Rheological Influence of Short-Chain Branching for Polymeric

May 26, 2017 - This negligible stress contribution from the branched backbone is ... The vertical dotted line specifies two characteristic flow regime...
0 downloads 0 Views 2MB Size
Download PDF
Article pubs.acs.org/Macromolecules

Rheological Influence of Short-Chain Branching for Polymeric Materials under Shear with Variable Branch Density and Branching Architecture Seung Heum Jeong, Jun Mo Kim, and Chunggi Baig* Department of Chemical Engineering, School of Energy and Chemical Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan 689-798, South Korea S Supporting Information *

ABSTRACT: An important objective in polymer science is to manipulate the material properties of polymers by altering their molecular architecture. To this end, understanding of the fundamental role of chain branches along the polymer backbone is crucial. Although the dynamics of linear and long-branched polymers have been thoroughly investigated over the past decades, a comprehensive understanding of branching effects has not yet been obtained, particularly because of a serious lack of knowledge on the role of short-chain branches, the effects of which have mostly been neglected in favor of the standard entropic-based concepts of long polymers. Here we have comprehensively studied the general effect of short-chain branching on the rheological properties of polymeric materials using Brownian dynamics simulations for a series of SCB polymers with systematically varied branch densities and branching architectures along the chain backbone. Our results demonstrate that the short branches, via their fast random motions, give rise to a more compact and less deformed chain structure in response to the applied flow, eventually reducing the shear-thinning behavior in viscosity and the first normal stress coefficient. Most importantly, by altering the distribution of short branches along the backbone, the structural and rheological properties of the SCB system are dramatically changed. We discuss the physical origins and molecular mechanisms underlying these effects and present a detailed interpretation using a molecular-level analysis of individual chain dynamics. This information is valuable, showing us how to systematically tune the material properties by controlling the molecular architecture of branched polymers under various flow conditions.

1. INTRODUCTION The many unique and useful physicochemical properties of polymers, in conjunction with a variety of monomeric aspects such as chemical composition, molecular weight, and molecular architecture, have enabled the application of polymeric materials to almost all areas of science and technology, including the large-scale use of plastic commodities in daily life, pharmaceutical biotechnology, and renewable (e.g., fuel and solar cells) energy engineering. Fundamentally, this versatile functionality of polymers lies in the intrinsic variety of their internal molecular structures. As such, the principles underlying the structure−property−phenomenon relationship, which has been a most challenging, long-standing issue in polymer science, cannot be established without comprehending the origins and the effects at the molecular level of varying the internal microstructure of a polymeric system under specific external conditions.1 Significant research into the structural and dynamical behaviors of linear and long-branched polymers under various flow conditions has been carried out in the past decades, and the fundamental aspects underlying the complex rheological properties and phenomena displayed by linear and long-branched polymers are now well understood.2−4 Despite these advances, our current understanding in the field of polymer rheology is still far from complete, mainly because of a © XXXX American Chemical Society

serious lack of information concerning the role of short-chain branches, which has been disregarded in favor of the customary entropic-based concepts of long polymers.5 Comprehending the fundamental role of the short branches in conjunction with that of the long branches is indispensable for achieving our major goal in polymer science to control the material properties by altering the molecular architecture of branched polymers. In our recent study,6 we identified the fundamental role of short branches in the polymer structure and dynamics and proposed that the fast random Brownian motions exerted by short branches, which arise from their very short characteristic relaxation times, have a significant effect on the structure and dynamics of short-branched polymeric materials. However, until now, there has been no systematic study quantifying the effects of short branches with respect to the material parameters such as branch density or branch distribution. Thus, our objective here is to make a comprehensive and systematic study of the role and effects of short-chain branching on the rheological properties of polymeric materials. To this end, we have conducted extensive mesoscopic bead−spring Brownian Received: March 14, 2017 Revised: May 9, 2017

A

DOI: 10.1021/acs.macromol.7b00544 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

SCB_dumbbell and the SCB_center polymer; see, e.g., Figures S2 and S3). Consequently, for clarity, only the results of the representative SCB systems are shown in the main text, and those of the other SCB systems are included in the Supporting Information. The three representative SCB polymers were set to have the same molecular weight (corresponding to a total of 70 beads) and branch density (33%), which was appropriate to analyze the effects of short branches and the construction of the various types of molecular architectures of SCB polymers. The magnitude of the branch mobility (specified by an adjustable parameter termed the “branch randomness factor (brfac)” that controls the characteristic relaxation time of short branches via the “τR × brfac” in the evolution equation of connector vectors; i.e., refer to eqs S5 and S6 in the Supporting Information) was set to 0.05,8 which was determined to match the structural and rheological properties between the full atomistic and coarsegrained BD simulations for a representative SCB_uniform polymer system under shear. Note that a smaller value of “brfac” corresponds to a shorter characteristic relaxation time and thus a higher mobility of short branches. The results derived for the SCB polymers were directly compared with those of the corresponding linear analogues. The bead−spring BD simulations for the linear and SCB polymers under shear flow were carried out using finite extensible nonlinear elastic (FENE) force law, without including hydrodynamic interactions between the beads. In conjunction with the well-known fact that the Rouse model without including any contributions of hydrodynamic interaction (HI) and excluded volume (EV) describes well the linear viscoelastic behaviors for unentangled dense polymer solutions or melts where the effects of HI and EV are essentially negligible,2 the results for polymeric systems modeled in this study can be qualitatively applied to unentangled polymer melts for which topological constraints (entanglement) between chains are ineffective.9−13 (Further details of the simulation methodology are described in the Supporting Information and also the Supporting Information of ref 6.) Because intermolecular interactions are not modeled, as in typical BD simulations for dilute solutions, the polymeric systems studied here are similar to untangled polymer melts, for which topological constraints (entanglement) between chains are ineffective.

dynamics (BD) simulations for a variety of well-defined shortchain branched (SCB) polymers with respect to the branch density and the branching architecture. Our results unambiguously confirm the proposition of our previous study6 that short branches, via their fast random motions, give rise to a compact and less deformed molecular structure in response to an applied flow field. We further show that these effects are associated with the structural resistance of the SCB polymers against the flow, which become stronger as the branch density increases, irrespective of the total molecular weight or the backbone length. Furthermore, we observed a dramatic influence of the branching architecture (by altering the distribution of short branches along the backbone) on the structural and rheological behaviors of the SCB system. We provide a detailed account of the physics behind these observations and reveal the distinctive molecular mechanisms for each type of SCB polymer via a molecular-level analysis on the individual chain dynamics in a wide range of flow strengths.

2. SYSTEM STUDIED AND SIMULATION METHOD To analyze the effect of branch density on the structural and dynamical properties of the SCB system, we devised two different types of SCB polymer: (i) one where the backbone length and the number of short branches for a chain were both varied according to a specified branch density while the total chain length was constrained to 70 beads and (ii) the second type where the backbone length was fixed at 47 beads, but the number of short branches for a chain was varied, corresponding to the given branch densities. The branch density (specified as the percentage of the backbone portion covered by short branches over the entire backbone) considered in this study spans the range of 6−100%. Furthermore, on the basis of the fact that in the case of the polyethylene melt system short branches usually contain either 4 or 6 carbon atoms and the Kuhn segment for polyethylene chain contains about 12 carbon atoms,7 we devised the present coarse-grained bead−spring model where one branch bead corresponds to one-half regular (backbone) bead; i.e., for the present SCB systems, we chose one Kuhn segment composed of 12 CH2 units for a bead in the backbone and one-half Kuhn segment composed of 6 CH2 units for each branch bead. To explore the influence of various branching architectures, we chose three representative types of the SCB polymer: (i) SCB_uniform where the short branches are uniformly distributed along the backbone, (ii) SCB_dumbbell where one-half of the short branches are located near one end of the backbone and the other half near another backbone end, and (iii) SCB_center where all the short branches are placed around the backbone center. We also studied other types of the SCB polymer, such as SCB_random where the short branches are randomly distributed along the backbone, SCB_half-dumbbell where the entire short branches are located near only one end of the backbone, and SCB_center-dumbbell where the two branched parts as in the SCB_dumbbell are shifted somewhat away from the chain ends toward the backbone center (refer to Figure S1 in the Supporting Information for their schematic images). However, we found that the structural and rheological properties of these additional types of SCB polymers can be readily understood from the results obtained for the three representative SCB polymers (i.e., the structural and rheological behaviors of the SCB_random polymer are found to be similar to those of the SCB_uniform, and the SCB_half-dumbbell and SCB_center-dumbbell polymers have rheological characteristics in between those of

3. RESULTS AND DISCUSSION We considered the effects of short-chain branching on the structural and rheological properties of the polymers by varying the branch density for two different cases: (i) one with the same total chain length (70 beads) and (ii) the other with the same backbone length (47 beads). In both cases, the short branches for all the SCB polymers were uniformly distributed along the backbone. For case i, Figures 1a and 1b show the effects of branch density on the shear viscosity η and the meansquare chain end-to-end distance ⟨R̃ ete2⟩ based on the longest linear chain dimension (backbone) as a function of the Weissenberg number Wi (defined as the product of the longest characteristic relaxation time of the system and the imposed strain rate). As shown in Figure 1a, the SCB systems studied exhibit a typical shear-thinning behavior, i.e., a decrease in the viscosity with increasing flow rate. However, the degree of shear thinning is smaller for systems with higher branch densities. Because shear-thinning behavior is ascribed mainly to chain alignment (alongside with structural deformation) in the flow direction, this result indicates that the degree of chain alignment and deformation in response to the flow weakens B

DOI: 10.1021/acs.macromol.7b00544 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

from their original (equilibrium) random orientation and random-coil conformation. The results presented in Figures 1a and 1b can be readily understood from this perspective. Figures 1c and 1d show the corresponding structural and rheological result for case ii where the chain backbone length is constant. Similar to the trends shown for case i, less shear thinning and structural deformation is evident for chains with higher branch densities for case ii. This result can also be understood using the physical interpretation given earlier. Interestingly, for both cases (Figures 1b and 1d), the magnitude of certain critical intermediate flow strength (where ⟨R̃ ete2⟩ begins to increase rapidly) increases for polymer systems with higher branch densities. This can be understood by considering that the total resistive force of a polymer chain against an external flow field is equal to the sum of all the individual contributions from each short branch along the chain and so increases with increasing branch density, resulting in an increase in the critical flow strength. We now focus on the systematic study of the structural and rheological influences of the molecular architecture of the SCB polymer by varying the distribution of the short branches along the backbone. Three representative SCB architectures are depicted schematically in Figure 2a: (i) SCB_uniform, (ii) SCB_dumbbell, and (iii) SCB_center. The SCB polymers with a nonuniform branching architecture (e.g., SCB_dumbbell and SCB_center) can be mapped into a sort of block copolymer, and the backbone part which is covered with short branches (the branched backbone) has distinct structural and dynamical characteristics from those of the backbone part without short branches (the bare backbone). Thus, representing the branched backbone as block “A” and the bare backbone as block “B”, various molecular architectures of the nonuniform SCB polymers can be described as types of block copolymer comprising two kinds of monomer. For instance, the SCB_dumbbell and SCB_center polymers can be represented as “A−B−A” and “B−A−B” triblock copolymers, respectively. Similarly, the SCB_half-dumbbell and SCB_center-dumbbell polymers depicted in Figure S1 are “A−A−B” and “B−A−B− A−B” triblock copolymers, respectively. Other types of SCB polymers with varying branch lengths and branch−branch frequencies are also possible, and the investigation of these types of polymers would be interesting. The short-branch density (33%) selected in this study was found to have a substantial effect on the structural and rheological responses of each type of SCB system studied. Two important rheological properties for the linear and SCB systems (shear viscosity η and the first normal stress coefficient Ψ1) are plotted as a function of Wi in Figures 2b and 2c. Notably, like the linear polymer, all the SCB polymers exhibit a typical shear-thinning behavior in both η and Ψ1. However, the degree of shear thinning was affected by the short branches and the branching architecture significantly. Specifically, the SCB_uniform and SCB_center polymers display a less shear thinning than the linear analogue. This behavior is associated with the lower chain stretching and alignment in the flow direction at a given field strength, which arise from the strong random motions of the short branches. However, we also found that the SCB_center polymer showed a less shear-thinning behavior than the SCB_uniform polymer, indicating weaker chain deformations in the former compared to the latter. In sharp contrast, the SCB_dumbbell polymer exhibited a stronger shear-thinning behavior than the linear counterpart, indicating a larger degree of chain deformation and alignment of the SCB_dumbbell chains in response to the

Figure 1. Structural and rheological properties of bead−spring BD simulations as a function of Wi for the SCB polymer systems with variable branch densities (the parentheses represent the percentage of the backbone portion covered by short branches). As for the numeric nomenclature shown in the figure legend, the first digit represents the number of beads contained in the backbone of a chain and the second digit the number of beads, each of which represents a short branch. Note that one branch bead corresponds to one-half backbone bead in the present coarse-grained model (see the main text); e.g., for the SCB polymer denoted by “48_44”, the total molecular weight of a chain corresponds to 70 (= 48 + 44/2) beads. The shear viscosity η is normalized by the equilibrium value η0 for each SCB system, and the mean-square chain end-to-end distance ⟨R̃ ete2⟩ is calculated based on the longest linear chain (backbone) dimension. The mean-square chain end-to-end distance is made dimensionless with ⟨R̃ ete2⟩ = ⟨Rete2⟩(K/kBT), where K, kB, and T denote the elastic spring constant, the Boltzmann constant, and temperature, respectively. (a, b) represent the results for case i where all the SCB systems have the same total molecular weight (70 regular beads) but with a variable branch density, and (c, d) show the results for case ii where all the SCB systems have the same backbone molecular weight (47 beads) but with a variable branch density.

with increasing branch density at a given field strength (see also the result Gyy of Figure S4 for the chain thickness δy in the velocity gradient (y-)direction δy = Gyy in conjunction with the following relation ηp ∼ Gyy based on the Giesekus stress expression14,15,23). The structural characteristics associated with this rheological behavior are presented in Figure 1b, which shows that the variation of ⟨R̃ ete2⟩ in response to the imposed flow strength becomes smaller with increasing branch density; i.e., chains with a higher branch density possess a more compact overall conformation and undergo less structural deformation against the external flow field.16 Altogether, these results demonstrate the strong influence of short branches on the overall structure and dynamics of polymer. As has been found in our previous study,6 the physical origin of the observed effects lies in the dynamical behavior of the short branches, which arises from their intrinsically high mobilities, corresponding to the fast random Brownian kinetics in association with their very short length that is no more than six atoms. (The longest characteristic (relaxation) time scale for such short branches is much smaller than that of the flow field commonly applied in practice; thus, their dynamics are essentially unaffected by the flow.) It is therefore physically plausible that the fast motions in all directions executed by the short branches along the backbone generally tend to reduce the flowinduced alignment and structural deformation of chains away C

DOI: 10.1021/acs.macromol.7b00544 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

respect to the imposed flow strength. The longest linear chain dimension (corresponding to the backbone length comprising 60 beads) was the same for all SCB polymers and the linear analogue. Figure 3a shows that for all systems ⟨R̃ ete2⟩ rapidly

Figure 3. (a) Mean-square chain end-to-end distance ⟨R̃ ete2⟩ as a function of Wi and (b) the probability distribution function (PDF) of the chain end-to-end distance |R̃ ete| (=|R ete| K /kBT ) at a weak (Wi = 2, inset) and an intermediate (Wi = 30) flow strength, for the simulated linear and SCB systems. See the note in ref 18 for the relative magnitude of the chain radius of gyration for the SCB polymers in comparison with the corresponding linear polymer. Figure 2. (a) Simple sketches of the molecular architectures for the linear and the three representative SCB polymer systems. (b) Shear viscosity η (normalized by the equilibrium value η0) and (c) the first normal stress coefficient Ψ1 (normalized by the equilibrium value Ψ1,0) as a function of applied flow strength for the linear and SCB systems. All the linear and SCB chains have the same total molecular weight (70 beads). The error bars are smaller than the size of the symbols unless otherwise specified.

increases from low to intermediate values of Wi (indicative of the significant chain deformation induced by the applied flow) and attains a plateau value at high values of Wi via chain rotation and tumbling under shear. 19,20 However, the magnitude of ⟨R̃ ete2⟩ is different for the polymer systems at a given flow strength. The ⟨R̃ ete2⟩ values of the SCB_uniform and SCB_center polymers are smaller than those of the corresponding linear polymer. Notably, the ⟨R̃ ete2⟩ value of the SCB_center polymer is only about half that of its linear counterpart in the intermediate-to-high flow regime. This is a remarkable structural characteristic of SCB_center polymer. In contrast, the SCB_dumbbell polymer exhibits much higher ⟨R̃ ete2⟩ values than the linear polymer over the whole range of Wi, indicative of its significantly larger chain stretching. Further structural information can be obtained from the analysis of the PDF of |R̃ ete| for each system. At low Wi numbers [i.e., Wi = 2, shown in the inset of Figure 3b], the PDFs for all systems can be characterized as Gaussian-like around the equilibrium value of |R̃ ete|. However, the PDF of the SCB_center polymer has a larger peak at a smaller value of | R̃ ete| with a narrower distribution compared to that of the corresponding linear polymer; this indicates that the SCB_center polymer has a stiffer chain conformation compared to the linear polymer. In stark contrast, the PDF of the SCB_dumbbell polymer has a smaller peak at a larger value of | R̃ ete| with a broader distribution than that of the linear polymer, indicating the more flexible structural characteristics of the SCB_dumbbell polymer. In addition, the SCB_uniform system

applied flow (this result will be discussed in more detail later). All these findings indicate the significant effect of the short branches on the macroscopic rheological properties of polymers and provide useful information concerning the tuning of material properties by controlling the branching architecture. By fitting the data within the shear-thinning regime to the standard power-law expression η ∼ Wi−b, the power-law index (b) was calculated to be 0.41 ± 0.02, 0.43 ± 0.02, 0.46 ± 0.01, and 0.51 ± 0.01 for the SCB_center, SCB_uniform, linear, and SCB_dumbbell systems, respectively (these values are in accordance with the typical range (0.4 < b < 0.9) obtained from experimental measurements for polymer melts17). Similarly, the power-law index for Ψ1 was found to be 0.95 ± 0.03, 1.06 ± 0.02, 1.16 ± 0.02, and 1.23 ± 0.01 for the SCB_center, SCB_uniform, linear, and SCB_dumbbell systems, respectively. To further understand the structural characteristics underlying the macroscopic rheological responses of the system, we analyzed the variation in ⟨R̃ ete2⟩ and the probability distribution function (PDF) of the chain end-to-end distance, |R̃ ete|, with D

DOI: 10.1021/acs.macromol.7b00544 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules has a somewhat stiffer chain conformation than that of the linear system. At an increased flow strength (Wi = 30 in Figure 3b), the PDF profile of each system becomes substantially distorted from the Gaussian shape near equilibrium, resulting from two effects: (i) chain extension in the flow direction and (ii) chain rotation and tumbling mechanisms on the vorticity plane. At this intermediate flow strength, the PDF of the linear polymer has a broad, flat distribution with two characteristic peaks: a rotation peak at low |R̃ ete| and a stretch peak at high | R̃ ete|. In comparison, the PDF of the SCB_uniform polymer shows a declining ramp-shaped distribution without any distinctive stretch peak, and the overall weight of the PDF is skewed toward lower values of |R̃ ete|, indicative of the more compact chain structure, which is again ascribed to the fast random motions of the short branches, rendering the chain backbone less stretchable against the applied flow. This feature is even more evident in the case of the SCB_center polymer, which exhibits a much tighter chain conformation with a particularly large rotational peak. In contrast, the PDF of the SCB_dumbbell system is characterized by a relatively weaker rotation peak at a higher value of |R̃ ete| and a stronger stretch peak at a larger value of |R̃ ete| than that of the corresponding linear system, indicating its less coiled and more stretched chain conformation with respect to the imposed flow field. These structural characteristics of the SCB polymers in comparison with the linear analogue are (at least semiquantitatively) closely linked with the macroscopic rheological properties shown in Figure 2. That is, compared to that of the linear polymer, the reduced shear-thinning behaviors in the shear viscosity and the first normal stress coefficient displayed by the SCB_uniform and SCB_center polymers are associated with their less deformed chain structures in response to the applied flow, and the greater shear-thinning behavior of the SCB_dumbbell system is associated with larger chain deformations. To obtain additional structural information on polymer chains in conjunction with their rheological responses, we performed a further structural analysis, the so-called brightness method, which categorizes the mesoscale chain structure into several classes based on the monomer distribution along the chain (Figure 4). Although the original brightness method21,22 involved the classification of various chain configurations into six representative configurations (namely, coil, fold, kink, dumbbell, half-dumbbell, and stretched), here we reduced the number of categories to four (coil, fold, dumbbell, and stretched) by combining some dynamically similar classes (i.e., stretched = stretched + kink + half-dumbbell) to better account for the temporal variation of the large-scale chain structures as a function of flow strength. Only the beads in the backbone were included in this analysis because we focused on the dynamical role of the short branches on the overall chain configuration. In addition, the brightness method places an emphasis on depicting the overall chain configuration or shape without regard to the actual chain size. Figure 4 presents the results for all the studied linear and SCB polymers with respect to the imposed flow strength. The linear polymer exhibits an entropically driven coiled configuration near equilibrium. In a relatively weak flow regime (Wi < 10), as Wi increases, the chains begin to deform from the coiled configuration due to chain stretching and alignment in the flow direction, leading to a rather sharp decrease in the proportion of coil configuration and an increase in the dumbbell configuration, corresponding to an extended chain conformation. In an intermediate flow

Figure 4. Probability distribution function (PDF) of the four representative configurations (fold, dumbbell, coil, and stretched) for the chain backbone evaluated by the brightness method for the simulated linear and the SCB systems as a function of Wi (see also Figure S5 for different representation of the same data).

regime (10 < Wi < 100), the amount of polymer in the stretched configuration significantly increases with increasing flow strength, as the polymer chains with the dumbbell configuration are further extended into the stretched configuration. In a strong flow regime (Wi > 100), the stretched portion decreases and the fold portion increases with increasing Wi; this is ascribed to chain rotation and tumbling dynamics under shear. Thus, there appears to be a specific interconversion between two different chain configurations depending on the flow range: (i) coil → dumbbell in the weak flow regime, (ii) dumbbell → stretched in the intermediate flow regime, and (iii) stretched → fold in the strong flow regime. The maximum behavior displayed by the dumbbell and stretched portions is a consequence of the above interconversions. The variation of the PDF profile for each configuration as a function of flow strength appears to be quite similar between the SCB_uniform and the linear system. A further examination shows that the PDF profile for each configuration for the SCB_uniform polymer is more extended and shifted to higher Wi values compared to that of the linear polymer; this is associated with the compact molecular structures of the SCB_uniform polymer that are less vulnerable to unraveling on exposure to flow because of the strong resistance exerted by the highly mobile short branches against the imposed field. By comparison, the configurational changes for the SCB_dumbbell system as a function of Wi number are substantially different from those for the linear and SCB_uniform systems, a difference that arises from the intrinsically distinct molecular architecture of the SCB_dumbbell polymer. As mentioned earlier, representing the branched backbone as block “A” and the bare backbone as block “B”, the E

DOI: 10.1021/acs.macromol.7b00544 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules SCB_dumbbell polymer can be represented as an “A−B−A” type triblock copolymer. The two “A” blocks located at either end of the two chain ends are believed to have a tightly coiled compact structure due to the random motions exerted by the short branches. As such, the compact “A” blocks can be considered as superbeads with a large hydrodynamic bead friction, producing an effective tensile force on the “B” block positioned in the middle of the chain backbone, thus leading to a large-scale chain stretch. This physical aspect is consistently mirrored in the results of calculations on the SCB_dumbbell. Compared to the linear and SCB_uniform polymers, the SCB_dumbbell exhibits a considerably larger proportion of the stretched and dumbbell configurations, with the coil configuration occupying a much smaller percentage. We note that in the intermediate-to-high flow regime, the branched backbone also unravels, which contributes to a further increase in the stretched portion. These structural features of the SCB_dumbbell polymer having a largely extended chain conformation with the high stretched and dumbbell portions are manifested in the results of ⟨R̃ ete2⟩ and the PDF as a function of |R̃ ete| (Figure 3). Furthermore, concerning the rheological properties, such an increase in the extended chain structure, together with the chain alignment to the flow direction, results in a high shearthinning behavior for the SCB_dumbbell polymer with respect to the imposed flow strength, as shown in Figure 2. In contrast, an important observation is emerged for the SCB_center polymer (which is a type of “B−A−B” type of triblock copolymer). Over the whole range of applied flow strengths, while the coil portion of the SCB_center polymer exhibits a similar tendency to that of the linear and SCB_uniform polymers, there is a significant decrease in the proportions of the stretched and dumbbell configurations and a remarkable increase in the fold portion. This large amount of the fold configuration is due to the strong dynamical role of the short branches located around the backbone center. The highly mobile short branches tend to compact the middle part of the chain backbone, making it less amenable to structural deformation induced by the applied flow; in contrast, the (two) sides of the backbone further from the middle are likely to be more susceptible to chain stretch and alignment to the flow direction. The contraction of the middle section, which is connected to the two stretched side parts along the backbone, results in a high proportion of the fold configuration for the SCB_center polymer in the intermediate-to-high range of flow strengths. As directly linked with the intrinsic molecular mechanisms in polymer dynamics, these unique configurational characteristics of the SCB_center polymer (very different from those of the linear and other types of the SCB_polymer) are supposed to make distinctive structural and rheological behaviors for the SCB_center polymeric materials under an external flow field, e.g., the lower shear-thinning behaviors in viscosity and the first normal-stress coefficient (Figure 2) and the overall compact chain structures measured in terms of the chain end-to-end distance and the corresponding PDF (Figure 3). In this respect, we closely examined the characteristic molecular mechanisms of the SCB_dumbbell polymer in response to the applied shear. We analyzed the contributions of the bare and branched backbone parts to the structural and rheological properties. Figure 5a plots the variation of the root mean-square chain end-to-end distance, ⟨R̃ ete2⟩1/2, with the imposed flow strength. For Wi < 30, while there is a steep increase in ⟨R̃ ete2⟩1/2 for the bare backbone with increasing

Figure 5. (a) Root mean-square chain end-to-end distance ⟨R̃ ete2⟩1/2 of the whole backbone (black filled circles), the bare backbone part (black open circles), and the branched backbone part (orange open circles). The vertical dotted line specifies two characteristic flow regimes: in the first regime (Wi < 30), only the bare backbone stretches, whereas in the second regime (Wi > 30), both the bare and branched backbone parts stretch. The horizontal arrows indicate the proper y-axis of data points that have same color. (b) Shear stress, σ̃xy, contributions of the whole chain (black filled circles), the bare backbone (black open circles), and the branched backbone (orange open circles). The shear stress is made dimensionless with σxỹ = σxy kBT /K3 . The error bars are smaller than the size of the symbols unless otherwise specified.

shear rate, very little change in ⟨R̃ ete2⟩1/2 is seen for the branched backbone, suggesting a rather compact configuration of the branched part, similar to those observed under equilibrium conditions. Therefore, the overall chain stretch of the SCB_dumbbell polymer under shear in this flow regime is mainly associated with the extension of the bare backbone rather than the branched part. This behavior of the SCB_dumbbell system is in good agreement with the result from the brightness analysis in the corresponding flow regime (Figure 4): the stretched bare backbone located in the middle section and the two tightly coiled branched backbones located near the backbone ends together give rise to a high proportion of the dumbbell configuration. In the flow range of Wi > 30, ⟨R̃ ete2⟩1/2 for the bare backbone increases rather slowly with increasing shear rate, reaching a plateau at Wi ≈ 200. In contrast, the branched backbone begins to stretch considerably with increasing flow field, approaching a plateau region at much higher field strengths (i.e., Wi ≥ 1000). Therefore, in this intermediate-to-high flow regime, the overall chain extension of the SCB_dumbbell polymer arises from the contributions of both the stretched bare- and branched backbone sections. This structural characteristic of the SCB_dumbbell system are clearly reflected in the brightness analysis shown in Figure 4 where it is seen that at Wi > 30 the dumbbell portion rapidly decreases F

DOI: 10.1021/acs.macromol.7b00544 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

Figure 6. (a) Schematic illustrations for the fundamental molecular mechanisms underlying the rotational and tumbling dynamics of an individual chain for the linear and the SCB polymer systems. (b) Temporal variation of the chain end-to-end distance |R̃ ete| and the chain orientation angle, θ, with respect to the flow direction as a function of time t, for a selected representative chain for each polymer system at Wi = 1000. The black solid line represents the |R̃ ete| result and the orange circle the θ result. The vertical lines with the Roman numerals (i)−(iv) indicate the instantaneous chain configurations corresponding to those specified in (a) at a certain time during chain rotation and tumbling. The time is made dimensionless with t ̃ = t(ζ/4K), where ζ is the bead friction coefficient.

and is largely converted into the stretched portion with increasing flow strength. The results from a similar analysis of the shear stress σxy are presented in Figure 5b. For Wi < 30, most of the total shear stress of the SCB_dumbbell system is from the bare backbone with only a minor contribution from the branched part. [Note that the trend of increasing σxy for the branched backbone is largely associated with the chain orientation (the xy-orientational correlation) in the flow direction without any significant chain stretching, as shown in Figure 5a.] This negligible stress contribution from the branched backbone is again ascribed to its compact structure being less vulnerable to deformation in response to the imposed flow because of the strong resistance against the field

exerted by the highly mobile short chains. In addition, the largescale hydrodynamic friction carried by each of the two branched backbone parts located around the backbone ends creates a tensile force pulling the middle part of backbone in opposite directions, resulting in orientation in the flow direction and increased stretching of the bare backbone, thus increasing the total stress of system. As the flow strength increases further (i.e., Wi > 30), the contribution of the branched backbone to the total stress, though still smaller than that of the bare backbone, becomes more and more substantial, i.e., roughly 25% of the total at the highest flow strength (Wi = 6000) applied here. G

DOI: 10.1021/acs.macromol.7b00544 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

structure of the backbone center and the stretched configuration of the two outside bare backbone parts for the SCB_center polymer results in a hairpin-like fold (instead of an S-shaped) configuration of the chain during the tumbling process [(ii) and (iii) for SCB_center in Figure 6b, where θ oscillates significantly with a rather collapsed chain structure in terms of |R̃ ete|]. Additionally, we expect a dynamical situation that because of the stiffer (and thus dynamically slower) chain structure in the middle of the backbone, the freely moving chain ends undergoing Brownian fluctuations often result in a negative orientation of the end-to-end vector, but without causing an actual chain tumbling cycle. To visualize the underlying molecular mechanisms of the tumbling dynamics of an individual chain, the readers are referred to the Supporting Information, in which movies of the linear and three representative SCB polymers under shear are presented.

Lastly, we examine the rotational and tumbling dynamics of individual chains for each polymeric system under shear flow. Knowledge of such molecular mechanisms is crucial to understanding the fundamental physics behind the apparently complex rheological properties and phenomena displayed by various polymeric materials with different molecular architectures. On the basis of the structural and rheological characteristics revealed in this study, in Figure 6a, we illustrate the representative rotational and tumbling mechanism characteristic of each polymer system schematically. The Roman numerals (i−iv) and the vertical lines in Figures 6a and 6b indicate the characteristic molecular configuration of the chains for each polymer system at a certain time during the tumbling process. In addition, during the chain tumbling cycle, |R̃ ete| varies from a collapsed coiled configuration to a nearly stretched configuration, and the chain orientation angle θ measured with respect to the flow direction (θ = 0°) oscillates dramatically from 90° to −90°, as shown in Figure 6b. First, as for the linear polymer, we start from the most stable state of the chain conformation with orientation in the flow (x-)direction [indicated as (i) for the linear polymer in Figure 6a; correspondingly, Figure 6b shows the fully extended chain configuration with a large value of |R̃ ete| and a small value of θ]. When one or two (free) chain ends come into the shear xzplane layered along the velocity gradient (y-)direction, which results in the directional change of the chain end-to-end vector from (favorable) positive to (unfavorable) negative orientation with respect to the flow direction [indicated as (ii) in Figure 6a], the chain undergoes a rotational movement that has an Sshaped or a hairpin-like conformation due to the different flow strengths along the velocity gradient (y-)direction. Eventually, this leads to an end-over-end tumbling dynamics [(ii) → (iii) in Figure 6a, for which a fast drop in |R̃ ete| and a change in the sign of θ occur, as shown in Figure 6b].22−25 After that the chain end-to-end vector comes back into the region of stable positive orientation and aligns with the flow direction until the next rotation [(iii) → (iv) in Figure 6a; |R̃ ete| becomes larger, and θ becomes smaller during this transition, as shown in Figure 6b]. The SCB_uniform polymer undergoes rotation mechanism similar to that of the linear polymer. However, the fast random motion of short branches in all directions results in a rather collapsed chain structure during the tumbling process [(ii) for SCB_uniform in Figure 6a; correspondingly, the value of |R̃ ete| in Figure 6b appears relatively small for SCB_uniform compared to that of the linear polymer, and also shown is a certain plateau region of θ]. In addition, these random movements of the short branches are considered to frequently disturb the stable chain orientation, resulting in an unstable negative chain orientation and promoting the chain tumbling dynamics [indicated as (iii) in Figure 6b, where frequent changes in the sign of θ occur, maintaining a rather collapsed chain configuration in terms of |R̃ ete|]. At the same time, the random motion of the short branches is supposed to interrupt, to some extent, the chain rotation during the tumbling process. For the SCB_dumbbell polymer, we also expect an S-shaped or a hairpin-like tumbling behavior of the chain. However, the compact molecular structure of the two branched backbones near the backbone ends results in a stretched dumbbell configuration of the chain during the tumbling process [(iii) for SCB_dumbbell in Figure 6b, where a relatively large value of | R̃ ete| is seen]. In the case of SCB_center, the situation is very different because of the location of the branched backbone in the middle of the backbone. We believe that the compact

4. CONCLUSIONS In this work, we carried out a comprehensive and systematic study on the effect of short-chain branching on the structural and rheological properties of polymeric materials using bead− spring Brownian dynamics simulations for various types of short-chain branched polymers. Our results show that in comparison with linear polymers, the SCB polymers with uniform or random distribution of short branches have a compact and less deformed molecular structure in response to an imposed flow field. This resistive structural characteristic of the SCB polymer against the flow is found to increase with increasing branch density, irrespective of the total molecular weight or backbone length. The physical origin behind this behavior lies in the dynamical role of the short branches via their intrinsically fast random Brownian kinetics, which correspond to their very short characteristic relaxation time. These highly mobile short branches moving in all directions, essentially unaffected by the flow field, reduce the degree of chain deformation and alignment with respect to the flow. This structural rigidity against an external field leads to a less shearthinning behavior in the shear viscosity and the first normal stress coefficient for the SCB_uniform system compared with the linear analogue. In addition to the above aspects associated with the compact molecular structures, according to our brightness analysis, the SCB_uniform polymer has a distinct PDF shape for each molecular configuration, being more extended and shifted to the higher Wi numbers compared to the corresponding linear polymer. To explore the effect of the molecular architecture on the structural and rheological behaviors of the SCB system, we studied various types of SCB polymers with different distributions of short branches along the backbone. Compared with the linear and SCB_uniform polymers, the SCB_center polymer undergoes a significantly lower chain deformation in response to the applied flow field. This feature is consistently linked with the less shear-thinning behavior of the SCB_center system. More importantly, the brightness analysis indicates that the SCB_center polymer possesses a high proportion of the fold configuration, accompanied by a significant decrease in the stretched and dumbbell configurations over the whole range of flow strengths. This aspect inherent to the SCB_center polymer leads to distinctive chain rotation and tumbling mechanisms under shear, i.e., a hairpin-like configuration during tumbling process instead of an S-shaped configuration in the case of the linear and SCB_uniform polymers. In sharp contrast, the SCB_dumbbell polymer chains undergo signifiH

DOI: 10.1021/acs.macromol.7b00544 Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

■

ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea (2016R1D1A1A09916994, 10067706, and 2016R1C1B1014515). The simulations of this work were carried out on the computational resources of the UNIST Supercomputing Center.

cantly greater structural deformations (stretching and alignment) at a given flow strength than the corresponding linear and SCB_uniform polymers. This is supposed to arise from a tensile force (which pulls out the middle part of the backbone in opposite directions) exerted by the two branched backbone parts (armed with a large-scale hydrodynamic friction) separately located around the two ends of the backbone. Consistently, the SCB_dumbbell polymer exhibits a relatively weaker rotation peak and stronger stretch peak in the probability distribution with respect to the chain end-to-end distance, indicative of the flexible structural characteristic vulnerable to deformation by the flow. In conjunction with this, the SCB_dumbbell polymer has a high shear-thinning behavior compared to the linear analogue. In line with these structural and rheological aspects, in comparison with the linear and SCB_uniform polymers, the SCB_dumbbell polymer contains large portions of the stretched and dumbbell configurations and a significantly low content of the coil configuration over the whole range of flow strengths; this characteristic is consistently mirrored by the chain rotation and tumbling mechanisms. Considering that the backbone with or without short branches has distinct structural and dynamical characteristics, we propose that each SCB polymer can be modeled as a block copolymer. This representation may be applied in the physical analysis of the structural and dynamical behaviors of various multiblock copolymers with different molecular architectures, such as star, ring, and H-shaped,26 or biological molecules.27−29 Although, currently, the findings in this study cannot be compared against experiment because of the lack of synthesis methods to prepare the well-defined short-branched polymers studied here, we expect these polymers to have practical applications in the near future, especially considering the rapid experimental advances in catalysts and chemical synthesis that enable a precise control of the branch density and branching architecture.

■

■

REFERENCES

(1) Beris, A. N.; Edwards, B. J. Thermodynamics of Flowing Systems with Internal Microstructure; Oxford University Press: New York, 1994. (2) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986. (3) McLeish, T. C. B. Tube theory of entangled polymer dynamics. Adv. Phys. 2002, 51, 1379−1527. (4) Mai, D. J.; Marciel, A. B.; Sing, C. E.; Schroeder, C. M. Topologycontrolled relaxation dynamics of single branched polymers. ACS Macro Lett. 2015, 4, 446−452. (5) Bird, R. B.; Curtiss, C. F.; Armstrong, R. C.; Hassager, O. Dynamics of Polymeric Liquids, Kinetic Theory, 2nd ed.; WileyInterscience: New York, 1987; Vol. 2. (6) Kim, J. M.; Baig, C. Communication: Role of short chain branching in polymer structure and dynamics. J. Chem. Phys. 2016, 144, 081101. (7) Baig, C.; Mavrantzas, V. G.; Kröger, M. Flow effects on melt structure and entanglement network of linear polymers: Results from a nonequilibrium molecular dynamics simulation study of polyethylene melt in steady shear. Macromolecules 2010, 43, 6886−6902. (8) The “brfac” is an adjustable parameter that controls the characteristic relaxation time (or mobility) of short branches, and it is thus generally dependent on the chemical composition for branches, the branch length (below approximately the Kuhn length), and also partly the system condition. Therefore, the value of “brfac” is determined indirectly by comparing between the BD and atomistic NEMD results for representative structural or dynamical results. In our previous work (ref 6), there was a numerical error in the integration part of evolution equation for short branches, which incorrectly underestimated the effect of short branches. After fixing the error, the correct value of “brfac” was found to be equal to 0.05 instead of 0.003 of the previous work. (9) It is generally known that a coarse-grained bead−spring linear chain (unless the number of beads is too small) with HI exhibits a less deformed structure and a less shear thinning (with a notable reduction in the viscosity at low shear rates) than the one without HI in response to the same flow strength under shear or elongational flow, which is ascribed to the reduced effective hydrodynamic drag on the chain beads with HI between beads (see, e.g., refs 10−12). Similar to the linear polymer, such HI effects are expected to occur for the SCB polymers, irrespective of variable branch density and branching architecture. (10) Lopez Cascales, J. J.; Garcia de la Torre, J. Hydrodynamic interaction effects on the conformation of flexible chains in simple shear flow. Macromolecules 1990, 23, 809−813. (11) (a) Fetsko, S. W.; Cummings, P. T. Brownian dynamics simulation of bead-spring chain models for dilute polymer solutions in elongational flow. J. Rheol. 1995, 39, 285−299. (b) Petera, D.; Muthukumar, M. Brownian dynamics simulation of bead-rod chains under shear with hydrodynamic interaction. J. Chem. Phys. 1999, 111, 7614−7623. (12) Sendner, C.; Netz, R. R. Single flexible and semiflexible polymers at high shear: Non-monotonic and non-universal stretching response. Eur. Phys. J. E: Soft Matter Biol. Phys. 2009, 30, 75−81. (13) Kim, J. M.; Stephanou, P. S.; Edwards, B. J.; Khomami, B. A mean-field anisotropic diffusion model for unentangled polymeric liquids and semi-dilute solutions: Model development and comparison with experimental and simulation data. J. Non-Newtonian Fluid Mech. 2011, 166, 593−606.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b00544. Schemes S1 and S2; Figures S1−S5 (PDF) Movie S1: tumbling dynamics of linear polymer (AVI) Movie S2: tumbling dynamics of SCB_uniform polymer (AVI) Movie S3: tumbling dynamics of SCB_dumbbell polymer (AVI) Movie S4: tumbling dynamics of SCB_center polymer (AVI)

■

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Ph +82-52-217-2538; Fax +82-52217-2649 (C.B.). ORCID

Chunggi Baig: 0000-0001-8278-8804 Author Contributions

S.H.J. and J.M.K. contributed equally to this work. Notes

The authors declare no competing financial interest. I

DOI: 10.1021/acs.macromol.7b00544 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules (14) Doyle, P. S.; Shaqfeh, E. S. G; Gast, A. P. Dynamic simulation of freely draining flexible polymers in steady linear flows. J. Fluid Mech. 1997, 334, 251−291. (15) Schroeder, C. M.; Teixeira, R. E.; Shaqfeh, E. S. G.; Chu, S. Dynamics of DNA in the flow-gradient plane of steady shear flow: observations and simulations. Macromolecules 2005, 38, 1967−1978. (16) The relatively small effect of branch density on ⟨R̃ ete2⟩ at high Wi numbers (i.e., Wi > 500) is mainly associated with the fast chain rotational and tumbling dynamics, which lead to the convergence of ⟨R̃ ete2⟩ for the backbone to a rather similar value for all the SCB systems. (17) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of Polymeric Liquids, Fluid Mechanics, 2nd ed.; Wiley-Interscience: New York, 1987; Vol. 1. (18) Under equilibrium conditions, ⟨Rg2⟩1/2/⟨Rg2⟩linear1/2 is computed as 1.13 ± 0.3, 1.79 ± 0.4, and 0.92 ± 0.3 for the SCB_uniform, SCB_dumbbell, and SCB_center polymers, respectively. As consistent with the result of ⟨R̃ ete2⟩ under shear flow in Figure 3, the SCB_center and SCB_uniform polymer systems exhibit more compact structures than the corresponding linear polymer, whereas the SCB_dumbbell polymer possesses more stretched configurations than the linear polymer. (19) Baig, C.; Edwards, B. J.; Keffer, D. J.; Cochran, H. D.; Harmandaris, V. A. Rheological and structural studies of linear polyethylene melts under planar elongational flow using nonequilibrium molecular dynamics simulations. J. Chem. Phys. 2006, 124, 084902. (20) Kim, J. M.; Keffer, D. J.; Kröger, M.; Edwards, B. J. Rheological and entanglement characteristics of linear-chain polyethylene liquids in planar Couette and planar elongational flows. J. Non-Newtonian Fluid Mech. 2008, 152, 168−183. (21) Venkataramani, V.; Sureshkumar, R.; Khomami, B. Coarsegrained modeling of macromolecular solutions using a configurationbased approach. J. Rheol. 2008, 52, 1143−1177. (22) Kim, J. M.; Edwards, B. J.; Keffer, D. J.; Khomami, B. Dynamics of individual molecules of linear polyethylene liquids under shear: Atomistic simulation and comparison with a free-draining bead-rod chain. J. Rheol. 2010, 54, 283−310. (23) Teixeira, R. E.; Babcock, H. P.; Shaqfeh, E. S. G.; Chu, S. Shear thinning and tumbling dynamics of single polymers in the flowgradient plane. Macromolecules 2005, 38, 581−592. (24) Xu, Z.; Kim, S.; de Pablo, J. J. Anisotropic friction and excluded volume effects in freely jointed bead-rod polymer chain models. J. Chem. Phys. 1994, 101, 5293−5304. (25) Schroeder, C. M.; Teixeira, R. E.; Shaqfeh, E. S. G.; Chu, S. Characteristic periodic motion of polymers in shear flow. Phys. Rev. Lett. 2005, 95, 018301. (26) Bates, F. S.; Hillmyer, M. A.; Lodge, T. P.; Bates, C. M.; Delaney, K. T.; Fredrickson, G. H. Multiblock polymers: Panacea or Pandora’s box? Science 2012, 336, 434−440. (27) Alemdaroglu, F. E.; Safak, M.; Wang, J.; Berger, R.; Herrmann, A. DNA multiblock copolymers. Chem. Commun. 2007, 1358−1359. (28) Hao, Y.; He, J.; Zhang, M.; Tao, Y.; Liu, J.; Ni, P. Synthesis and characterization of novel brush copolymers with biodegradable polyphosphoester side chains for gene delivery. J. Polym. Sci., Part A: Polym. Chem. 2013, 51, 2150−2160. (29) Hao, Y.; He, J.; Li, S.; Liu, J.; Zhang, M.; Ni, P. Synthesis of an acid-cleavable and fluorescent amphiphilic block copolymer as a combined delivery vector of DNA and doxorubicin. J. Mater. Chem. B 2014, 2, 4237−4249.

J

DOI: 10.1021/acs.macromol.7b00544 Macromolecules XXXX, XXX, XXX−XXX