One of the fundamental goals of bioinspired synthetic systems is to achieve enhanced and controlled transport. Here, we demonstrate the transport of a colloidal cargo with an active filamentous engine. We explore the efficiency of the directional transport of the colloidal cargo by attaching it either at the front (pushing) or at the back (pulling) of the filament. The filament is chemo-mechanically active and acquires the activity by attaching chemically active beads that provide local tangential force along the chain. The effect of the size and location of the load, activity, and bending rigidity is comprehensively explored. We observe a transition from directional to rotational motion for the pushing load filament, whereas no such transition is observed for the filament pulling the load. The transition between different states is characterized using structural properties, such as the radius of gyration, order parameter, and tangent–tangent correlation. With the help of dynamical properties, we show that the modes of propulsion and their efficiency is different for pushing and pulling, which depend on the load size.

The entities having an in-built mechanism for consuming internal sources of energy to carry out directed motion fall under the category of active matter.1,2 These species include a variety of both biological and synthetic entities spanning from microscopic to macroscopic length scale2–5 and play an essential role in taking the rate of transport beyond diffusion limit. In the present context, it is important to emphasize that active self-propelled particles are not limited to rigid micro-swimmers, such as chemically active Janus spheres,6 magnetic propellers,7 bi-metallic rods,8 and nanomotors.9 Instead, there exists a class of polymeric motors motivated by the active biological systems, such as actin filaments,10 microtubules,11 and active polymeric gels,12 that are being synthesized in different groups worldwide.13–16 These active filaments exhibit fascinating phenomena, such as loop-formation,17 spontaneous oscillations and spiral formations,18,19 swelling,20–22 coil-globule transitions,20,23,24 and enhanced diffusion.22,23

In the biological realm, microtubule filaments are an important active polymer where the family of motor proteins moves along it to generate forces that affect the dynamics of the cytoskeletal network and transport cellular cargo to their destinations.25–27 Microtubules are also a major constituent of cilia found on the surface of many eukaryotic cells.28 The oscillatory motion of cilia is quite remarkable and acts as the propulsive component for the cell and the fluid around it. Outside the biological domain too, several attempts have been made to create such active polymer in laboratory with the help of active colloids.29–32 In recent years, active flexible and semi-flexible polymers have also received tremendous theoretical attention for both fundamental interest and also for designing new materials for possible applications.21,33–35

A potential application of such filament lies in active transport, wherein a load can be transported from one place to another after defeating the limit imposed by diffusion.36 A few theoretical attempts have been made in the past to design such active polymer systems that can carry out the active transport of the colloids attached to them. However, effective navigability is yet to be achieved using active filaments. In one of the pioneering works in this direction, Isele-Holder et al.37 have revealed versatile phase behavior for self-propelled filaments while pushing a finite sized rigid. The work tries to bridge the existing gap between the two extremes of freely swimming filaments and pinned or clamped filaments that correspond to filaments pushing the infinite load.37 It was shown that, based on the size and shape of the load, the filament can exhibit a variety of trajectories, such as straight-line, rotating, beating, and beat-and-circle motion, with increasing active force in the absence of hydrodynamic interactions in two-dimensions. Another work by Manna et al. shows the navigation of a colloidal particle by active filament composed of chemo-mechanically active beads. A generic model that includes hydrodynamic interactions to describe the active filament is used in this study. It was shown that the transport efficiency depends on the length of the filament, the strength of activity, and the colloid radius. It is noteworthy that the efforts to design active filament as a load carrier are so far concentrated on pushing a load, assuming that pushing may be the best strategy. However, it is clear from biology that the microtubule engages a push or pull mechanism based on the cell geometries and the distance over which the organelles have to be moved.27 The pushing mechanism by the microtubule is effective over short distances until the microtubules do not buckle, whereas, when the distances to be covered by the organelles is large or when the geometry is complex, the pulling mechanism that comes to the rescue.

In this work, we investigate the synthetic active filament as a carrier for load both pushing and pulling mode to find out the best suited approach for navigation. In our work, the filament is an active heteropolymer chain constructed from monomers with several chemically active dimers, comprising catalytic and noncatalytic monomers interspersed along its backbone made from inactive linker beads. The filament is then attached to a passive load either at the front or the back and the filament-load assembly is then immersed in an explicit solvent bath. In our model, the activity on the polymer will be derived from chemical reactions and diffusiophoretic mechanisms that operate along the polymer backbone to impart local tangential force to the filament. We find that the filament motion is decisively controlled by the placement as well as the size of the cargo. The activity and the bending rigidity of the filament too play an important role in the modes of navigation. The presence of the load in front of the active filament exhibits a versatile motion under the compressive force, whereas, when the load is attached at the back of the filament, the suppression of bending modes leads to only elongated states of the polymer. Our main focus in this article is to find various modes of propulsion emerging due to different load sizes and positions. To do so, we have calculated the order parameter (Q), radius of gyration (Rg), and persistence length of the filament for different configurations. The efficiency of transport was further estimated by finding the dynamical properties of the filament. The findings of our work not only demonstrate an enhanced directed transport of colloid by an active filament engine but also shows that, for smaller cargo, it is better to use the engine in the pushing mode, whereas, for larger load size, the pulling mode of the engine is more effective. The findings of our work clearly emphasize that the synthetic filaments should explore both push and pull strategies and use the appropriate one depending on the need.

This paper is organized as follows: In Sec. II, we discuss the coarse grained model for the chemically active filament-load assembly immersed in an explicit solvent. Various structural and dynamical behavior of filament-load assembly is elucidated in the results (Sec. III). Finally, the work has been summarized with in Sec. IV with some suggestions for experimental realization.

The polymer-load structure consists of Nm monomer beads of the filament and a load bead (L). The load of radius RL is attached to the monomers with positions ri (i = 1, …, Nm) and diameter σm either at the front or at the back of the polymer, as shown in Fig. 1. This heterogeneous polymer load structure is free to move in three dimensions. The total interaction potential on each bead due to all other beads of the filament-load assembly is given by

Um(ri)=bondUsp(qi)+angleUbe(qi,qi+1)+EVULJ(rij),
(1)

where the spring potential Usp(qi) = κs2(qiq0)2 acts between all the neighboring beads to maintain the bond length |qi| = |riri+1| at its equilibrium value q0. The three-body potential Ube(qi, qi+1) = κb(1 − cos ϕi) incorporates the rigidity of the polymer by maintaining the bond angle ϕi near the desired equilibrium value of zero. The parameters κs and κb regulate the stiffness of the spring and rigidity of the polymer, respectively. All the beads also interact via pair-wise excluded volume interaction ULJ(rij)=4ε[(σrij)12(σrij)6+14], if rij ≤ 21/6σ, where σ depends on the specific pair interaction under consideration.

FIG. 1.

The schematic diagram of the active filament carrying the load. The chemically active (C) monomer is in brown, non-catalytic (N) in purple, and inactive beads (I) in green color. The passive colloidal load (L) in blue can be attached either to the front (a) or to the back (b). The red arrow indicates the direction of local self-propulsion force.

FIG. 1.

The schematic diagram of the active filament carrying the load. The chemically active (C) monomer is in brown, non-catalytic (N) in purple, and inactive beads (I) in green color. The passive colloidal load (L) in blue can be attached either to the front (a) or to the back (b). The red arrow indicates the direction of local self-propulsion force.

Close modal

The property of self-propulsion is integrated into our filament carrying the load by virtue of its chemical heterogeneity. Our filament is composed of a linear sequence of catalytic (C), non-catalytic (N), and inactive (I) linker monomers, arranged along the chain and a passive load (L) attached to it either at the front or back, as shown in Fig. 1. The C, N, and I monomers differ in their chemical properties, as described below. The polymer is then immersed in a fluid containing a large number of point-like A and B type solvent particles (Ns = NA + NB). An irreversible chemical reaction A + CB + C takes place on all the C monomers whenever the fuel type solvent A encounters the C surface in a defined cutoff region rc ≤ 21/6(σ/2). On the other hand, N, I, and L do not participate in such a chemical reaction. The chemical reaction then generates a non-equilibrium gradient of the product (B) around the other beads in the chain, which is a maximum around N. The polymer acquires self-propulsion by responding to this chemical gradient, which is set up in the model by the different interaction energy parameters of N beads with different solvent types. Section 1 of the supplementary material describes in detail the origin of propulsive force due to dimer pairs CN.

The solvent molecules interact with the polymeric beads through the same excluded volume interaction as that given above with the energy parameter ɛαS, where α = C, N, L, I and S = A, B. The energy parameter of C, I, and L with solvent molecules have been taken to be the same (ɛA), while A and B solvent molecules have different energy parameters ɛA and ɛB, for their interaction with the N bead. It is the combined effect of non-equilibrium gradients around N (produced by C), and the potential asymmetry provides a local tangential force to the N bead, which then gets translated to self-propulsion of the polymer. We define the activity number Ac to characterize the strength of the self-propulsion force,

Ac=εA/εB.
(2)

Here, we tune the strength of activity by changing the value of ɛB keeping ɛA = 1.0. The solvent–solvent interaction, on the other hand, is through a coarse-grained collision technique, as discussed below.

In this article, we explore the dynamics of the polymer-load in an explicit solvent by using hybrid Molecular Dynamics (MD) and Multi-Particle Collision Dynamics (MPCD) simulation scheme, where the evolution of the system consists of streaming and collision steps.38–40 In the streaming, the positions and velocities evolve by Newton’s equation of motion governed by generated forces determined from the total potential energy. The collision step is performed at regular interval τc wherein all point-like solvent particles are sorted into cubic cells of size a, where they exchange their momenta. The post-collision velocity of solvent particle i is given by vi=Vζ+Rζ(α)(viVζ), where Vζ is the center-of-mass velocity of particles in cell ζ. Rζ(α) is the rotation matrix, where α is the angle by which the velocities of the solvent particles are rotated about a randomly chosen axis in a cell ζ. This scheme conserves the system’s mass, momentum, and energy and includes complete hydrodynamic interaction. To maintain the propulsion of the polymer, it is desired to preserve the non-equilibrium solvent gradient. This is achieved by converting the product B molecules back to the fuel A whenever they diffuse sufficiently far from the center of mass of the polymer chain.

In this system, all the physical parameters are scaled in units of thermal energy kBT, the mass of the solvent m, and MPC cell length a. Time is scaled in the units of τ=(ma2/kBT), velocities are in the units of a/τ, and temperature in terms of kBT/ɛA. A cubic simulation box of side L = 76 with periodic boundary conditions in all directions is used here. The integration time for MD has been taken to be Δt = 0.01, while the MPC collision time τc = 0.5. The mass of solvent particles is m = 1. The diameter of the monomers in the active chain is taken to be σm = 4.0 and mass 320m to ensure density matching with solvent. We have presented the results for Nm = 12 chain in which the load radius (RL) is varied in the range of 0.5–6. Here, we study the dynamics of the filament with different bending rigidity κb = 0 and 3, where κb = 0 represents a flexible polymer-load assembly. The rotation angle for the MPC simulation is taken to be α = π/2, and the average number density per cell N0 = 10. Therefore, the total number of solvent particles Ns ≈ 4 × 106. The temperature is fixed at 0.2, and the strength of the harmonic spring is ks = 30. The value of the energy parameter for LJ interactions between the beads is ɛ = 10. The strength of the activity is varied in the range of (2, 100).

The main focus of the paper is to design the appropriate filament-load assembly that imparts optimum navigability to the load using the active filament. Therefore, we explore distinct configurations of the assembly by placing the load at various sites along the polymer length. We find that the properties of the filament-like, rigidity, activity, and load size play a crucial role in deciding the fate of the navigation. We analyze the maneuvering capacity of the polymer by quantifying the trajectories using order parameter and by various conformational and dynamical investigations. Our calculations show that carrying the load at the front as done in previous studies37,41 may not be the best strategy for all load sizes to transport the colloid in case of active filament.

Our calculations show that the manner in which the active polymer transports the load depends on the physical properties of the load, such as its size and the position at which it is attached to the polymer. Also, the propulsion mode is very much controlled by the activity of the polymer.

1. Effect of size and location of the load

The two extreme aspects of active polymer propulsion, i.e., a freely propelling polymer and a clamped polymer (analogous to infinite load), have been studied in the past by different groups;18,19,42–44 however, any systematic study by varying the load size and location along the filament has not been explored. In this study, we intend to understand the effect of the load on the motility of the filament by varying the load size and position along the filament systematically. In the usual scenario, one would expect that increasing the load size will reduce the ability of the filament to move. Here, we show that increasing the load size not only slows down the filament but also dictates the propulsion style. Figure 2 shows the time evolution of the polymer-load assembly with a large load at the front and back. It is evident that when a large load is attached at the front of the filament (front load polymer), the filament starts showing a rotational motion. In contrast, when the large colloid is attached at the back of the active filament (back load polymer), it exhibits a straight line motion.

FIG. 2.

Few simulation snapshots of the filament-load assembly for RL = 6 at κb = 3.0 and activity Ac = 100: (a) front load polymer (FLP) and (b) back load polymer (BLP). The red arrow shows the direction of motion of the filament.

FIG. 2.

Few simulation snapshots of the filament-load assembly for RL = 6 at κb = 3.0 and activity Ac = 100: (a) front load polymer (FLP) and (b) back load polymer (BLP). The red arrow shows the direction of motion of the filament.

Close modal

To understand it further, we show typical trajectories followed by the filament carrying loads of different sizes at a different site of the filament in Fig. 3. The front-load polymer (FLP) exhibits a snaking motion when the size of the load is small RL = 2 [Fig. 3(a)], whereas on increasing the size RL = 6, the snaking motion gets converted to a rotational motion, wherein the filament mainly swirls around the load with intermittent openings [Fig. 3(b)] (see movies 1 and 2 in the supplementary material). On the other hand, for the back load polymer (BLP), the trajectory of the filament remains snaking for all the load sizes. The snaking motion becomes more persistent along the propulsion direction with increasing the load size as shown in Figs. 3(c) and 3(d) (see movies 3 and 4 in the supplementary material). It is important to note here that, for BLP, the filament never exhibits a rotational motion, as is the case for FLP. The color code of Fig. 3 displays the center of mass velocity Vcm of the filament, normalized by center of mass velocity Vcm0 when there is no load, indicating the expected decrease in velocity on increasing the load size. The inset of Fig. 3(d) shows the snaking motion of the filament having a large load (RL = 6) attached to the middle of the filament. The above observations indicate that a heavy load at the front leads to the filament losing the directional motion. On the other hand, the heavy load at the back enhances the directionality of the filament.

FIG. 3.

Typical trajectories of the center of the mass of the filament for FLP and BLP for load sizes RL = 2 and RL = 6. The bending rigidity κb = 3.0 and activity Ac = 100. The inset plot of (d) is for middle load polymer case of RL = 6. In all cases, the color code of the trajectory corresponds to the Vcm of the filament, normalized by velocity when there is no load attached.

FIG. 3.

Typical trajectories of the center of the mass of the filament for FLP and BLP for load sizes RL = 2 and RL = 6. The bending rigidity κb = 3.0 and activity Ac = 100. The inset plot of (d) is for middle load polymer case of RL = 6. In all cases, the color code of the trajectory corresponds to the Vcm of the filament, normalized by velocity when there is no load attached.

Close modal

To quantify the above stated snaking and rotational motion with respect to the load size and the flexibility of filament we define the order parameter Q as45 

Q=1V0Ttot|0TtotVcmdt|,
(3)

where Vcm is the center-of-mass velocity of the filament, V0 is the speed of polymer while taking a straight path, and Ttot is the total simulation time. According to this definition, Q = 1.0 represents a purely straight line motion, whereas Q = 0.0 is for purely rotational motion. Any intermediate value denotes the snaking motion of the polymer. Figure 4(a) shows the variation of Q with the different load size (RL) for various values of bending rigidity κb in both FLP and BLP configurations.

FIG. 4.

(a) The plot of order parameter Q with varying load radius RL for different rigidity κb = 3 and κb = 0 for both FLP and BLP as shown in main and inset plot, respectively, at Ac = 100. (b) The probability distribution of the averaged bond angle ⟨θ⟩ for FLP (main) and BLP (inset) at fixed value of κb = 3 and same activity for different load sizes. The schematic presents the definition of θ between successive monomers.

FIG. 4.

(a) The plot of order parameter Q with varying load radius RL for different rigidity κb = 3 and κb = 0 for both FLP and BLP as shown in main and inset plot, respectively, at Ac = 100. (b) The probability distribution of the averaged bond angle ⟨θ⟩ for FLP (main) and BLP (inset) at fixed value of κb = 3 and same activity for different load sizes. The schematic presents the definition of θ between successive monomers.

Close modal

The order parameter in the case of FLP displays a drastic decrease on increasing the load radius RL for a semiflexible filament with κb = 3. For lower RL, Q ≈ 0.6 indicates a snaking motion, which then decreases to Q ≈ 0.01 testifying to the transition from snaking to rotational motion of FLP. The BLP, on the other hand, does not exhibit any significant change in Q certifying the snaking motion of BLP even for high RL. A similar behavior is observed even for flexible polymer (κb = 0), as shown in the inset of Fig. 4(a). It must be noted that, for a semiflexible filament in FLP configuration, we do not see any sudden change in Q value by increasing RL; hence, this transition from snaking to rotational motion is continuous, which is also discussed in an earlier work.46 

The difference in the mode of propulsion is also visible when we plot the probability distribution [P(θ)] of the bond angle θ between successive monomers along the polymer chain. Figure 4(b) shows this distribution for FLP and BLP for different load sizes in the main plot and inset, respectively. The shifting and broadening of the peak toward the lower value of θ with increasing load radius RL in FLP again confirm the transition from extended to the bent state of the polymer. The inset plot of Fig. 4(b) does not exhibit such behavior indicating only the elongated shape for BLP.

2. Effect of activity

To understand the role of activity on the propulsion modes, we probe the dynamics of FLP with different activity strengths Ac. The dependence of average center-of-mass speed on Ac shows a monotonic increase in speed of the polymer with activity (see Table I); however, high center of mass speed does not translate to efficient transport. Figure 5 shows the typical trajectories of the filament, with activity number Ac increasing from left to right with load radius RL = 6. At smaller Ac, the filament exhibits a slower motion with snaking trajectories [Figs. 5(a) and 5(b)]. On the other hand, for larger activity (Ac ≥ 20), the competition between the propulsion force along the filament and a big load at the front causes compression in the filament resulting in the rotational motion, as shown in Fig. 5(c). The inset plot of Fig. 5(c) displays the calculation of Q as a function of load radius RL for Ac = 20, showing the decrease in Q, and hence, the rotational motion as the load size is increased.

TABLE I.

Center of the mass speed Vcm of the FLP with RL = 6 and κb = 3 for various activity strength Ac.

Ac 10 20 100 
Vcm 0.0064 0.0069 0.0072 0.0097 
Ac 10 20 100 
Vcm 0.0064 0.0069 0.0072 0.0097 
FIG. 5.

Trajectories of the center mass of the filament for RL = 6 at different activities Ac = (a) 2, (b) 10, (c) 20, and κb = 3 for FLP case only. Also, the snapshots of representative polymer conformations at different times are shown.

FIG. 5.

Trajectories of the center mass of the filament for RL = 6 at different activities Ac = (a) 2, (b) 10, (c) 20, and κb = 3 for FLP case only. Also, the snapshots of representative polymer conformations at different times are shown.

Close modal

1. Radius of gyration

To further quantify the snaking to rotational transformation in polymer-load assembly, we characterize the filament conformation by computing the average radius of gyration Rg of the filament without considering the load,

Rg=1Nmi=1Nm(riRcm)2,
(4)

where Rcm is the center-of-mass of the chain (without the load) and angular brackets indicate the ensemble average. The prevalence of snaking and rotational motion for FLP and directed motion for BLP is also very clear from the Rg calculations. Figure 6(a) displays the dependence of the Rg on the load size (RL) for different flexibility of the polymer. κb = 0 indicates a flexible filament, whereas κb = 3 is for a semi-flexible filament. The plots in Fig. 6(a) shows the behavior of the Rg with increasing load radius (RL). For a semi-flexible polymer, we observe a polymer stretching in case of load at the back, indicating a directional motion for BLP. However, a drastic decrease in FLP can be attributed to the transition from snaking to swirling motion with increasing load size. On the other hand, the flexible filament with load at the front does show a decrease in Rg [inset of Fig. 6(a)], which is not due to rotational motion. Rather it collapses into a globular structure (similar to one reported in an earlier work24). The inset of Fig. 6(a) shows an extension of filament while moving due to a heavier load at the back. This elongation is more pronounced for the flexible filament, as it has more scope of extension. Therefore, the filament’s flexibility and the load’s size are crucial to dictate the final conformations acquired by the filament.

FIG. 6.

(a) The Rg as a function of load radius (RL) for different polymer flexibility κb = 3 (main plot) and κb = 0 (inset plot) in the FLP and BLP mode. (b) The time evolution of Ree (main plot) and Rg (inset plot) for RL = ∞ for the FLP and BLP mode. The activity parameter Ac is fixed to 100 in all these cases.

FIG. 6.

(a) The Rg as a function of load radius (RL) for different polymer flexibility κb = 3 (main plot) and κb = 0 (inset plot) in the FLP and BLP mode. (b) The time evolution of Ree (main plot) and Rg (inset plot) for RL = ∞ for the FLP and BLP mode. The activity parameter Ac is fixed to 100 in all these cases.

Close modal

Figure 6(b) reveals the Ree and Rg for FLP and BLP with RL = ∞, i.e., the polymer is clamped at one end. Here, the end-to-end distance Ree=r1rL2 with r1 and rL being the position of first monomer and load, respectively. The oscillation in Ree and Rg for FLP demonstrates the opening and closing of polymer while performing the swirling motion. Interestingly, the clamping at the back stalls the filament in a straight configuration as opposed to oscillations with clamping at the front. The above observations clearly illustrate that positioning the load along the filament greatly impacts the polymer’s configuration, and therefore, it is important to choose the load location correctly. Furthermore, it is clear that flexible filament (κb = 0) is more prone to collapse; therefore, for all the subsequent analyses, we will focus on the semi-flexible filament with κb = 3.

2. Local effects on the polymer structures

To understand how the deformation induced by the load propagates along the filament, we calculate the average distance between the load and each monomers of the filament, ds = ⟨|rsrL|⟩, where s = 1, 2, …Nm is the monomer site s as we move away from the load. Here, rL is the position of the load.

It is evident from Fig. 7(a) that the final FLP configuration depends on the load size at the steady state. Figure 7(a), in the regime of small load size, is such that the average distance ds linearly increases with the distance from the load, indicating a straight conformation of FLP for RLσm/2 (σm is the diameter of the monomer). However, for a larger load size RL > σm/2, the ds initially increases linearly, followed by a tendency to saturate at larger s, indicating the propensity of formation of swirls. For RL = 6, the ds exhibits a saturation indicating the formation of a stable swirling state of the filament around the load due to the compressive force the load provides to the filament. The BLP, on the other hand, shows a systematic linear increase of ds for all load sizes [inset of Fig. 7(a)], demonstrating the elongated states for BLP.

FIG. 7.

(a) Average distance between load and each monomers ds, as a function of the monomer index s for different values of the load radius RL for FLP in the main plot and inset is for BLP. (b) The time evolution of ds for RL = 6 (only the FLP case). The activity Ac = 100 and rigidity κb = 3 is fixed in these plots.

FIG. 7.

(a) Average distance between load and each monomers ds, as a function of the monomer index s for different values of the load radius RL for FLP in the main plot and inset is for BLP. (b) The time evolution of ds for RL = 6 (only the FLP case). The activity Ac = 100 and rigidity κb = 3 is fixed in these plots.

Close modal

The time evolution of the average distance between load and monomers for a large load (RL = 6) in Fig. 7(b) provides insight into the route of forming swirls. At initial times (t1, t2, t3), the linear increase in ds manifests the stretched state of the filament, but at later times (>t3), the compression from the load at front leads to swirl formation with saturation in ds.

The swirling, snaking, and straight configuration of the polymer with front and back load further suggests an effective change of rigidity of the filament with load size and position. To measure this effective change in rigidity, we calculate the persistence length of the filament. The persistence length of the polymer is computed from the tangent–tangent correlation C(k) of the active filament C(k)=β̂(ri)β̂(r1), where i = 1, 2, 3 …, Nm − 1, k is the contour length and β(ri) = (ri+1ri)/|r(i+1)ri| is the unit tangent vector on the ith monomer of the filament. The decay of C(k) is roughly exponential as C(k) ∼ exp(−k/lp), with lp as persistence length of the filament. We evaluate the value of lp from the best exponential fit of each curve of C(k) shown in Fig. 8. The decay in C(k) for FLP and BLP are depicted in Figs. 8(a) and 8(b), respectively. It is important to note that the decrease in the correlation function is faster for higher load size in the case of FLP, leading to a decrease in lp [inset of Fig. 8(b)], whereas an opposite trend is observed for BLP. Therefore, both the load size and position can be the tuning parameter to change the effective rigidity of the polymer.

FIG. 8.

(a) Bond–bond spatial correlation C(k) along the contour length of the polymer for different RL, κb = 3.0, and activity Ac = 100 for the FLP mode. (b) C(k) of BLP for various values of RL in the main plot and the persistence length lp as a function of RL in the inset plot for same parameter (κb = 3.0 and Ac = 100). Black dashed lines show the exponential decay.

FIG. 8.

(a) Bond–bond spatial correlation C(k) along the contour length of the polymer for different RL, κb = 3.0, and activity Ac = 100 for the FLP mode. (b) C(k) of BLP for various values of RL in the main plot and the persistence length lp as a function of RL in the inset plot for same parameter (κb = 3.0 and Ac = 100). Black dashed lines show the exponential decay.

Close modal

The objective of the paper is to identify the best-suited forcing method, i.e., push or pull for transportation of a load by an active filament. Therefore, we next calculate the polymer’s directed velocity and mean square displacement (MSD) to find the effectiveness of transport in these two cases.

1. Directional velocity

The conformations achieved by the filament carrying the load directly influence the dynamics of the filament. To understand the influence of the load position on the dynamics, we estimate the directed velocity47 of the filament-load assembly by computing the center of mass velocity Vcm projected along the unit vector ẑ, defined as the vector joining the first CN dimer (as shown in Fig. 1) Vz=Vcmẑ(t). The double angular brackets denote an average over time and realizations. The decrease in ⟨Vz⟩ with the increasing load radius RL is shown in Fig. 9 for FLP and BLP conformations. The probability distribution P(⟨Vz⟩) for the same is given in the inset.

FIG. 9.

The main plot shows Vz as a function of load radius (RL) for κb = 3 and Ac = 100 for the FLP and BLP mode. The probability distribution P(Vz) of different load radius (RL) for FLP at same Ac and κb is shown in the inset plot.

FIG. 9.

The main plot shows Vz as a function of load radius (RL) for κb = 3 and Ac = 100 for the FLP and BLP mode. The probability distribution P(Vz) of different load radius (RL) for FLP at same Ac and κb is shown in the inset plot.

Close modal

The overall directed velocity of the polymer decreases monotonically on increasing the load size. However, it is interesting that the decrease in FLP is much more drastic than in BLP. For lower load size RLσm/2 FLP that possess higher ⟨Vz⟩ than BLP, whereas the ⟨Vz⟩ for BLP is higher for RL > σm/2. This indicates pulling is more efficient than pushing for a larger load size.

The dynamics of the filament can be further characterized by estimating the mean-square displacement (MSD) of the center of mass. We compute mean square displacement (MSD) of the filament ΔRcm2(t)=(Rcm(t)Rcm(0))2, where Rcm(t) is the center-of-mass position of the filament-load assembly at time t. In equilibrium, the MSD of a filament consists of two main regimes.48 In the short-time limit, ΔRcm2(t)t2 known as ballistic motion. However, at the long-time scales, it possesses a diffusive limit with ΔRcm2(t)t.

Figure 10 shows the MSD for the filament in both FLP and BLP configurations. The straight structures in the case of BLP help in the directed motion of the filament, and hence the MSD exhibits a ballistic motion for BLP throughout our simulation time scale. On the other hand, the load at the front inhibits the straight line motion of the filament, and hence the polymer tends toward a diffusive regime at large time scales.

FIG. 10.

Mean square displacement (MSD) of the center of the mass of the filament-load assembly with time. The MSD for FLP of different RL is shown in main plot and inset shows the MSD for BLP. The ballistic motion ∼t2 and diffusive motion ∼t of the assembly is observed. The activity Ac = 100 and rigidity κb = 3 is fixed.

FIG. 10.

Mean square displacement (MSD) of the center of the mass of the filament-load assembly with time. The MSD for FLP of different RL is shown in main plot and inset shows the MSD for BLP. The ballistic motion ∼t2 and diffusive motion ∼t of the assembly is observed. The activity Ac = 100 and rigidity κb = 3 is fixed.

Close modal

As we know, the dimer pair CN is the main source of propulsion force. Therefore, its location, number, and the sequence in which it is attached to the polymer will also play a crucial role in the dynamics. To understand the effect of the orientation of the dimer, we reverse the sequence of one of the dimers along the polymer chain for FLP, as shown in Fig. 11. In Figs. 11(a) and 11(b), we reversed the sequence of the second and third dimer, respectively, and hence the local propulsion force for this reverse dimer is pointing in the opposite direction with respect to others. It is interesting to note that such reversal alters the conformational states and dynamics of FLP.

FIG. 11.

A schematic representation of polymer-load assembly with the reversed sequence of one of the dimer. (a) Second dimer pair (middle) CN is reversed; (b) last dimer pair is reversed.

FIG. 11.

A schematic representation of polymer-load assembly with the reversed sequence of one of the dimer. (a) Second dimer pair (middle) CN is reversed; (b) last dimer pair is reversed.

Close modal

The original rotational dynamics of FLP when all the three dimers force the filament in the same direction changes drastically upon reversal of one dimer. The typical trajectories for configurations in Figs. 11(a) and 11(b) are shown in Figs. 12(a) and 12(b). A comparison of Figs. 3(b) and 11(b) shows that the rotation motion of FLP can be converted to the directed motion just by reversing the sequence of one dimer. When the last dimer exerts force in the opposite direction polymer tends to achieve a straight configuration that helps it to perform directed motion. Such elongation of the polymer due to orientation reversal of dimer has also been shown recently in the context of active filament network.49 

FIG. 12.

Typical trajectories of the center of mass of the filament (RL = 6) by varying the sequence of one dimer. (a) The sequence of second dimer is reversed, and (b) sequence of last dimer is reversed (Ac = 100 and κb = 3).

FIG. 12.

Typical trajectories of the center of mass of the filament (RL = 6) by varying the sequence of one dimer. (a) The sequence of second dimer is reversed, and (b) sequence of last dimer is reversed (Ac = 100 and κb = 3).

Close modal

In this work, we have studied the colloidal transport driven by a chemically active filament, wherein the activity is introduced by the chemical reaction taking place on the monomers distributed along its length. These chemically functional monomers act as a local active force generation source on the polymer that drives the filament and hence load clamped to it. Since the engine is a long filamentous object, it is susceptible to thermal fluctuations, the main focus of the work is to identify the appropriate strategy, i.e., push or pull, for transporting a rigid spherical load.

We show that the active filament can impart a directional velocity to the colloid, which is clamped to it, in a viscous fluid. However, the mode of propulsion and transport efficiency depends on various factors, such as activity of the filament, load size, position of the load with respect to the engine and the flexibility of the filament. For a semi-flexible filament, we show that the load in the front of polymer (FLP) significantly favors rotating states upon increasing the shipment size. The buckling instabilities caused by the compressive loads result in a versatile mode of propulsion, including straight line motion and swirling around the load. In contrast, when the load is attached to the back of the filament (BLP), the filament always stays in an extended state and is able to impart higher directional velocity to the load as compared to FLP. The noticeable difference between FLP and BLP, i.e., a transition from the elongated state of the polymer to the rotational or swirling state in FLP and no such transition in BLP, is well described by the order parameter and the radius of gyration of the polymer. Furthermore, the higher efficiency of the polymer to impart directional velocity to large loads in the pulling mode is visible in the dynamical properties of the polymer.

Our work attempts to unravel the interplay between the active force and the shape deformation in the extended system, which can give rise to interesting dynamical behavior. Although our discussions in the paper are limited to one filament and one load, the method used here can extend the work for more complex geometries closer to natural swimmers.

See the supplementary material for Part 1 of the supplementary material it describes the origin of the active force on the polymer. Part 2 consists of four supplementary movies depicting different modes of propulsion.

The computational work was performed at the HPC facility in IISER Bhopal, India.

The authors have no conflicts to disclose.

Namita Jain: Data curation (equal); Formal analysis (equal); Writing – original draft (equal). Snigdha Thakur: Conceptualization (equal); Formal analysis (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

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Supplementary Material