Conformation and Dynamics of Individual Star in Shear Flow and

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Conformation and Dynamics of Individual Star in Shear Flow and Comparison with Linear and Ring Polymers Wenduo Chen,† Kexin Zhang,§ Lijun Liu,‡ Jizhong Chen,‡ Yunqi Li,*,† and Lijia An‡ †

Key Laboratory of Synthetic Rubber, Changchun Institute of Applied Chemistry, and ‡State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, 5625 Renmin Street, Changchun, P. R. China 130022 § School of Environmental Science, Northeast Normal University, 5268 Renmin Street, Changchun, P. R. China 130024 S Supporting Information *

ABSTRACT: How polymers with different architectures respond to shear stress is a key issue to develop a fundamental understanding of their dynamical behaviors. We investigate the conformation, orientation, dynamics, and rheology of individual star polymers in a simple shear flow by multiparticle collision dynamics integrated with molecular dynamics simulations. Our studies reveal that star polymers present a linear transformation from tumbling to tank-treading-like motions as the number of arms increases. In the transformation region, the flow-induced deformation, orientation, frequency of motions, and rheological properties show universal scaling relationships against the reduced Weissenberg number, independent of the number and the length of arms. Further, we make a comprehensive comparison on the flow-induced behaviors between linear, ring, and star polymers. The results indicate that distinct from linear polymers, star and ring polymers present weaker deformation, orientation change, and shear thinning, either contributed by a dense center or without ends. shear flow an individual polymer is stretched and oriented along the flow direction, complementary with the shrink in the gradient and vorticity directions. The deformation of polymers in gradient direction is characterized by the ensemble-averaged radius of gyration projected on the flow-gradient direction, which follows the scaling relationship ⟨Gyy⟩ ∼ γ̇−0.52. The orientation of a polymer along the flow direction is noted by the orientational angle between the principal vector of a polymer and the flow direction, which obeys ⟨θ⟩ ∼ Wi−0.46.22 Here the Weissenberg number Wi is a product of shear rate γ̇ and the longest relaxation time τ of a polymer chain. At high shear rates, an individual chain continually undergoes the endover-end tumbling (TB) motion associated with large conformational fluctuations, and the characteristic frequency of TB motion ( f tb) follows the scaling law f tb ∼ γ̇2/3. Gerashchenko and Steinberg studied the orientation of λDNA as a function of shear rate by particle image velocimetry.23 They found that the orientational angle strongly deviates from the Gaussian distribution due to the nonequilibrium characteristics of polymers in shear flow.24,25 Recently, novel J- and Uturn-like motions were reported for semiflexible polymers like F-actins, which are speculated from the competition of the intrinsic bending energy and the extrinsic flow-induced stresses.26 For linear polymers in the absence of excluded volume and hydrodynamic interactions, the scaling rules

1. INTRODUCTION The influence of architecture on the flow-induced behaviors of an individual polymer is a fundamental problem in polymer physics.1 The response of polymer chains with different architectures to shear stress, such as deformation, orientation, and the adaptation of their dynamical behaviors, is directly related to the mechanical and rheological behaviors of polymer solutions and melts.2−5 Unveiling how polymers with different architectures respond to shear flow at the molecular level also can advance practical applications in molecular separation,6 purification,7 and fractionation,8 etc. Representative architectures, including linear, ring, and star, result in characteristic properties of particular interest.9−11 Unfortunately, the systematic view and comparison of the conformation and the dynamics for these polymers in simple shear flow are still not available yet. The conformation, orientation, and dynamics of linear polymers in shear flow have attracted considerable attention since 1974, and continuously novel details are discovered.12−19 De Gennes first proposed the coil−stretch conformation transition for an isolated linear polymer under shear.12 When a critical velocity gradient is reached, the fluid viscous forces overwhelm the entropic elastic retraction and polymer chains turn to be stretched. Conventional experiments involving birefringence as well as light and neutron scattering techniques have observed such transition according to the averaged properties.20,21 Chu and co-workers have shed light on conformation and dynamics of an individual DNA chain via fluorescence microscopy.13,15,22 They found that in simple © XXXX American Chemical Society

Received: December 6, 2016 Revised: January 18, 2017

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Macromolecules changed to f tb ∼ γ̇3/4 and ⟨Gyy⟩ ∼ γ̇−2/3 from a Brownian dynamics (BD) simulation.27 Meanwhile, the scaling exponents of −6/11 and −14/11 for the viscosity and the normal stress contributed by polymers against shear strength have also been reported.28 Whether these scaling exponents are universal for polymers with different architectures, e.g., ring and star, is still unknown yet. Ring polymer is a classical model for biomacromolecules, such as plasmid, genome, actins, and polysaccharide.29,30 It has distinct properties comparing with its linear counterpart due to the absence of chain ends. In recent years, the conformations and dynamics of ring polymers have attracted intensive scientific interest.31−36 We previously found that individual ring polymers in a simple shear flow exhibit two primary motions,37 i.e., TB motion similar to that of linear polymers and tank-treading (TT) motion like the rolling of fluid droplets17,19 and capsules.18 These two motions can be distinguished using the time evolution of the orientational angle.38 The TT frequency (f tt) determined according to the angular autocorrelation function follows a scaling relationship f tt ∼ Wi0.62.37 We also found the scaling relationship ⟨Gyy⟩ ∼ γ̇−0.41 and tan(2θ) ∼ Wi−0.40 for deformation and orientation, respectively. In the presence of excluded volume interaction, the ring polymer prefers a loop rather than a two-strand linear conformation, and TT motion becomes dominant to relax shear strain.39 Alternatively, Cifre et al. demonstrated that the shear dependence of the average extension of ring polymers in simple shear flow is analogous to that of linear polymers.40 Recently, Hsiao et al. directly observed a coil-to-stretch transition and a large loop conformation stabilized by the hydrodynamic interaction at a high shear rate via single molecule fluorescence microscopy and BD simulation.41,42 Furthermore, the chain stiffness also plays an important role on the deformation and motion of ring polymers in shear flow. For example, the radius of gyration of rigid ring polymers decreases at high shear rates, distinct from the monotonical increase for flexible ring polymers.43 Besides linear and ring polymers, star polymers, which consist of a number of linear arms covalently joined to a central core,9 also exhibit a unique response to shear flow.44 The architecture of star polymers implies that the monomer density is high in the core region and decreases toward the corona.2 With the number of arms increasing, star polymers experience a continuous change from flexible linear polymers to spherical colloidal particles.45−47 In shear flow, such change is contributed from the dynamical relaxation of arms and the structural rearrangements of the whole polymer48−51 and preliminarily understood as a function of the number (f) and the length (Lf) of arms.52,53 Hitherto, lots of work focus on the flow-induced properties of semidilute solutions and melts,54,55 while little attention is paid to the behaviors of an individual star polymer.46,47,56 These behaviors are directly related to many applications, such as motor oil viscosity modifier,57 polymer brushes,58 and drug delivery agents.59 It is noteworthy that the characteristic behaviors of star polymers to relax shear strain are not straightforward or analogous to linear and ring polymers.46 Ripoll et al. demonstrated that star polymers may experience TB motion similar to linear polymers when f equals 2; only TT-like motion presents when f is no less than 10; in the transformation region from 2 to 10, TT-like motion gradually increases accompanied by the reducing of TB motion.46

In this work, we performed a multiparticle collision dynamics (MPCD) integrated with molecular dynamics (MD) simulation to study an individual star polymer under simple shear flow in the transformation region and make a comprehensive comparison in conformation, orientation, dynamics, and rheology with linear and ring polymers from our recent progresses.37,39,43,60 The outline is as follows: In section 2, we present the model for star polymers and details of the hybrid MPCD method. In section 3, the deformation, orientation change, dynamics, and rheological properties of star polymers in shear flow as a function of f and Lf are presented. A comparison of typical scaling exponents for linear, ring, and star polymers against the reduced Weissenberg numbers Wi/f is discussed and followed by the conclusions.

2. MODEL AND SIMULATION METHOD The schematic description for a star polymer in a simple shear flow is shown in Figure 1. It consists of f linear arms with equal

Figure 1. Schematic diagram of a star polymer in simple shear flow. The orientation angle θ is between the principal vector of a star polymer and the flow direction. G1 and G3 are the largest and the smallest eigenvalues of the gyration tensor in an equivalent ellipsoid for a star polymer, respectively.

length Lf connected to a common center. The size and mass of the center particle are the same as any other monomer in arms. The star polymers have N (N = f × Lf + 1) beads and N − 1 bonds. Analogous to the bead−spring model for star polymers, linear polymers have N beads and N − 1 bonds, and ring polymers have an additional bond connecting two ends. The shear flow is adjusted by the shear rate γ̇ and recorded as the Weissenberg number Wi. Associated with the coarse-grained model, interactions between beads have simply an excluded-volume and a bond terms. The excluded-volume interactions are taken into account by a truncated and shifted Lennard-Jones potential ULJ39,61 ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σ σ ⎪ ⎪ 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ + ε r ≤ rcut ⎝ ⎠ ⎝ r⎠ ⎦ ULJ(r ) = ⎨ ⎣ r ⎪ ⎪0 r > rcut ⎩

(1)

where r = |ri − rj| is the spatial distance between beads i and j located at position ri and rj. The parameters ε and σ are the units of energy and length, respectively. The short-range, purely repulsive interactions are taken into account by choosing rcut = 21/6σ. The consecutive beads are connected by a bead−spring model, and the bond potential Ub follows Hooke’s law61,62 B

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Macromolecules Ub(r ) =

k (r − r0)2 2

solvent particles m = 1.0, and the mass of solute particles M = ρm. The collision time interval is 0.1τp, and MD time step is

(2)

0.005τp, with time unit τp = mσ 2/(kBT ) . The viscosity of solvent fluid is η = 3.96(εm)1/2/σ2, and the Schmidt number is 12.0.65 The equilibrium bond length r0 = σ, and the spring constant k = 10000ε/σ2. The large spring constant k ensures the small fluctuation of the equilibrium bond length.67 In all reported simulations, Lf is varied from 10 to 20 and f is varied from 2 to 10, and the according sizes of simulation boxes change from 60σ × 40σ × 40σ to 120σ × 40σ × 40σ. Systems are equilibrated for at least 2 × 106 collision time steps, and the mean properties are represented by statistical averages over 4 × 106 collision time steps with eight parallel samples.

where r0 is the equilibrium bond length and k is the spring constant. The velocity-Verlet algorithm is used to integrate Newton’s equations of motion of beads. Explicit solvents are simulated by the MPCD method, consisting of streaming and collision steps.63,64 Solvents are modeled as Nst point-like particles of mass m. In the streaming step, the solvent particles propagate ballistically and their positions are updated according to65 ri(t + h) = ri(t ) + h vi(t )

(3)

where i = 1, ..., Nst and h is the time interval between collisions. In the collision step, the particles are sorted into cubic cells with the size of unit length. Their relative velocities, relative to the center-of-mass velocity of each cell vcm(t), are rotated by an angle φ around a random axis Ω(φ),63 i.e. vi(t + h) = vcm(t ) + Ω(φ)[vi(t ) − vcm(t )]

3. RESULTS AND DISCUSSION 3.1. Conformation. We use the ensemble-averaged gyration tensor Gαβ to measure the conformation of star polymers in shear flow62

(4)

where vi(t) is the velocity of particle i at time t and the centerof-mass velocity vcm(t ) =

1 Nstc

Gαβ =

Nstc

∑ vi(t )

Ncst

is the total number of solvent particles within the collision cell. The coupling of solute and solvent particles is achieved by taking the solute into account in the collision step. The velocity of the center-of-mass in a cell is63,64 Nc

vcm(t ) =

Nc

∑i =st1 m vi(t ) + ∑ j =se1 M vj(t ) mNstc + MNsec

(6)

where m is the mass of solvent particles, M is the mass of solute particles, and Ncse denotes the number of monomers in the cell. The collision rule conserves mass, momentum, and energy within each cell. A random shift is performed to ensure Galilean invariance at every collision step.60,66 To achieve a simple shear flow,the Lees−Edwards boundary condition is imposed using a linear velocity profile,61 i.e., vx = γ̇y, vy = 0, vz = 0. A local Maxwellian thermostat is used to maintain a constant temperature T.34 The relaxation time of star polymer chain, τ, is determined based on the exponential decay of autocorrelation function of center-to-end vector,56 CR(t): C R (t ) = C0 exp( −t /τ ) = (⟨R(t )R(0)⟩ − ⟨R⟩2 )/(⟨R2⟩ − ⟨R⟩2 )

N

∑ Δri , αΔri ,β i=1

(8)

where Δri is the distance between particle i and the center-ofmass of a star polymer, and α, β ∈ (x, y, z) are Cartesian components. The diagonal components ⟨Gαα⟩ are the squared radii of gyration in the α direction. The three eigenvalues of the average gyration tensor Gαβ give the principal axes of polymer chains, which denote the shape of polymer chains. The sum of the largest eigenvalue G1, the intermediate G2, and the smallest G3 is the mean-square radius of gyration ⟨Rg2⟩.68 Under the condition of zero shear, a star polymer presents a statistically spherical conformation, i.e., ⟨Gxx⟩ = ⟨Gyy⟩ = ⟨Gzz⟩ = ⟨Rg2(0)⟩/3. Only when the shape of a star polymer is exactly spherical, G1 = G2 = G3 = ⟨Rg2(0)⟩/3. Here, ⟨Rg2(0)⟩ is the gyration radius of a star polymer under zero shear conditions. The mean-square radius of gyration ⟨Rg2⟩ and its three components in the flow, gradient, and vorticity directions ⟨Gαα⟩ as a function of Wi/f for star polymers at different f and Lf are shown in Figure 2. At low shear rates (Wi/f < 1), star polymers have no significant deformation and only orient along the flow direction. The average gyration tensor in the flow direction follows the scaling law of ⟨3Gxx⟩/⟨Rg2(0)⟩ − 1 ∼ (Wi/f)2.0. The exponent 2.0 indicates that the stretching is directly proportional to the shear strength, in good agreement with theoretical predictions for linear polymer in good solvents.69,70 With the increase of Wi/f, when the hydrodynamic drag force is larger than the entropic restoring force that keeps the polymer in the random coil (Wi/f > 1), star polymers are stretched along the flow direction with each arm temporally shaking between an extended state and a collapse state. The deformation becomes remarkable at high shear rates (Wi/f > 10), especially for star polymers with longer Lf or smaller f. The stretching along the flow direction is similar to the whole size, while the “necking” converges to a master curve with fixed scaling exponents of −0.42 and −0.29 against Wi/f in the gradient and vorticity directions, respectively. The shrink in the gradient direction is much faster than that in the vorticity direction. This behavior is believed to be responsible for the shear-thinning behavior of polymers at the molecular level.22,71 When the shear rate is high enough, the deformation expressed by ⟨Rg2⟩ and ⟨Gxx⟩ levels off, as a result of the finite chain stretching.72 It is worthy to note that the bond length may be extended infinitely because it

(5)

i=1

1 N

(7)

where R is the center-to-end vector for a single arm and C0 is the adjustable fitting parameter. The relaxation time for star polymers increases as f increases. It is worthy to emphasize that the conformational, dynamical, and rheological properties follow universal behaviors when τ is reduced by f (τs = τ/f) in the transformation regime in this work. Thus, the following results are presented against the reduced Weissenberg number Wi/f for star polymers. All simulations are performed with the rotation angle φ = 130° as well as length and energy unit of σ = 1.0 and ε = 1.0kBT, where kB is the Boltzmann constant. The average number of solvent particles per collision cell ρ = 5. The mass of C

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3.2. Orientation. The orientation of a star polymer can be quantified by the orientation angle θ, defined as15 tan(2θ ) =

2Gxy Gxx − Gyy

(9)

The dependence of orientation angle θ on the reduced Weissenberg number Wi/f is presented in Figure 4. With the

Figure 2. (a) Mean-square radius of gyration normalized by the radius of gyration under zero-shear ⟨Rg2⟩/⟨Rg2(0)⟩, (b) the flow component ⟨3Gxx⟩/⟨Rg2(0)⟩, (c) the gradient component ⟨3Gyy⟩/⟨Rg2(0)⟩, and (d) the vorticity component ⟨3Gzz⟩/⟨Rg2(0)⟩ as a function of Wi/f. The solid lines indicate the scaling relationships of ⟨3Gxx⟩/⟨Rg2(0)⟩ − 1 ∼ (Wi/f)2.0 in the flow direction as shown in the inset of (b) and the scaling exponents of −0.42 and −0.29 for the gradient and vorticity directions, respectively.

Figure 4. Orientation angle tan(2θ) as a function of the reduced Weissenberg number Wi/f. The solid lines indicate the scaling exponents: −1.0 for Wi/f < 1 and −0.37 for Wi/f > 10.

shear increasing, star polymers are oriented along the flow direction. At low shear rates (Wi/f < 1), a universal scaling relationship of tan(2θ) ∼ (Wi/f)−1.0 is observed. It is a nearequilibrium state, according to the results from Ripoll et al.,46 where Gxy ∼ γ̇ and (Gyy − Gzz) ∼ γ̇2. At high shear rates (Wi/f > 10), a scaling exponent −0.37 is observed, which agrees with the scaling −0.35 for flexible star polymers with f = 15 and 50.46 3.3. Motion. The dynamical properties of star polymers strongly depend on the number of arms f. With the increase of f, the whole star polymers show the continual transformation from TB motion ( f = 2) to TT-like motion (f > 10) with each linear arm undergoing collapsed and stretching conformations to relax the shear stress. Such evolution of motions as a function of f can be directly observed in the animations in the Supporting Information. In order to analyze the remarkably large conformational changes in the flow-gradient plane during TB motion, we determine the normalized cross autocorrelation C̅ xy between flow and gradient directions,16,22 defined as

is maintained by a harmonic spring force. We accumulated the probability distribution of bond lengths at different shear strains. All bond lengths have limited extension, as shown in Figure S1 of the Supporting Information. It is further confirmed that the level off of ⟨Rg2⟩ and ⟨Gxx⟩ against Wi/f is a result of finite chain stretching. Furthermore, the ratio G1/G3 is presented to trace the deformation of star polymers from nearly spherical coils to stretched shapes. If the ratios G1/G3 are equal to unity, it means that the distribution is spherical, while it diverges in the limit of a long rod.68 Figure 3 shows the ratios G1/G3. At Wi/f < 1, the

Cxy ̅ (t ) =

Cxy(t ) 2

2

⟨δGxx (t0)⟩⟨δGyy (t0)⟩

=

⟨δGxx(t0)δGyy(t0 + t )⟩ ⟨δGxx 2(t0)⟩⟨δGyy 2(t0)⟩ (10)

Here δGαα = Gαα − ⟨Gαα⟩ and ⟨...⟩ denotes the time average. As shown in Figure 5, each curve of Cxy exhibits a valley at time t+ and a peak at time t−. Such sinusoidal profile clearly shows the stretching and the coupling shrinkage in flow and gradient directions.16,22 The time interval between t+ and t− is the characteristic time of TB motion. While the difference between the maxmium and minimum (hpeak) denotes the amplitude of the deformation, which decreases with f increasing at the same Wi/f. It clearly demonstrates the transformation of TB to TTlike motion as the number of arms increases. According to the inset in Figure 5a, we find that hpeak linearly decreases with f increasing. It agrees with previous observation that more arms leads to less deformation of star polymers in shear flow. Furthermore, the height of C̅ xy is independent of f, as shown in Figure 5b.

Figure 3. Ratio of the largest (G1) and smallest (G3) eigenvalues of the gyration tensor as a function of Wi/f for star polymers with different lengths and numbers of arms.

star polymers are close to their equilibrium conformations. With the increase of Wi/f, star polymers are stretched. The lengths of arms have significant impact on the deformation response to shear flow, and the scaling exponents against Wi/f are 0.47 and 0.58 for Lf = 10 and 20 with the same f (f = 3), respectively (see Figure S2). Further, the less number of arms shift the curve to higher G1/G3 values, as a result of lower monomer density in the center region. Therefore, unsurprisingly, star polymers with more and shorter arms will have less deformation in shear flow. D

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which the rotational frequency ωz is independent of γ̇,18 due to the deformation and orientaiton of star polymers. 3.4. Rheology. Clarify rheological behaviors of polymers in shear flow at the molecular level from the dynamical and conformational aspects are always plausible. We use the modified Giesekus model to calculate the stress tensor ταβ40,74 N

ταβ =

∑ ⟨ri ,αfi ,β⟩ i=1

(12)

where fi is the total force on bead i. Then the dimensionless viscosity contributed by polymers is defined as −τxy η= (13) Wi Figure 7 shows the shear evolution of the viscosity η relative to zero shear viscosity η0 contributed by star polymers with Figure 5. (a) Cross-autocorrelation function Cxy and (b) its normalized form C̅ xy against simulation time t/τs as a function of f for star polymers with a fixed length Lf = 10 in shear flow for Wi/f = 17. The inset displays the linear dependence of the peak height hpeak on f.

The frequency for TB motion can be expressed by f tb = τ/ 2(t+ − t−), and the scaled rotation frequency for TT-like motion defined as73 ftt = |ωz| /γ ̇ =

⟨Gyy⟩ M(⟨Gxx⟩ + ⟨Gyy⟩)

Figure 7. Relative viscosity contributed by polymers η/η0 as a function of Wi/f for various star polymers. The solid line indicates the dependency η/η0 ∼ (Wi/f)−0.40. The dashed line (η/η0 = 1.0) is a guide to the eye.

(11)

where ωz is the rotation frequency. Both the characteristic frequencies for TB and TT motions of star polymers are presented in Figure 6. The frequency for TB motion follows a

various f and Lf. There is a Newtonian plateau at low shear rate (Wi/f < 1). Then the viscosity monotonically decreases as the shear increases. The shear thinning behavior follows the scaling relationship η/η0 ∼ (Wi/f)−0.40 when Wi/f > 10, nearly the same scaling as exhibited by Gyy ∼ (Wi/f)−0.42. These results suggest that the decrease of viscosity at high shear rates is mainly contributed from the deformation of polymer chains in the gradient direction.67 In addition to the shear viscosity, the first and second normal stress coefficients are calculated as τxx − τyy Φ1 = (14) Wi 2 and Φ2 =

τyy − τzz Wi 2

(15)

A plateau in small shear flow (Wi/f < 1) can be seen from Figure 8. In high shear flow, these two coefficients have scaling exponents of −1.08 and −1.14 as a function of Wi/f. The decrease of Φ1 and Φ2 also indicates the shear thinning behaviors.67 3.5. Comparison of Linear, Ring, and Star Polymers in Simple Shear Flow. Table 1 summarizes the typical scaling exponents for linear, ring, and star polymers in simple shear flow and the changes in conformation, orientation, dynamics, and rheology from our results and several related reports.15,22,23,25,27,28,37,39,43,46,47,56,60,69,70 Most of the scaling exponents are significantly different associated with the polymer architectures. In the shear direction, the deviation from

Figure 6. (a) Frequency of tumbling motion f tb. (b) Scaled rotation frequency of TT-like motion f tt as a function of Wi/f for various f and Lf.

scaling relationship of f tb ∼ (Wi/f)0.64 in high shear flow (Wi/f > 10). Meanwhile, the frequency for TT-like motion holds a constant of 1/2 at low shear rates, which is a typical value for vesicles.18 At high shear rates, it follows a scaling relationship f tt ∼ (Wi/f)−0.52. It is different from the power law of vesicles, in E

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interesting to find that the scaling of the frequencies of TB motions f tb is independent of architecture. For star polymers with the number of arms more than 10, star polymers almost roll in simple shear flow; in the transformation region from 2 to 10, the less exponent of the scaled rotation frequency f tt is due to the deformation and orientation of star polymers. Further, to provide clear understanding of the physics behind these scaling relationships, we compare the deformation and the conformational restoring force as presented in Figure 9.

Figure 8. (a) Relative first normal stress coefficients Φ1 /Φ10 contributed by various star polymers as a function of Wi/f. The solid line indicates the dependency Φ1/Φ01 ∼ (Wi/f)−1.08. (b) Relative second normal stress coefficients Φ2/Φ02 as a function of Wi/f. The solid line indicates Φ2/Φ02 ∼ (Wi/f)−1.14. Figure 9. End-to-end vector R with projections (a) ⟨Rx⟩, (b) ⟨Ry⟩, (c) ⟨Rz⟩, and (d) the restoring force Fx as a function of Wi/f for linear, ring, and star polymers. R0 is the end-to-end vector under zero-shear state.

spherical symmetry Gxx for all polymers shows a power law dependence Wi2 at low shear rates and a linear extension for star polymers at high shear rates. In the gradient direction, the shrink Gyy of linear polymer is significantly larger than ring and star, while the latter two have similar scaling exponents of Gzz in the vorticity direction. Our results indicate that either the loop or the dense core structure for ring and star polymers can retard the deformation. Since viscosity η and Gyy have the same scaling exponent, it can be used to understand the smaller shear thinning for ring and star polymers comparing with linear polymers, which has been experimentally well explored.71,75 The scaling of orientation angle tan(2θ) presents at high shear rates. The orientation change of linear polymers is more than ring and star polymers, which is attributed to TB motions.22 The latter two kinds of polymers have less scaling exponents against the increase of shear rates because of the constant orientation of TT-like motions.46 In addition, it is

Here ⟨Rx⟩, ⟨Ry⟩, and ⟨Rz⟩ are the projection of the end-to-end vectors along the flow, gradient, and vorticity directions, respectively. The conformational restoring force in the flow direction can be estimated according to the Langevin equation1 ⎤ ⎡ ⎛ Fxr0 ⎞ 1 ⎥ ⎢ ⟨R x⟩ = 2Lf ⎢coth⎜ ⎟ − Fr ⎥ x0 ⎝ kBT ⎠ ⎣ kBT ⎦

(16)

In this equation, besides the values for star polymers, f = 2 and Lf = 10 are taken for linear polymers, and f = 1 and Lf = 10 for ring polymers with the two strands.

Table 1. Scaling Exponents of Different Physical Quantities for Linear, Ring, and Star Polymers as a Function of Wi/f in a Simple Shear Flow (Here, f = 2 for Linear and f = 1 for Ring Polymers)a parameter

linear

ring

star

conformation

3Gxx/Rg2(0) − 1

2.0 (Wi/f < 1)69,70

2.0 (Wi/f < 1)43

orientation

Gyy Gzz tan(2θ)

−0.5022 −0.3422 −0.4622

−0.4143,60 −0.3243,60 −0.4037,43

dynamics

f tb f tt

0.6715,23,25,27

0.6637 0.6237,43

rheology

η Φ1

−0.5222,28 −1.2722,28

−0.4343,60 −0.9760

2.0 (Wi/f < 1)46 1.0 (Wi/f > 1)46,56 −0.42 ( f ≤ 10) −0.29 ( f ≤ 10) −0.37 ( f ≤ 10) −0.35 ( f = 15 and 50)46 −0.42 ( f < 15)47 0.64 ( f ≤ 10) 0.48 ( f ≤ 10) 1.0 ( f ≥ 10)46 −0.40 ( f ≤ 10) −1.08 ( f ≤ 10)

Gxx, Gyy, and Gzz are the flow component, the gradient component, and the vorticity component of the mean-square radius of gyration, respectively. Rg2(0) is the gyration radius of a star polymer in zero-shear state. tan(2θ) is orientation resistance parameter. f tb is the scaled TB frequency. f tt is the scaled rotation frequency. η is the viscosity contributed by polymers, and Φ1 is the first normal stress coefficient. a

F

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At low shear rates, all polymers have no obvious deformation. With the increase of Wi/f, an individual polymer is stretched in the flow direction, companied with its shrinking in the sheargradient direction and, less pronounced, in the vorticity direction. It is emphasized that for polymers with the same contour length, the stretching of ring polymers (N = 20) is much smaller than the linear polymers, even smaller than the star polymers with the length of arms (Lf = 10), due to the loop structure of the two strands. Accordingly, ring polymers have much larger restoring force than the linear and star polymers. At high shear rates, the restoring forces of polymers with different architectures level off at a constant.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work was supported by National Natural Science Foundation of China (21374117 and 21404105), Major State Basic Research Development Program (2015CB655302), and One Hundred Person Project of the Chinese Academy of Sciences. We are grateful to Computing Center of Jilin Province for essential support.

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4. CONCLUSION In this work, the effects of the length and number of arms for individual star polymers on the conformation, orientation, dynamics, and rheology in the transformation regime under simple shear flow are studied by a hybrid MPCD simulation method. For star polymers with typical nonequilibrium features, we found star polymers experienced a linear transformation from TB motion to TT-like motion with the increase of the number of arms to relax shear stress, while universal scaling relationships are present as a function of the reduced Weissenberg number Wi/f, independent of the number and the length of arms. Polymers with linear, ring, and star architectures exhibit different scaling relationships against shear rates in all aspects, and the latter two types of polymers behave differently from linear polymers. Because of either the absence of chain ends or the presence of dense core, ring and star polymers have significantly smaller deformation in the gradient direction, responding to less shear thinning than linear polymers. Complementarily, ring and star polymers take TTlike motion to relax the shear stress. Further, ring polymers have the largest restoring force and so forth the least deformation. This work presents a systematic study and comparison on polymers with different architectures in shear flow. It should be helpful to setup an overview of flow-induced behaviors for an individual polymer and complex rheological behaviors for polymers with different architectures at the molecular level.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b02636. Figures S1 and S2 (PDF) Animation S1: dynamics of a star polymer with Lf = 10 and f = 3 (MPG) Animation S2: dynamics of a star polymer with Lf = 10 and f = 5 (MPG) Animation S3: dynamics of a star polymer with Lf = 10 and f = 10 (MPG)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Phone +86 (0)431 85262535 (Y.L.). ORCID

Wenduo Chen: 0000-0002-4828-4706 Yunqi Li: 0000-0002-5190-3037 G

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DOI: 10.1021/acs.macromol.6b02636 Macromolecules XXXX, XXX, XXX−XXX