Siteng
Zhang
,
Jiuling
Wang
and
Ting
Ge
*

Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina 29208, USA. E-mail: tingg@mailbox.sc.edu

Received
2nd June 2022
, Accepted 6th August 2022

First published on 8th August 2022

Recent advances in the functional material and biomedical applications of nanorods call for a fundamental understanding of the active motion of nanorods in a viscoelastic medium. Molecular dynamics simulations are performed to investigate a model system consisting of force-driven active thin nanorods in a melt of unentangled polymers. The activeness of a thin nanorod arises from a constant external force applied uniformly along the rod. The simulations demonstrate that the active force overcomes the randomness of the diffusive motion and results in a ballistic motion along the direction of the applied force at long timescales. The constant speed of the force-driven ballistic motion is determined by the balance of the active force and the friction from the coupling of the nanorod with the polymer viscosity. The friction coefficient, which is computed as the ratio of the active force and the speed, decreases as the active force increases. The origin of the reduction in the friction coefficient is the high speed that allows the nanorod to renew its local environment faster than the relaxation time of melt chains. A scaling theory is developed to quantify the dependence of the friction coefficient on the strength of the active force. The simulations also demonstrate that the force-driven ballistic motion suppresses the rotational diffusion of the rod and cuts off the de-correlation of the rod axis with time. On the scaling level, the long-time trajectory of a force-driven active nanorod piercing through unentangled polymers may be described as a stretched array of “active blobs”, where the short-time random-walk trajectory within an active blob is unperturbed by the active force.

The passive dynamics of thin nanorods in viscoelastic polymers are characterized by a length-scale- and shape-dependent breakdown of the Stokes-Einstein relation in the continuum theory.^{40} The diffusion coefficients for different modes of the nanorod dynamics, including the translational motion along and normal to the rod axis, the overall translational motion, and the rotational motion of the rod axis, are all related to the bulk viscosity of surrounding molecules in the Stokes–Einstein relation.^{41,42} Nevertheless, recent experiments^{28,29} and computer simulations^{30,31,43} have revealed the coupling of a nanorod only to a fraction of the bulk viscosity, indicating the breakdown of the Stokes–Einstein relation. Molecular dynamics simulations of monomerically thin nanorods in unentangled polymer melts^{43} have shown the anisotropy in the translational diffusion along and normal to the rod axis. The parallel diffusion in the body frame of the thin nanorod is coupled to the local viscosity on the monomer length scale, which is independent of the rod length l. The coupling to the local monomeric viscosity reflects a full screening of the hydrodynamic interactions. By contrast, the normal diffusion is coupled with an effective viscosity that depends on l with respect to the polymer chain size, corresponding to unscreened hydrodynamic interactions. The same effective viscosity for the normal diffusion also dictates the rotational diffusion of the rod axis. For the overall translational diffusion in the lab frame, the effective viscosity corresponds to a partial screening of the hydrodynamic interactions, as both the parallel and normal components contribute to the overall diffusion.

Activeness is anticipated to alter the interplay between the nanorods and the viscoelasticity of the surrounding medium, as the ballistic motion induced by the activeness has been shown to enhance the mobility of nanoscale objects in experiments^{27} and computer simulations.^{44,45} To investigate the active dynamics of nanorods in a viscoelastic medium, we perform molecular dynamics simulations and scaling analysis of a model system consisting of force-driven active thin nanorods in a melt of unentangled polymers. The thickness of the nanorod is comparable to the monomer size, and the aspect ratio of the nanorod is ≥4. The unentangled polymer chains represent a viscoelastic medium with internal chain relaxation dynamics. The activeness of a thin nanorod is introduced by applying an active force uniformly along the initial rod axis, which facilitates the arrival of the steady state in the simulations. We find that the active force overcomes the randomness of the diffusive motion and results in a ballistic motion of the thin nanorod along the force direction at long timescales. The constant speed of the force-driven ballistic motion is determined by the balance of the active force and the friction from the coupling of the nanorod with the polymer melt. The friction coefficient decreases as the strength of the active force increases. We determine the origin of the reduction in the friction coefficient is the fact that the high speed renews the local environment of the nanorod faster than the relaxation time of polymer chains. We develop a scaling theory that successfully describes the dependence of the friction coefficient on the strength of the active force as well as the onset of the reduction in the friction coefficient in the simulations. We also find that the active force suppresses the rotational diffusion of the rod and cuts off the de-correlation of the rod axis with time.

Nanorods and polymers were placed in a cubic box with side length L and periodic boundary conditions in all three directions. Each sample was equilibrated at temperature T = 1.0ε/k_{B} and pressure P = 0. The equilibration was performed with a Nosé–Hoover thermostat/barostat applied to the polymer chains. A Nosé–Hoover thermostat was applied to both the translational and rotational degrees of freedom of the rigid nanorods. The characteristic damping time of the thermostat was 10τ, and that of the barostat was 100τ. The equilibration run lasted for a time that is much longer than the diffusion time of the polymer chains. There are two sets of simulation samples. In one set, the polymer chain length N = 16, and the rod length l = 4σ, 8σ, 16σ, and 32σ. In the other set, N = 64, while l = 4σ, and 8σ. The first set corresponds to l comparable to and larger than the root-mean-squared end-to-end size of the polymer chain, R = 4.7σ. The second set corresponds to l < R = 10.1σ. In all samples, the number of nanorods is N_{r} = 27, while the number of polymer chains N_{c} varies. Multiple nanorods are simulated simultaneously for a better statistical average of the dynamics data. As shown in Table 1, the volume fraction of the nanorods ϕ_{r} = N_{r}lσ^{2}/L^{3} is less than 0.07%, and the volume fraction of the excluded volume ϕ_{exl} = N_{r}l^{2}σ/L^{3} is less than 2.1% in all samples. Recently, it has been shown that the organization of nanorods in polymer nanocomposites depends on the details of the nanorod model.^{49,50} For the specific models of nanorods and polymers used here and the low volume fraction of nanorods in the simulations, we have confirmed that the nanorods are well dispersed without any aggregation in all samples.^{43} Additionally, because of the low volume fraction of nanorods in the simulations and the lack of a theory for the long-range correlation of nanorod pairs, we do not attempt to quantify and correct for the effects of any long-range pair correlation on the dynamics of individual nanorods.

Chain length N | Num. of chains N_{c} |
Box size L (σ) | Nanorod length l (σ) | Num. of rods N_{r} |
Vol. fraction ϕ_{r} (%) |
Excluded vol. fraction ϕ_{exl} (%) |
---|---|---|---|---|---|---|

16 | 73818 | 110.5 | 4 | 27 | 0.008 | 0.032 |

16 | 73818 | 110.5 | 8 | 27 | 0.016 | 0.128 |

16 | 73818 | 110.5 | 16 | 27 | 0.032 | 0.512 |

16 | 73818 | 110.5 | 32 | 27 | 0.064 | 2.048 |

32 | 36909 | 110.2 | 4 | 27 | 0.032 | 0.516 |

32 | 36909 | 110.2 | 8 | 27 | 0.065 | 2.065 |

The dynamics of nanorods in polymer melts were simulated at a fixed volume and a constant temperature T = 1.0ε/k_{B}. The temperature of polymer chains was controlled using a Nosé–Hoover thermostat with a characteristic damping time of 10τ. The activeness of a nanorod was modeled by applying a constant active force along the initial rod axis such that the driven motion of the nanorod along the active force direction is commensurate with the preferential thermal diffusion of the nanorod along the rod axis,^{43} facilitating the arrival of the steady state in the simulations. The active force is uniformly distributed along the rod with each bead subjecting to an active force f^{a}_{0}u(0), where u(0) is the unit vector along the initial rod axis. Fig. 1 illustrates a force-driven active thin nanorod in an unentangled polymer melt. f^{a}_{0} varied from 0 to 2.0ε/σ. To enable the activeness, no thermostat was applied to the thin nanorods. The time step for integrating the equations of motion was 0.01τ. The net linear momentum and net angular momentum of the polymer melt were zeroed after each step of integration, so there was no drift or rotation of the melt. All simulations were performed using the LAMMPS simulation package.^{51,52}

The scenario with f^{a}_{0} = 0 corresponds to the passive dynamics of nanorods. The corresponding MSD exhibits a crossover from the initial thermal ballistic motion with α = dlog〈Δr^{2}(t)〉/dlogt = 2 to the sub-diffusive motion with α < 1 at intermediate timescales and eventually to the terminal diffusion with α = 1, in agreement with the previous simulation result.^{43} For f^{a}_{0} > 0, the MSD is characterized by the new force-driven ballistic regime with α = 2 at long timescales. The overlap of the MSD for the active nanorods prior to the force-driven ballistic regime with the MSD for the passive dynamics of nanorods indicates that the active force is not sufficiently strong such that the nanorod dynamics are dictated by the thermal motion at short timescales. With increasing f^{a}_{0}, the force-driven ballistic regime begins at a shorter time. Eventually, for the largest f^{a}_{0} = 2ε/σ, the force-driven ballistic regime covers the entire range of timescales explored in the simulations.

For the simulation samples in Fig. 3, we further decomposed the overall MSD into two components that are parallel and normal to the direction of the applied force, respectively. Note that the direction of the force, which is aligned with the initial rod axis, is fixed in the lab frame. The results from the decomposition are presented as ESI.† The parallel component is almost identical to the overall MSD in the force-driven ballistic regime, indicating the force-driven ballistic motion is along the force direction. The normal component exhibits rich time-dependent features that need further study, but its contribution to the overall MSD in the force-driven ballistic regime is very minor. See ESI† for a more detailed discussion.

In all cases, the force-driven ballistic motion is quantified by a constant speed v^{a}, which is computed from the long-time value of . The increase of v^{a} with increasing f^{a}_{0} for the samples in Fig. 3 is shown in Fig. 4. v^{a} for f^{a}_{0} = 2ε/σ is slightly above the thermal speed of the nanorod with 32 beads of mass m, which is . v^{a} > v^{thermal} indicates that the ballistic motion dominates the thermal motion at all timescales, as the displacement v^{a}t from the active motion is sufficiently large that it surpasses the displacement v^{thermal}t from the undamped thermal ballistic motion. The increase of v^{a} with increasing f^{a}_{0} can also be observed in the representative trajectories of active nanorods in Fig. 2, where the displacement over the same time of 200τ increases with increasing f^{a}_{0}.

Fig. 4 The speed v^{a} for the force-driven ballistic motion (right axis) and the monomeric friction coefficient ζ^{a}_{0} = f^{a}_{0}/v^{a} (left axis) as functions of the active force per monomer f^{a}_{0} for the same systems in Fig. 3. |

As the speed v^{a} increases with the strength of the active force, the friction coefficient for the force-driven ballistic motion of an active nanorod decreases. In the force-driven ballistic regime, there is a balance between the active force f^{a}_{0}(l/σ) and the frictional force ζ^{a}v^{a}, where ζ^{a} is the friction coefficient. From f^{a}_{0}(l/σ) = ζ^{a}v^{a}, the monomeric friction coefficient ζ^{a}_{0} = ζ^{a}/(l/σ) = f^{a}_{0}/v^{a}. The dependence of ζ^{a}_{0} on f^{a}_{0} for the simulation samples in Fig. 3 is shown in Fig. 4. In the limit of vanishing f^{a}_{0}, ζ^{a}_{0} approaches a plateau that corresponds to the passive dynamics of the nanorod, for which the monomeric friction coefficient is ζ_{0} = 22.5ετ/σ^{2}.^{43} As f^{a}_{0} increases, ζ^{a}_{0} deviates from ζ_{0} and decreases as the force-driven ballistic motion dominates over the thermal motion at decreasing timescale.

To understand the dynamics of an active nanorod in unentangled polymers, we develop a scaling theory. Consider a nanorod of length l and thickness comparable to the Kuhn monomer size b in a polymer melt. Each polymer consists of N monomers, and the polymer chains are unentangled. The active force along the initial rod axis is f^{a}_{0} per monomer. Refer to Fig. 1 for a schematic illustration of the force-driven nanorod in polymer chains.

We first review the scaling description of the passive dynamics of the nanorods with f^{a}_{0} = 0 in unentangled polymers. The nanorod in diffusion experiences an effective viscosity η_{eff} that is comparable to the geometric mean of the viscosity η_{0} on the monomer length scale and the maximum viscosity η_{max} that the rod can experience. The scaling relation η_{eff} ≈ (η_{0}η_{max})^{1/2} corresponds to a partial screening of the hydrodynamic interactions in the coupling between the nanorod and the melt. Note that if η_{eff} ≈ η_{0} and η_{eff} ≈ η_{max}, there is a full screening and no screening of the hydrodynamic interactions beyond the monomer length scale, respectively. When l is larger than the average size R ≈ N^{1/2}b of the melt chain in the random-walk conformation, η_{max} for unscreened hydrodynamic interactions is comparable to the melt viscosity η_{melt} ≈ η_{0}N. However, when l < R, η_{max} is instead only comparable to the viscosity η_{g} ≈ η_{0}g ≈ η_{0}(l^{2}/b^{2}) of local chain segments, which contain g monomers in each segment and have an average chain size g^{1/2}b ≈ l.

A length-scale dependent η_{eff} has been used to describe the diffusion of spherical nanoparticles in various polymer systems,^{40,53–57} and it is consistent with the length-scale dependent nanoparticle dynamics in alternative theoretical approach.^{58–60} For nanorods, the rule of geometric mean reflects the additional effects of the anisotropic particle shape on η_{eff}. While the diffusion along the rod axis is determined by the monomeric viscosity η_{0}, the diffusion normal to the rod axis is determined by η_{max}. Such a difference between the parallel and normal components with respect to the rod axis, i.e., in the body frame, has been shown by the recent simulations on the diffusion of nanorods.^{43} The rule of geometric mean composites the viscosity for the diffusion parallel and normal to the rod axis to obtain η_{eff} for the diffusion in the lab frame. The rule was first used by Aponte-Rivera and Rubinstein^{61} in the scaling theory for the dynamical coupling in oppositely charged polyelectrolytes, where η_{eff} for the polyanion correlation segments is the geometric mean of the values for the full screening and no screening of the hydrodynamic interactions. Note that in the theoretical description throughout the paper, ≈ indicates the equation is on the scaling level with an order-of-unity coefficient dropped.

The friction coefficient for the passive dynamics of a nanorod ζ ≈ η_{eff}l, and the diffusion coefficient D = k_{B}T/ζ, according to the Stokes law and Einstein relation, respectively. As a result, the MSD is 〈Δr^{2}(t)〉 ≈ Dt ≈ (k_{B}T/η_{eff}l)t in the terminal diffusive regime. If l ≥ R, the time scale τ_{d} at which the terminal diffusion begins is comparable to the relaxation time of the melt τ_{melt}. Using the N-dependence of the relaxation time in the Rouse dynamics model,^{41}τ_{melt} ≈ τ_{0}N^{2}, where τ_{0} is the monomeric time scale. If l < R, τ_{d} is comparable to the Rouse time of the g-monomer segments, τ_{g} ≈ τ_{0}g^{2} < τ_{melt}.

For a force-driven active nanorod, the friction coefficient ζ^{a} is comparable to ζ in the limit of vanishing f^{a}_{0}, as shown by the simulation result in Fig. 4. The force-driven ballistic motion of an active nanorod with a small speed v^{a} begins at a time scale τ^{a}_{ballistic} > τ_{d}, where the motion of a passive nanorod with f^{a}_{0} = 0 is already in the terminal diffusive regime. The ballistic motion with τ^{a}_{ballistic} > τ_{d} renews the local environment of the nanorod sufficiently slow such that the relaxation dynamics that determine the thermal diffusion of the passive nanorod is complete. As a result, the same friction that affects the thermal diffusion resists the force-driven ballistic motion, i.e., ζ^{a} ≈ ζ. An example of this scenario in the simulations is the active dynamics with the smallest f^{a}_{0} = 0.05ε/σ in Fig. 3.

The reduction of ζ^{a} with respect to ζ results from the strong activeness that makes the force-driven ballistic motion begins at τ^{a}_{ballistic} < τ_{d}. The ballistic motion with τ^{a}_{ballistic} < τ_{d} renews the local environment of the nanorod at a rate that only allows the relaxation dynamics of chain segments with g^{a} monomers each to be complete, i.e.

τ^{a}_{ballistic} ≈ τ^{a}_{g} ≈ τ_{0}(g^{a})^{2} | (1) |

(2) |

η^{a}_{max} ≈ η_{0}g^{a} | (3) |

η^{a}_{eff} ≈ (η_{0}η^{a}_{max})^{1/2} ≈ η_{0}(g^{a})^{1/2} | (4) |

ζ^{a} ≈ η^{a}_{eff}l | (5) |

For a large f^{a}_{0} that makes τ^{a}_{ballistic} ≈ τ^{a}_{g} < τ_{d}, we calculate the friction coefficient ζ^{a} of a force-driven active nanorod by matching the MSDs for the active and passive motion at τ^{a}_{ballistic} ≈ τ^{a}_{g}. The constant speed of the force-driven ballistic motion

(6) |

(7) |

〈Δr^{2}(τ^{a}_{g})〉^{a}_{ballistic} ≈ (v^{a}τ^{a}_{g})^{2} | (8) |

(9) |

(10) |

(11) |

(12) |

(13) |

ζ_{0} ≈ η_{0}b | (14) |

(15) |

(16) |

Matching the two MSDs at τ^{a}_{g} corresponds to the work performed by the active force f^{a} at a speed v^{a} over a time of τ^{a}_{g} is comparable to the thermal energy k_{B}T, i.e.,

w^{a} ≈ f^{a}v^{a}τ^{a}_{g} ≈ k_{B}T | (17) |

(18) |

The regime with the reduced friction coefficient ζ^{a}_{0} in eqn (15) occurs only when f^{a}_{0} is above the threshold , at which the number of monomers per segment for the chain segments that affect the nanorod dynamics is

(19) |

(20) |

(21) |

The scaling theory is compared with the simulation data for the systems with varying l and N. As shown in Fig. 5a, the plateau of ζ^{a}_{0} and the decrease of ζ^{a}_{0} with increasing f^{a}_{0} are observed in different simulation samples. The results for different l and N can be collapsed to a master curve

(22) |

Fig. 5 (a) The monomeric friction coefficient ζ^{a}_{0} for the force-driven ballistic motion of an active thin nanorod in an unentangled polymer melt as a function of the active force per monomer f^{a}_{0}. The rod length l and the polymer melt chain length N are indicated in the format l − N. (b) Rescaling of ζ^{a}_{0} by the plateau value ζ_{0} and f^{a}_{0} by , which is threshold active force for the reduction in ζ^{a}_{0}, results in the collapse of the data points in (a) to a master curve. The thick dashed line indicates the master curve (eqn (22)). The inset shows as a function of l for the nanorods in the melts of N = 16 and N = 64. The dashed lines in the inset indicate two scaling regimes in the theory. |

As shown in the inset to Fig. 5b, the l-dependence of for the force-driven active nanorods in the polymer melt of N = 16 exhibits a crossover from to as l increases from l = 4σ slightly below R ≈ 4.7σ to l = 32σ > R. For the nanorods in the other polymer melt of N = 64, l = 4σ and l = 8σ are both smaller than R ≈ 10.1σ, and the l-dependence of is close to the scaling relation . These results agree with the scaling theory for the onset of the reduction in the friction coefficient (eqn (21)), further validating the theory.

Along with the decrease of ζ^{a}_{0}, there is a suppression of the rotation of the rod. We track the unit vector u(t) along the rod axis as a function of time, as visualized in Fig. 6. The range spanned by the tip of u(t) on the unit sphere decreases with increasing strength of the active force. The time-correlation function of the unit vector along the rod axis 〈u(t)·u(0)〉 is computed. The results for the simulation samples in Fig. 6 are shown in Fig. 7. Here, as in the calculation of MSD (Fig. 3), the average is taken over different rods as well as different time periods of the same length in the steady state. For vanishing f^{a}_{0}, the correlation function approaches 0 over a time scale comparable to the rotational relaxation time τ_{rot} = 7.4 × 10^{4}τ for the passive thermal motion, which was obtained by fitting the decay of 〈u(t)·u(0)〉 to exp(−t/τ_{rot}).^{43} This indicates that the strength of the active force is weak and cannot keep the rod axis around its initial orientation or equivalently the force direction. As f^{a}_{0} increases, the correlation function levels off at a finite value up to the longest time scale in the simulations, indicating a stronger active force that suppresses the rotational diffusion of the rod axis. For the largest active force with f^{a}_{0} = 2ε/σ, the correlation function levels off at 0.93, indicating a much reduced rotational diffusion of the nanorod. The suppression of the rotational diffusion and the cutting off of the de-correlation of rod axis with time are consistent with the force-driven nanorod being coupled to local chain segments rather than entire polymer chains.

Fig. 6 Rotational trajectories of the unit vector (t) along the rod axis for the force-driven active nanorods of l = 32σ in polymer melts of N = 16. |

Fig. 7 Time-correlation function of the unit vector (t) along the rod axis for the same systems in Fig. 6. |

The reduction in rotational diffusion originates from the active force being in a fixed direction. If a self-propelled active force f^{a}_{0}u(t) were applied instead, the force would change its direction at timescales above the rotational relaxation time of the rod axis τ_{rot}. As a result, one would expect the terminal dynamics of the self-propelled active nanorod to be different from that with a constant driving force f^{a}_{0}u(0). Recent Brownian dynamics simulations of self-propelled active filaments^{44} showed that the force-driven ballistic motion only persists to the timescale of τ_{rot}, after which the rotational diffusion of the filament randomizes the direction of self-propulsion, leading to a diffusive translational motion again.

Regarding the connections to experiments,^{28,29,62} the model of thin nanorods in the present work captures the effects of the anisotropic particle shape. Nevertheless, the model has yet to be extended to cover more aspects such as the rod thickness, which may be larger than the monomer length scale b, as well as the boundary conditions at the nanorod and polymer interface, which may involve attractive interactions between the rod and surrounding polymers and a grafted polymer brush layer that prevents the aggregation of nanorods. The steady-state motion of an active nanorod under constant external force as in the simulated model may be realized by the implementation of a nanorod with a uniformly distributed charge under an external electric field. Another way to extend the present computational and theoretical work to connect to a broader range of experiments is to include more disparate polymer matrices, such as entangled polymer melts, polymer solutions, and hydrogels.

Conceptually, the effect of the active force on the center-of-mass trajectory of a thin nanorod in an unentangled polymer melt may be compared with the effect of an external force on the conformation of a polymer chain. For a polymer chain with force of magnitude f applying at both chain ends, the random-walk conformation of an unperturbed chain is replaced by an extended array of tension blobs.^{63} Each tension blob contains g monomers and the blob size ξ ≈ bg^{1/2} is the threshold length scale that chain segments are able to adopt the random-walk conformation without being perturbed by the external tensile force. The tension blob size is determined by fξ ≈ kT, reflecting the competition between the external perturbation and the thermal energy. Similarly, for a thin nanorod under the active force of magnitude f^{a}, the center-of-mass random-walk trajectory in the time domain for the passive dynamics of the nanorod is replaced by an extended array of “active blobs”, as illustrated in Fig. 8. Each active blob corresponds to a time scale τ^{a}_{ballistic} ≈ τ^{a}_{g} (eqn (1)) and a length scale that is comparable to the MSD〈Δr^{2}(τ^{a}_{g})〉^{a}_{ballistic} ≈(v^{a}τ^{a}_{g})^{2} (eqn (8)). The random-walk trajectory at time scales below τ^{a}_{g} is not perturbed by the active force and follows the MSD for the thermal motion of the nanorod. τ^{a}_{g} and hence 〈Δr^{2}(τ^{a}_{g})〉^{a}_{ballistic} are determined by w^{a} ≈ f^{a}v^{a}τ^{a}_{g} ≈ k_{B}T (eqn (17)), which is a balance between the external perturbation and the thermal energy. Note that the random-walk trajectories for the passive dynamics and for the force-driven active dynamics at timescales below τ^{a}_{ballistic} are not ideal 3-dimensional random walks, as the translational diffusion along the rod axis is faster than that normal to the rod axis.^{43} The description of the active dynamics trajectories using the language of blobs in polymer physics, as illustrated in Fig. 8, may provide a new perspective on the theory of active particles and active polymers.^{64}

The microscopic insights provided by the molecular simulations and scaling theory may aid in the control of nanorods in functional nanorod–polymer composites using external force fields^{65} and the manipulation of the trajectories of one-dimensional nanomotors in the applications of bio-imaging and drug delivery.^{27} Additionally, the microscopic insights may also help understand the active filaments propelled by motor proteins in the cytoskeleton, RNA and DNA molecules in the transcription process within a cell, and also rod-like bacteria and viruses in biological settings.^{64,66,67}

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2sm00731b |

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