Effects of Chain Stiffness on Conformational and Dynamical Properties

Aug 30, 2013 - We report the effects of chain stiffness on the conformational, dynamical, and rheological properties of ring polymers. When the Weisse...
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Effects of Chain Stiffness on Conformational and Dynamical Properties of Individual Ring Polymers in Shear Flow Wenduo Chen, Jizhong Chen,* Lijun Liu, Xiaolei Xu, and Lijia An* State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People’s Republic of China ABSTRACT: Individual semiflexible ring polymers in a steady shear flow are studied by the multiparticle collision dynamics method combined with molecular dynamics simulations. We report the effects of chain stiffness on the conformational, dynamical, and rheological properties of ring polymers. When the Weissenberg numbers are smaller than unity, the behavior of semiflexible ring polymers is consistent with that of flexible ones. For larger Weissenberg numbers, the effects of chain stiffness are observed. We find that the radius of gyration tensor elements, the orientation resistance parameter, and the alignment distribution functions show strong stiffness dependences. The scaling behavior of tumbling motion corresponding to large conformational changes is found independent of chain stiffness, while the scaling behavior of tank-treading motion corresponding to the motion of monomers moving along the contour of the chain exhibits a chain stiffness dependence in the crossover regime from a flexible to a rigid ring polymer. Chain rigidities are found to have negligible effects on the polymer shear viscosity. The simulations reveal the similarities and differences in the nonequilibrium behavior of flexible and semiflexible ring polymers.

1. INTRODUCTION The semiflexible ring polymer (SRP) is one of the common forms in the nature environment, such as plasmid, genome, actin, and polyose.1−5 The shapes of SRPs exhibit planar, elliptical configurations due to bending rigidities with large persistence length and strong constraints without ends.6 Recently, the equilibrium properties of SRPs have intensively been studied with the goal of understanding the role of chain stiffness in conformational changes.7−12 In contrast, the conformational and dynamical properties of SRPs in dilute solutions under shear flow have remained elusive. These properties are generally related to biological processes and polymer physics, e.g., plasmid DNA manufacturing,13 DNA molecules transport through a capillary,14 virus injecting their DNA into living host,15 and polymers flowing near surfaces,16 and hence the understanding of the dynamical behavior of SRPs in a shear flow is of great practical interest in both biological systems and technical applications. Shear flow usually plays a key role in separation required for identification, quantification, purification, or fractionation. For ring polymers, macrocyclic products need be separated from the residual linear precursor and linear polycondensates byproducts by size exclusion chromatography and liquid chromatography in the synthesis of ring-shaped polymers.17 Winkler and co-workers used tangential flow filtration to purify plasmid DNA from RNA and protein.18 Zheng and Yeung separated circular ϕX174 RF DNA from λDNA, based on their radial migration in capillary electrophoresis with applied hydrodynamic flow.19 In addition, the rheological properties of ring polymers are also of great interest in polymer physics. Roovers used high-resolution size exclusion chromatography to separate polybutadiene ring polymers from linear ones, analyzed their viscoelastic properties, and observed that melt © 2013 American Chemical Society

viscosity of a pure ring is smaller than that of the linear polymer with the same molecular weight.20 When subjected to shear flow, polymer chains may orient and deform, and these changes of the microscopic conformational and dynamical properties may affect the macroscopic rheological behavior of the polymer solution. The capacity of deformation and orientation depends not only on the strength of an applied flow but also on the intrinsic chain flexibility. For SRPs at equilibrium, Alim and Frey reported that the shapes transit from prolate, crumpled structures to planar, rigid rings as the stiffness is magnified.6 These different conformations can be attributed to the complex competing interplay among configurational entropy, bending energy, and excluded volume. The varieties of equilibrium conformations suggest that the responses of SRPs with different persistence lengths to an applied flow may be different, and so in this work, a detailed description of the effects of chain stiffness on the nonequilibrium behavior of SRPs is presented. Rather extensive experimental,21−28 theoretical,29−31 and numerical32−40 efforts have been devoted to understand the dynamical behavior of flexible/semiflexible polymers under shear. Chu and co-workers had done comprehensive studies on the dynamics of an individual DNA via fluorescence microscopy.25−28 They observed directly that DNA molecules continually undergo end-over-end tumbling motion as reflected in large conformational fluctuations. Winkler studied the conformational and rheological properties of semiflexible polymers in shear flow using a mean-field semiflexible chain model.30,31 The results are found to be in qualitative and partially quantitative agreement with the description of Received: June 1, 2013 Revised: August 20, 2013 Published: August 30, 2013 7542

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experimental studies. Their calculations also give a good explanation to the similarities in the behaviors of flexible and semiflexible polymers.31 Besides experiment and theory, simulation usually plays an important role in studying instantaneous dynamics of a single polymer in shear flow since it can provide a bridge between theoretical analysis and experimental observation. The nonequilibrium behavior of individual polymers observed experimentally has been confirmed by simulations,32−34,36,38,39,41 including both the conformational and dynamical properties, such as polymer deformation and alignment along the flow direction as well as tumbling motion, and macroscopic rheological characteristics, such as shear thinning at sufficiently large shear rates. However, most of these works focus on linear polymer systems, and we are far from a similar understanding of the dynamics of ring polymers. A Brownian dynamics simulation study on the effects of flow on the static properties of individual ring polymers, by Cifre et al.,42 demonstrated that linear and ring polymers present a same shear dependence of deformation at small Weissenberg numbers Wi < 1 (Wi = γ̇τ0, where γ̇ is the shear rate and τ0 is the longest polymer relaxation time). For large Weissenberg numbers (Wi ≫ 1), our more recent simulation revealed that the dependence of orientation of ring polymers on shear is quantitatively different from that of linear ones.43 In addition, ring polymers were found to exhibit two types of motion: tumbling (TB) motion, corresponding to the chain large conformational changes, which is well-known for linear polymers under shear, and tank-treading (TT) motion, corresponding to the motion of monomers moving along the contour of the chain, which is similar to vesicles.43 Further detailed analysis revealed that the TT motion results from the unique circular architecture that can form an ellipse in the flowgradient plane. In this work, we report the simulation results of the effects of chain stiffness on the conformational, dynamical, and rheological properties of individual SRPs in a steady shear flow. Here, we apply a hybrid simulation approach, combining the multiparticle collision dynamics (MPCD) method describing the solvent with molecular dynamics simulation (MD) for the polymers. As has been shown, the MPCD method, taking into account hydrodynamic interactions and thermal fluctuations, is very well suited to study the nonequilibrium properties of polymers under shear flow.43−47 The outline of the paper is as follows: In section 2, the model and simulation approach are described. The conformational properties of the system under shear flow are presented in section 3.1. Section 3.2 is devoted to the dynamical properties of the system. In section 3.3, results are presented for the polymer contribution to the shear viscosity. A summary and conclusions are given in section 4.

repulsive interaction is taken into account by choosing rcut = 21/6σ. Attractive interactions between each pair of adjacent bonded beads are described by a finitely extensible, nonlinear elastic (FENE) potential:

2. MODEL AND SIMULATION METHOD In our model system, a ring polymer is composed of N beads of mass M, and nonbonded interactions are represented by a truncated and shifted Lennard-Jones potential ULJ:

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

⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σ σ ⎪ ⎪ 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ + ε r ≤ rcut ⎝ ⎠ ⎝ r r⎠ ⎦ ⎨ ULJ(r ) = ⎣ ⎪ ⎪0 r > rcut ⎩

⎧ k ⎪− rbond 2 ln[1 − (r /rbond)2 ] r ≤ rbond UFENE(r ) = ⎨ 2 ⎪∞ r ≥ rbond ⎩ (2)

with the maximum bond length rbond = 1.5σ and the spring constant k = 30ε/σ2. The interplay between ULJ and UFENE yields the average bond length ⟨l⟩ ≈ 0.97σ; in the simulation, the contour length, L = (N/2)⟨l⟩, undergoes small fluctuations in the considered range of shear rates. In this work, chain rigidities are concerned and described by a bond bending potential between two consecutive bonds. The bond bending potential is48 Ubend(γ ) = (ω/2)(cos γ − cos γ0)

(3)

with cos γ =

(ri + 1 − ri) ·(ri − ri − 1) |ri + 1 − ri||ri − ri − 1|

(4)

where γ is the angle between two adjacent bonds and γ0 is the equilibrium bond angle. The value of persistence length Lp can be extracted from the initial exponential decay of the bond orientation correlation function ⟨uiuj⟩ = exp[−(|i − j|⟨l⟩)/Lp] in which ui and uj hold for the unit vectors tangent to the chain contour at positions of ith and jth beads, respectively.53 In the simulation, the strength of the bond bending potential ω is varied in the range 1−300, corresponding to the ratio L/Lp changing from 20 to 1.5. The velocity Verlet algorithm with time step hp is used to integrate Newton’s equations of motion of beads. In MPCD, the solvents are modeled by Ns pointlike particles of mass m. The algorithm consists of alternating streaming and collision steps. In the streaming step, the solvent particles move ballistically with the time interval h between collisions and their positions are updated according to ri(t + h) = ri(t ) + h vi(t )

(5)

where i = 1, ..., Ns. In the collision step, all the particles (here every monomer is also taken to be a point-particle) 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 ))

N

vcm =

(6)

Nm

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

(7)

is the center-of-mass velocity of the particles in the cell. Here, Nc and Nm c are the number of solvent particles and monomers in the collision cell, respectively. In addition, a random shift is performed to ensure Galilean invariance at this step.49 All simulations are performed with ϕ = 130°, σ = ε = a = 1, the average number of solvent particles per cell ρ = 5, M = ρm, τ = (ma2/kBT)1/2 (where T is the temperature and kB is the

(1)

where r = |ri − rj| denotes the distance between beads i and j located at ri and rj. The parameter ε governs the strength of the interaction and σ defines a length scale; the short-range, purely 7543

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Figure 1. Normalized mean-square radius of gyration ⟨Rg2⟩ (a), the flow component of the gyration tensor Gxx (b), the gradient component Gyy (c), and the vorticity component Gzz (d) as a function of the Weissenberg number for SPRs with L/Lp = 11 (■), L/Lp = 7.4 (●), L/Lp = 3.2 (▲), and L/ Lp = 1.5 (▼). The solid lines indicate the dependences Gyy ∼ Wi−0.45 in (c) and Gzz ∼ Wi−0.34 in (d) for Wi > 20, respectively. The solid line in the inset of (b) is the fit.

eigenvalues of ⟨Gαβ(γ̇)⟩ are denoted by the largest eigenvalue G1(γ̇), the middle G2(γ̇), and the smallest G3(γ̇), and their sum (⟨G1(γ̇)⟩ + ⟨G2(γ̇)⟩ + ⟨G3(γ̇)⟩) is just the mean-square radius of gyration ⟨Rg2(γ̇)⟩. In the absence of flow, the statistical conformation of a SRP is spherical, i.e., ⟨Gxx(0)⟩ = ⟨Gyy(0)⟩ = ⟨Gzz(0)⟩ = ⟨Rg2(0)⟩/3. Figure 1 displays simulation results for the mean-square radius of gyration ⟨Rg2⟩ and three diagonal elements of the gyration tensor ⟨Gαα⟩ as a function of the Weissenberg number. At low shear rates (Wi < 1), it can be seen that ⟨Rg2⟩ remains practically unaffected by the applied flow, implying that the chains have no detectable deformation and align along the flow direction. Theoretical and numerical studies for single linear and branched polymers in dilute solutions also predict that the deviation from spherical symmetry (though such deviation is really quite small) exhibits a Wi2 power-law dependence.47,53 The inset of Figure 1b shows that our simulations reproduce this dependence for individual SRPs. For Wi < 1, Figure 1 also indicates that the scaling behavior of SRPs is essentially independent of chain stiffness, since the data for each conformational quantity of SRPs with different rigidities collapse onto a universal curve. For higher Weissenberg numbers, on the other hand, the effects of chain stiffness significantly appear. The values of ⟨Rg2⟩ for relatively flexible SRPs (L/Lp = 11 and 7.4) increase rapidly with the applied flow (Figure 1a), a behavior similar to fully flexible polymers, which indicates that such SRP molecules are not only aligned along the flow direction but also assume a stretched conformation. Clear evidence for this change is given by the variations of the values of ⟨Gαα⟩ with Wi as shown in Figure 1b−d, where the

Boltzmann constant), h = 0.1τ, hp = 0.005τ, m = 1, and kBT = 1. The chain length is 40, and accordingly the system size is 60a × 40a × 40a. Lees−Edwards boundary conditions are used to impose a shear flow,50,51 and a local Maxwellian thermostat is used to maintain the temperature at the desired value.52

3. RESULTS AND DISCUSSION We will now discuss the properties of individual SRPs in shear flow. The flow strength is characterized by the Weissenberg number. For Wi ≪ 1, the conformation remains essentially unchanged compared to the equilibrium state, while for Wi ≫ 1, the chains may be deformed by the flow and they are not able to relax back to the equilibrium conformation. Similarly, the characteristic value, Wi = 1, plays an essential role in characterizing the microscopic dynamical properties of an individual chain as well as the macroscopic rheological properties of polymer solutions under shear flow.26,31 3.1. Conformational Properties. The shape of polymers in flow can be conveniently measured by the gyration tensor, Gαβ, defined as Gαβ (γ )̇ =

1 N

N

∑ Δri ,αΔri ,β i=1

(8)

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 quantity ⟨Gαβ(γ̇)⟩ (here ⟨•⟩ denotes ensemble average) is directly accessible in scattering experiments, and its diagonal components, ⟨Gαα(γ̇)⟩, are the squared radii of gyration in the α direction. In addition, the three 7544

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SRPs are considerably stretched in the flow direction and are compressed in the gradient and vorticity directions for high Weissenberg numbers. With increasing the stiffness of SRPs, the increase of the values of ⟨Rg2⟩ with shear rate gradually slows down. Furthermore, when the Weissenberg number is increased larger than unity, the average sizes of more rigid SRPs (L/Lp = 3.2 and 1.5) remain practically unaffected and then decrease (Figure 1a). This can be explained by the fact that even stiff SRPs behave increasing flexible with increasing shear rate. When the shear rates are relatively low, the applied flow is not strong enough to deform these stiff SRPs; the values of ⟨Rg2⟩ remain almost constant in this moderate range of Weissenberg numbers, and the increase of the values of the flow component of the gyration tensor ⟨Gxx⟩ is mainly due to the ring orientation while the vorticity component is practically unaffected by the applied flow and the gradient component drops rapidly, as shown in Figure 1, indicating that the rings prefer to flat in the flow−vorticity plane. When the strength of flow is stronger, the stiff SRPs become flexible; the values of ⟨Rg2⟩ decrease, and this decrease gradually slows down, leading eventually to a plateau mainly because of the competitive equilibrium between the intrinsic bending energy of SRP chains and the applied flow. Similarly, for relatively flexible SRPs, like the two cases considered here (SRPs with L/Lp = 11 and 7.4), a asymptotic plateau is also expected at large Weissenberg numbers, which can be observed typically at Wi > 104,31,54 larger than the maximum reported in this work. The shear dependence of the gradient component of the average gyration tensor ⟨Gyy⟩ describing the mean thickness of polymers under flow is of interest because this quantity is tightly linked with shear thinning. Different with other two diagonal components, the values of ⟨Gyy⟩, for a given Wi, are practically independent of chain stiffness, and a power law ⟨Gyy⟩ ∼ Wi−0.45±0.02 is obtained at large Weissenberg numbers (Wi > 20), as shown in Figure 1c. A similar scaling behavior was also observed for linear polymers under shear flow both in experiments27 and theoretical simulations.55 In addition, Figure 1d shows that the values of ⟨Gzz⟩, when shear rates are not large enough, hold constant, reflecting the fact that the deformation of SRPs is undetectable in this range of shear rates. With the increase of shear rate, the values of ⟨Gzz⟩ drop significantly from a plateau value, indicating that the deformation of SRPs occurs, for which the Wi depends on chain stiffness, and surprisingly, the slopes of the curves for SRPs with different persistence lengths predict a universal dependence, ⟨Gzz⟩ ∼ Wi−0.34, as shown in Figure 1d. Although the three elements of the gyration tensor, ⟨Gαα⟩, well describe the statistical conformational properties of SRPs under shear flow, they cannot fully satisfy identifying the deformation of the shapes of SRPs subjected to flow. For example, these conformational components of a rigid polymer (in the limit of L/Lp → 0) having no deformation in the considered range of shear rates may continually change due to the orientation of the chain with increasing shear rates. In order to gain a deeper understanding of the deformation of SRPs under a shear flow, the three eigenvalues of the gyration tensor are calculated in our simulations. Figure 2 shows the ratios ⟨G1/ G3⟩ as a function of the Weissenberg number, and if they are equal to unity, it means that the mass distribution of the chain is spherical, while they diverge in the limit of a prolate ring; a plateau is found over a wide range of shear rates, depending on chain rigidities. Considering the sum of these eigenvalues ⟨Rg2⟩ shown in Figure 1a, the fact that the ratio ⟨G1/G3⟩ is essentially

Figure 2. Ratios of the largest (G1) and smallest (G3) eigenvalues of the average gyration tensor as a function of the Weissenberg number for various chain rigidities.

constant at low shear rates implies that SRPs do experience only weak deformation and also confirms that the changes of ⟨Gαα⟩ are mainly due to the orientation of SRPs in this range of shear rates. Only when the strength of flow is larger than some shear rate which again depends on chain stiffness, the apparent deformation for SRPs can be observed, and the increase of ⟨G1/ G3⟩ with respect to its plateau slows down as increasing persistence lengths (see Figure 2). The average alignment of a polymer can be characterized by the orientation resistance parameter mG, straightforwardly obtained in the simulations via mG =

2⟨Gxy⟩ ⟨Gxx⟩ − ⟨Gyy⟩

Wi (9)

It has been shown for several systems including linear, flexible circular,43 branched,47 and rodlike53 polymers that close to equilibrium mG is independent of Wi. Our results for the orientation resistance are presented in Figure 3 for SRPs of various persistence lengths. At small shear rates (Wi < 1), data for different rigidities collapse onto a universal curve, which approaches an asymptotic plateau, as expected, shown in the bottom inset of Figure 3. For larger shear rates (Wi > 20), a 31

Figure 3. Orientation resistance parameter mG as a function of the Weissenberg number for SPRs with L/Lp = 11 (■), L/Lp = 7.4 (●), L/Lp = 3.2 (▲), and L/Lp = 1.5 (▼). The dashed lines indicate the dependences mG ∼ Wiμ for Wi > 20. The top inset shows the scaling exponent μ as a function of persistence length, and the bottom displays the same data, in which the dashed line is a guide to the eyes. 7545

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power law mG ∼ Wiμ is used to describe the orientation resistance. The scaling exponent μ shows a weak stiffness dependence and increases from μ ≈ 0.60 for flexible polymer rings43 to an asymptotic plateau for rigid ones, as shown in the top inset of Figure 3. This result is slightly different with that obtained by the theoretical calculations of single semiflexible linear polymers for which a power law mG ∼ Wi2/3 is obtained and independent of chain stiffness.31 Further insight into the orientational behavior of SRPs is gained by probability distribution functions (PDFs) P(θ) and P(φ), where θ is the angle between the eigenvector of the gyration tensor with the largest eigenvalue and its projection onto the flow-gradient plane and φ the angle between this projection and the flow direction, respectively. At equilibrium, no angle is preferred, and hence both P(θ) and P(φ) are uniform PDFs. When exposed to shear flow, the statistical conformation deviates from spherical symmetry and both P(θ) and P(φ) from uniform distribution. The results for probability distribution functions P(θ) are displayed in Figure 4a. Theoretical,30 experimental,56 and numerical57,58 studies for linear polymers predict a crossover for a Gaussian shape of the distribution function to a power-law decay according to P(θ) ∼ θ−2 with increasing shear rate, within a certain range of angles, which is reproduced by our simulation for SRPs (see the inset of Figure 4a). For a given Weissenberg number, Figure 4a

displays that the width of P(θ) increases with the increase of chain rigidities, consistent with the result for semiflexible linear polymers.30 Similar to the PDF P(θ), P(φ) exhibits a significant shear dependence. The inset of Figure 4b displays that the peak for the PDF P(φ) shifts to smaller values with increasing shear rate and, at the same time, the width of P(φ) decreases; at high shear rates, the PDF of angle φ can be described by a power law P(φ) ∼ (sin φ)−2, which is in accord with the depiction for semiflexible linear polymers.30 The results of these two PDFs for SRPs imply that the orientation behavior has no qualitative difference with that reported for linear polymers. 3.2. Tumbling and Tank-Treading Motions. Polymer molecules may undergo large conformational changes due to TB motion; i.e., a polymer stretches and recoils in the coarse of time. It is quite different from linear polymers that, apart from TB motion, rings may exhibit a unique motion, TT, since a ring can present an elliptical shape in the flow-gradient plane and the velocity across the chain is substantial. The dynamical behavior of individual SRPs is directly reflected in chain conformation and orientation. As has been shown in our previous work,43 when a ring exhibits a TT motion, it adopts a stretching conformation without large fluctuations in chain extension and orients with a small positive orientation angle; when tumbling, in contrast, the ring stretches, compresses and restretches with large fluctuations in orientation, consistent with linear polymers exhibiting a TB motion.28 When the shear rates are small (1 < Wi < 20), at which the effects of thermal fluctuations are still significant, the simulations reveal that TT and TB generally coexist. As increasing shear rates, the two motions can occur independently and be identified in terms of their corresponding conformational changes. For relative flexible SRPs (e.g., L/Lp = 11 and L/Lp = 7.4), their dynamical behaviors are similar to the flexible rings. The apparent effects of chain stiffness are observed for stiffer SRPs (L/Lp = 3.2 and L/Lp = 1.5). The time traces of three eigenvalues of the gyration tensor, their sum Rg2 and the orientation angle for a SRP with L/Lp = 3.2 at Wi = 132 are shown in Figure 5. It is clear that the SRP almost has no deformation during TT, where the three eigenvalues and Rg2 hold constant with small fluctuations, and the orientation angles are positive and close to zero. In contrast, when the SRP tumbles, these quantities

Figure 4. Probability distribution functions P(θ) (a) and P(φ) (b) for SPRs with persistence lengths L/Lp = 11 (■), L/Lp = 7.4 (●), L/Lp = 3.2 (▲), and L/Lp = 1.5 (▼). Inset in (a): P(θ) with L/Lp = 7.4 is shown for various Weissenberg numbers; the solid line is the theoretical prediction,30,56,57 indicating the exponent decay P(θ) ∼ |θ|−2 for large Weissenberg numbers. Inset in (b): P(φ) with L/Lp = 7.4 for various Weissenberg numbers; the solid line describes the power law P(φ) ∼ (sin φ)−2.

Figure 5. Time trajectories of the tangent of orientation angle tan(2φ), the mean-square radius of gyration ⟨Rg2⟩, and the three eigenvalues G1 (pink), G2 (black), and G3 (green). The parameters used in simulations are Wi = 132 and L/Lp = 3.2. The vertical solid lines (blue) are guides to eyes for the motion regimes, TT and TB. The upper horizontal dashed line (red) represents tan(2φ) = 0, and the lower denotes the mean-square radius of gyration at equilibrium ⟨Rg2(0)⟩. 7546

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excluding G2 significantly and regularly fluctuate. Comparing G1 with G2 and G3, we can find that the major contribution to their sum Rg2 is from G1. It is interesting that, at this given Weissenberg number Wi = 132, Rg2 is practically equal to the average value at equilibrium ⟨Rg2(0)⟩ when the SRP exhibits a TT motion while it decreases when the ring compresses during TB. Note that the average size of this SRP with L/Lp = 3.2 at Wi = 132 is smaller than that at equilibrium (see Figure 1a), for which the reason is that the strong flow softens rigid rings. Therefore, these instantaneous results for Rg2 indicate that the softening of SRPs in Figure 1a is mainly due to TB motion. TT motion describes the motion of monomers moving along the contour of a ring adopting an elliptical shape in the flowgradient plane and is characterized quantitatively by an angular autocorrelation function,43 defined as Cangle(t ) =

Wi = 52. Twice the secondary peak time can be used to define the characteristic TT time τtt, in which monomers can make one complete TT cycle.43 Obviously, the more rigid a SRP of considered persistence lengths, the smaller the time for the secondary peak (the first peak at t = 0); i.e., the faster monomers move along the contour of the chain. The height of peaks (of course excluding the first one) is related to the ability of monomers making continuous TT cycles.43 The simulations show that the more rigid a SRP, the higher the height of peaks, which indicates that the probability of a ring exhibiting a TT motion increases with the increase of chain stiffness. The scaled TT frequencies f tt = τ0/τtt are shown in Figure 6b as a function of the Weissenberg number for various chain rigidities. The shear dependence of f tt can be described by the power law f tt ∼ Win. The inset of Figure 6b presents that the exponent n exhibits a chain stiffness dependence in the crossover regime from a flexible to a rigid polymer ring. For small L/Lp, the exponent n approaches an asymptotic plateau, for which the reason is that, considering the results of the gyration tensor shown in Figures 1 and 5, the more rigid a SRP, the more difficult the deformation, especially for a ring exhibiting TT motion. With increasing L/Lp, indicating that the SRP becomes flexible, the exponent n decreases toward the value n = 0.62 that is obtained for individual flexible rings under shear flow.43 In contrast to TT motion, TB motion is used to describe the conformational changes of polymers under shear flow and characterized by the scaled TB frequency f tb. It has been wellknown that individual linear polymers follow the power law f tb ∼ Wi0.66 for Wi ≫ 1, which is independent of chain stiffness.30 The simulations reproduce this dependence for SRPs of different chain rigidities. 3.3. Shear Viscosity. The polymer shear viscosity can be directly accessed in simulation. In our work, the shear stress σαβ is calculated using the virial formulation of the stress tensor42,59

⟨A(t0)A(t0 + t )⟩ ⟨A2 (t0)⟩

(10)

where A(t) = sin[2ψ(t)]; ψ (0 < ψ ≤ π) is the angle between the angle between the instantaneous longitudinal axis and the line connecting the marker monomer of the ring chain with the center of mass. Figure 6a displays the curves of Cangle for a SRP of various persistence lengths at a given Weissenberg number

N

σαβ ∝

∑ ⟨Δri ,α·Fi ,β ⟩ i=1

(11)

where Fi the total force on bead i. The polymer contribution to the shear viscosity are defined as σxy η=− (12) Wi The SRP contribution to shear viscosity is shown in Figure 7 as a function of the Weissenberg number for various persistence lengths. Data for different L/Lp collapse onto a universal curve, which implies that the effects of chain stiffness are undetectable. These curves approach a plateau for small shear rates, as expected. For larger shear rates (Wi > 20), the shear viscosity can be described by a power law η ∼ Wi−0.43±0.01. This scaling behavior is consistent with simulation results for linear polymers as well as experiments.27 Apart from directly calculating the shear dependence of shear viscosity in simulation, the Giesekus stress tensor formula26,27 can be used to describe the relation between the shear viscosity η and the gradient component of the gyration tensor ⟨Gyy⟩ in a steady shear flow, which is preferable in single chain experiments where the shear viscosity is difficultly accessible,27 and yields η ∼ ⟨Gyy⟩. As has been shown in Figure 1c, ⟨Gyy⟩ obeys the power law ⟨Gyy⟩ ∼ Wi−0.45, indicating the relationship η ∼ Wi−0.45. Note that hydrodynamic interactions between chain beads are included in direct calculations44 but neglected in the Giesekus stress tensor formula.26,27 The slight difference between the

Figure 6. (a) Angle autocorrelation functions Cangle with Wi = 52 for different chain rigidities as indicated. (b) Scaled tank-treading frequencies f tt as a function of the Weissenberg number with various stiffness as indicated. The dashed lines show the dependences f tt ∼ Win. The inset shows the scaling exponent n as a function of persistence length. 7547

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flexible to a rigid ring polymer. In addition, no apparent effect of chain stiffness on the scaling behavior of shear viscosity is found. Furthermore, the simulation reveals that the effects of hydrodynamic interactions are almost undetected in the scaling behavior of the viscous properties. Our simulations reveal the similarities and differences in the nonequilibrium behavior of flexible and semiflexible ring polymers. Especially, the simulation indicates that when a SPR exhibits a TT motion the deformation of the chain is almost undetectable; in other words, for a SPR in shear flow the shear effects on the structure of the chain will be minimized if the motion of the chain is restricted to only TT. It has been well-known that shear forces can convert the desirable supercoiled, circular form of the plasmid DNA into undesirable forms,13 such as open circular and linear DNA, and the findings may therefore shed some light on plasmid DNA manufacturing.

Figure 7. SRP contribution to viscosity as a function of the Weissenberg number for various chain rigidities. The solid line indicates the dependence η ∼ Wi−0.43±0.01 for Wi > 20, and the dashed is a guide to eyes with η/η0 = 1.

■

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*E-mail [email protected]; Ph +86 (0)431 85262137; Fax +86 (0)431 85262969. *E-mail [email protected]; Ph +86 (0)431 85262137; Fax +86 (0) 431 85262969.

scaling exponents obtained in the two ways implies that the effects of hydrodynamic interactions on the shear dependence of the shear viscosity are weak.

Notes

4. CONCLUSIONS In this work individual SRPs in dilute solutions under shear flow are studied via a hybrid simulation approach. The effects of chain stiffness on the conformational, dynamical, and rheological properties are analyzed over a wide range of shear rates in detail. When exposed to shear flow, the polymer rings exhibit deformation, which depends on the competitive interactions between shear flow and chain stiffness. In the linear flow regime (e.g., Wi < 1), a SRP is able to undergo conformational changes before the local strain has changed by a detectable amount, and hence the applied flow practically has no effect on the mean-square radius of gyration. Consistent with linear and branched polymers, the scaling behavior of SPRs deviating from spherical symmetry exhibits a Wi2 powerlaw dependence. SPRs begin to orient in the flow direction, and correspondingly PDFs P(θ) and P(φ) deviate from uniform distributions. In this range of Weissenberg numbers, thermal fluctuations are dominant, and the changes of conformation are random; the effects of chain stiffness are practically negligible. Apparent stiffness effects on the conformational and dynamical properties of SPRs are observed for large Weissenberg numbers Wi > 1. The chain stiffness dependence of ⟨Rg2⟩ shows that, with increasing shear rates, the values of ⟨Rg2⟩ do not increase but decrease for rigid polymer rings, which is quite different with flexible molecules where ⟨Rg2⟩ increase monotonically. The calculations of the three eigenvalues of the gyration tensor reveal that the apparent deformation of SPRs can be detectable when the strength of flow is larger than some shear rate depending on chain stiffness. The values of Gyy are found to follow the scaling relation Gyy ∼ Wi−0.45, which is independent of chain stiffness. Although the decrease of Gzz exhibits a same power-law dependence Gzz ∼ Wi−0.34 for different chain rigidities, the initial value of shear rate for this decrease depends on chain stiffness. The effects of chain stiffness on the alignment of SPRs are reflected in the scaling behavior of the orientation resistance parameter mG. SPRs can exhibit both TB and TT motions. The TB motion is found independent of chain stiffness while the TT motion shows a chain stiffness dependence in the crossover regime from a

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was subsidized by the National Basic Research Program of China (973 program 2010CB631102 and 2012CB821500) and supported the National Natural Science Foundation of China (Grants 20974110 and 21274153). L.J.A. is also grateful for the support of the National Natural Science Foundation of China (Grant 21120102037).

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REFERENCES

(1) Semlyen, J. A., Ed.; Cyclic Polymers, 2nd ed.; Kluwer Academic Publishers: Dordrecht, 2001. (2) Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Walter, P., Eds.; Molecular Biology of the Cell, 4th ed.; Garland Science: New York, 2002. (3) Trabi, M.; Craik, D. J. Trends Biochem. Sci. 2002, 27, 132−138. (4) Sanchez, T.; Kulic, I. M.; Dogic, Z. Phys. Rev. Lett. 2010, 104, 098103. (5) Micheletti, C.; Orlandini, E. Macromolecules 2012, 45, 2113− 2121. (6) Alim, K.; Frey, E. Phys. Rev. Lett. 2007, 99, 198102. (7) Camacho, C. J.; Fisher, M. E.; Singh, R. R. P. J. Chem. Phys. 1991, 94, 5693−5700. (8) Odijk, T. J. Chem. Phys. 1996, 105, 1270−1286. (9) Robertson, R. M.; Laib, S.; Smith, D. E. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 7310−7314. (10) Norouzi, D.; Mohammad-Rafiee, F.; Golestanian, R. Phys. Rev. Lett. 2008, 101, 168103. (11) Ostermeir, K.; Alim, K.; Frey, E. Phys. Rev. E 2010, 81, 061802. (12) Dolgushev, M.; Berezovska, G.; Blumen, A. J. Chem. Phys. 2011, 135, 094901. (13) Levy, M. S.; O’Kennedy, R. D.; Ayazi-Shamlou, P.; Dunnill, P. Trends Biotechnol. 2000, 18, 296−305. (14) Junfang, S.; Kaifu, L. Soft Matter 2012, 8, 367−374. (15) Marshall, E. Science 2001, 294, 1640. (16) Romanowski, G.; Lorenz, M. G.; Wackernagel, W. Appl. Environ. Microbiol. 1991, 57, 1057−1061. (17) Lepoittevin, B.; Dourges, M.; Masure, M.; Hemery, P.; Baran, K.; Cramail, H. Macromolecules 2000, 33, 8218−8224. 7548

dx.doi.org/10.1021/ma401137c | Macromolecules 2013, 46, 7542−7549

Macromolecules

Article

(18) Kahn, D. W.; Butler, M. D.; Cohen, D. L.; Gordon, M.; Kahn, J. W.; Winkler, M. E. Biotechnol. Bioeng. 2000, 69, 101−106. (19) Zheng, J.; Yeung, E. S. Anal. Chem. 2003, 75, 3675−3680. (20) Roovers, J. Macromolecules 1988, 21, 1517−1521. (21) LeDuc, P.; Haber, C.; Bao, G.; Wirtz, D. Nature 1999, 399, 564−566. (22) Doyle, P. S.; Ladoux, B.; Viovy, J.-L. Phys. Rev. Lett. 2000, 84, 4769−4772. (23) Douville, N.; Huh, D.; Takayama, S. Anal. Bioanal. Chem. 2008, 391, 2395−2409. (24) Gerashchenko, S.; Steinberg, V. Phys. Rev. Lett. 2006, 96, 038304. (25) Smith, D. E.; Babcock, H. P.; Chu, S. Science 1999, 283, 1724− 1727. (26) Teixeira, R. E.; Babcock, H. P.; Shaqfeh, E. S. G.; Chu, S. Macromolecules 2005, 38, 581−592. (27) Schroeder, C. M.; Teixeira, R. E.; Shaqfeh, E. S. G.; Chu, S. Macromolecules 2005, 38, 1967−1978. (28) Schroeder, C. M.; Teixeira, R. E.; Shaqfeh, E. S. G.; Chu, S. Phys. Rev. Lett. 2005, 95, 018301. (29) Sing, C. E.; Alexander-Katz, A. Macromolecules 2011, 44, 9020− 9028. (30) Winkler, R. G. Phys. Rev. Lett. 2006, 97, 128301. (31) Winkler, R. G. J. Chem. Phys. 2010, 133, 164905. (32) Jendrejack, R. M.; de Pablo, J. J.; Graham, M. D. J. Chem. Phys. 2002, 116, 7752−7759. (33) Liu, S.; Ashok, B.; Muthukumar, M. Polymer 2004, 45, 1383− 1389. (34) Kröger, M. Phys. Rep. 2004, 390, 453−551. (35) Ryder, J. F.; Yeomans, J. M. J. Chem. Phys. 2006, 125, 194906. (36) Lee, J. S.; Shaqfeh, E. S. G.; Muller, S. J. Phys. Rev. E 2007, 75, 040802(R). (37) Ziebert, F.; Aranson, I. S. Phys. Rev. E 2008, 77, 011918. (38) Kobayashi, H.; Yamamoto, R. Phys. Rev. E 2010, 81, 041807. (39) Dalal, I. S.; Albaugh, A.; Hoda, N.; Larson, R. G. Macromolecules 2012, 45, 9493−9499. (40) Slater, G. W.; Holm, C.; Chubynsky, M. V.; de Haan, H. W.; Dubé, A.; Grass, K.; Hickey, O. A.; Kingsburry, C.; Sean, D.; Shendruk, T. N.; Zhan, L. Electrophoresis 2009, 30, 792−818. (41) Pasquali, M.; Shankar, V.; Morse, D. C. Phys. Rev. E 2001, 64, 020802(R). (42) Cifre, J. H.; Pamies, R.; Martínez, M. L.; de la Torre, J. G. Polymer 2005, 46, 267−274. (43) Chen, W.; Chen, J.; An, L. Soft Matter 2013, 9, 4312−4318. (44) Malevanets, A.; Kapral, R. J. Chem. Phys. 2000, 112, 7260−7269. (45) Ripoll, M.; Mussawisade, K.; Winkler, R. G.; Gompper, G. Europhys. Lett. 2004, 68, 106−112. (46) Ripoll, M.; Mussawisade, K.; Winkler, R. G.; Gompper, G. Phys. Rev. E 2005, 72, 016701. (47) Ripoll, M.; Winkler, R. G.; Gompper, G. Phys. Rev. Lett. 2006, 96, 188302. (48) Rapaport, D. C., Ed.; The Art of Molecular Dynamics Simulation; Cambridge University Press: New York, 2004. (49) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986. (50) Ihle, T.; Kroll, D. M. Phys. Rev. E 2001, 63, 020201(R). (51) Allen, M. P., Tildesley, D. J., Eds.; Computer Simulation of Liquids; Oxford University Press: New York, 1987. (52) Kikuchi, N.; Pooley, C. M.; Ryder, J. F.; Yeomans, J. M. J. Chem. Phys. 2003, 119, 6388. (53) Huang, C. C.; Chatterji, A.; Sutmann, G.; Gompper, G.; Winkler, R. G. J. Comput. Phys. 2010, 229, 168−177. (54) Baig, C.; Mavrantzas, V. G.; Kröger, M. Macromolecules 2010, 43, 6886−6902. (55) Huang, C. C.; Winkler, R. G.; Sutmann, G.; Gompper, G. Macromolecules 2010, 43, 10107−10116. (56) Chertkov, M.; Kolokolov, I.; Lebedev, V.; Turitsyn, K. J. Fluid Mech. 2005, 531, 251−260. (57) Puliafito, A.; Turitsyn, K. Physica D 2005, 211, 9−22.

(58) Huang, C. C.; Sutmann, G.; Gompper, G.; Winkler, R. G. Europhys. Lett. 2011, 93, 54004. (59) Bird, R. B.; Hassager, O.; Armstrong, R. C.; Curtiss, C. F. Dynamics of Polymeric Liquids, 2nd ed.; John Wiley & Sons: New York, 1987.

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