Article pubs.acs.org/Macromolecules
Tube Diameter of Stretched and Compressed Permanently Entangled Polymers Jian Qin,* Jungseob So, and Scott T. Milner Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States ABSTRACT: Entangled ring polymers behave like gels. By applying uniaxial compression and stretching to such gels initially equilibrated with molecular rebridging Monte Carlo moves, and using isoconfigurational ensemble simulations to explore the tube confinement, we studied the strain dependence of the tube contour and tube diameter. It is found that the tube contour length increases with both compression and stretching, the tube diameter increases with strain monotonically, and the strain dependences of both tube contour length and tube segment orientation are well described by the independent alignment approximation (IAA). For strains ranging between 0.5 and 2.0, fractional changes of the tube contour length and the tube diameter are less than 10%. The orientation dependence of the tube diameter is also analyzed and found to be weak. These results should prove useful for constructing theories of gel elasticity.
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nonaffinely: a(λα) = λα1/2a(1), where α = x, y, or z is the Cartesian component index, λα the corresponding strain component, and a(1) the tube diameter in the nondeformed state. The same result was later rederived in a more heuristic way by Rubinstein and Panyukov (RP)12 and was used to construct a molecular theory for network elasticity.13 More recently, the predicted strain dependence of the HS and RP theories was incorporated into the double tube model14 by Mergell and Everaers (ME) to describe the collective confining effects caused by molecular entanglements and cross-links in networks. Furthermore, Svaneborg, Grest, and Everaers15,16 conducted molecular dynamics simulations for networks created with both end-linking and random cross-linking, studied the strain dependence of the confining effects on both cross-links and on bead separation vectors, and found that the results can be well described by the ME model. Recently, we have developed a noninvasive method for studying various statistical properties of tubes, the isoconfiguration ensemble simulation, and have used it to study the effects of free surface on tube diameter.17 In this work, we will apply the same method to deformed ring polymer melts which have been pre-equilibrated to study how various statistical properties of tube vary with strain. Studying rings instead of linear polymers offer several advantages: first, the entangled ring polymers behaves like a network but do not have cross-links; second, molecular entanglements, once formed, are permanent; third, the absence of free ends excluded the unwanted molecular relaxation mechanisms, CR (constraint release)18,19 and CLF (contour length fluctuations).1,20,21
INTRODUCTION The confining tube is the key concept in modern theories of polymer rheology for dense melts and concentrated solutions.1 It accounts for the collective constraints imposed upon a test chain by its many neighboring molecules (entanglement), by replacing them with a tube-like confining potential. The confining potential effectively restricts the motion of the test molecule into a tube-like region. The tube diameter a, which is a material property, measures the strength of confinement and is one of the two input parameters needed by molecular theories of polymer rheology. As an emergent length scale, the tube diameter should depend on microscopic properties of materials. However, since a predictive theory of this length scale is still absent, practically the tube diameter is obtained by comparing experimental measurements, e.g., plateau modulus, against rheology theory predictions. At the empirical level, progress has been made to correlate experimentally inferred tube diameters with various material properties,2,3 e.g., density and chain flexibility. This correlation was also confirmed by the simulation studies of Everaers and Kremer et al.,4 in which a chain-shrinking algorithm is introduced to identify the tube diameter from the simulation snapshots. Following this work, several closely related algorithms for finding the tube diameter have been proposed.5−8 These studies suggest that computer simulation can be a useful tool for investigating the tube diameter. In this work, we are interested in how the tube diameter varies with external conditions, such as flow and deformation. This question is important, e.g., for constructing a microscopic theory for the network elasticity in the nonlinear deformation regime. The first theory for predicting the strain dependence of tube diameter is the Heinrich−Straube (HS) theory.9−11 It predicts that tube diameter changes anisotropically and © 2012 American Chemical Society
Received: August 31, 2012 Revised: December 3, 2012 Published: December 12, 2012 9816
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imation is normally used to describe how tube sections deform right after the application of a step strain. It treats the tube as connected straight segments and assumes the deformation of each segment can be described by the macroscopic deformation tensor. For linear polymers, the deformed tube can restore its equilibrium contour length within a Rouse time. The IAA further assumes that this restoring procedure occurs along the contour of the deformed tube, so after the Rouse time, the deformed tube segment orientations can still be described by the affine deformation theory. For ring polymers, this restoring procedure is absent, so the IAA is identical to the affine deformation approximation. Although the IAA is the simplest approximation for tube deformation, it has proven to be surprisingly successful for describing viscoelastic behaviors of polymer melts, even in the nonlinear regime (see examples given in ref 1, e.g., the nonlinear stress relaxation curve given by Figure 7.16). Here we examine the IAA predictions for how the length of tube primitive path or centerline and the orientation distribution of the tube segments vary with deformation. Throughout this work, we are interested in uniaxial volumeconserving deformations, for which the deformation tensor has the form
The paper is organized as follows: in the next section, we explain the model, simulation detail, deformation protocol, and analysis methodology; in the third section, we derive the predictions of the affine deformation assumption as applied to individual tube segments; in the fourth section, we present the main results; the last section is the summary.
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SIMULATION AND METHODOLOGY The system we studied contains three ring polymers inside a tetragonal cell, with periodic boundary condition applied along each direction. Similar to our previous work,17 polymers are modeled as bead−spring rings. Bonded beads interact via a harmonic potential with a constant rest length, and nonbonded beads interact via the shifted purely repulsive L-J pair potential. The interaction parameters we used are the same as in our previous work,17 for which the polymers can be thought as essentially uncrossable. Each polymer has N = 800 beads, and the simulation cell dimension is chosen to yield a bead number density of ρ = 0.7σ, where σ is the length unit for in the L-J pair potential. This density has been shown to produce the melt behavior of polymers.22 We performed four independent sets of simulations, each starting with a different initial configuration, but all following the same deformation protocol composed of three stages. In the first stage, the systems are equilibrated using Monte Carlo methods. Various Monte Carlo moves including bead random displacement moves are used to equilibrate the system pressure initially, and hybrid MC/MD (molecular dynamics) moves are used to equilibrate various kinds of local modes. More importantly, several ring rebridging moves were used to allow the rings to cross, which are essential for equilibrating the system entanglement topology. The details of these ring rebridging moves were explained in our previous work.23 In the second stage, the equilibrated systems are deformed (stretched or compressed) affinely and uniaxially, in a sequence of steps. At each deformation step, the systems are equilibrated with MD simulation for a time much longer than the Rouse time of the ring, before the next deformation is applied. Since the ring rebridging moves are turned off during this deformation process, the entanglement topology is fixed. In our simulation, the maximum stretching ratio is 2.0 and the maximum compression ratio is 0.5. In the third stage, isoconfigurational ensemble (“ICE”) averaging is applied to the deformed systems. This averaging was used in our previous work.17 Essentially, we run multiple MD simulations, all starting from the same initial condition, but each with a different velocity distribution, to allow the beads to explore different portions of the tube confinement. The results of isoconfigurational ensemble averaging are used to obtained tube contour, tube diameters, and their dependence on the deformation rate. Following the previous study, we set the number of independent MD runs (trajectories) to be 100 and recorded 250 configuration frames along each trajectory. For the parameters we used, the frame rate is 4 per τe, so each recorded isoconfiguration trajectory is 62.5τe long.
⎛1/ λ 0 0⎞ ⎜ ⎟ E = ⎜0 1/ λ 0 ⎟ ⎜ ⎟ ⎝0 0 λ⎠
(1)
where λ is the deformation rate along the z direction and is a positive number greater or less than 1, depending on whether the system is stretched or compressed. Within IAA and under the application of E, an arbitrary unit vector u is deformed to a new vector E·u, which has a length |E·u| = (λ−1 + (λ2 − λ−1) cos2 β)1/2, where cos β ≡ uz is the z component of vector u. We first consider how the tube contour length L(λ) varies with deformation. It is calculated by the summation of the lengths of the deformed tube segments, |E·u|. Since the tube segments are oriented randomly in the undeformed state, the ratio of L(λ) and L(1) equals the average of |E·u| with respect to isotropic orientations of u (see eqs 7.103, 7.104, and 7.139 of ref 1), i.e. L(λ)/L(1) = ⟨|E·u|⟩ =
1⎛ arccos λ 3/2 ⎞ ⎜λ + ⎟ 2⎝ (λ − λ 4)1/2 ⎠
(2)
This function has a minimum at λ = 1 and can be approximated by 1 + 2(λ − 1)2/5 for λ values close to 1. Its full shape will be given in the next section, when simulation results are presented. We then consider the variation of tube segment orientation distribution. For uniaxial deformation, the distribution only depends on the polar angle θ of the deformed vector E·u, i.e., the angle between the vector E·u and the z-axis. θ is related to the polar angle β of undeformed vector u by (see Figure 1) cos θ =
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IAA PREDICTIONS Before presenting the simulation results, we summarize the predictions of the simplest assumption we can make regarding how tube deforms, the independent alignment approximation (IAA) (section 7.7.2 in ref 1), which is built upon the affine deformation approximation. The affine deformation approx-
λ cos β |E · u|
(3)
This implies that cos θ always has the same sign as cos β and that cos θ approaches 0 as λ3/2 cos β when cos β vanishes. In the undeformed state, the unit vector orientation is isotropic, so the distribution of cos β is p(cos β) = 1/2. In the deformed state, the distribution of cos θ, p(cos θ), is related to p(cos β) by (see eq 7.111 in ref 1) 9817
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smoother the primitive path. Here, we fix the value of τa and examine how the tube contour varies with strain. The tube contours of a single polymer ring at strains λ = 0.5, 0.66, 1.0, 1.52, and 2.0 are shown in Figure 2. From left to right, the strain values are increased. The curve in the middle (blue) is the undeformed tube contour. The red curves to its left and right are tube contours for the compressed and stretched cases. The other blue curves are affine deformation theory predictions obtained by applying the tensor E to the undeformed tube contour. These visualizations suggest that, except for the λ = 2.0 case, the deformation of tube contour for strains between 0.5 and 2.0 can be well approximated with the affine deformation theory. We then examine the tube contour length L(λ), which is calculated by summing up bond lengths of the tube contour. The result of L(λ)/L(1) for different strain values are shown in Figure 3 (curves with error bars). The two curves are obtained
Figure 1. Deformation of a unit vector u by a uniaxial deformation tensor E.
p(cos θ) d cos θ = C|E·u|p(cos β) d cos β
(4)
which states that the probability of the deformed vector pointing to θ direction must equal that of the undeformed unit vector pointing to β direction. Here |E·u| accounts for the vector length change, which implicitly changes the fraction of tube segment pointing to θ direction, and C is the normalization constant. Substituting the differential d cos θ/d cos β = |E·u|−3 derived from eq 3 into eq 4, we get p(cos θ ) =
C C λ4 |E · u| 4 = 2 2 2(λ 3 − (λ 3 − 1) cos2 θ )2
(5)
The normalization constant C equals C = 2/(λ + arcsh((λ3 − 1)1/2)/(λ(λ3 − 1))1/2). For small strains, the distribution can be approximated by 1/2 + (3 cos2 θ − 1)(λ − 1).
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RESULTS We present in this section the isoconfigurational ensemble simulation results, including the shape of tube contour, the contour length, the tangent orientation distribution of tube contour, the tube diameter, and the correlation between tube diameter and tube orientation. In all cases, we focus on the deformation dependence. When possible, the results are compared to IAA theory predictions. Shape and Length of Primitive Path. Following our previous work,17 we define the tube primitive path as the average of bead positions recorded from all 100 isoconfiguration ensemble trajectories, each of which contains 250 frames. The number of frames used in the average defines a time τa. As shown previously,17 the shape of the primitive path weakly depends on the choice of τa. The higher the value of τa, the
Figure 3. Strain dependence of the tube contour length. Lines with error bars are simulation results; blue for τa = 50 frames and red for τa = 250 frames. Solid curve is the IAA theory prediction.
from τa = 50 (blue) frames and 250 (red) frames. The data values are calculated as the average of 12 independent ratios L(λ)/L(1), i.e., 3 molecules in all 4 simulation sets. The magnitudes of error bars are standard deviations divided by √12. The green solid curve in Figure 3 is the prediction of IAA theory, eq 2, which describes qualitatively the trend of simulation results but is only quantitatively consistent with the case of τa = 50 frame. Both the data and theory suggest that the contour length increases upon deformation. (It can be
Figure 2. Strain dependence of the tube contour. From left to right: λ = 0.5, 0.66, 1.0, 1.52, and 2.0. Results from isoconfigurational simulation (red) and from the affine deformation prediction (blue) are show. The tube contours are defined using data from all 250 frames (τa = 250 frames). 9818
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approximation to the tube diameter. More quantitatively, for each bead cloud, we can evaluate the radius of gyration tensor, calculate its eigenvalues, and use the average of the two smaller eigenvalues as an estimate to the tube diameter. Details are given in ref 17. This method yields one value of tube diameter from each bead in the system. The values from all 2400 beads in all 4 independent simulation sets can be used to generate an distribution of tube diameters, which measures the spatial inhomogeneity of the tube. The distributions we obtained for the maximally compressed (λ = 0.5) case, the undeformed (λ = 1.0) case, and the maximally stretched (λ = 2.0) case are shown in Figure 7. The figure on the left and right are for τa = 50 (12.5τe) frames and 250 (62.5τe) frames, respectively. Because of the limited sample size, the distribution is a bit noisy, but it is clear that as τa increases, the distribution becomes broader, and that as strain increases, the distribution shifts toward right (larger tube diameter). To alleviate the dependence on τa, we calculated the average tube diameters at different strains and normalized them by the undeformed value. The results versus strain are plotted in Figure 8. The data values in the figures are obtained as follows: we first computed the ratios a(λ)/a(1) for each bead, then averaged the ratios for all beads, all molecules, and all sets. The error bar is calculated by the standard deviation, normalized by the square root of the number of independent values, which we choose as 10 × 3 × 4 (10 entanglement strands per chain for Ne ≃ 80, times 3 molecules, times 4 data sets). Two curves in Figure 8 are for τa = 250 frames (red) and τa = 50 frames (blue). The results suggest that the dependence on τa, visible in Figure 7, is nearly absent for this quantity and that the tube diameter increases with λ. For λ = 0.5 and 2.0, the changes are of order 10%. This is definitely weaker than that predicted by the HS11 and RP12 theories, which predicts a(λ)/ a(1) equal to(1/2)1/2 and √2 for λ = 0.5 and λ = 2 if the orientation dependence is neglected (see the next section for more discussion on the subtle difference between the two). Tube Diameter versus Orientation. In Figure 9, we plot the tube diameter versus the tube segment orientation (τa = 250 frames). The diameter is computed using the method of the previous section, and the orientation is given by the cosine of tube tangent polar angle. To improve statistics, we also combined results from positive and negative cosine values. The three panels from left to right are results for λ = 0.5, 1.0, and 2.0, respectively. In each panel, the cloud of green dots are diameter−orientation pairs found from all beads. The red bars indicate the standard deviation for cloud points in different ranges of orientations. The denser the cloud, the more beads have the corresponding tube diameter and orientation. It is clear that the tube segments tend to align along directions perpendicular to the z-axis in the compression case and that they tend to align with the z-axis in the stretching cases. But the
shown that for volume-conserving deformations the IAA theory always predict the contour length to increase; see eq 7.105 in ref 1.) For strains up to 0.5 and 2.0, the contour length is increased by about 10%. The results obtained from two different τa values are consistent but do not agree quantitatively. The origin of this unwanted dependence on τa will be discussed in the section on using cloud peak to define the primitive path. There we also introduce an alternative method to define the tube centerline, which can be used to alleviate this effect. Tangent Distribution. We present results on the distribution of tube tangent vector orientations in this section. Since we focus on uniaxial deformations in this study, the orientation is represented by the cosine of polar angle θ of the tangent vector. For the undeformed system, the tangent vector orientation is random, so the distribution of cos θ is flat. In systems to which stretch is applied, as λ increases, the tangent vectors tend to be aligned with the z-axis, so the distribution should have two peaks near values cos θ ± 1. For systems that have been compressed, as λ decreases, the tangent vectors tend to be aligned within the xy plane, so the distribution should have one peak near cos θ = 0. The λ dependence of the distribution predicted by the IAA theory is given by eq 5. The results from the isoconfigurational ensemble simulations are shown in Figures 4, 5, and 6 (solid lines; the results shown
Figure 4. Tangent vector distribution in the undeformed case (τa = 250 frames).
are for τa = 250 frames; those for τa = 50 frames are similar). Different plots are results for different strain values. For each strain, we have used the tube contour data of all 3 rings in all 4 independent data sets to generate the distribution. The dashed lines are predictions of the IAA theory, which describes the distribution of the tube tangents nearly quantitatively, in accord with observation in the previous section that the tube contour shape is well described by the IAA theory. Tube Diameter. To estimate the tube diameter, we use the method developed in our previous work.17 In our simulation, the bead positions recorded from all isoconfigurational ensemble trajectories up to time τa form a cloud around the mean positions. The cloud of a single bead has the shape of ellipsoid, whose longest axis points along the tube tangent direction because the bead motions along the perpendicular directions are restricted by the tube confinement. So the extent of cloud spreading along the perpendicular directions is a first
Figure 5. Tangent vector distribution in the stretched case (τa = 250 frames). From left to right: λ = 1.15, 1.32, 1.52, 1.74, and 2.0. 9819
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Figure 6. Tangent vector distribution in the compressed case (τa = 250 frames). From left to right: λ = 0.87, 0.76, 0.66, and 0.5.
Figure 7. Tube diameter distributions. Left: τa = 50 frames; right: τa = 250 frames. Blue: maximally compressed case (λ = 0.5); red: undeformed case (λ = 1); yellow: maximally stretched case (λ = 2.0).
estimate the tube diameter, HS and RP found that the tube diameter scales with λα1/2. This predicted nonaffinely varied tube diameter and anisotropy were later examined by simulated networks with short (N ≤ 200) entanglement strands between cross-links.15,16 To exclude the effects of the cross-links (which are important for short bridging strands) and confirm the prediction of HS and RP theories, refs 15 and 16 must rely on the double tube model14 to indirectly test the HS and RP theory prediction. Our work is different from those earlier works. First, by studying entangled rings, we have eliminated the cross-linking effects completely, and we can use much larger rings (N = 800) to examine the entanglement phenomena. Second, the sense of orientation dependence in our work is also different from theirs. They all focused on the effects of anisotropic confinements on the bead motion, while neglecting the local (actual) tube orientations. We however can look at the tube orientations directly and examine how the strength of confinement depends on orientations, which is different from what were studied in previous work. Results Based on Cloud Peak. Many results presented in the previous sections depend on the choice of τa. This dependence is generic as long as the tube contour is defined as the average of bead positions from isoconfigurational trajectories and is more pronounced for tube sections with
Figure 8. Strain dependence of the average tube diameter. Blue: τa = 50 frames; red: τa = 250 frames.
dependence of average tube diameter on the orientation is very weak. Only for the λ = 0.5 case, one can see that the tube diameter for tube sections along the z-axis is slightly higher than those perpendicular to the z-axis. The orientation dependence we found from simulation is much weaker than that predicted by the HS11 and RP12 theories. Those authors analyzed the bead mean-squared displacements (MSD) along different directions. By introducing assumptions on how the strength of the confining potential varies with deformation, it was predicted that the bead MSD along direction α (α = x, y, z) should be proportional to the deformation rate λα. Using the square root of bead MSD to
Figure 9. Orientation dependence of tube diameter in the deformed case (τa = 250 frames). From left to right: maximally compressed (λ = 0.5), undeformed (λ = 1.0), and maximally stretched (λ = 2.0). Cloud points are values of tube diameters for tube segment with the specific polar angle; the probability of tube segments orienting along a particular direction is proportional to the local cloud density. Data points with error bars are binaveraged tube diameters, the error bar given by the standard deviation. 9820
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higher curvature.24 To illustrate this point, in Figure 10 we plotted the cloud of a test bead at a curved tube section. As
Figure 10. Effects of tube curvature on the transverse spreading of bead cloud.
Figure 11. Effects of τa on the strain dependence of the tube contour length, and the predictions of cloud peak contour. Thick green curve: IAA theory prediction. Thick red curve: results from tube contour defined using cloud peak. Thin dashed lines: results from tube contour defined using τa = 50, 100, 150, 200, and 250 frames.
averaging time increases, the number of cloud points penetrating the tube sections of neighboring beads will increase. Since those tube sections are curved downward, effectively, the transverse spreading of the test bead cloud increases. This spreading prevents us from using bigger values of τa. At the same time, the averaging time cannot be taken to be too small since the beads need time to explore the tube confinement.17 This problem, however, can be mitigated by using an alternative method to define the tube contour. We notice that each isoconfiguration trajectory represents one specific path for the beads to explore the tube region. If we look at a single bead, in one such path, it may move along the forward direction (forward in the sense of along the tube contour); in another path, the direction may be reversed. On averaging, the beads are wandering around their own location at the primitive path, spending half the time moving forward and the other half backward. So if we think of the cloud of bead positions recorded from all paths as a sample of an underlying smooth function, the cloud peak or the maximum of that smooth function, must be the place where the bead visited most frequently, i.e., its position along the tube contour. This suggests immediately that the cloud peak can be used to define the tube contour. To find the cloud peak, we need to first define a cloud density function by dividing the cloud into several boxes and counting the number of beads inside and then interpolate the density function, smooth it, and locate the maximum. The results show that the primitive path defined in this way saturate as τa is greater than about 10τe. Using this “invariant” primitive path, we reanalyzed all the isoconfigurational ensemble simulation data and obtained the strain dependence of the contour length and tube diameter. The results for the normalized tube diameter are nearly identical to those presented in previous sections, so we will not show results from this “cloud peak” procedure here. The result for the normalized tube contour length are shown in Figure 11. The qualitative features are similar to those obtained from the tube contour defined as the cloud mean path. More quantitatively, it shows that the IAA theory gives a satisfactory description to the increase of tube contour length for the compression case but overestimates the effect for the stretching case.
simulation17 to investigate tube properties, and studied the effects of uniaxial stretching and compression. Isoconfigurational ensemble simulations effectively generate a cloud of bead positions that reveals the effects of tube confinement (section). Using the cloud to study tube properties consists of two steps: (1) define the tube centerline or primitive path with averages of bead positions in the cloud; (2) estimate the tube diameter from the transverse spreading with respect to the centerline. This enables us to visualize the tube contour, calculate the tube centerline length and tube diameter, and study their dependence on applied strain. We studied deformations with modest strains, ranging from 0.5 to 2.0. Within this range, we found that the deformation of tube contour and distribution of the tube tangent orientations are well described by the independent alignment approximation given in section. This is consistent with previous results that the IAA theory provide a very good description to various viscoelastic properties of linear polymer melts, even in the nonlinear regime.1 The values of tube centerline length and tube diameter depend on the choice of averaging time τa used to generate the bead cloud, as shown in Figures 3 and 7. To circumvent this dependence, we normalized these two quantities by their values in the nondeformed state. The fractional change of the average tube diameter with strain is nearly independent of τa, and varies monotonically with strain, up to ±10% for the range of strain studied. In contrast, the dependence on the averaging time for the tube centerline length is still visible even after this normalization (Figures 3). To fix this and, more importantly, to find an “invariant” contour to the tube diameter, in the section on cloud peak, we propose to use the cloud peak density instead of the bead mean position to define the tube contour. The results of tube diameter calculated from the two definitions of tube contour are consistent. The result of tube centerline length from the “invariant” tube contour (Figure 11) suggests that the contour length increases with both stretching and compression. For systems that have been compressed, the fractional increase is consistent with that given by the IAA prediction; for stretched systems, it is less than the IAA prediction. The changes are moderate on both sides, up to about 10%. The isoconfigurational ensemble simulations yield tube orientation and tube diameter simultaneously, allowing us to investigate the orientation dependence of tube diameter. The results shown in Figure 9 indicate that within the range of strain studied this dependence is surprisingly weak. This is in
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SUMMARY Understanding how the confining tube varies with deformation is crucial for constructing a theory of gel elasticity.13 In this work, we used topologically equilibrated ring polymers to represent gels, applied the isoconfigurational ensemble 9821
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apparent contrast to the much stronger orientation dependence suggested previously,9−12 which was derived from the analyses of anisotropic bead mean-squared displacements. We note that the sense of “orientation” in our work is quite different from theirs: in their case, the orientation refers to the direction of bead motion; in our work, it refers to the orientation of actual tube sections.
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AUTHOR INFORMATION
Corresponding Author
*E-mail
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank NSF DMR-0907370 and ACS-PRF 49964-ND7 for support. REFERENCES
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