuel concentration is low, the motors show little tendency to cluster and execute Brownian motion (Figure 5a); however in portions of the oscillatory cycle where the fuel concentration is high, motors form dynamic clusters (Figure 5b). The clusters gradually evaporate and break apart as the fuel concentration decreases; thus, the dispersion−aggregation process also occurs periodically in the oscillating medium. Similar cluster dynamics has also been observed in experiments.57
5. PINNED MOTORS Since phoretically propelled motors produce fluid flows to generate motion, when such motors are pinned to prevent motion, they pump fluid. Systems with multiple motors may then interact through their hydrodynamic and chemical fields to produce orientational collective motion. Consider a chain of sphere-dimer motors with equal-sized spheres immersed in a fluid with their centers of mass pinned to a wall (Figure 6a).58 The chain of motors is contained in a rectangular simulation box with confining walls in the z direction and periodic boundary conditions in the other directions. The motor spheres interact with the lower wall through potential functions that restrict their motions to lie in the plane parallel and close to the wall. The motors are far enough apart that the finite-range direct intermolecular forces among motor spheres play no role; thus, interactions among the motors arise only from hydrodynamic and chemical fields. The system is maintained in a nonequilibrium steady state by bulk-phase reactions. Under these conditions, the pinned motors can undergo rotational motion that is strongly
NR
S2(j) = NR −1 ∑ ⟨P2(cos θji)⟩ i=1
(3)
which measures the average over all NR rotors of the second Legendre polynomial of the cosine of the angle between the directors of rotors j units apart. As shown in Figure 6c, the orientational order parameter is high when both hydrodynamic and chemical interactions are included (perfectly aligned or antialigned rotors have the value S2(j) = 1), while for simulations with only hydrodynamic interactions, where the chemical interactions have been turned off (as described above), S2(j) ≈ 0.25, consistent with random orientations. If instead chains of rotors that move backward (−û direction, UAN < UBN, Λ < 0) when unpinned are considered, the rotors align parallel to one another in the y direction in the form of stacks.58 A simple Langevin model that approximately accounts for diffusiophoretic coupling among rotors but neglects hydrodynamic interactions can be written for this system as F
DOI: 10.1021/acs.accounts.8b00239 Acc. Chem. Res. XXXX, XXX, XXX−XXX
Article
Accounts of Chemical Research d û i = [ζR −1·(Tdi + Tfi)] × û i dt
repulsive interactions between motor spheres were used to ensure that depletion forces would play no role in motor collective motion. Attachment of protein machines to biofilaments mitigates some of the effects of strong thermal noise that could have adverse effects on active transport. Attachment of oligomeric motors to filaments serves a similar function. Rotational Brownian motion is one of the main detrimental effects to the directed motion of diffusiophoretic motors. For example, the oligomeric motor in solution has an orientational relaxation time (τR) of ∼245 (simulation units). When the motor is attached to a filament, τR is ∼104 000, which is longer by a factor of more than 400.14 When many oligomeric motors are attached to the filament, as shown in Figure 7 (top), they behave differently from isolated motors when the line motor density (n = nM/lf, where nM is the average number of motors attached to the filament and lf is the length of the filament) is high. Strong positional correlations are seen in the one-dimensional position distribution function,
(4)
where Tdi is the torque acting on rotor i due to chemical interactions from all other rotors, Tfi is the random torque, and ζR is the rotational friction tensor. This model is able to capture the major features of the collective orientational ordering, again pointing to the importance of chemical interactions. Other simulations incorporating active rotation through chemistry have exhibited phase locking and other selforganized behavior.59 The placement of rotors on a wall is not restricted to a line and can be generalized to two dimensions. In Figure 6d the rotors are placed at the vertices of a hexagon and adopt a specific conformation with the noncatalytic spheres pointing outward. In even larger arrays, frustrated stable states occur in which rotors have multiple close neighbors, and thermal fluctuations may induce transitions to other stable states.
6. MOTORS ON FILAMENTS In future applications one might expect chemically propelled nanomotors to perform tasks in complex media crowded by obstacles of various kinds. Ultimately one might even envision applications inside the cell, a highly crowded environment controlled by complex chemical networks. Molecular machines have evolved to function in such an environment, and they carry out diverse biochemical and transport tasks by moving on biofilaments or operating in membranes. Motivated by these considerations, we consider how synthetic motors might operate in complex environments containing filaments.14 Figure 7 (top) depicts oligomeric motors attached to a filament, and Figure 7 (bottom) shows the self-generated concentration fields produced by the catalytic reactions at the C sphere. The system we study contains a stiff filament made from linked spheres. The three-bead oligomers attach to this filament through solvent depletion forces, whereas strong
g (x ) =
(lf /nM 2)
Non
∑
δ(xL , ij − x) (5)
j
