Letter pubs.acs.org/macroletters
Structural Mechanism for Viscosity of Semiflexible Polymer Melts in Shear Flow Xiaolei Xu and Jizhong Chen* State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People’s Republic of China S Supporting Information *
ABSTRACT: The viscosities of semiflexible polymers with different chain stiffnesses in shear flow are studied via nonequilibrium molecular dynamics techniques. The simulation reproduces the experimentally observed results, giving a complete picture of viscosity as chain stiffness increases. Analysis of flow-induced changes in chain conformation and local structure indicates two distinct mechanisms behind a variety of viscosity curves. For polymers of small stiffnesses, it is related to flow-induced changes in chain conformation and, for those of large stiffnesses, to flow-induced instabilities of nematic structures. The four-region flow curve is confirmed for polymers of contour length close to persistence length and understood by combining the two structural mechanisms. Thus, these findings clarify the microscopic structures indicated by the macroscopic viscosity. the S-pN-S three-region flow curve.27−30 Since the pN region in the S-pN-S curve is too narrow to be identified in some experiments, only two shear thinning regions having different slopes31,32 or even one shear thinning region20,28 were observed. Unfortunately, from the experimental side, the explanations of these viscosity curves are difficult, most likely because of the uncertainty in stiffness and the absence of microscopic structural properties. To understand semiflexible polymers in shear, it is desirable to obtain a complete picture of the evolution of viscosity curves and the corresponding microscopic structure changes as chain stiffness varies, which is currently only possible by computer simulations. In this Letter, we employ nonequilibrium molecular dynamics to study the melt viscosity of semiflexible polymers with different chain stiffnesses in shear flow. The main questions we want to address are What shape does the viscosity curve present with increasing chain stiffnesses? Do the plateaus exhibited by polymers with different stiffnesses have the same microscopic origin? What are the similarities in the shear thinning regime of polymers with different stiffnesses? It will be shown that experimentally observed results are reproduced. As it turns out, a complete picture of the viscosity is given as chain stiffness increases. The analysis of structural properties indicates two microscopic mechanisms for understanding the viscosity. For polymers of small chain stiffnesses,
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he relationship between structure and viscosity of polymers in shear is of great practical interest, because this type of flow is omnipresent in technical applications.1−5 Extensive experimental,6,7 theoretical,8−10 and numeric11−15 studies have focused on flexible polymers and provided a consistent, albeit in part, explanation of the shear-induced changes in viscosity. For example, the well-known Newtonian plateau−shear thinning (N-S) curve is directly related to the changes in chain conformation.9−11 In contrast, a comparable understanding of semiflexible polymers is lacking. Real polymers have more or less stiffness and, in fact, the conformational changes are completely suppressed if the persistence length is far larger than its chain length. The stiffness makes the theoretical treatment more complex, because the orientation interaction with each other has to be taken into account.2,16,17 This is particularly true for semiflexible polymers, where both deformation and orientation interaction should be considered. When exposed to shear flow, the coupling of flow and chain stiffness further complicates the conformational changes.18,19 Rather extensive experimental studies on semiflexible polymers in shear flow have revealed a rich shear dependence of viscosity. Similar to flexible polymers, they can exhibit a N-Slike curve in shear flow.20−23 Interestingly, a plateau-like region is found in the shear thinning region; this region is in fact somewhat indefinite, just like a gentle crossover between two shear thinning regions.22−26 For very stiff polymers usually forming the nematic/smectic phase at equilibrium, the first plateau is absent and the second plateau, named as pseudoNewtonian (pN) region, is apparent, which is well-known as © XXXX American Chemical Society
Received: December 28, 2016 Accepted: March 10, 2017
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DOI: 10.1021/acsmacrolett.6b00979 ACS Macro Lett. 2017, 6, 331−336
Letter
ACS Macro Letters
The zero-shear viscosity η0 depends on the equilibrium structure. As chain stiffness increases, the structural quantities, both ⟨R20⟩ and ⟨S0⟩ (where the subscript zero refers to equilibrium values), exhibit qualitative transitions at chain length close to persistence length, κ ≈ 1.0 (see Figure 1). At
the viscosity depends on the changes in chain conformation; for polymers of large chain stiffnesses, the formation and change of nematic structure are responsible for the viscosity. In this work, the polymer chain is described by the classical Kremer-Grest model,14 in which the adjacent beads are connected by finitely extensible nonlinear elastic springs with the maximum bond length 1.5σ and the spring constant 30ϵ/σ2, σ and ϵ being the reduced MD units of length and energy, respectively. Chain stiffness is introduced by a bond bending potential: Ubend(θ) = K[cos(θ) − cos(θ0)]2, where prefactor K determines the bending strength, θ denotes the angle between two adjacent bonds, and the equilibrium angle θ0 is set to be 180°. In general, chain stiffness is quantitatively characterized by the ratio between persistence length and contour length: κ = Lp/L, where persistence length Lp is obtained through the exponential decay of the bond orientation correlation function and contour length L is related to chain length Nm, L ≈ (Nm − 1)l0, where l0 = 0.96σ is the bond length. The Lees-Edwards boundary condition is applied to impose a flow velocity field (vx, vy, vz) = (γż, 0, 0) with a tunable shear rate γ̇, where x, y, and z are the flow, vorticity, and gradient directions, respectively. Dissipative particle dynamics thermostat is used to keep the temperature at a constant value T = 1.0, as in our previous work.11,12,33 In our systems, chain length Nm is set to be 10. When κ ≪ 1, the polymer can be practically viewed as a random walk chain and exhibits the N-S curve in shear flow;15 when κ ≫ 1, the polymer is stiff and has a large length/ diameter ratio (≈9). K studied ranges from 0.0 to 30.0, and correspondingly, κ from 0.11 (for which Lp decays to bond length l0) to 2.89, covering the range from flexible to stiff polymers. A monomer density of ρ = 0.85/σ3 and the time step of 0.01τ (τ is the unit of time) are considered. The largest cubic simulation box studied consists of more than 6000 chains to minimize finite-size effects, of which the box side length is about 5L. Increasing stiffness would result in locally ordered structures, which requires longer time for equilibrating the systems than chain relaxation time. Each run of the stiffest polymer system uses over 108 time steps before sampling; to ensure the system arrives at its stable state, both a run of 5 × 108 time steps and over 10 separate runs with different initial velocities and positions are used. It is also found that a longer chain length Nm = 20 shows no qualitative influence on the results. Moreover, the time for each run increases with chain length, especially for rigid polymers. A polymer may exhibit a coil−rod transition as chain stiffness increases, resulting in the local ordering in structure. The changes in the polymer conformation can be characterized by its mean-square end-to-end distance ⟨R2⟩ = ⟨|rNm − r1|2⟩. An order parameter quantifying the structure of the system can be 1 N defined as ⟨S⟩ = 2N ∑i =c 1 (3 cos2 φi − 1), where Nc is the
Figure 1. Mean-square end-to-end distances ⟨R02⟩, zero-shear viscosities η0, and order parameters ⟨S0⟩ at equilibrium as a function of chain stiffness κ. Vertical lines indicate four systems with different chain stiffnesses: from left to right, κ = 0.23, 0.97, 1.74, and 2.89, respectively.
small stiffnesses (κ < 1.0), ⟨R20⟩ increases rapidly, indicating that the polymer is stretched and deviates from the coil conformation. In this range of stiffness, the system is practically isotropic and ⟨S0⟩ ≈ 0. As a result of the increase of chain size, a rapid increase is found in η0. When κ increases above approximately 1.0, the polymer assumes a rod-like conformation, of which ⟨R20⟩ approaches its maximum value of L2 = 74.65σ2. Strong orientation interactions between polymers make them orient well along a common direction, leading to a nematic phase (⟨S0⟩ > 0.6). Such an isotropic−nematic transition results in an apparent decrease in η0, which can be qualitatively understood by considering that the orientation along a common direction makes diffusion easier for semiflexible polymers. On the basis of the equilibrium results, we choose four systems, as indicated in Figure 1, to investigate the effects of stiffness on the shear-dependent viscosity of polymers. As shown above, these four representative systems exhibit the diversity of structures and viscosities from well-studied flexible polymers to stiff polymers at equilibrium. For each system, the longest chain relaxation time at equilibrium is extracted from the initial exponential decay of the end-to-end correlation function,2,34 and 230τ, 1680τ, 12900τ, and 54600τ are obtained for polymers of κ = 0.23, 0.97, 1.74, and 2.89, respectively. Another important point to make is that, even for the polymer of the largest stiffness studied, κ = 2.89, ⟨R20⟩ = 63.62σ2 is smaller than 74.65σ2, corresponding to the fully stretched chain configuration, and hence, the chain is not fully stiff and a considerable bending is still allowed. For flexible polymers in shear flow, the viscosity curve is known to consist of two regimes, Newtonian plateau and shear thinning. Polymers of κ = 0.23 present such a viscosity curve, as shown in Figure 2; the Newtonian and shear thinning regimes are divided by a critical shear rate γ̇c ≈ 4 × 10−3τ−1, essentially equal to the reciprocal value of the relaxation time 230τ. A striking result is found for polymers of κ = 0.97 close to the critical stiffness for the isotropic−nematic phase transition in equilibrium. It is shown that the viscosity curve has two plateaus (γ̇ < 5 × 10−5τ−1 and 2.5 × 10−4τ−1 < γ̇ < 10−3τ−1) and two shear thinning regimes (5 × 10−5τ−1 < γ̇ < 2.5 × 10−4τ−1 and γ̇ > 10−3τ−1); the second plateau is just like a gentle
c
number of polymers and φi is the angle between the direction of polymer i and the director n⃗ defined as the average direction of all the polymers. Here, the direction of a polymer is determined by the eigenvector of the gyration radius tensor with the largest eigenvalue. For a completely random and isotropic system, ⟨S⟩ = 0, whereas for a perfectly aligned system, ⟨S⟩ = 1. Under shear flow, the shear viscosity is obtained via η(γ̇) = −σxz/γ̇, where σxz denotes shear stress and is calculated from the viral formulation of stress tensor,34 and the zero-shear viscosity η 0 is determined by a linear extrapolation to zero shear rate. 332
DOI: 10.1021/acsmacrolett.6b00979 ACS Macro Lett. 2017, 6, 331−336
Letter
ACS Macro Letters
the larger stiffness strengthens the topological constraints between neighboring rods, and the dynamical correlation length will dramatically increase; when the correlation length is comparable to or even larger than the box size, the finite-size effect emerges. To understand the observed results of viscosity, we investigate the flow-induced changes in structure. It is wellknown that in the Newtonian regime the flexible polymer holds a random conformation as in equilibrium, and hence, ⟨R2⟩ keeps unchanged.11 As shown in Figure 3a, ⟨R2⟩ of polymers of κ = 0.23 essentially maintains a constant value for γ̇c < 4 × 10−3τ−1. Figure 3b shows ⟨S⟩ ≈ 0 in this range of shear rate, indicating that the system is still isotropic. In the shear thinning regime, the polymer is stretched and oriented along the flow direction and ⟨S⟩ increases rapidly with shear rate. This conformation transition from linear to nonlinear regime is also illustrated in Figure 3c. Since the flexible polymer deforms and tumbles under shear,11 the end-to-end vector cannot orient steadily along the flow direction and the value of ⟨R2⟩ is much less than that of the fully stretched chain conformation, even at the highest shear rate. In contrast to polymers of κ = 0.23, those of κ = 1.74 and 2.89 represent another limit that chain conformation is difficult to deform. In the whole range of shear rate, the values of ⟨R2⟩ almost keep unchanged, indicating that the stiff polymer maintains the rod-like shape as in equilibrium. We find that shear flow almost has no influence on the vales of ⟨S⟩ of stiff polymers in the plateau regime. The analysis of the transient structural properties reveals that the director n⃗ changes regularly with small fluctuations in order parameter (see more details in SI). This evidence indicates that stiff polymers in this regime make a collective rotation, which has also been observed in other simulations35 and theoretical works.16,17,36 In addition, the rotation period dramatically decreases with increasing shear rate, for instance from around 1.4 × 105τ to 1.0 × 105τ when shear rate increases from 1.0 × 10−4τ−1 to 2.0
Figure 2. Shear viscosities η for polymers of various chain stiffnesses, as a function of shear rate. The lower inset: η scaled by η0. The upper inset: time traces of shear stress of polymers of κ = 2.89 at γ̇ = 0.0002.
crossover between the two shear thinning regimes. This finding is in good agreement with the experimental observations of thermotropic liquid crystalline polymers:22−24 a hesitation in the shear thinning regime. The two stiffer polymers (κ = 1.74 and 2.89) form a nematic phase in equilibrium. We find that they in shear flow present a plateau regime and a following shear thinning regime, similar to flexible polymers but of a smaller γ̇c (≈ 4 × 10−4τ−1 for κ = 1.74 and ≈10−3τ−1 for κ = 2.89), which is consistent with experimental results.20,21,23 γ̇c for stiff polymers deviates from the reciprocal value of the longest chain relaxation time significantly, while the transient shear stress for these two stiff polymers undergoes a cyclic oscillation in the plateau regime. These facts may imply a different structural mechanism underlying the viscosity curve from flexible polymers. In addition, it is clear that data for polymers of κ = 0.97, 1.74, and 2.89 collapse on a common curve at high shear rates. In our simulation, the experimentally observed S-pN-S threeregion flow curve is reproduced only for polymers of larger stiffnesses (e.g., κ = 6.4). Unfortunately, these results suffer a finite-size effect (see more details in Supporting Information (SI)), which might be understood by the following mechanism:
Figure 3. Mean-square end-to-end distances ⟨R2⟩ (a) and order parameters ⟨S⟩ (b) for various chain stiffnesses as a function of shear rate. Schematic illustrations of polymers of κ = 0.23, 2.89, and 0.97 in different viscosity regimes are shown in (c). 333
DOI: 10.1021/acsmacrolett.6b00979 ACS Macro Lett. 2017, 6, 331−336
Letter
ACS Macro Letters × 10−4τ−1 for κ = 1.74. Clearly, although flexible and stiff polymers have a similar plateau viscosity regime at low shear rates, the corresponding structural mechanisms are completely distinct; such a regime for stiff polymers is reasonable to be named as the pseudo-Newtonian regime. At higher shear rates, ⟨S⟩ shows a slight decrease at the beginning of shear thinning, which is most likely due to the decrease in the cooperativity of stiff polymers when rotating. At the largest shear rates, stiff polymers tend to orient steadily along the flow direction and ⟨S⟩ increases accordingly. Since stiff polymers are difficult to bend, the tumbling motion disappears, and they mainly slide past with each other at high shear rates, as illustrated in Figure 3c. Further confirmation of the origin of viscosity arises from examination of the flow-induced structures of polymers of κ = 0.97. In the first plateau regime, the small values of ⟨S⟩ (
