Linking Catalyst-Coated Isotropic Colloids into “Active” Flexible

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Linking Catalyst-Coated Isotropic Colloids into “Active” Flexible Chains Enhances Their Diffusivity Bipul Biswas,† Raj Kumar Manna,‡ Abhrajit Laskar,¶ P. B. Sunil Kumar,‡,§ Ronojoy Adhikari,*,¶,∥ and Guruswamy Kumaraswamy*,† †

Complex Fluids and Polymer Engineering, Polymer Science and Engineering, CSIR-National Chemical Laboratory, Dr. Homi Bhabha Road, Pune 411008, India ‡ Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India ¶ The Institute of Mathematical Sciences−Homi Bhabha National Institute (HBNI), CIT Campus, Chennai 600113, India § Department of Physics, Indian Institute of Technology Palakkad, Palakkad 678557, India ∥ DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom S Supporting Information *

ABSTRACT: Active colloids are not constrained by equilibrium: ballistic propulsion, superdiffusive behavior, or enhanced diffusivities have been reported for active Janus particles. At high concentrations, interactions between active colloids give rise to complex emergent behavior. Their collective dynamics result in the formation of several hundred particle-strong flocks or swarms. Here, we demonstrate significant diffusivity enhancement for colloidal objects that neither have a Janus architecture nor are at high concentrations. We employ uniformly catalyst-coated, viz. chemo-mechanically, isotropic colloids and link them into a chain to enforce proximity. Activity arises from hydrodynamic interactions between enchained colloidal beads due to reaction-induced phoretic flows catalyzed by platinum nanoparticles on the colloid surface. This results in diffusivity enhancements of up to 60% for individual chains in dilute solution. Chains with increasing flexibility exhibit higher diffusivities. Simulations accounting for hydrodynamic interactions between enchained colloids due to active phoretic flows accurately capture the experimental diffusivity. These simulations reveal that the enhancement in diffusivity can be attributed to the interplay between chain conformational fluctuations and activity. Our results show that activity can be used to systematically modulate the mobility of soft slender bodies. KEYWORDS: active matter, colloidal assembly, diffusivity, Brownian motion, ice templating fields, for example, through electrohydrodynamic phenomena that result in colloidal rotation about an axis normal to the imposed electric field.20 Symmetry breaking can also arise from hydrodynamic interactions of colloids with boundaries.29 In colloidal systems that convert chemical to mechanical energy, symmetry breaking typically arises from the preparation of chemically anisotropic colloids, such as Janus colloids.7,19,27,28,30 One can also induce symmetry breaking by interactions between colloids in concentrated systems. For example, recent work has demonstrated the formation of self-assembled swarms, bundles, and strings through interactions between active colloidal moieties.19−24,31−34 In these reports, asymmetry at the level of the individual colloidal particle renders them

ctive systems are characterized by an energy flux through them, therefore, they are out-of-equilibrium.1 All living organisms are active in this sense, for example, motility within living cells relies on molecular motors that use chemical reactions to power mechanical motion and effect directed transport.2−4 Synthetic systems that mimic active biological systems hold promise for applications ranging from microrobots and micromixing to targeted drug delivery.5−11 Several schemes to realize active colloidal systems have been demonstrated.12−18 One class of active colloids relies on the imposition of external fields: electric,19−21 magnetic,8,22,23 or light12,24 to deliver energy to the colloidal system. In another class, chemical transformations are carried out on the colloidal surface to convert chemical energy to mechanical motion.9,25−28 To effect self-propulsion of individual colloids in dilute systems, it is necessary to break symmetry. This symmetry breaking can arise from interactions with external

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© 2017 American Chemical Society

Received: June 19, 2017 Accepted: September 12, 2017 Published: September 12, 2017 10025

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ACS Nano active. The strength and directionality of interactions between these active colloids governs emergent assembly in concentrated colloidal dispersions. Another way to induce symmetry breaking is to bond together colloidal particles, each of which is isotropic, to form chains. Recent simulations have demonstrated that semiflexible chains comprising active beads can exhibit autonomous motion,35−39 reminiscent of flagellar beating. Two aspects were important for observing active effects in these simulations: (i) the semiflexible nature of the chain allows noise to continually break symmetry, and (ii) covalent bonding of individual active beads along the chain allows for spatial proximity so that there is strong hydrodynamic coupling between them. Thus, while each of the beads is isotropic, symmetry breaking happens due to Brownian fluctuations in chain conformations. In such systems, out-ofequilibrium dynamics results from bead−bead interactions due to active flows. However, an experimental realization of this system has not been reported. One advantage of such a system is the ease of preparation of isotropic colloids relative to Janus entities. These synthetic systems also hold promise as models for studying the mechanisms that govern biological flagellar dynamics.40−44 Combining these mechanistic insights with the ease of fabrication of such active systems might provide a way to address the challenge of enhancing transport phenomena at small length scales. Here, we report the facile preparation of active colloidal chains that exhibit over 60% enhancement in diffusivity over their passive counterparts. The colloids that are enchained to form necklaces are uniformly coated with platinum nanoparticles (PtNPs) that catalyze the decomposition of hydrogen peroxideunlike self-motile Janus colloids. We combine experiments and simulations to show that the “active” enhancement in diffusivity arises from interactions between enchained colloids due to phoretic hydrodynamic flows.

Figure 1. Ice templating route to synthesis of active colloid chains. (A) Polyethylenimine (PEI) coated 1 μm-sized polystyrene particles (PS) and cross-linker (PEG diepoxy) are mixed in aqueous suspension. (B) When the suspension is cooled to −18 °C, PEI-coated PS and cross-linker accumulate along the grain boundaries of crystalline ice. The PEI forms a cross-linked mesh, so that the colloids are linked into chains. (C) The suspension containing cross-linked colloid chains is thawed, and PtNPs are added and mixed by slow shaking for 1 min. The suspension is left undisturbed for 2 h to allow PtNPs to absorb and the chains to sediment. The suspension is decanted and rediluted with DI water, with the process being repeated twice, to remove excess PtNPs. (D) 100 μL of 30% H2O2 is added to 100 μL of chain suspension, with slow shaking for 1 min, to render the chains active. (E) Representative images from fluorescent microscopy of rigid and semiflexible colloidal chains (N = 3−12). The scale bar in all the images is 5 μm.

Colloidal chains are electrostatically complexed with negatively charged 1.5 nm PtNPs, resulting in an average coverage of 5.9 × 103 PtNPs per PS colloid (viz. average interPtNP spacing ≈25 nm). Chains are rendered active by addition of hydrogen peroxide. PtNPs on the surface of the PS colloids catalyze peroxide decomposition into water and oxygen and set up diffusophoretic flows.27,28 The activity is proportional to reaction rate which is, in turn, proportional to the available peroxide concentration. We use a high peroxide concentration and work with dilute chain dispersions so that there is no measurable change in peroxide concentration (SI, Figure S2; Section I.C) over a time scale of ∼30 min, significantly longer than the period over which chain motions are studied. Thus, all our experiments are conducted at constant peroxide concentration or, equivalently, at constant value of activity. The microscope slide is carefully sealed to eliminate possible artifacts due to convective effects47 (SI, Figure S3). Finally, we note that our experiments are at sufficiently dilute chain concentrations so that there is no observable evolution of oxygen bubbles. We use fluorescence microscopy to resolve the center, ri, i = 1, ..., N, of each bead in the colloidal chain. We ensure that colloidal chains are at least several tens of microns from the top and bottom slide surfaces during the time scale of observation (on the order of several minutes). Over this time scale, sedimentation effects due to gravity are unimportant. Thus, wall effects on chain dynamics can be neglected. The positions, {ri(t)}, of the N beads are used to estimate the configurational

RESULTS AND DISCUSSION Experimental Results. We modify a previously reported45 protocol (Figure 1, details in Methods) to prepare active colloidal chains. Briefly, we freeze a dilute aqueous dispersion of colloidal particles, polyethylenimine (PEI), and diepoxy crosslinker so that the colloids assemble into chain-like and sheetlike aggregates.45 We note in passing that above a threshold concentration, ice templating results in the formation of percolated colloidal aggregates.46 Here, we work at concentrations that are over 2 orders of magnitude below this threshold. Cross-linking PEI in the frozen state results in the formation of a polymer mesh that traps colloidal particles in the aggregated state. The aggregates formed in this manner are polydisperse in size, and the aggregate size distribution obtained from ice-templating has been investigated in detail previously.45 We harvest the aggregates by thawing the mixture and work with dilute dispersions so that our investigations are restricted to isolated linear chain-like colloidal assemblies. We vary the stiffness (SI, Vidoes SI-V1 and SI-V2) of the chains by varying the cross-linking duration, and the variation of normalized end to end distances of rigid and semiflexible chains is shown in SI (Figure S7). We characterize the variation in center-to-center distance for bonded colloids and in the angle between three consecutive colloids (SI, Figure S5). From these, we estimate the spring constant that characterizes colloidal bonding and the barrier for bond bending. Both these increase systematically with increase in cross-linking duration. 10026

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ACS Nano distributions and chain diffusivity. Due to configurational fluctuations, the magnitude of the end-to-end distance RN(t) = rN(t) − r1(t) differs from the contour length L = (N − 1)b, where b ≈ 1.1 μm, is the mean bond length. We define the chain flexibility ξ as ⟨(RN(t) − ⟨RN⟩)2⟩ = ξ2L2. We investigate chains with N = 3, ..., 12, with ξ varying 10-fold, from ξ = 4 × 10−4 for stiff chains to ξ = 5 × 10−3 for semiflexible chains. We use ξ as a measure of flexibility since the persistence length is difficult to estimate reliably in short chains. We measure chain diffusivity in the usual manner by calculating the mean square 1 displacement (MSD) of the center of mass r(t ) = N ∑i ri(t ) and extracting the coefficient of its linear variation with time. We begin by presenting control experiments on passive colloid chains, as a baseline for comparison with active chains (Figure 2). The left panel shows the normalized diffusivity of

Figure 3. Variation of the mean-squared displacement of passive (left panel) and active (right panel) chains with flexibility ξ for N = 5. The mean-squared displacement is comparatively enhanced for active chains, with greater enhancement for more flexible chains.

data for chains with N = 5 as an example in Figure 3, similar trends are observed for all chain lengths investigated. MSD data for passive and active chains for N = 3, 4, and 6 are shown in SI (Figure S4). We now plot normalized diffusivity of the active chains as a function of flexibility for chains of different size in the left panel of (Figure 4). It is clear that diffusivity increases

Figure 2. Control experiment on passive colloid chains in different solvents. The left panel shows the variation of chain diffusivity D for an N bead chain, normalized by the diffusivity D0 of a single bead. The dotted line D ∼ Nν, where ν = −0.58 is a least-squares fit to the data. Zimm scaling from our simulations predicts ν = −0.6. The right panel shows the variation of the chain diffusivity with flexibility ξ. The slopes of the least-squares lines are close to zero, indicating that the diffusivity does not vary with flexibility. Figure 4. Variation of the diffusivities of active chains (D ( ) normalized by the diffusivity D0 of a single bead, with flexibility (ξ) and number of beads N, from experiments (left panel) and Brownian dynamics simulations with active hydrodynamic interactions (right panel). The general trend, discernible through the experimental noise and supported by simulation, is that the diffusivity ratio increases with flexibility.

colloid chains without the PtNP coating in both water and 15% w/v aqueous hydrogen peroxide and with the platinum coating in water. Each of these combinations is passive. The diffusivity is fitted well by a power law in N with an exponent ν = −0.58. In the right panel we show that, within experimental resolution, there is no variation of diffusivity with chain flexibility ξ. Our results are consistent with Zimm’s treatment48 of the dynamics of chains that exhibit Gaussian fluctuations. The colloidal chains that we investigate are not flexible. Rather, they can be considered as semiflexible assemblies that exhibit harmonic fluctuations around the rigid rod limit. As these fluctuations are governed by a Gaussian measure (independent of N), the Zimm pre-averaging strategy can be employed to obtain a power law scaling for the diffusivity. We observe that in the range of parameters considered, passive colloidal chains exhibit Zimm dynamics and their stiffness has no measurable influence on their diffusivity. Zimm scaling for the length dependence of semiflexible chains has also been reported previously.49 We now compare (Figure 3, left panel), the MSD of passive chains with active chains in 15% w/v aqueous hydrogen peroxide, for fixed chain size but varying flexibility. Two features immediately stand out: first, the MSD for flexible active chains is greater than that for passive chains and, second, there is significant variation of the MSD with chain flexibility for active chains, but not for passive. For active chains, there is an increase in MSD with increasing flexibility. While we present

with flexibility for all chain sizes (compare with left panel of Figure 2), which shows the flexibility-independent diffusivity of passive chains. The uniform coating of PtNPs distinguishes the active colloids in our experiment from Janus particles, in which the catalyst is distributed nonuniformly over the colloid surface. Thus, the individual active colloids in this work are chemomechanically isotropic in contrast to Janus active colloids.27,28,50 We observe no enhancement in the diffusivity of active “monomeric” colloidal particles when compared with the corresponding passive particles (SI, Figure S6). Clearly, proximity between enchained colloids provides the source of anisotropy for the active enhancement in diffusivity. Simulations. To rationalize these experimental results, we perform Brownian microhydrodynamic simulations using a well-established mathematical model.35,36 The model describes the Brownian motion of beads interacting through conservative forces, that enforce connectivity and stiffness, and dissipative forces, from Stokes drag and activity. The motion of the N 10027

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breaks detailed balance (eq 1) and, therefore, the nonequilibrium stationary state has to be determined from a balance of conservative, dissipative, and thermal forces.35,36,38 To estimate the influence of departure of the chain conformations from equilibrium, we first compute the RHS of eq 2 over the nonequilibrium trajectories. The hydrodynamic radius estimated from this nonequilibrium extension of Kirkwood’s formula does not differ appreciably from that of passive chains nor does it show a strong dependence on chain flexibility. This is consistent with the estimates of the hydrodynamic radius from experiment (SI, Figure S8). The trends observed for the chain diffusivity, then, are clearly not observed in the hydrodynamic radius (see Figure 5 and more

beads is described by the following coupled stochastic differential equations: d ri = (μij ·Fj + πij·Sj)dt + ( 2D )ij ·dWj

(1)

with repeated indices being summed over. Here Fj = −∇jU are the conservative forces derived from the interaction potential U(r1, ..., rN) (that includes the bending elasticity) and μij is the mobility tensor connecting the velocity of particle i due to the sum of all nonhydrodynamic forces on particle j, Sj is the active stresslet on the j-th bead, and πij are propulsion tensors connecting the velocity of particle i to the stresslet on particle j. The thermal forces are described by the Wiener processes Wj, where √D is a square-root of the diffusion matrix D whose matrix elements, by the Einstein relation, are Dij = kBTμij. The functional forms of the potential, the mobility, and the propulsion matrices are provided in the SI (Section II). We note that though each bead has isotropic activity, the presence of neighboring beads breaks this symmetry and induces anisotropic effects whose leading term is the stresslet. We set the sign of the stresslet as extensile. This is purely phenomenological and is based on a comparison of simulation results with experiments. The flowfields around an active colloidal chain are displayed in the SI (Figure S10), and trajectories of chain motion for active and passive chains, with matched flexibility, are shown in the SI (Video SI-V3). As in the experiments, we conduct control simulations of passive chains by setting Sj = 0. We compute the diffusivity for varying flexibility and chain size. The power law exponent (ν = −0.6) for the scaling of diffusivity with N from simulation (SI, Figure S11a) is in excellent agreement with experimental data (Figure 2). There is essentially no dependence on chain flexibility of the chain center of mass MSD (SI, Figure S9a, compare with Figure 3a). We find approximately one part in a thousand variation of the diffusivity with flexibility, well below the experimental resolution (SI, Figure S11b, compare with Figure 2b). The scaling of diffusivity with chain size is consistent with experiment and theory. The diffusivity as obtained from the MSD is in very good agreement with Kirkwood’s formula:51 DK =

kBT 3N 2

Figure 5. Comparison of the diffusivity as obtained from the MSD and from the Kirkwood formula for passive and active chains for N = 4. Both these estimates are in very good agreement for passive chains but differ widely for active chains. Activity-induced change of the hydrodynamic radius of the chain cannot account for the enhancement of the diffusivity.

detailed comparisons with the Kirkwood formula in SI, Figure S12a), which leads us to conclude that active reduction of chain size is not the mechanism underlying the enhanced diffusivity. Rather, the enhanced diffusivity arises primarily due to active hydrodynamic interactions from phoretic flows. For chain lengths much smaller than the persistence length, the enhancement of the center of mass diffusion results from an interplay of thermal fluctuations and activity (SI, Figure S13). The former generate fluctuations in the chain curvature, while the latter drive the chain locally in the direction opposite to this induced curvature. Since the thermal curvature fluctuations are unbiased, the active displacements resulting from such fluctuations are random and, when integrated over several chain relaxation time scales, produce an additional diffusive contribution to the chain motion.

Tr⟨∑ μij ⟩ (2)

ij 52,53

with the expected small overestimation. The angle brackets represent averages over equilibrium trajectories of the chain, and the trace is over the Cartesian indices which are implied by the boldface notation. With our theoretical model thus validated we turn, now, to results for active chains. The diffusivity of active chains from simulations is shown in the right panel of Figure 4. It is not easy to make a direct comparison between the experimental results and simulations, since we cannot establish an exact relation between the elasticity in our experimental chain and the elastic Hamiltonian in the simulations. This results in discrepancies between the Kirkwood average54 and experimental results. However, the trends from our simulation results agree well with those observed in the experiments: the diffusivity increases with flexibility. The most rigid chains (ξ ∼ 1.0 × 10−3) exhibit diffusivities similar to the passive chains. Finally, the enhancement in diffusivity for active chains with higher flexibility (ξ ∼ 5 × 10−3) relative to their rigid counterparts is the greatest for the longest chains. To understand the mechanism behind this enhancement, we turn to eq 1. It is clear that the active term

CONCLUSIONS In summary, our work demonstrates that large diffusivity enhancements, comparable to those demonstrated for Janus colloids, can be achieved for colloidal systems that are uniformly coated with catalytic nanoparticles. In our work, individual colloids along the chain are chemo-mechanically isotropic. Phoretic flows from catalytic reactions on the surfaces of colloidal monomers result in active hydrodynamic interactions between enchained colloids. Brownian fluctuations in the conformations of the semiflexible chains break the symmetry of the active fluid flow. The combination of active hydrodynamic flows and Brownian fluctuations results in an enhancement in chain center-of-mass diffusivity. Thermal forces more readily induce conformational transitions in flexible chains. Therefore, the enhancement in chain diffusivity is larger 10028

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ACS Nano for more flexible chains. Experimental trends are captured accurately by Brownian microhydrodynamic simulations that incorporate phoretic flows due to activity. Our results indicate how chain flexibility and activity can be used to tune diffusivity of colloidal objects. Faster than equilibrium dynamics of active chains have implications for their use in drug delivery and mixing. We have previously demonstrated the preparation of completely biocompatible icetemplated colloidal aggregates.46 Biocompatible active colloidal chains powered by enzymatic catalysts7,28 represent fascinating constructs with potential for directed therapeutic delivery. Hydrodynamically driven active chains grafted on the walls of microfluidic devices could have implications for technologically important problems such as mixing at low Reynolds number. The active colloidal chains presented here are well represented by a mathematical model and, therefore, represent an attractive model system for fundamental investigations of active flexible fibers that are ubiquitous motifs in biology, such as bacterial flagellae.

there is no significant asymmetry in the coverage of PtNP over the PS colloid surface. We estimate a PtNP coverage of 5.9 × 103 PtNP adsorbed per PS colloid (SI, Experimental Details). A dispersion of PtNP-coated colloidal chains is added to an equal volume of 30% (w/v) hydrogen peroxide. The PtNPs on the particle surface catalyze decomposition of the hydrogen peroxide, rendering the chains active. There is no chance in concentration of hydrogen peroxide during the experimental duration when the chains are visualized (SI, Experimental Details). Video Recording and Data Analysis. Laser scanning fluorescence microscopy (Carl Zeiss Axio Observer) was used with a 63× objective (NA = 1.4 and working distance = 0.19 mm). Samples were taken in a cavity slide. Great care was taken to seal the slides to eliminate convective effects (SI, Experimental Details). Fluorescent PS beads were excited with laser light (wavelength = 543 nm). We recorded images at 20fps for more than 40 s. Image analysis was performed using ImageJ software and Matlab. To find the center of mass of the colloidal chain, we used ImageJ. We subtract a constant background and then thresholded the images. We used the inbuilt Particle Analysis function in ImageJ to obtain the center of mass of the chain. A sequence of images was then used to obtain the MSD for the chain center of mass. We calculate the mean square displacement in two-dimensions as, MSD(t) = ⟨(x(t) − x(0))2⟩ + ⟨(y(t) − y(0))2⟩ , where x(t) and y(t) are the center of mass coordinates at time t, obtained from ImageJ. The MSD is linear in time with a slope = 4D, where D is diffusion coefficient (since we track the colloidal chains in 2 dimensions). As a check on the ImageJ analysis, we used the particle tracking function in Matlab to obtain the center of mass for each colloidal monomer and calculated the chain center of mass based on these coordinates. Analysis using ImageJ was consistent with that using Matlab. We used the Matlab center of mass positions of the colloidal beads in a chain to estimate chain flexibility. Simulation Details. The active colloidal chain is modeled as a series of N beads of radius a that can generate spontaneous flows in the surrounding fluid.35,37,38 We model the generation of spontaneous flow by the beads through extensile stresslets (to leading order) that are oriented along the local tangent to the chain. We derive a set of update equations by considering an instantaneous force balance between dissipative, thermal, and conservative forces. Together, these equations represent a set of coupled stochastic equations of motion for N beads of the model chain. In eq 1, ri is the position of the i-th bead. The mobility matrix μij and the propulsion matrix πij are defined as

METHODS Materials. Polyethylenimine (PEI) (50% w/v solution; Mw = 750 kg/mol, specified by the supplier) and polyethylene glycol diepoxide (PEG-diepoxide, Mw = 500 g/mol, specified by the supplier), chloroplatinic acid hexahydrate, and horseradish peroxidase (HRP) were obtained from Sigma-Aldrich and were used as received. A 2.5% w/v dispersion of fluorescent (Rhodamine-B labeled) polystyrene (PS) beads with a size of 1.08 μm and 4% polydispersity were used as received from Microparticle GmbH, Germany. Sulfonate groups present on the surface rendered the PS particles negatively charged. Hydrogen peroxide (30% aqueous solution), 4-aminoantipyrine (4AAP) 98%, and phenol 98% were used as received from Avra Synthesis Pvt. Ltd. Synthesis of Active Colloidal Chains. One μL of polystyrene (PS) latex comprising ∼108 particles was added to 200 μL of DI water. After sonication for 2 min to disperse the particles, this was added to 5 μL of PEI stock solution (comprising 100 mg PEI in 1 mL DI water), and the mixture was vortexed for 5 min. This dispersion was centrifuged, and the supernatant was removed to separate unabsorbed PEI. This was diluted with DI water and sonicated for 2 min to redisperse the PEI coated PS colloids. The zeta potential of the PS particle changes from −30 mV to +57 mV after treatment with PEI, confirming coating of the PS surface.To this was added 1 μL of PEGdiepoxide diluted in 1 mL of DI water, and the mixture was then vortexed and subsequently sonicated for 1 min. This mixture was then stored in a refrigerator maintained at −18 °C. The water is transformed into ice on freezing, and the colloids, polymer, and cross-linker are concentrated at the boundaries between ice crystals. PEI is cross-linked by PEG-diepoxide in the frozen state. We observe that allowing the cross-linking to proceed for 12 h results in the formation of a rigid chain (SI, Video SI-V1). Decreasing the crosslinking time to 8 h yields semiflexible chains (SI, Video SI-V2) and highly flexible chains are obtained when we cross-link for 4 h. After cross-linking, we thaw the frozen sample at room temperature. We synthesized PtNPs according to a previously reported protocol.55 An aqueous dispersion of colloidal chains, comprising PS colloids in a cross-linked PEI mesh, is treated with an excess of PtNPs. PtNPs are negatively charged and adsorb on the surface of the colloidal chains. Elemental mapping (SI, Figure S1b) confirms the presence of platinum on the surface of extensively washed PEI-coated PS colloids, confirming adsorption of PtNPs on the colloid surface. We verify that the colloids are isotropically coated with PtNP catalyst as follows. In control experiments, we measure the diffusion of single PS colloids coated with PEI and with PtNP adsorbed on their surface. The diffusivity of such colloids in the presence of hydrogen peroxide is experimentally indistinguishable from that in water (after accounting for differences in solvent viscosity, SI, Figure S6). This suggests that

8πημij (r, r′) = - i0- 0j G(r, r′) 8πηπij(r, r′) = - i0- 1j∇jG(r, r′)

Gij(r, r′) =

(

- li = 1 +

δ ij ρ

+ 2

ρi ρj ρ3

a ∇2 4l + 6 i

is the Oseen tensor with ρ = r − r′ and

). The diffusion matrix D is defined as D

ij

=

kBTμij. The body force, Fi is obtained from the gradient of the potential, U = UC + UE + US which, in sequence, are potentials enforcing connectivity, stiffness, and self-avoidance. The connectivity potential is 1 a two-body harmonic spring potential U C(ri, ri + 1) = 2 k(r − b0)2 , where b0 refers to the equilibrium bond length and r = |ri − ri+1|. The stiffness potential, UE = κ(1 ̅ − cos ϕ), penalizes any departure of the angle ϕ between consecutive bond vectors from its equilibrium value of zero. The rigidity parameter κ̅ is related to the bending rigidity κ, as κ̅ = κ/b0. Steric effects are incorporated through the Weeks− Chandler−Andersen potential which vanishes if the distance between any two beads rij = |ri − rj| exceeds rmin = 2a. The stresslet tensor of jth bead, Sj, which is aligned along the local tangent tj of the chain with its strength S0 is defined as

Sj = S0(tjtj − δ/3) . 10029

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ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.7b04265. Experimental details: PtNPs synthesis and characterization, video data collection and analysis, and control experiments. Simulation details: model for passive and active chains, effect of activity on enhancement of diffusion (PDF) Videos of rigid and semiflexible chain motions from experiments (SI−V1 and SI−V2); trajectories of chain motion from simulations of active and passive chains (SI−V3) (ZIP)

AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Guruswamy Kumaraswamy: 0000-0001-9442-0775 Author Contributions

Experiments were conducted by B.B. R.K.M. and A.L. performed simulations. The research was supervised by P.B.S.K., R.A., and G.K. Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS B.B. acknowledges an Inspire fellowship from the Department of Science and Technology, India. G.K. thanks the Chemical Engineering PAC of the SERB, DST for funding this project through grant number SB/S3/CE/072/2014. We thank Dr. Omkar Deshmukh for helping with analysis of the microscopy data and Dr. Jhumur Seth and Dr. B. L. V. Prasad for providing us with PtNPs. Numerical simulations were performed on the HPCE clusters at IIT Madras. REFERENCES (1) Fodor, É.; Nardini, C.; Cates, M. E.; Tailleur, J.; Visco, P.; van Wijland, F. How Far from Equilibrium is Active Matter? Phys. Rev. Lett. 2016, 117, 038103. (2) Schliwa, M.; Woehlke, G. Molecular Motors. Nature 2003, 422, 759−765. (3) Vale, R. D.; Milligan, R. A. The Way Things Move: Looking under the Hood of Molecular Motor Proteins. Science 2000, 288, 88− 95. (4) Sanchez, T.; Chen, D. T.; DeCamp, S. J.; Heymann, M.; Dogic, Z. Spontaneous Motion in Hierarchically Assembled Active Matter. Nature 2012, 491, 431−434. (5) Patra, D.; Zhang, H.; Sengupta, S.; Sen, A. Dual StimuliResponsive, Rechargeable Micropumps via ’Host−Guest’ Interactions. ACS Nano 2013, 7, 7674−7679. (6) Wang, J.; Gao, W. Nano/Microscale Motors: Biomedical Opportunities and Challenges. ACS Nano 2012, 6, 5745−5751. (7) Baraban, L.; Makarov, D.; Streubel, R.; Monch, I.; Grimm, D.; Sanchez, S.; Schmidt, O. G. Catalytic Janus Motors on Microfluidic Chip: Deterministic Motion for Targeted Cargo Delivery. ACS Nano 2012, 6, 3383−3389. (8) Ghosh, A.; Fischer, P. Controlled Propulsion of Artificial Magnetic Nanostructured Propellers. Nano Lett. 2009, 9, 2243−2245. (9) Dey, K. K.; Zhao, X.; Tansi, B. M.; Mendez-Ortiz, W. J.; Cordova-Figueroa, U. M.; Golestanian, R.; Sen, A. Micromotors Powered by Enzyme Catalysis. Nano Lett. 2015, 15, 8311−8315. 10030

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