Conformational Properties of Active Semiflexible Polymers

The conformational properties of flexible and semiflexible polymers exposed to active noise are studied theoretically. The noise may originate from the interaction of the polymer with surrounding active (Brownian) particles or from the inherent motion of the polymer itself, which may be composed of active Brownian particles. In the latter case, the respective monomers are independently propelled in directions changing diffusively. For the description of the polymer, we adopt the continuous Gaussian semiflexible polymer model. Specifically, the finite polymer extensibility is taken into account, which turns out to be essential for the polymer conformations. Our analytical calculations predict a strong dependence of the relaxation times on the activity. In particular, semiflexible polymers exhibit a crossover from a bending elasticity-dominated dynamics to the flexible polymer dynamics with increasing activity. This leads to a significant activity-induced polymer shrinkage over a large range of self-propulsion velocities. For large activities, the polymers swell and their extension becomes comparable to the contour length. The scaling properties of the mean square end-to-end distance with respect to the polymer length and monomer activity are discussed.

Various features are common to all active systems [28], and the challenge of a theoretical description is to find a suitable approach capturing these characteristics.Generically, the activity-induced hydrodynamic flow field of a microswimmer is described by a force dipole [1,29,30].Experiments, theoretical calculations, and computer simulations, e.g., for E.
In this article, the conformational properties of flexible and semiflexible active Brownian polymers (ABPO) are studied analytically.Thereby, we consider a polymer composed of active Brownian particles, which are assembled in a linear chain.The diffusive motion of the propulsion velocity of the monomers is described by a Gaussian but non-Markovian process.The emphasize is on the conformational properties due to the intimate coupling of the entropic polymer degrees of freedom and the activity of the monomers.We adopt the Gaussian semiflexible polymer model [82,94], which allows us to treat the problem analytically.As an important extension to previous studies, we account for the finite polymer extensibility and demonstrate that it strongly affects the out-of-equilibrium properties of an active polymer.Evaluation of the polymer relaxation times shows a drastic influence of that constraint on the polymer dynamics.In general, the relaxation times decrease with increasing activity, whereby the decline is more pronounced for stiffer polymers.Here, activity induces a transition from semiflexible polymer behavior, determined by bending elasticity, to entropy-dominated behavior of flexible polymers with increasing activity.Correspondingly, the conformational properties depend on activity.In the simpler case of flexible polymers, activity leads to their swelling over a wide range of activities.Thereby, the dependence on activity is very different from the theoretical prediction of a Rouse model [68].Interestingly, semiflexible polymers exhibit an activity induced shrinkage.However, for large activities the polymer conformations are ultimately comparable with those of flexible polymers.Shrinkage of active polymers in two dimensions has been observed by simulations in Ref. [68].How- ever, that shrinkage is due to excluded-volume effects and is unrelated to our observations for semiflexible polymers, where excluded-volume interactions are negligible.
Our theoretical considerations shed light on the nonequilibrium properties of semiflexible polymers and underline the importance of an adequate description already for moderate activities.Models without the constraint of a finite contour length, e.g., the standard Rouse model [95], would by no means be able to reproduce and capture the correct structural and dynamical aspects.

II. MODEL OF ACTIVE POLYMER
We adopt a mean-field model for a semiflexible polymer [82,94,[96][97][98][99], which is denoted as Gaussian semiflexible polymer (GSFP), complemented by the activity of the monomers (GSFAP).We describe the GSFP as a continuous, differentiable space curve r(s, t), where s (−L/2 ≤ s ≤ L/2) is the contour coordinate along the chain of length L and t is the time.
Activity is added by assigning the self-propulsion velocity v(s, t) to every point r(s, t), as typical for active Brownian particles (cf.Fig. 1) [6-8, 38, 39, 41].The equation of motion of the GSFAP is then given by the Langevin equation [78,[100][101][102][103]] with the boundary conditions The terms with the second and fourth derivative in Eq. ( 1) account for the entropic degrees of freedom and bending restrictions, respectively.Formally, the entropic part looks like a stretching energy due to harmonic bonds along the polymer contour with λk B T and λ 0 k B T as the Hookean spring constants [79,104] of the continuous chain.In the following, we will denote λ and λ 0 as stretching and as bending coefficient.Note that λ and λ 0 are in general different due to the broken symmetry at the chain ends.The stochastic force Γ(s, t) is assumed to be stationary, Markovian, and Gaussian with zero mean and the second moments where T is the temperature, k B the Boltzmann constant, γ the translational friction coefficient per length, and α, β ∈ {x, y, z}.The Lagrangian multipliers λ, λ 0 , and are determined by constraints [80,82].In general, we find = 3/4p and λ 0 = 3/4 for a polymer in three dimensions, where p is related to the persistence length l p via p = 1/2l p [80,82], i.e., the bending coefficient = 3l p /2 is solely determined by the persistence length as is well known [103,105,106].In Eq. ( 1), we apply a mean-field value for the Lagrangian multiplier λ.
Strictly, we expect the Lagrangian multiplier to depend on the contour coordinate for the active system, because, as shown in Refs.[76,78,80,82,83], λ strongly depends on the presence of an external force, i.e., λ = λ(s), since it is determined by the local inextensibility condition (∂r/∂s) 2 = 1.However, in Eq. ( 1), we neglected this aspect and assume that λ is constant along the polymer contour.Hence, we imply the global constraint of a finite contour length ∂r(s, t) ∂s corresponding to a mean-field approach.As a consequence, the polymer conformations may be inhomogeneous along its contour as, e.g., in the stretching of the GSFP [82].However, the full solution of a discrete free-draining polymer model with individual Lagrangian multipliers for every bond and bond angle [80,82,94], yields expectation values for global quantities such as viscosity which deviate only very little from those determined with the constraint (4) in the limit of a nearly continuous polymer.Hence, the solution of the equations of motion with the constraint (4) suffices for many practical purposes.
We regard the self-propulsion velocity v(s, t) as a non-Markovian stochastic process in time with the correlation function Here, v 0 the magnitude of the propulsion velocity and γ R the damping factor of the rotational motion.The velocity correlation function arises, on the one hand, from the independent stochastic process for the propulsion velocity where η(s, t) is a Gaussian and Markovian stochastic forces with zero mean and the second moment On the other hand, the correlation function ( 5) also follows for the active force γv 0 e(s, t), with a constant selfpropulsion velocity v 0 and the unit vector e of the propulsion direction, where e performs a random walk according to [6,8,28,51] ∂ ∂t e(s, t) = η(s, t) × e(s, t).
Here, η(s, t) is a Gaussian and Markovian stochastic process with zero mean and the second moment Since we will need and apply only the correlation function (5) in the following, the exact nature of the underlying process is irrelevant and our considerations apply for both type of processes.
Note that the continuum representation of the semiflexible polymer requires to introduce a length scale l in Eqs. ( 5) and (7).With a touching-bead model in mind for a discrete polymer, this minimum length corresponds to the bead diameter and bond length of that model (cf.Fig. 1).Strictly speaking, l is a free parameter in the continuum model.For a flexible polymer, we regard l = 2l p = 1/p as the Kuhn length [107,108].
In the above description, we consider the velocity v as an intrinsic property of the active polymer.However, we may also consider v as an external stochastic process with an exponential correlation (colored noise) [6,8,28,71].Such a correlated noise may be exerted by active Brownian particles on an embedded polymer [63,69,70].

III. SOLUTION OF EQUATION OF MOTION
To solve the equation of motion (1), we apply an eigenfunction expansion in terms of the eigenfunctions of the eigenvalue equation [76,100] The resulting eigenfunctions are given by [76,100] , n even , with Inserting the eigenfunction expansions into Eq.( 1) yields the equation of motion for the mode amplitudes χ n with the relaxation times The stationary-state solution of Eq. ( 16) is The time correlation functions of the mode amplitudes, which are useful in the further analysis, are obtained as IV. RESULTS

A. Center-of-Mass Motion
The center-of-mass position is given by [100,102] With the solution of Eq. ( 16) for the zeroth's mode we obtain the center-of-mass mean square displacement As for an active Brownian particle, the term linear in time on the right-hand side accounts for the translational Brownian motion [6].As a generalization, the total friction coefficient γL appears.The second term represents the contribution of activity.Again, it is similar to the term appearing for ABPs, aside from the ratio L/l.We can identify the latter as the number of frictional sites or monomers N of diameter l, i.e., L = N l.Then, N = 1 corresponds to an ABP with the friction coefficient γl, and N = 2 to a dumbbell [28,109].
The long-time diffusion coefficient follows as with the diffusion coefficient D L = k B T /γL of a passive polymer, the Péclet number P e, and the ratio ∆ of the diffusion coefficients [6,28,110] Here, we introduce the diffusion coefficient D T = k B T /γl as the diffusion coefficient of a segment of length l (cf.description of the model on page 3).In the following, we use the thermal translational and rotational diffusion coefficients of spherical particles of diameter l in solution, which yields ∆ = 1/3.

B. Lagrangian Multiplier-Stretching Coefficient
Inextensibility is a fundamental property of a polymer and determines its conformational and dynamical characteristics.Hence, we have to calculate the Lagrangian multiplier λ first in order to relate other polymer aspects to the constraint Eq. ( 4).Insertion of the eigenfunction expansion (15) for the position r(s, t) into Eq.( 4) yields which determines the Lagrangian multiplier λ.In terms of the Péclet number P e = v 0 /D R l and ∆ of Eq. ( 24), this equation can be expressed as with the abbreviation Here, we introduce the Lagrangian multiplier µ via the relation λ = 3pµ/2, i.e., µ is the ratio between the stretching coefficients of the active and the passive polymer.In the integral, we substituted s by x = s/L.represents the solution of the asymptotic equation (31).The straight lines indicate the power-law dependencies µ ∼ P e 2 for pL < 10 −1 and P e < 1, and µ ∼ P e 4/3 (cf.Eq. ( 32)), respectively.
pronounced dependence on P e already for moderate Péclet numbers.In the limit P e → 0, the multiplier assumes the value of a passive polymer µ = 1.Over the considered range of Péclet numbers, the curves exhibit the asymptotic dependence µ ∼ P e 4/3 for large P e, independent of the polymer stiffness.For polymers with pL 10, an intermediate regime appears, where µ ∼ P e κ , with κ > 3. Very stiff polymers (pL < 10 −1 ) even exhibit another power-law regime for small P e, where µ ∼ P e 2 .The various activity-induced features reflected in the Lagrangian multiplier imply pronounced effects on the conformations and internal dynamics of an active polymer.
Flexible-Polymer Limit-An analytical solution of Eq. ( 25) can easily be obtained for a flexible polymer, where pL 1.In this case, the wavenumbers are given by ζ n = nπ/L and Eqs. ( 32) and (33), respectively) the eigenfunctions reduce to trigonometric functions [100] such that Hence, Eq. ( 25) turns into including modes up to order n 2 .Evaluation of the sum yields or in terms of the Péclet number P e and ∆ [Eq.( 24)], The solution of this equation is compared with the exact solution of Eq. ( 25) in Fig. 2.
Evidently, we find good agreement for pL 1 and P e 10.Taking into account modes of order n 4 or even n 6 , leads to a better agrement between the results of the two equations.
• In the limit pL → ∞ and P e < ∞, i.e., 1 µ < ∞, Hence, in the asymptotic limit pL → ∞, µ ∼ P e 4/3 /pl (cf.Figs. 2 and 3).Note that when we set l = 1/p, i.e., identify l with the Kuhn length, µ is independent of the polymer length in the considered scaling regime.This is illustrated in Fig. 3.
In the asymptotic limit P e → ∞, we find a crossover of the Lagrangian multiplier from the power-law dependence µ ∼ P e 4/3 to µ ∼ P e.In the latter regime, the Lagrangian multiplier depends on polymer length.The crossover behavior is illustrated in Fig. 3.The figure presents results for flexible polymers of various lengths, where the Kuhn segment length is identified with l, i.e., pL = L/l.The power-law dependence µ ∼ P e 4/3 is specific to the large number of internal degrees of freedom of a polymer.This applies to flexible as well as semiflexible polymers.As is discussed in the next section, activity changes the properties of semiflexible polymers and they exhibit flexible polymer behavior at large Péclet numbers.However, in the asymptotic limit P e → ∞, activity causes a stretching of the polymer and a crossover to the dependence µ ∼ P e appears.The same relation is obtained for a finite-extensible active dumbbell, which lacks internal degrees of freedom [28].Hence, the dynamical properties of active polymers are not only determined by the longest relaxation time, as is often the case for passive polymers, but the internal degrees of freedom play a much more significant role than for passive polymers.

C. Relaxation Times
The relaxation times [Eq.( 17)] depend via µ on the activity v 0 (or P e).We like to emphasize once more that this is a consequence of the finite extensibility of a polymer [28].Neglecting this intrinsic property implies µ = 1 and the relaxation times are independent of the activity [68,71].The presence of the factor µ gives rise to a particular dynamical behavior, specifically for semiflexible polymers.
In the limit of a flexible polymer, the relaxation times become with the Rouse relaxation time τ R = γL 2 /3πk B T p [95,100].Since, µ 1 is a monotonically increasing function of P e, activity accelerates the relaxation process and the relaxation times become shorter.However, the mode-number dependence is not affected.
The influence of activity on semiflexible polymers is much more substantial.For such polymers, pL < 1 and the ζ 4 -dependence (bending modes) typically dominates the relaxation behavior.However, with increasing activity, and hence µ, the flexible modes (ζ 2 n ) in Eq. ( 34) dominate over the bending modes.Thus, the contribution µζ 2 n determines the relaxation behavior of the polymer for n 2 4(pL) 2 µ/π 2 .Only for larger modes, semiflexibility matters.
As a consequence, starting from the large length-scale dynamics, activity induces a transition from semiflexible to flexible polymer behavior, which extends to smaller and smaller length scales with increasing P e.This behavior is illustrated in Fig. 4 for the longest polymer relaxation time τ 1 .For pL 1, τ 1 exhibits the predicted 1/µ behavior [cf.Eq. ( 35)], with τ 1 ∼ P e −4/3 for large P e.At P e 1, the relaxation times of the stiffer polymers are determined by the bending modes, and τ 1 approaches the persistence-length and P e independent value with decreasing pL.The increase of µ with increasing Péclet number causes a decrease of the relaxation time τ 1 , and in the limit P e 1, the relaxation times assume the same asymptotic value of Eq. ( 17) independent of the stiffness.Quantitatively, τ 1 ∼ 1/µ as soon as µ (π/2pL) −2 .The latter is already satisfied for rather moderate Péclet numbers on the order of P e ∼ 10 1 − 10 2 .yields the condition n > 2pL √ µ/π for the dominace of bending modes.Hence, active polymers at large Péclet numbers appear flexible on large length and long time scales and only exhibit semiflexible behavior a small lengths scales.

D. Mean Square End-to-End Distance
To characterize the conformational properties of the polymers, we consider the mean square end-to-end distance r 2 e = (r(L/2) − r(−L/2)) 2 , which is given by in terms of the eigenfunction expansion (15), where If the stretching coefficient λ and, hence, the relaxation times were independent of the activity, the average mean square mode amplitudes (38) would increase quadratically with the Péclet number for P e → ∞ (cf.second term in the right-hand side of Eq. ( 38)).Thus, the mean square end-to-end distance would increase quadratically with P e [68].As shown in Fig. 6, the constraint of a constant contour length drastically changes the activity dependence of the polymer conformations.In the limit of a flexible polymer (bottom curve of Fig. 6), r 2 e increases with increasing Péclet number as P e 2/3 from the passive equilibrium value r 2 e = L/p.The mean square end-to-end distances of passive polymers itself increases with increasing persistence length, until the limit r 2 e = L 2 is reached for pL → 0. For bending stiffnesses pL 1 and P e > 1, activity causes a significant shrinkage of the polymer over a wide range of Péclet numbers.Above a certain Péclet number, the actual value depends on the stiffness, the polymer swells again, but now similar to a flexible polymer and the asymptotic value r 2 e = L 2 /2 is assumed for P e → ∞.This reflects the above mentioned activity-induced transition from semiflexible to flexible-polymer behavior.
In the passive case P e = 0, r 2 e increases quadratically with increasing pL for pL < 1 (α = 1, rodlike scaling).In the limit pL 1, the flexible Gaussian polymer scaling is obtained, where r 2 e = L/p (α = 1/2), as is well know.In an active system, the local slope assumes the asymptotic value α = 1 for pL → 0, independent of the Péclet number P e < ∞.
At a given P e > 0, the mean square end-to-end distance exhibits a monotonic progression with increasing pL , but the local slope is non-monotonic.Starting from the asymptotic value α = 1, the local slope decreases first with increasing flexibility, i.e., pL, passes through a minimum, which depends on P e, and increases again.This is illustrated in Fig. 7(b) for P e = 3, 10, and 30.The intermediate regime is rather broad, with local slopes almost as small as the value 1/2 for simple Gaussian polymers.In terms of scaling, we can identify a pL-regime for pL > 1, the actual range depends on P e, where α is gradually increases with increasing Péclet number from the flexible polymer value α = 1/2 to the rod limit α = 1.
In addition, (smaller) scaling regimes exist in the crossover region, which shift to smaller pL values with increasing P e, with local slopes increasing from α = 1/2 with increasing Péclet number.The slopes for P e 3 decrease for large pL values.This is related to the selected density of active sites N = L/l = 10 3 along the polymer.For pL < 10 3 , a polymer is stiff on the length scale p = 1/l.In contrary, for pL > 10 3 , the polymer becomes flexible on lengths scales smaller than l, which gives rise to the decrease of the local slope.
Flexible-Polymer Behavior-Evaluation of Eq. ( 37) in the limit of flexible polymers taking into account modes up to n 4 , but neglecting all terms, yields This equation exhibits the asymptotic behaviors: • For finite pL and P e → ∞, the argument of the hyperbolic tangent function becomes small and Taylor expansion gives Insertion of the asymptotic behavior of Eq. ( 33) for the Lagrangian multiplier yields r 2 e P e→∞ −→ L 2 /2.Hence, the polymers assume nearly stretched conformations independent of the persistence length.This is visible in Fig. 6.
• For P e 1, such that 1 µ ∞, and pL → ∞, the argument of the hyperbolic tangent function becomes large.By setting the hyperbolic tangent to unity, we obtain Insertion of the asymptotics of Eq. ( 32) for the stretching coefficient yields r 2 e ≈ lLP e 2/3 .This dependence on the Péclet number is shown in Fig. 6 for the polymer with pL = 10 3 .

V. SUMMARY AND CONCLUSIONS
We have presented an analytical approach to study the conformational and dynamical properties of active semiflexible polymers.We have adopted a continuum representation of a polymer with a certain number of active segments.Each of the segments is considered as an active Brownian particle whose orientation changes independently in a diffusive manner.Alternatively, the active random process can be considered as an additional external correlated (colored) noise acting on the polymer [6,8,28,71].Active polymers have been considered before, both by theoretically and simulations [52,53,56,57,68,71].As an important extension of the previous studies, we have taken into account the finite polymer extensibility due to its finite contour length.As has been shown, this constraint changes the dynamical behavior of active dumbbells drastically [28].Taking into account the constraint by a Lagrangian multiplier leads to a linear equation, which is analytically tractable.
Evaluation of the polymer relaxation times shows a major influence of the finite contour length on the polymer dynamics.Models without such a constraint, e.g., the standard Rouse model [95], would not be able to reproduce and capture the correct dynamics, as reflected in the strong dependence of the stretching coefficient (Lagrangian multiplier) on the Péclet number already for moderate P e values.In particular, the relaxation times decrease with increasing activity (Péclet number).Thereby, the influence of activity on stiff polymers is much more sever.Here, activity induces a transition from semiflexible polymer behavior, characterized by bending modes, to flexible polymer behavior, characterized by stretching modes, with increasing activity.Thereby, the affected length scale depends on the activity.For activities P e 20, large length-scale and low-mode number properties are altered.With increasing P e, an increasing number of modes and hence smaller length scales are affected.Due to the continuous nature of the considered polymer model, the (very) small-scale properties will always be dominated by bending modes.
The effect on the relaxation times translates to the conformational properties.In the simpler case of flexible polymers, activity leads to a monotonous swelling of the polymers over a wide range of Péclet numbers in a power-law manner, which is dictated by the constraint.Hence, our theoretical prediction is very different from the relation r 2 e ∼ P e 2 of a Rouse model derived in Ref. [68] for any flexibility and Péclet number.For semiflexible polymers, with pL 10, activity leads to shrinkage over a wide, stiffness-dependent range of Péclet numbers.At large P e, the polymer conformations are comparable with those of flexible polymers.An activity-induced shrinkage of semiflexible passive polymers embedded in a fluid of ABPs has been observed in simulations of two-dimensional systems [69,70], in qualitative agreement with our theoretical predictions.This supports the equivalence between intramolecular activity and the impact of external colored noise on the properties of semiflexible polymers (cf.Sec.II).
The simulation studies of Ref. [68] for two-dimensional ABPO predict an activity induced shrinkage of self-avoiding polymers.These kind of shrinkage may be particular for 2D ABPS in combination with self-avoidance.As stated in Ref. [68], the polymer shrinkage at moderate Péclet numbers can be attributed activity-induced encaging by neighboring ABPs.
The particular relevance of excluded-volume interactions in 2D systems is also reflected in other studies, e.g., in Refs.[57,69,70].The activity-induced shrinkage of our 3D semiflexible polymers is of different origin.Here, self-avoidance does not play any role.In general, selfavoidance is less important in 3D than in 2D systems.Nevertheless, we expect interesting collective dynamical effects in 3D systems based on our studies of suspensions of 3D ABPs [41].Moreover, the 2D simulations of Ref. [68] suggest that the scaling relation of the mean square end-to-end distance with polymer length is unperturbed by the activity.However, this should only apply to (very) small Péclet numbers, as is evident from Fig. 7, which suggest swelling of the polymer already for P e 1 and an activity-induced modified scaling behavior for large pL values.Note that the Péclet number of Ref. [68] is larger than ours due to the different definitions in terms of translational and rotational diffusion coefficient, respectively.We definitely find for P e > 10 a wide crossover regime to the asymptotic scaling behavior of rodlike polymers, namely r 2 e ∼ L 2 (cf.Fig. 7).Our studies illustrated the usefulness of basic polymer models for the understanding of the complex interplay between polymer entropy, stiffness, and activity.Extension of the current studies toward further dynamical properties and other propulsion preferences, e.g., along the tangent of the polymer contour, are under way.
Experimentally, chains of ABPs can be synthesized by linearly connecting self-propelling Janus particles [7] by a flexible linker.A random distributed of linker sites on the colloid surface yields a random orientation of the propulsion directions of the individual "monomers".
The ensemble average over various realizations corresponds to our description.
The c n s follow from the normalization condition, and the wave numbers ζ n and ζ n are determined by the boundary conditions (2).ϕ 0 describes the translational motion of the whole molecule.

Figure 2 displays 2 ∝ 4 / 3 FIG. 2 .
Figure 2 displays Lagrangian multipliers as function of the Péclet number for various bending stiffnesses pL = L/2l p (at constant polymer length L, variation of pL corresponds to a variation of the polymer persistence length).Evidently, activity leads to an increase of the multiplier µ with increasing P e. Thereby, semiflexible polymers with pL 10 exhibit a

Figure 5 1 FIG. 5 .
Figure5displays the dependence of the relaxation times τ n of a stiff polymer on the mode number for various Péclet numbers.At low P e, we find the well-know dependence τ n /τ 1 ∼ (2n − 1) −4 valid for semiflexible polymers[100,103,106].With increasing P e, the relaxation times increase, and for P e 50 the small-mode-number relaxation times exhibit the dependence τ n /τ 1 ∼ n −2 of flexible polymers.At larger n, the relaxation times cross over to the semiflexible behavior again.However, the crossover point shifts to larger mode numbers with increasing activity.Taking the wavenumbers for flexible polymers, Eq.(34)