Scaffold Structures by Telechelic Rodlike Polymers: Nonequilibrium

Article pubs.acs.org/Macromolecules

Scaffold Structures by Telechelic Rodlike Polymers: Nonequilibrium Structural and Rheological Properties under Shear Flow Farzaneh Taslimi, Gerhard Gompper,* and Roland G. Winkler* Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany ABSTRACT: The nonequilibrium structural and rheological properties of suspension of rodlike polymers with attractive ends are investigated by mesoscale hydrodynamic simulations. The hybrid simulation approach combines the multiparticle collision dynamics method for the fluid with molecular dynamics simulations for the polymers. In equilibrium, the rodlike polymers self-organize into scaffold-like network structures above a critical density and attraction strength. Shear flow induces significant structural changes. We identify three major regimes. At low Weissenberg numbers, the scaffold structure is compactified accompanied by a phase separation into a fluid domain and a dense scaffoldlike structure. The shear viscosity shows a Newtonian plateau, which is determined by the fluid layer. At intermediated shear rates, shear bands appear with bundles of polymers. Again, we observe a nearly Newtonian viscosity regime governed by the fluid bands. At high shear rates, any structure is dissolved and a paranematic phase appears, with rods well aligned with the flow, and shear thinning appears.

1. INTRODUCTION Soft solids or gels are viscoelastic materials that resist an external force by either undergoing deformationsolidlike behavioror flowingfluidlike behavior. Various types of these materials are able to withstand certain external stress and therefore behave macroscopically like solids under certain loads. Typically, they are amorphous and exhibit anisotropic structures.1−3 Many consumer products, such as food, gelatins, pectins, cosmetics, paints, inks, pharmaceutics, contact lenses, and many others, are composed of gels.1,4 Gels are applied in a broad range of technological applications, e.g., photonics crystals, tissue engineering, nanocomposites, optoelectronic devices, organic transistors and sensor technology, and lightweight structures.4−6 Furthermore, in biological systems, a variety of these solid networks can be found, such as bone tissue and cytoskeleton networks.7 Plant cell walls are networks of cellulose fibers.8 Colloidal gels belong to the class of soft materials, which typically possess amorphous or disordered structures, and are often composed of supermolecular aggregates in a liquid phase. Aggregates may be formed via chemical reactions or physical interactions. In chemical colloidal gels, strong covalent bonds keep the particles together.1 In physical gels, particles may either self-assemble into a networkdirect assemblyor aggregation is induced by an external field or a template indirect assembly.6 Thereby, physical aggregation is reversible. Often, electrostatic interactions between particles in such aggregates are present, which are relatively weak.9,10 Clustering is achieved by a balance between electrostatic interactions, hydrophobicity, Brownian motion, and hydrodynamics.4,11 Telechelic polymers, which are triblock copolymers with the same end block on both sides,11−13 and patchy colloids are two © XXXX American Chemical Society

examples of particles with hydrophobic (or hydrophilic) interactions.11,14−17 Associating groups form aggregates in form of thermoreversible fractal structures in solution.4 Since their mechanical properties strongly depend on the degree of association (independent of the chemical composition18), they can be applied as nanosensors; e.g., the physical properties of DNA hybrids with dissolvable end groups are sensitive to the salt concentration of the environment.11 Alternatively, shape-induced capillary forces drive aggregation at the interface of fluids. Hence, they form quasi-twodimensional structures. Surface properties, volume fraction, shape, and aspect ratio of nonspherical particles are essential factors for self-assembling at the liquid interface.6 External fields, e.g., shear flow, can induce gel-like structures in a colloidal solution.19 Shear rate and strain, concentration and polydispersity of the particles, and interparticle interactions affect the network structure.6 In this article, we elucidate the nonequilibrium properties of end-functionalized rodlike polymers under shear. Rodlike polymers and colloids are omnipresent in nature. Examples comprise viruses such as the tobacco mosaic virus or the semiflexible fd virus, filaments such as filamentous actin or microtubules, or even short fragments of DNA. State-of-the-art chemical synthesis provides colloidal rods like boehmite and silicon needles,20,21 cylindrical dendrimers,22 and polymeric rods.23 These structures can be complemented by functionalized ends. Computer simulations of end-functionalized rods demonstrate that originally isotropic rods self-organize into novel scaffold-like structures above a critical end-attraction Received: June 12, 2014 Revised: August 21, 2014

A

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strength and density at equilibrium.24−26 These structures are distinctively different from structures formed by spherical particles, insofar as large cavities appear whose size depends on the rod length. Hence, rods can provide large stable structures at low polymer content. Our simulations reveal a strong influence of shear flow on the scaffold structure, and novel shear-rate-dependent structures appear. This comprises specific network-like structures as well as the formation of a paranematic phase above a certain shear rate. We apply a mesoscale simulation approach with an explicit solvent by combining the multiparticle collision dynamics (MPC) method for the fluid27−31 with molecular dynamics simulations of coarse-grained rodlike polymers.30,32 As has been shown, the MPC method properly accounts for hydrodynamic interactions28,30,31,33 and has been applied in a broad spectrum of equilibrium and nonequilibrium simulations of complex systems such as polymers,30,31,34−40 colloid suspension,41−48 and other soft materials such as vesicles and blood cells.49,50 An explicit solvent is essential for the intended study, since we want to avoid any bias on the induced flow profile. The latter excludes simulation methods, which would impose the shear profile locally. By the explicit solvent, bands with different shear rates are permitted and indeed appear, as is demonstrated in the article. The paper is organized as follows. In section 2, the simulation method as well as the polymer model are described. Results for structure formation at equilibrium are presented in section 3. The influence of shear on structure formation and the rheological behavior is discussed in section 4. Section 5 summarizes our findings.

Ub(R i) =

Ubend(ri) =

1 κbend 2

Nm − 2

∑

(R i + 1 − R i)2

i=1

(4)

where the bending rigidity κbend describes the stiffness and is related to the persistence length lp via κbend = lpkBT/lb3. Intermolecular interactions are taken into account by the Lennard-Jones potential ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ ⎪ ⎢⎜ σ ⎟ σ⎟ ⎥ ⎜ ε 4 U − + ⎪ ⎢⎜ ⎟ 0 ⎥ , rij < rc ⎜r ⎟ r ULJ(rij) = ⎨ ⎣⎝ ij ⎠ ⎝ ij ⎠ ⎦ ⎪ ⎪ 0, rij > rc ⎩

(5)

where ε represents the potential strength. We consider intramolecular interactions between monomers with the indices i = 2, ..., Nm−1 as purely repulsive. Thus, we set rc = 21/6σ and U0 = 1/4. The end monomers are mutually attractive. Here, we set the cutoff radius as rc = 2.5σ and choose U0 such that the Lennard-Jones potential is continuous at rij = rc. The dynamics of the monomers is described by Newton’s equations of motion, which are solved by the velocity-Verlet algorithm53,54 with the time step h/20. The solvent−polymer coupling is efficiently achieved by including the pointlike monomers in the collision step.34,35,38 We set the monomer size equal to the collision cell size, i.e., σ = a. Consequently, in a collision cell there is typically only a single monomer. Three-dimensional periodic boundary conditions are applied and shear flow is imposed by Lees−Edwards boundary conditions.53,55 We consider rodlike polymers of Nm = 10 monomers and the bond length lb = a. The spring constant is chosen as kb = 5000kBT/a3 to prevent stretching of bonds under nonequilibrium conditions, and the persistence length is set to lp = 50a. The size of the cubic simulation box is L = 80a. Up to Np = 3793 polymers are considered, i.e., Nm = 37 930 monomers, and 5.12 × 106 fluid particles. Temperature is maintained locally by a Maxwellian thermostat as described in ref 51. The shear rate is characterized by the Weissenberg number Wi = γ̇τ, where γ̇ is the shear rate and τ is the rotational relaxation time of a rodlike polymer in dilute solution. Simulations yield the relaxation time τ = 1389[ma2/(kBT)]1/2 for a polymer of length Nm = 10. Large-scale simulations became possible by employing an optimized code on graphics processing units (GPUs).56

(1)

In the collision step, the simulation box is divided into cubic collision cells of size a. In the stochastic rotation dynamics (SRD) version of MPC, the particle velocities in the center-ofmass reference frame of a cell are rotated around a randomly oriented axis by an angle α, i.e. vi(t + h) = vcm, i(t ) + 9(α)[vi(t ) − vcm, i(t )]

(3)

where kb is the spring constant and lb the equilibrium bond length.32,38 In order to control the stiffness of the polymers, we additionally apply the bond bending potential

2. SIMULATION METHOD AND MODEL 2.1. Multiparticle Collision Dynamics. The MPC fluid is composed of Ns point particles of mass m with the positions ri and the velocities vi (i = 1, ..., Ns), which interact with each other by a stochastic process. The algorithm comprises two steps: a streaming and a collision step. In the streaming step, the particles move ballistically during a time increment h, which is called collision time. Hence, the positions of the particles during streaming change as ri(t + h) = ri(t ) + h vi(t )

1 k b(R i − lb)2 2

(2)

where 9(α) is the stochastic rotation matrix and vcm,i(t) = ∑Nj cmjvj(t)/Nc is the center-of-mass velocity of the Nc particles of the cell containing particle i.30,31,51 The algorithm conserves particle number, energy, and momentum. To establish Galilean invariance, a random shift of the collision lattice is applied at any collision step.29 In our simulation, we consider h = 0.1[ma2/(kBT)]1/2, α = 130°, and ⟨Nc⟩ = 10, which corresponds to a Schmidt number of fluids rather than gases.51,52 2.2. Polymer Model. A rodlike polymer is composed of Nm mass points connected by harmonic bonds. The bond potential between subsequent monomers with the bond vector Ri = ri+1 − ri (i = 1, ..., Nm−1) is

3. STRUCTURE FORMATION AT EQUILIBRIUM We start the simulation from an equilibrated solution of isotropically orientated rods without attraction of the end monomers. Turning on attraction leads to aggregation by the formation of bonds between polymer ends. Eventually a scaffold-like structure is formed, as shown in Figure 1, which is similar to the structure obtained in ref 25. It is a homogeneous and isotropic network on large scales,57,58 with the space between the rods filled by fluid particles. B

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Figure 3. Distribution of angles between polymer end-to-end vectors for the packing fraction ϕ = 0.039 and ε/kBT = 3.5. The most probable angle is θij = 60° (cos θij = 0.5), which illustrates that a triangular lattice and, hence, a scaffold structure is formed. The pronounced peak at θij = 0° (cos θij = 1) is due to bundles of parallel rods.

Figure 1. Scaffold-like structure at equilibrium. The end-functionalized particles self-assemble into a complex space-spanning network at equilibrium (ϕ = 0.039 and ε = 3.5 kBT).

The interaction between the end monomers is short-ranged. We consider two end monomers to form a temporary bond, when their distance is smaller than 1.22a. To characterize the appearing structure, the following structural elements are introduced: • Bundle: a bundle consists of several polymers whose respective ends form bonds; i.e., the end monomers on each side of a bundle are bound with each other (Figure 2a).

that we can rarely find two rods with an angle less than 45° (cos θij > 0.7) in the structure. The sharp peak at θij = 0° (cos θij = 1) arises from parallel rods forming bundles, which contribute in the node formation. At equilibrium, three main parameters govern structure formation: the volume fraction of rods ϕ, the adhesion strength ε, and the attraction range.26,60,61 3.1. Density Effect. At low densities, structure formation is governed by rod diffusion. As an example, we consider ε/(kBT) = 3.5. Aggregation at low concentrations ϕ < 0.02 is correspondingly slow, and polymers mostly form bridges rather than nodes. Nodes are formed for higher densities, and for ϕ > 0.03 scaffold-like structures appear, which span almost the whole available volume. Figure 4a displays the ratio between the number of nodes Nnode and the number of rods for various concentrations. Starting from dilute systems, an increase of ϕ implies an increase in Nnode, and a maximum appears for ϕ ≈ 0.035. Addition of further rods leads to a decrease of Nnode/Np. This is related to a saturation of the node number, as indicated in Figure 4b, and the increase of the population of rods in nodes, which implies a decrease of the ration Nnode/Np. 3.2. Attraction Strength. In the following we will focus on the polymer packing fraction ϕ = 0.039. The attraction strength ε of the end monomers determines the structure of the network. The appearing structures can be characterized by various quantities. Here, we use the global clustering coefficient and network diameter, quantities typically used in small-world network models.62−65 The global clustering coefficient is defined asa

Figure 2. Different types of temporary bonds: (a) In a bundle of polymers, their respective ends are bound. (b) A bridge is formed by two bundles of polymers (red bound end monomers). (c) A node is formed by three or more bundles of polymers (red bound end monomers). The monomers of a bridge/node are displayed by the same color.

• Bridge: the bound end monomers of two joining bundles are denoted as a bridge (Figure 2b). • Node: when three or more distinguishable bundles join together, we denote the bound end monomers a node (Figure 2c). • Link: a bundle that connects two nodes is a link. During scaffold formation, bundles, bridges, and nodes are formed gradually. The final structure is rather stable and does not significantly change with time anymore, although the number of nodes and bridges fluctuates. After turning off the adhesive potential, the rods separate again and the initial isotropic polymer solution is recovered. Therefore, this is a stable and reversible polymer−gel aggregate.59 As indicated in Figure 1, the rods orient randomly; i.e., their end-to-end vectors are uniformly distributed in all directions, but there is a preferred angle between the end-to-end vectors of two neighboring rods. Figure 3 depicts the distribution of the angles between rods whose ends share at least one node. The most probable angle is 60°, which indicates that the system is composed of equilateral triangles of rods. The figure also shows

C=

3 × number of triangles number of closed paths = number of paths of length two number of connected triplets

(6)

The network diameter of a small-world network is calculated as62−65 DN = 1 +

log(Nnode/⟨NBN⟩) log([⟨NBN 2⟩ − ⟨NBN⟩2 ]/⟨NBN⟩2 )

(7)

where ⟨NBN⟩ is the average number of bundles per node. DN measures the average shortest path-length in the network C

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jumps to a finite and almost constant value. This indicates a transition from a polymer solution to a connected, gel-like structure. However, it is not clear whether it is a phase transition. As stressed in ref 25, in case the system would undergo a genuine phase transition, the abrupt increase of C suggests a first-order transition. This aspect deserves further investigations. The network diameter, displayed in the inset of Figure 5, confirms the formation of clusters for ε > 3kBT. Note that the smaller the network diameter, the higher the connectivity. The network diameter cannot be calculated for systems with ε < 2.5kBT, since the structure is still not formed in the system and nodes are not connected. Hence, Figure 5 provides a quantitative measure for the transition from a disordered to a self-organized structure consistent with the visual impression.

4. NONEQUILIBRIUM BEHAVIOR 4.1. Morphology. The response of the system to an applied shear flow is illustrated in Figure 6. Evidently, shear leads to major changes of the structures. For attraction strengths ε < 3kBT, the original isotropic rodlike molecules align with the shear flow and form a paranematic phase at hight shear rates (cf. Figures 6a−e).44,66 Thereby, individual rods continuously undergo a tumbling motion. A similar behavior has been obtained for nonattractive flexible and starlike polymers.38,67,68 Figure 6 represents snapshots for systems with the attraction strengths ε = 2kBT, 3.5kBT, and 4kBT, at a long time after imposing the flow. The flow direction is parallel to the x-axis. As discussed before, a scaffold-like network can be formed in systems with adhesion strength larger than ε = 3kBT; hence, the system of ε = 2kBT corresponds to a polymer solution under good solvent conditions. At low shear rates, Wi ≤ 1.4, flow leads to polymer tumbling and weak polymer alignment.51 Hence, these snapshots are quite similar to those at equilibrium (Figure 6, first row, (a)−(c)). At higher shear rates, Wi ≥ 7, polymers perform a faster tumbling motion and are aligned along the flow direction. Hence, the system changes from an isotropic to a paranematic phase (Figure 6d,e). The equilibrium scaffold-like structures for ε = 3.5kBT and 4kBT resists the shear flow, and we observe various structural changes. At low shear rates the soft network becomes more compact with a pronounced scaffold-like structure, which phase separates from a nearly polymer-free fluid phase with a sharp interface (cf. Figures 6f and 6k).69 The rods inside this compressed network hardly feel the flow. Polymers that desorb from the network experience a high shear rate in the fluid and perform a fast tumbling motion. Higher shear rates exert stronger fores on the network which breaks and separates into distinct bands (cf. Figures 6g and 6m). The bands are mainly composed of bridges with polymers aligned with the flow direction. The critical shear rate for breakup depends on the attraction strength, and hence, for the system with ε = 4kBT, breakup appears at higher shear rates only. A further increase of the shear rate leads to rod alignment along the flow direction (cf. Figures 6h and 6n). Thereby, almost all nodes and bridges in the bands join, and thick bundles are formed of parallel rods, which look like thick ropes (Figure 6h). Very high shear flows destroy the network and promote the dissociation of the bridges and nodes (Figure 6i,o). Large bridges break into smaller ones with short lifetimes, and we can find only a few small temporary clusters in the system.

Figure 4. Illustration of node formation for various packing fractions ϕ. (a) The ratio between the number Nnode of nodes and the number Np of rods exhibits a maximum between 0.026 < ϕ < 0.04. (b) The number of nodes increases with increasing density and saturates for ϕ > 0.05. The solid lines are guides for the eye.

connecting two randomly selected nodes. A finite, small value indicates the presence of a percolating structure. Figure 5 illustrates the dependence of the global clustering coefficient on the interaction strength. Below a critical value of ε ≈ 3kBT, essentially no clusters are present, whereas above that ε value C

Figure 5. Global clustering coefficient C as a function of the attraction strength for ϕ = 0.039. Networks are formed for ε > 2.5kBT. The inset illustrates the network diameter or the average shortest path length in the network. For ε < 2.5kBT no network is formed; hence, the network diameter is not defined. D

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Figure 6. Shear-induced structures for various attraction strengths and shear rates for the concentration ϕ = 0.039. The flow direction is parallel to the x-axis, and the gradient direction is along the y-axis. The snapshots of the top, middle, and bottom rows correspond to ε = 2kBT, 3.5kBT, and 4kBT, respectively. Each column represents the results for the same shear rate in the range Wi = 0.14−14, as indicated. End monomers of the same node/bridge are indicated by the same color. The conformations correspond to the stationary state with the strain γ = 100.

Numerous nonbonded rods are present, which tumble in the flow. The network and clusters completely disappear for Wi ≳ 14 in the system of ε = 3.5kBT, and the system behaves like a nonattractive polymer solution at high shear rates (Figure 6e,j). A few cross-links may be formed, but the flow is strong and destroys them shortly after formation.59,70 4.2. Rheology. The shear-induced structural changes strongly affect the system’s rheological properties.71−73 Figure 7 shows monomer velocity profiles for various attraction strengths and shear rates. The profiles represent averages over the last few hundred thousand time steps. For ε = 2kBT, the shear profile is linear at all shear rates, as we expect for a polymer solution under good solvent conditions (Figure 7a). For the higher attraction strengths ε = 3.5kBT and 4kBT, the profiles become nonmonotonic for shear rates Wi < 7. The separation into low- and high-density polymer bands leads to corresponding velocity bands,74 where plateaus correspond to high-density polymer regions. In the plateau regions, the actual shear rates can be significantly smaller than the applied shear rate. Between plateaus, there are regions, in which the shear rates are much higher than the applied one. The constant

plateaus suggest that the bands of polymers move as a whole. The internal connectivity is strong enough to resist the fluidinduced shear gradient. Contrary, the fluid in-between moves much faster such that the (no-slip) boundary conditions are satisfied. In the low-velocity bands, the flow is weak and the viscosity is high. While in the high shear regions, the particles remain aligned and form a paranematic phase with low viscosity (see snapshots in Figure 6). The nonuniformity in the shear profile is reminiscent of shear banding.75 Shear banding is the response of the system to the shear flow with strain localization in narrow zones. When the structure separates into the regions with different concentrations and shear rates, the strain in the zones with higher shear rates is much higher than that in other regions in the system.76,77 Indeed, the strain zones, which contain dangling network rods and nonbonded tumbling polymers, are driven by instabilities caused by structure resistance in the flow. At high shear rates, a special case of shear bands has been reported in experiments76,78−80 and simulations72,81 for networks of telechelic copolymers, which is called shear fracture. Shear fracture occurs when the system bears high and sudden stress such that the system does not have enough time to relax; E

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curves are continuous at low and medium shear rates (Wi ≤ 1.4), and the sizes of the shear regions, which are at least on the order of a polymer length, are not that narrow to be recognized as fracture zones. At higher shear rates, Wi > 1.4, the strong flow produces a linear and monotonic velocity profile. Here, the system behaves like a Newtonian fluid as expected for a paranematic fluid. Figure 8 shows the velocity profile for various adhesion strengths and the fixed shear rate Wi = 1.4. The velocity profile

Figure 8. Monomer velocity profiles for various adhesion strengths and the shear rate Wi = 1.4.

for ε = 2kBT is linear like the velocity profile of a polymer solution under shear flow. In the other systems, two bands can be recognized that occupy approximately the same ratio of the system volumeapproximately 75% in the gradient direction. In systems with periodic boundary conditions, shear bands can appear anywhere along the shear gradient direction. In experiments, however, shear bands or fractures always form at the moving plates, and the jammed phases appear at the stationary walls.72,78 The width of a shear band, or the average distance between the clusters, is called mesh size. It is on the order of the polymer length in our systems (Figure 8). The shear rate in the high-shear-rate bands is 4−6 times that of the applied shear rate. The polymer contribution to the shear stress σxy is displayed in Figure 9 as a function of the strain γ̇t. Thereby, we calculate the flow gradient component of the stress tensor by the virial expression as described in refs 83 and 84. The systems show a pronounced response to the shear flow. For γ < 0.5, the shear stress increases drastically and the system exhibits strain hardening.85,86 The network stiffness depends on the system structure; thereby shear flow supplies the necessary energy. This energy is stored and increases the stress. In this situation, the network is not deformed, yet. Bonds broken by the flow are re-formed, since the duration is much shorter than their relaxation time. Shear stress passes through a maximum for γ ≳ 1. The position and height of the maximum depend on the shear rate. For the lowest shear rate Wi = 0.14, the weakest maximum occurs at γ ≈ 1, while for the highest Wi = 14 the largest value appears at γ ≈ 2.5. The polymers gain more energy at higher shear rates, which leads to a larger maximum value. On time scales exceeding the rod relaxation time, the polymers release their energy, σxy decreases, and the system exhibits strain softening. Strain softening is related to a

Figure 7. Monomer velocity profiles for various shear rates: (a) ε = 2kBT, (b) ε = 3.5kBT, and (c) ε = 4kBT. Plateaus in the profiles represent polymer aggregates.

i.e., the strain induced by the flow is large compared to the characteristic time scales of the polymers.79,82 Unlike the smooth response in a plastic manner in shear banding, the network ruptures in shear fracture and the system responds as a solidlike material. In shear fracture, shear strain is localized in a definite narrow zone (narrower than in case of shear banding). Therefore, the shear profile is not continuous.72 Recognizing that characterization of shear fracture requires a very accurate shear profile, in practice, rather small shear bands are already considered as fractured bands. Although polymer networks are apt to undergo fracture, they do not always do so.72 In the shear profiles in Figure 7, the flow F

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Figure 11. Viscosity as a function of Weissenberg number for various attraction strengths. Instabilities between 1 < Wi < 5 are related to shear bands.

Figure 9. Shear stress as a function of strain for various shear rates. Applying a shear flow imposes a high stress on the system at the onset of deformation (ε = 3.5kBT).

are not really solidlike with respect to shear, but rather soft and fluidlike. However, the viscosities for ε = 3.5kBT and 4kBT show various features which are not present for the fluidlike systems with smaller ε. The small increase in the viscosity for ε = 3.5kBT at Wi ≈ 0.15 could indicate a shear thickening behavior of the network for small shear rates. However, it is rather demanding to achieve more accurate data in this shear-rate regime, and we are unable to provide a clear answer on a possible shearthickening behavior. One might suspect such a behavior because the network resists the flow to some extent. The structural rearrangements presented in Figure 6 are reflected in the viscosities. In particular, we find extended “Newtonian” plateaus between 1 < Wi < 6. They are somewhat smaller than those for lower shear rates. The viscosity values agree with those extracted from the shear profiles of Figure 7 in the fluid shear band. The bands of aggregated polymers seem not to contribute to the viscosity. It is essentially determined by the low-polymer−high-fluid density bands. The phases of aggregated polymers possess different viscoelastic properties. Specifically, their viscosity is larger. Increasing the shear rate above Wi ≈ 6, which breaks the cross-links, leads to shear thinning as expected for a paranematic fluid.

Figure 10. Shear stress as a function of shear rate for various attraction strengths. The system with ε = 2 kBT does not undergo shear banding.

deformation of the network. Since at higher shear rates the structural changes are more sever, the reduction in σxy is more pronounced in this case than at the lower shear rates.87 The shear stress levels off at high strains. The magnitude of the stationary-state value depends on the shear rate; the higher the shear rate, the higher the shear stress. Figure 10 shows the dependence of the shear stress on the shear rate at large strains (γ = 100), i.e., in the stationary state. For all attraction strengths, we observe a nonlinear increase of σxy with the shear rate. For small γ̇, the curve can be approximated by a power law with the an exponent of order unity and for larger Weissenberg numbers with an exponent of approximately 2/3. This nonlinear dependence is also reflected in the shear viscosity, which is defined as η=

5. SUMMARY AND CONCLUSIONS We have performed coarse-grained nonequilibrium simulations of suspensions of end-functionalized telechelic rodlike polymers. For end-attraction strengths above ε = 3kBT and polymer concentrations ϕ > 0.026, the rods self-organize into scaffoldlike structures. Thereby nodes are formed whose number increases with polymer concentration and saturates for ϕ > 0.045. Further addition of polymers leads to more crowded bundles, which connect the nodes. Depending on the attraction strength, networks of rods exhibit distinctly different responses to shear flow. For low attraction strengths ε < 3kBT, the systems behave like polymer solutions in good solvent with a Newtonian regime for Wi < 1 and shear-thinning behavior for larger Weissenberg numbers. The scaffold-like structures exhibit three distinctly different flow regimes for low (Wi < 1), medium (1 < Wi < 6), and high (Wi > 6) shear rates. At low shear rates, the structure resists the flow by becoming more compact along the gradient direction (with a sharp

⟨σxy⟩ γ̇

(8)

As indicated by Figure 11, the system with ε = 2kBT is Newtonian for Wi < 1 and exhibits shear thinning for Wi > 1, where the rods preferentially align along with shear direction. A similar behavior has been found for flexible polymers.38 The viscosities for ε = 3.5kBT and 4kBT are larger than those of the rodlike polymer solution due to scaffold formation. However, quantitatively the difference is only about a factor 2 and thus rather small. This suggests that the formed scaffold structures G

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interfaces) and by forming a narrow fluid band which is strongly sheared. Thereby, the whole compact structure moves with the same velocity, and the polymers inside the block do not experience the flow. The shear viscosity is determined by the narrow fluid band. Under medium shear rates, the system undergoes shear banding; i.e., the system is divided into two or more bands of aggregated polymers, which are parallel aligned with the flow direction. Between the bands, a few free tumbling rods are found and dangling polymers at the interfaces. In this regime, shear flow can break bonds and detach polymers from the aggregated bands and induce tumbling, but the nonbonded rods may readsorb to the bands and form new bonds. Here, the viscosity is also determined by the fluid bands, which are somewhat wider and hence the viscosities are smaller, but they are essentially independent of the shear rate. At high shear rates, the bonds and the network disappear, and the polymers behave as in a semidilute polymer solution. In particular, they show shear-thinning behavior. The band formation is partially reversible; i.e., when shear flow is turned off, homogeneous and isotropic structures are reformed. However, for strong attraction strengths (ε > 3kBT), shear leads to the formation of compact structures, which no longer relax back into the original scaffold structures. Comparing the potential energies of the original scaffold structures and the shear induced and relaxed structure at zero shear shows that the latter is somewhat smaller than the original value. Hence, care has to be taken on the equilibration of the self-organized system. Our simulations shed light on the nonequilibrium properties of self-organized scaffold networks. We hope that the results will stimulate experimental studies along the same lines.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (G.G.). *E-mail [email protected] (R.G.W.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support of the work by the EU through the FP7Infrastructure project ESMI (Grant No. 262348) is gratefully acknowledged.

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ADDITIONAL NOTE In a network, when node v is connected to node u and node u is connected to node w, they form a path of two edges (vuw) or a triplet. If node v and node w are connected to each other, as well, they form a triangle or a closed triplet.

a

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