Shape and Diffusion of Circular Polyelectrolytes in Salt-Free Dilute

Aug 15, 2017 - ABSTRACT: The shape and diffusion of circular polyelec- trolytes in salt-free dilute solutions are investigated over a large range of t...
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Shape and Diffusion of Circular Polyelectrolytes in Salt-Free Dilute Solutions and Comparison with Linear Polyelectrolytes Lijun Liu, Wenduo Chen, and Jizhong Chen* State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin 130022, China ABSTRACT: The shape and diffusion of circular polyelectrolytes in salt-free dilute solutions are investigated over a large range of the Bjerrum length lB by mesoscale hydrodynamic simulations, where lB characterizes the strength of electrostatic interactions (EIs). A comprehensive comparison of linear and circular polyelectrolytes is also made to gain a deep understanding of the effects of topological constraints on the conformational and dynamical properties. As lB increases, counterions become increasingly important due to their condensations on the polyelectrolyte backbone. The shape of a circular polyelectrolyte changes from a prolate coil to an oblate ring at small lB, then to a prolate coil at intermediate lB, and finally to a dense coil at large lB; in contrast, the shape of a linear polyelectrolyte changes from a prolate coil to a rod, then to a prolate coil, and finally to a dense coil. By switching on/off hydrodynamic interactions (HIs), the simulations clarify the complex coupling effects of hydrodynamic and electrostatic interactions on the diffusion of polyelectrolytes. With increasing lB, the diffusion coefficient with HIs decreases rapidly and then increases gradually, but the diffusion coefficient without HIs displays an almost monotonically decreasing behavior and eventually approaches a plateau. The significant, quantitative but not qualitative difference in diffusion coefficient in the presence of HIs is found between linear and circular polyelectrolytes with an identical chain length, but there are only slight differences between their diffusion coefficients in the absence of HIs. By exploitation of the changes in chain size and the number of condensed counterions, we show that the diffusion of polyelectrolytes can be still qualitatively understood within the framework of Zimm and Rouse models.

1. INTRODUCTION Polyelectrolytes have attracted considerable attention in recent decades because of their practical biological and industrial applications.1 Similar to neutral polymers, polyelectrolytes have a variety of architectures, such as linear, circular, and starlike molecules, and the conformational and dynamical properties depend on their architecture. Rather extensive experimental,2−5 theoretical,6−10 and numerical11−24 efforts have been conducted with the goal of understanding linear polyelectrolyte dilute solutions. These studies addressed the roles of EIs, HIs, and their coupling in conformation transition and diffusion and illustrated significant differences between charged and neutral polymers. In contrast, a similar understanding of circular polyelectrolytes remains elusive, although they are highly relevant to biological systems.25,26 For example, most of short genomes as well as plasmids are well-known circular. In addition, many unique behaviors originate from the circular architecture, e.g., tank-treading motion.27,28 EIs strongly affect or even dominate the conformational properties of polyelectrolytes in dilute solution.12,14,29,30 The increase of EI strength results in significant changes in the shape of such molecules such as the coil−rod transition behavior, which is particularly important for biological systems. For instance, the shape of DNA might enhance or reduce the possibility for proteins locating their specific targets on DNA.31 Unfortunately, the prediction of the shape of a polyelectrolyte © XXXX American Chemical Society

chain is by no means an easy task. From a theoretical point of view, the major difficulty is the long-range nature of EIs that add more than one new length scale making scaling theories to those of neutral polymers.32−34 Furthermore, for high EI strengths, the counterions will concentrate on the polyelectrolyte backbone, which significantly affects its conformation. Numerical studies had proved that beyond a certain EI strength the chain with the condensed ions collapses into a dense coil.12,14,29 In comparison with linear polymers, the circular architecture imposes a stronger constraint on the polymer’s shape.35,36 This constraint leads to a smaller size for a circular polymer than its linear counterpart at the same degree of polymerization, making the monomers distribute more tightly around the center of mass.37 Thus, the shape of a circular polymer should be more sensitive to EIs than its linear counterpart. To understand the effects of topological constraints on chain conformation, it is desirable to obtain a picture of any circular polyelectrolyte in dilute solution as EI strength varies. HIs play a key role in the dynamical behavior of dilute polymer solution19,24,29,30,38−40the motion of one part of the chain perturbs the surrounding flow and modifies the Received: January 24, 2017 Revised: July 28, 2017

A

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technique is used for handling the long-range EIs in the simulation. The excluded volume interaction is described by a truncatedshifted purely repulsive Lennard-Jones (LJ) potential:

hydrodynamic force exerted on another part. For neutral polymers, scaling relations, e.g., for the dependence of dynamical quantities such as diffusion coefficient on chain size, are well predicted by the Zimm model.41 However, the size of a polyelectrolyte chain depends on the EIs between charged monomers.12,14,29,30 This coupling of hydrodynamic and electrostatic interactions complicates scale theories and other theoretical approaches considerably. In addition, the influences of counterions on the dynamical behavior of polyelectrolytes have to be taken into account because when moving the chain drags not only solvents but also counterions, particularly for high EI strengths.29 In this work, we use a hybrid mesocale simulation approach to study the shape of a circular polyelectrolyte chain over a large range of EI strengths. The solvent is treated explicitly by employing the multiparticle collision dynamics (MPCD) techniques.42 Standard molecular dynamics describes the monomers and counterions. As has been shown, this hybrid approach is very suited for studying the conformational and dynamical properties of polymers in solution, in which HIs and thermal fluctuations are important.27,28,43−45 It is easy to switch off HIs in this method,45 which allows us for investigating the influences of HIs on the dynamics of a polyelectrolyte chain in the same method framework. In order to gain a deep understanding of topological constraints, a comparison of linear and circular polyelectrolytes is also made. We show that the circular chain transits from a prolate coil to an oblate ring, then to a prolate coil, and finally collapses into a dense coil as the strength of EIs increases; in contrast, the linear chain transits from a prolate coil to a rod, then to a prolate coil, and finally collapses into a dense coil. The simulation reveals that the overall polyelectrolyte shape as well as the counterion condensation is important for its mobility in salt-free dilute solutions. The paper is organized as follows: The simulation method and model details are presented in section 2, and results and discussion are given in section 3. In section 3.1, we provide the radius of gyration, asphericity, prolateness, and normalized average number of counterions as a function of the strength of EIs for various chain lengths. Moreover, the diffusion of polymers and counterions and the binding dynamics of counterions are presented with and without HIs in section 3.2. Finally, we summarize our conclusions in section 4.

12 ⎧ − (σ /r )6 ] + ϵ r ≤ rc ⎪ 4ϵ[(σ / r ) ULJ = ⎨ ⎪ r [[mml:gt]] rc ⎩0

Here, the cutoff radius is chosen as rc = 21/6σ and ϵ represents the unit energy. Adjacent monomers interact with each other through a finite extensible nonlinear elastic (FENE) potential to ensure the connectivity of the chain: ⎡ ⎛ r ⎞2 ⎤ 1 UFENE(r ) = − κR 0 2 ln⎢1 − ⎜ ⎟ ⎥ ⎢⎣ 2 ⎝ R 0 ⎠ ⎥⎦

ri(t + h) = ri(t ) + vi(t )h

(4)

where ri(t) denotes the position of solvent particle i at time t and vi(t) is the velocity. In the collision step, all the particles (here both monomers and counterions are also taken to be point-particles) are sorted into cubic cells of side length a, and their relative velocities, with respect to the center-of-mass velocity of the cell, are rotated around a randomly oriented axis by a fixed angle ϕ, i.e. vi(t + h) = vi(t ) + (R(ϕ) − E)(vi(t ) − vcm(t ))

(5)

where vi(t) denotes the velocity of particle i at time t, R(ϕ) is the rotation matrix, E is the unit matrix, and Ns

vcm =

Nm

∑i =c1 m vi(t ) + ∑ j =c 1 M vj(t ) mNcs + MNcm

(6)

Nsc

is the center-of-mass velocity of all particles in the cell. is the number of solvent, Nm c is the number of monomers and counterions, and m is the mass of the solvent. Under this rotation operation, all the particles in a cell rotate with the same angle. The rotation axis and orientation are chosen randomly for every cell and can be different at every time step. The mass, local momentum, and kinetic energy are conserved in the collision. In addition, a random shift is performed at each collision step to ensure the Galilean invariance.46 An elaborate description of the implementation can be seen in previous papers.42,47 HIs can be conveniently switched off in MPCD.45 To do this, the stream step is eliminated, and in the collision step, the contributions of solvents in eq 6 are replaced with an effective momentum K, which is directly calculated from a Maxwell− Boltzmann distribution of variance mρkBT and a mean value of zero. Thus, eq 6 is rewritten as

zizj r

(3)

where R0 = 1.5σ is the maximum bond length and κ = 30ϵ/σ2 is the spring constant. The velocity-Verlet algorithm is used to integrate the Newton’s equation of motion of monomers and counterions. To take HIs into account, we employ the MPCD simulation techniques to describe the solvent.42 The MPCD algorithm consists of two alternating steps: streaming and collision. In the streaming step solvent particles move basically during the time interval h between collisions, and the position of solvent i is updated according to

2. SIMULATION METHOD AND ALGORITHM The polyelectrolyte chain is composed of N monomers with mass M and electric charge e each. To guarantee the charge neutrality of the system, an equal number of counterions, each carrying a charge −e, are taken into account explicitly with the same mass as the chain monomer. The EIs between the charged particles are given by Ucoul = kBTlB

(2)

(1)

where r = |ri − rj| denotes the spatial distance between particle i and j and zi is the valency. The Bjerrum length lB = e2/4πεε0kBT is utilized to characterize the EI strength, which is defined as the distance at which the Coulomb interaction between two charged particles is comparable in magnitude to the thermal energy scale. Here, ε0 is the vacuum permittivity, ε is the dielectric constant of medium, kB is the Boltzmann constant, and T is the temperature. The standard Ewald summation B

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vcm =

K + ∑ j =c 1 M vj(t ) mρ + MNcm

(7)

All simulations are performed using the reduced units, in terms of which all physical quantities are expressed. The simulation parameters are chosen as follows: ϕ = 130°, σ = ϵ = a = 1, the average number of solvent particles per cell ρ = 10, m = 1, M = ρm, h = 0.1(ma2/kBT)1/2, hp = 0.005(ma2/kBT)1/2, and kBT = 1. A cubic simulation box with size L and periodic boundary conditions are applied. The chain length N is chosen from 20 to 120, and the total number of counterions is Nion. To reduce finite-size effects, N changes from 20 to 120, and correspondingly L lies in the range 22σ−40σ, keeping the density of monomers N/L3 around 0.0018/σ3. The simple global velocity rescaling technique is applied in every collision step to maintain the temperature at the desired value.48 For each data presented in this work, 50 parallel runs with different initial states are performed to improve the statistics. In each run, the system is equilibrated over 4 × 106(ma2/kBT)1/2 before sampling.

3. RESULTS AND DISCUSSION 3.1. Conformational Properties. Measurements of the conformational properties of a polymer are based on the radius of gyration tensor, Gαβ = (1/N)∑Ni=1Δri,αΔri,β, where Δri,α is the distance between monomer i and the center of mass of the polymer and α, β ∈ x, y, z denote Cartesian components. The three eigenvalues of Gαβ are denoted by the largest eigenvalue λ1, the middle λ2, and the smallest λ3; their sum is just the mean-square radius of gyration ⟨Rg2⟩, where ⟨...⟩ indicates a statistical average. It has been shown that the effects of HIs on the conformational properties are practically negligible,29 and hence we only present the results without HIs. 3.1.1. Size of Polyelectrolytes. The mean-square radius of gyration ⟨Rg2⟩ of a polymer characterizes its size. In comparison with its neutral counterpart, it has been shown that the polyelectrolyte size depends on EIs in dilute solution.12−14,30 Figure 1a shows the mean-square radius of gyration ⟨Rg2⟩ of circular polyelectrolytes, scaled by the value at lB = 0, ⟨Rg02⟩, as a function of the Bjerrum length lB. Data for different N exhibit a sharp increase in the small lB regime and then decrease gradually with increasing lB further. The increase in ⟨Rg2⟩ at small lB indicates the fact that even very weak EIs may lead to a significant chain expansion. At small lB, the condensation of counterions can be negligible, and thus the electrostatic repulsion between charged monomers dominates the changes in the polyelectrolyte’s size. The nature of the long-range of EIs requires the distance between charged monomers as far as possible, but the connectivity of monomers inhibits such separation. As a result, one monomer suffers a increasing repulsion from other monomers as N increases, which also explains that the ratio of ⟨Rg2⟩/⟨Rg02⟩ increases dramatically with increasing N at a given lB in this range (see Figure 1b). ⟨Rg2⟩/⟨Rg02⟩ arrives its maximum at a certain interaction strength that decreases with increasing N as shown in Figure 1c. For our current cases, the maximum value of ⟨Rg2⟩/⟨Rg02⟩ increases from 1.29 for N = 20 to 3.94 for N = 120. At larger lB, the values of ⟨Rg2⟩/⟨Rg02⟩ decrease with increasing lB. Different from the situations at small lB, circular polyelectrolytes of larger N exhibit a faster decrease in the ratio of ⟨Rg2⟩/⟨Rg02⟩. At large lB, the decrease of ⟨Rg2⟩/⟨Rg02⟩ indicates that circular polyelectrolytes begin to collapse. This is mainly due to the condensation of counterions on the polyelectrolyte backbone,

Figure 1. Normalized mean-square radius of gyration ⟨Rg2⟩ of circular polyelectrolytes as a function of the Bjerrum length lB for various chain lengths (a) and as a function of chain length N at various lB (b). Colored symbols correspond to various chain lengths N: 20 (square), 40 (circle), 60 (up-triangle), and 120 (down-triangle). (c) Value of lB for ⟨Rg2⟩ arriving at its maximum, as a function of N. Inset: dependence of normalized mean-square radius of gyration ⟨Rg2⟩ on lB for circular and linear polyelectrolytes with N = 60.

where the electrostatic attraction between monomers and their counterions weakens the repulsive interactions between monomers. In addition, Figure 1b also shows ⟨Rg2⟩/⟨Rg02⟩ < 1.0 at the largest Bjerrum length studied (lB = 10) for N > 20, indicating that these circular polyelectrolytes may have smaller sizes than their neutral counterparts at very strong EIs. A comparison of ⟨Rg2⟩/⟨Rg02⟩ of circular and linear polyelectrolytes with chain length N = 60 is presented in Figure 1a. Because of the topological constraint, the uncharged circular polymer has a smaller size than its linear counterpart, and in the current case of N = 60, ⟨Rg02⟩ = 16.05σ2 and 29.65σ2 for circular and linear polymers, respectively. When charged, ⟨Rg2⟩/⟨Rg02⟩ of circular polymers has a smaller value than those of linear polymers for lB < 8. This can be attributed to the difference in extensibility between circular and linear architectures. At larger lB, the values of ⟨Rg2⟩/⟨Rg02⟩ for circular polyelectrolytes become larger than those for their linear counterparts, for which the possible reason is that the unchanged circular polymer has a more compact conformation than its linear counterpart. The nonmonotonous dependence of ⟨Rg2⟩/⟨Rg02⟩ of linear polyelectrolytes on lB is consistent with the previous studies.12,14,30 In order to investigate the effects of chain architecture, we determine the ratio CR = ⟨R gL 2⟩/⟨R gR 2⟩ C

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Macromolecules between the mean-square radii of gyration of linear ⟨RgL2⟩ and circular ⟨RgR2⟩ chains with an identical chain length, which is of great interest in experiment. It is well-known that CR ≈ 1.8 for the neutral polymer system in the good solvent condition, which is independent of chain length.37 As shown in Figure 2,

Figure 2. Ratio of the mean-square radius of gyration of linear polyelectrolytes to that of circular counterparts as a function of the Bjerrum length lB for various chain lengths as indicated.

we reproduce this result for various chain lengths. CR for different chain lengths approaches its maximum at very weak EIs. Interestingly, it seems that the maximum value of CR (≈2.5) is independent of chain length when considering statistical errors. To the best of our current knowledge, the largest CR reported in experiments of DNA is about 2.49, close to the result in this work.49 Unfortunately, we cannot conclude that CR ≈ 2.5 is the characteristic value for polyelectrolytes as CR ≈ 1.8 for neutral polymers on the basis of the current results, and more evidence, particularly in theory, is highly desirable. Given a chain length, the critical value of lB at which CR arrives at its maximum is smaller than that for the ratio ⟨Rg2⟩/⟨Rg02⟩ in Figure 1. Further increasing lB leads to a monotonic decrease in CR for various chain lengths. Figure 2 also shows that CR is larger than unity in the considered range of lB, indicating that the circular architecture leads to a denser configuration than its linear counterpart. 3.1.2. Shape of Polyelectrolytes. To gain a deeper understanding the effects of EIs on chain conformation, we investigate the shapes of polyelectrolytes. Two characteristic quantities are usually used to characterize the shape of a polymer: asphericity, A, and prolateness, P, both of which are determined by the three eigenvalues of the radius of gyration tensor, λ1, λ2, and λ3. A denotes the deviation from a fully symmetric object, defined as18 A=

1 2

Figure 3. Asphericity A (denoted by empty symbols) and prolateness P (denoted by filled symbols) as a function of the Bjerrum length lB for (a) circular polyelectrolytes with chain length N = 40 (square), N = 60 (circle), and N = 80 (triangle) and (b) linear polyelectrolytes with chain length N = 60. Snapshots: from left to right, the circular polyelectrolyte of N = 40 and the linear polyelectrolyte of N = 60 at lB = 0.0, 1.0, 6.0, and 10.0; monomers and condensed counterions are highlighted in red and green, respectively.

topological constraint, as lB increases, the circular polyelectrolyte chain exhibits more intricate changes in shape than its linear counterpart. For the neutral circular polymer, our simulation results show that the values of aspericity and prolateness are about 0.26 and 0.16, respectively, for all considered chain lengths. With increasing lB, both A and P show a nonmonotonic behavior for all chain lengths studied. Compared with the neutral polymer, the circular polyelectrolyte exhibits a more extended oblate structure in the small lB region, reflecting in the larger value of A and the smaller value of P. The decreases in A and P at small lB indicate the polymer tends to change its configuration from a prolate coil to an fully rigid ring. The increase of lB leads to chain collapsing. As a result, A increases slowly and P increases rapidly in the intermediate range of lB, implying that the polymer configuration changes from the oblate ring to the comparatively prolate coil. At large lB, the changes in A and P arise from the fact that the polymer collapses into a dense coil configuration. The changes in shape are also illustrated in Figure 3a, which displays snapshots for various lB. For the purposes of comparison, we also present A and P for linear polyelectrolytes with chain length N = 60 in Figure 3b. For the neutral linear polymer, the simulations show the values of A and P around 0.48 and 0.55, respectively, which are consistent with the previous studies.18 At lB close to zero, the data for A and P exhibit a sharp increasing behavior, indicating the linear polymers tend to form a rod at small lB due to electrostatic repulsive interactions between monomers. As lB increases, both A and P show a monotonic decreasing behavior, suggesting the configuration of linear chain changes from rod to

(λ1 − λ 2)2 + (λ 2 − λ3)2 + (λ1 − λ3)2 (λ1 + λ 2 + λ3)2

(9)

The nature of asphericity is given by P=

(λ1 − λ ̅ )(λ 2 − λ ̅ )(λ3 − λ ̅ ) λ̅

3

(10)

where λ̅ = (λ1 + λ2 + λ3)/3 is the average of the three eigenvalues. The value of asphericity A ranges from 0 to 1. A = 0 indicates that the polymer is a spherically symmetric object. For A = 1, the polymer forms a rigid rod. Prolateness P takes a value between −0.25 and 2. The polymer chain forms an oblate structure for −0.25 < P < 0 and a prolate structure for 0 < P < 2. We present A and P as a function of lB in Figure 3a for circular polyelectrolytes of various chain lengths. Because of the D

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Macromolecules prolate coil and finally collapses into the dense coil, which is also illustrated by snapshots at various lB in Figure 3b. 3.1.3. Condensation of Counterions. The conformational change of a polyelectrolyte chain is substantially relevant with the condensed counterions. Since there is an attractive interaction between counterion and monomer, the counterions can condense onto the polyelectrolyte backbone in particular at large interaction strengths. Previous studies demonstrated that the counterions influence not only the conformation but also the diffusion of the polymer.13,19,29 So, it is worth presenting the distribution of counterions with respect to the chain backbone. Figure 4a shows the average number of condensed counterions NC scaled by the total number of counterions

attract more counterions around its backbone in this range. For small (or large) lB, few (or most of) counterions condense and NC/Nion is about zero (or unit) for various chain lengths. For the purposes of comparison, we also show the results of linear polyelectrolytes with an identical chain length in Figure 4. Given a chain length, NC/Nion of circular polymers is slightly larger than that of linear ones at a fixed lB in the intermediate range as shown in Figure 4b. This discrepancy can partly be attributed to the difference in chain architecture. For a fixed chain length, the circular polymer has a smaller size than its linear counterpart, resulting in a more powerful attractive force on the counterions. At relatively small lB, such as lB = 3.0, Figure 4b also indicates that this discrepancy exhibits a dependence on chain length, which gradually becomes smaller with increasing chain length. On the basis of molecular dynamics simulations, Chremos and Douglas also made a similar comparison.53 They found that the counterion condensation for linear and circular polyelectrolytes is in essence identical. However, there is a significant discrepancy between their and our simulations; i.e., their simulations are salt, but ours are salt-free. Considering the significant effect of salt on the counterion condensation,52 it is reasonable that there are some discrepancies between their and our results. 3.2. Dynamic Properties. 3.2.1. Long-Time Self-Diffusion of Polyelectrolytes. The long-time self-diffusion of the polyelectrolyte chain is determined by the Einstein expression D = lim

1

t →∞ 6t

⟨[rcm(t ) − rcm(0)]2 ⟩

(11)

where rcm(t) denotes the position of the center of mass at time t. It is well-known that HIs have significant influences on the diffusion of polymers in dilute solution. In this work, such influences are also investigated by switching off HIs in MPCD, which is helpful to clarify the coupling effects of HIs and EIs. More importantly, the results without HIs in essence reflect the effects of EIs on the diffusivity of polyelectrolytes and hence provide important details about the dependence of the diffusion on the polymer size and the condensation of counterions. Diffusion coefficients of circular and linear polyelectrolytes with chain length N = 60, both with and without HIs for the purposes of comparison, are shown as a function of lB in Figure 5a. It is clear that the discrepancy in the diffusion coefficient in the presence of HIs between linear and circular polyelectrolytes is significant, quantitative but not qualitative, but there are very slight differences between their diffusion coefficients in the absence of HIs. When considering HIs, it means the complex coupling effects of HIs and EIs. For simplicity, we first discuss the results without HIs. Since HIs have been eliminated, the changes in the diffusion coefficient can been safely attributed to the existence of EIs. Figure 5a shows that a small increase in EIs results in a sharp decrease in D without HIs. At lB close to zero, the EIs are too weak for the polymer to drag the counterions, which is confirmed by the small number of condensed counterions around its backbone, as shown in Figure 4a. And so, this decrease should be attributed to the long-range repulsive interaction between chain monomers. Repulsive interactions prevent monomers in a polyelectrolyte chain from approaching each other, which is reinforced by the chain connectivity. This is partly supported by the data of ⟨Rg2⟩ shown in Figure 1a, where ⟨Rg2⟩ presents a sharp increase when lB is slightly larger than zero. For small 0 < lB < 0.5, there is a small plateau in the curve of D without HIs as shown in the

Figure 4. Comparison of linear and circular polyelectrolytes for the normalized average number of condensed counterions as a function of the Bjerrum length lB for various chain lengths (a) and as a function of chain length N at various lB (b).

Nion as a function of lB for several chain lengths. In this work, condensed counterions are defined as counterions that lie within a distance Rd of the polymer backbone. The choice of Rd is somewhat arbitrary in simulation. If chosen appropriately, different Rd will give a qualitatively consistent result.14,15 In this work, we choose Rd = 2.0σ in line with that used in previous studies by other groups.14,15 It is clear that with increasing lB, NC/Nion increases rapidly until all counterions are condensed onto the polymer backbone at large lB. In experiment, researchers could obtain the solution with a large value of lB by selecting solvents with low dielectric constant.50 The high percentage condensation of counterions at large lB has been reported experimentally.51,52 The percentage of condensed counterions can be as high as 90%, essentially consistent with our simulations. In addition, several previous simulations also presented the high percentage of condensed counterions.14,30 NC/Nion is found to depend significantly on chain length in the intermediate range of lB, indicating that the longer chain can E

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decreasing behaviors of both D and DC without HIs. Although it has been shown in Figure 4 that the number of condensed counterions of the circular polymers is slightly larger than that of its linear counterpart in the intermediate range of lB, there are almost no differences between their diffusion coefficients in this range. At large lB, almost all the counterions condense on the polymer backbone, implying that the total friction coefficient is close to 2Nξ. Figure 5a shows that D without HIs approaches a plateau value of 1.17 × 10−4a2/h which is around half the value of 2.45 × 10−4a2/h for both the neutral circular and linear polymers with N = 60, as expected by the Rouse model. When HIs are switched on, Figure 5a shows that as lB increases circular and linear polyelectrolytes exhibit a similar diffusion behavior that is qualitatively different from that without HIs: D with HIs shows a sharp decrease at small lB but a gradual increase with increasing lB further unlike D without HIs which gradually approaches a plateau. Furthermore, the small plateau in the curve of D without HIs become unclear for D with HIs. It is clear that at a fixed lB the value of D with HIs is larger than that without HIs, consistent with results reported by Winkler et al.40 From a theoretical point of view, the diffusion of a polymer in dilute solution is well described by the Zimm model. This model demonstrates that the diffusion coefficient of a polymer is inversely proportional to its chain size. As mentioned in the above section, the repulsive interactions between monomers of a polyelectrolyte result in a sharp increase in its size at small lB where almost no counterions condense onto its backbone. The value of D with HIs exhibits a sharp decrease in this range, as expected. Similar to the situations that HIs are eliminated, the repulsive interactions between monomers also hinder the diffusion of the polymer. As shown in Figure 5a, the drop of D with HIs is obviously faster than that without HIs, and no plateau region is observed for D with HIs at small lB. As lB increases, the counterions begin to condense onto the polymer backbone. This condensation leads to two important effects: the polymer and condensed counterions diffuse together and the chain size becomes smaller. The former will result in a decrease of DC with HIs as shown in Figure 5b. The latter implies that there should be an increase in D according to the Zimm model. In fact, the quantitative differences in D between circular and linear polymers at the whole range of lB can be also accounted for by this explanation, where the circular polymer has a smaller chain size than its linear counterparts even at the largest Bjerrum length studied, lB = 10 (at which ⟨Rg2⟩ = 12.75σ2 for circular polymers with N = 60 and ⟨Rg2⟩ = 15.05σ2 for their linear counterparts). We find that the value of lB for the minimum value of D with HIs is different from that for the maximum value of ⟨Rg2⟩. For example, the minimum value of D with HIs for the circular polymer of N = 60 is found at lB ≈ 1.0 (see Figure 5a), but its largest ⟨Rg2⟩ at lB ≈ 1.2 (see Figure 1). This suggests that the condensed counterions might hinder the diffusion of the polymer backbone. In contrast, the diffusion of counterions in the presence of HIs is essentially independent of chain architecture, similar to the situation in the absence of HIs, as shown in Figure 5b. 3.2.2. Binding of Counterions. The shape and diffusion of a polyelectrolyte show a significant dependence on counterions condensing on its backbone. Different from monomers connected by the bond potential, the binding time of a condensed counterion is finite, transient at small lB and relatively durable at large lB. To find the characteristic time for

Figure 5. (a) Diffusion coefficient D of the circular and linear polyelectrolyte of N = 60 and (b) DC of their condensed counterions as a function of the Bjerrum length lB, with and without HIs. Colored symbols correspond to D and DC of the circular polyelectrolyte with (square) and without HIs (down-triangle) and linear polyelectrolyte with (circle) and without HIs (up-triangle). Inset: the same data without HIs at small Bjerrum lengths.

inset of Figure 5a. Interestingly, the plateau value for circular polyelectrolytes is somewhat smaller than that for linear polyelectrolytes. It is possible that the circular architecture reinforces the repulsive interaction between chain monomers, which results in this slight discrepancy. Similarly, the diffusion coefficient of counterions DC without HIs is also independent of lB in this range as shown in Figure 5b. As lB increases, D without HIs drops monotonically. It has been shown in Figure 4 that the increase of lB leads to an increase in condensed counterions because of the increasing attractive interactions between counterion and monomer. Furthermore, the next section will show that the binding time of counterions also increases with lB. This implies that the polymer backbone has to drag the condensed counterions during moving. As lB increases, the condensation of counterions becomes the dominant factor for hindering the diffusion of the polymer, even if the attractive interactions between counterion and monomer may counteract the repulsive interactions between monomers, which reflects in the decrease in chain size as shown in Figure 1. Another powerful evidence is given by the decrease in the values of the diffusion coefficient of counterions as shown in Figure 5b. In this work, the counterion has the same size and mass with the monomer. In this sense, a condensed counterion can be treated as an additive monomer to the polymer. Hence, the increase in condensed counterions might be viewed as the increase in chain length N. It is well-known that the neutral polymer dynamics in the absence of HIs can be described by the Rouse model.41 In this model, the total friction coefficient of the chain equals the summation of the friction coefficient of N connected beads, each bead moving with a friction coefficient ξ, and hence the diffusion coefficient D is reciprocally proportional to Nξ, i.e., D ∼ 1/Nξ. Therefore, the increase in condensed counterions indicates the increase in the total friction coefficient of the polymer, which gives the most reasonable understanding of the F

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understanding of the binding dynamics of counterions, we also present the results with HIs of linear and circular polyelectrolytes with N = 60 in Figure 6c. At a fixed lB, our results show that the binding times are almost same, suggesting the effects of chain architecture on the binding dynamics of counterions are trivial.

the binding dynamics, we determine the autocorrelation function21 CQQ (t ) =

⟨Q (t )Q (0)⟩ − ⟨Q (0)⟩2 ⟨Q 2(0)⟩ − ⟨Q (0)⟩2

(12)

where Q(t) is the number of condensed counterions at time t and ⟨Q(0)⟩ is just Nc. The condensed counterions are defined again as the counterions within Rd = 2.0σ with respect to the polymer backbone. At small lB, the counterions are easy to depart from the backbone due to the random collision of solvent particles and condense again through attractive interactions between counterion and monomer, and hence Q(t) fluctuates with time. Figure 6a shows autocorrelation

4. CONCLUSION We have calculated the shape and diffusion of circular polyelectrolytes in salt-free dilute solutions by employing mesoscale hydrodynamic simulations. To gain a deep understanding of the effects of topological constraints on the properties of polyelectrolytes, a comprehensive comparison of linear and circular polyelectrolytes is also made. The chain conformational properties depend on the long-range EIs between charged particles. The size of a polyelectrolyte expands for weak EIs due to repulsive interactions between monomers and collapses for strong EIs due to the condensation of counterions. Corresponding to the changes in chain size, the shape of the circular polyelectrolyte changes from a prolate coil to an oblate ring at weak EIs, and then a prolate coil at intermediate EIs, and eventually appears as a dense coil at the strong strength of EIs. The major discrepancy in shape between circular and linear polyelectrolytes is that the linear chain exhibits the coil−rod transition at weak EIs, rather than the coil−ring transition. As one of important results of the paper, we have shown that the ratio of linear and circular polymer chain size CR has an identical maximum value for various chain lengths. Since the circular polyelectrolyte has a smaller size than its linear counterpart, we find that the counterions more easily condense onto its backbone. The diffusion of circular polyelectrolytes in dilute solution has a strong dependence on electrostatic and hydrodynamic interactions. The ability to switch off HIs in the simulation directly reveals the effects of EIs on the diffusion coefficient D. The repulsive interactions between monomers lead to a sharp drop in D from the value for neutral analogues. As the Bjerrum length lB increases, the counterions begin to condense onto the polymer backbone, which implies that the polyelectrolyte has to drag the condensed counterions when moving. As a result of the condensation of counterions, D without HIs exhibits an almost monotonically decreasing behavior with increasing lB. We find that D without HIs for a polyelectrolyte arrives at a plateau value that is about half the value for its neutral counterpart at large lB where most of counterions condense. Considering the condensed counterions effectively increases the total polyelectrolyte backbone, the Rouse model can qualitatively explain these changes in D without HIs. Furthermore, evidence from linear polyelectrolytes confirms such an explanation. It is also shown that the slight differences in the number of condensed counterions between the circular and linear polyelectrolyte might account for their slight quantitative differences in D without HIs. When HIs are taken into account, the coupling of EIs and HIs determines D. The curve of D versus lB is divided into two different regimes. D with HIs exhibits a monotonic decreasing behavior at small lB and then gradually increases with lB. These changes in D with HIs agree well with the changes in the reciprocal of ⟨Rg2⟩, suggesting that the Zimm model can still qualitatively explain the diffusion of polyelectrolytes in dilute solution. In addition, the simulation also shows that the value of D with HIs for the circular polyelectrolyte is larger than its

Figure 6. (a) Autocorrelation functions (eq 12) of condensed counterions CQQ for circular polyelectrolytes with N = 60 with HIs. Inset: plot of log(CQQ) vs time t. The solid lines denote the corresponding fit of linear parts of log(CQQ). (b) Average binding time of counterions τb for circular polyelectrolytes with N = 60 as a function of the Bjerrum length lB with HIs and without HIs. (c) Comparison of linear and circular polyelectrolytes τb with HIs.

functions of polyelectrolytes of N = 60 for several lB. It can be found that CQQ(t) decreases, typically exponentially, with time. The binding time τb is determined by fitting CQQ(t) with an exponential function of CQQ(t) = Ω exp(−t/τb) as shown in the inset of Figure 6a, where Ω is the fitting parameter. The plot of log(CQQ) vs time t together with the corresponding fit of the linear parts of log(CQQ) is utilized to show the quality of the exponential decay. Note that the exponential function gives a good fit to data in the intermediate range of t. At large lB, the condensed counterions are however difficult to escape from the polymer backbone because of strong EIs, leading to ⟨Q2(0)⟩ − ⟨Q(0)⟩2 approaching zero, and thus CQQ(t) diverges. Therefore, the function of CQQ(t) is the way only to calculate the binding time τb for small lB. Binding times τb extracted from the correlation functions with HIs and without HIs are presented in Figure 6b. τb increases with increasing lB, as expected. Although HIs show no influence on the average number of condensed counterions (no shown), we find that τb without HIs is apparently larger than that with HIs for a given chain length at a fixed lB. Nevertheless, it seems that there is no qualitative difference between binding times with and without HIs. Moreover, to obtain a deeper G

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Macromolecules linear counterpart at a fixed lB, which is in good agreement with the discrepancy between their chain sizes. Unlike the polyelectrolyte backbone, HIs imposes quantitative but not qualitative effects on the diffusion of counterions. The diffusion coefficient of counterions decreases monotonically with increasing lB and is in essence independent of chain architecture. Finally, we have also calculated the binding dynamics of counterions. The simulation shows that both EIs and HIs determine the life of counterions condensing on the polyelectrolyte backbone. In addition, it seems that chain architecture has weak effects on the binding dynamics. Our simulations reveal a complex interplay between circular polyelectrolyte’s shape and EIs, which is difficult to grasp by analytical theory. Although the coupling of electrostatic and hydrodynamic interactions complicates the dynamics of circular polyelectrolytes in dilute solution, their diffusion can still be qualitatively understood in the framework of the Rouse model for the without HIs cases or the Zimm model for the with HIs cases.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (J.C.). ORCID

Lijun Liu: 0000-0003-0379-0194 Wenduo Chen: 0000-0002-4828-4706 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported the National Natural Science Foundation of China (Grants 21504093 and 21574134) and Key Research Program of Frontier Sciences, CAS (Grant QYZDY-SSW-SLH027). We are grateful to the Computing Center of Jilin Province for essential support.

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