Relation between Cross-Link Fluctuations and Elasticity in Entangled

Article pubs.acs.org/Macromolecules

Relation between Cross-Link Fluctuations and Elasticity in Entangled Polymer Networks M. Lang* Leibniz-Institut für Polymerforschung, Hohe Straße 6, 01069 Dresden, Germany ABSTRACT: Computer simulation data on junction fluctuations in end-linked model networks are analyzed using the framework of the constraint junction model and the slip tube model for entangled networks. It is argued that cross-linking freezes in conformational fluctuations of chains and cross-links, which creates a small preload of entangled strands. This can be modeled using a “confinement degree of polymerization” Nc(N) with a weak dependence on the degree of polymerization N. The confined motion of network junctions with f active connections to the network is well described by using the exact solution of the constrained junction model with a constraining term that is given by the superposition of f/2 entanglement tubes with confinement Nc(N). It is shown that the modulus of end-linked model networks is the sum of a phantom and an entanglement contribution, whereby the entanglement contribution shows the same correction Nc(N) as both the cross-link fluctuations and the time average bond orientations. The results for the cross-link fluctuations allow to compute instantaneous or time average segment orientations and, thus, are necessary to understand birefringence experiments or double quantum nuclear magnetic resonance data of entangled networks.

1. INTRODUCTION

Gc =

The restricted fluctuations of network junctions are one of the key quantities that need to be understood in order to model the properties of a polymer network. For instance, the factor relating the prediction for modulus of the phantom model,1 Gph, to the prediction of the affine model2 of rubber elasticity, Ga, for a perfect f-functional network of ideal chains Gph =

f−2 f−2 Ga = νkT f f

(1.2)

since only the N segments of the real chain out of the N + 2n segments of the combined chain contribute to stress. These combined chains are a convenient tool to understand orientational properties of network strands. For instance, the time average bond orientationsas measured by double quantum nuclear magnetic resonance (NMR)depend on the length of the combined chains.5,6 Instantaneous bond orientations (e.g., birefringence) depend on average end-to-end vectors of the chains and the fluctuations around these,7,8 which can be modeled using combined chains.9 While the role of cross-link fluctuations in the phantom model1 or the constrained junction model10,11 is straightforward, the situation seems to be different for models of entangled networks. Here, a “cross-link contribution” Gc is added to an entanglement contribution Ge

(1.1)

results from considering junction fluctuations in the phantom model. Here, ν is the number density of network strands, k is the Boltzmann constant, and T the absolute temperature. A convenient way to model cross-link fluctuations in a phantom network is to attach virtual chains of n > 0 Kuhn segments between the ends of “network” strands of N Kuhn segments and the affinely deforming nonfluctuating “elastic background”. These virtual chains model the confinement of the surrounding network structure on the motion of the network junction, if the N-mer under observation were removed from the network; see ref 3 for more details. This allows to reduce a complex network connectivity to a single “combined” chain, where the fixed end-points of this combined chain deforms affinely. The “cross-link” contribution to elasticity is then written using combined chains of N + 2n segments as4 © XXXX American Chemical Society

N νkT N + 2n

G = Gc + Ge

(1.3) 3,9

While in some models an ideal phantom model contribution Gc = Gph is used, there are other works, for instance refs 12 and 13, that introduce a cross-link contribution larger than the phantom limit Gc > Gph, which is derived from considering constrained junction fluctuations. Received: December 14, 2016 Revised: March 3, 2017

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Table 1. Virtual Chains n As Determined from Monomer Fluctuations as Part of a Combined Chain (See Ref 6 for Details)a N

16

32

64

128

256

n( f = 3) n( f = 4) Nc

4.35 ± 0.14 2.32 ± 0.05 6.5 ± 0.1

6.09 ± 0.28 3.40 ± 0.09 9.1 ± 0.1

6.84 ± 0.44 4.47 ± 0.17 11.4 ± 0.1

7.68 ± 0.64 5.53 ± 0.31 12.5 ± 0.1

8.69 ± 1.12 6.16 ± 0.47 13.2 ± 0.2

Error bars for n reflect the statistical error due the finite set of cross-links with f active connections. The confinement degree of polymerization, Nc, was measured for the strands between 4-functional junctions in ref 6.

a

fluctuation data are readily converted into virtual chains and vice versa by matching the fluctuations in space. For a combined ideal chain made of two virtual chains of n segments and an ideal chain of N segments in between, the cross-link fluctuations are restricted by two ideal chains of n and n + N segments, respectively. This confinement is modeled by a virtual chain of nc segments with

In the present paper, we want to understand which of these approaches to compute G is in accordance with the available microscopic simulation data on both confined cross-link fluctuations and rubber elasticity. This is achieved by developing a quantitative model for the magnitude of crosslink fluctuations that is based upon network parameters such as N, f, and the tube diameter dT in section 2. The model of crosslink fluctuations provides an explicit prediction for a Gc > Gph and thus allows to decide how entanglements and the cross-link contribution add up to the full modulus of the network (see section 3). It turns out that using Gc = Gph is consistent with the available simulation data, while the same fluctuation correction needs to be considered for the entanglement contribution of modulus as for the cross-link fluctuations or the residual bond orientations. A brief summary of the main results is provided in section 4.

1 1 1 = + nc n N+n

(2.1)

One end of this virtual chain is attached to the nonfluctuating elastic background, while the other end models the long time mean-square fluctuations Δr2 = ⟨(R(t) − R)2⟩ of the cross-link position vectors R(t) at time t around an average position R. For ideal chains, we find n(N + n) Δr 2 = nc = N + 2n b2

2. CROSS-LINK FLUCTUATIONS One requirement to understand cross-link fluctuations in entangled networks is to compare with data that allow for an alternate estimate of entanglement properties, for instance, by comparison with the plateau modulus or residual segment orientations (as accessible in NMR). There are several works published5,6,14−18 that provide simulation results on cross-link fluctuations, sufficient information about the network structure to correct for network defects, and at least one alternative estimate of entanglement properties for the same samples. These references are preferred, since additional uncertainties due to comparing data from different studies can be avoided. Two of these studies show cross-link fluctuations that were not fully averaged14,18 (see Figures 5 and 10a of these references, respectively). Therefore, only the data of the remaining works5,6,15−17 based upon the bond fluctuation model19 (refs 5 and 6) or the standard bead spring model for polymers20 (refs 15−17) are used for comparison. The second requirement is to compare with data for networks made of a different f to check a possible dependence on junction functionality. References 5, 6, and 15−17 provide only data for 4-functional cross-links or averages of networks that are dominated by 4-functional junctions. In previous work,5,6 we also recorded cross-link fluctuations for junctions with f = 3 active connections to the network (data not published previously), which is used as a basis for model development. The data for f = 3 were determined in the same way as the data for f = 4, except of the point that the junction attached to the other end of the chains was allowed to have either 3 or 4 active connections. This results in a weak systematic underestimation of the virtual chains n (of less than 10% that is largest for smallest N; see also discussion below) at a significantly reduced statistical error of the data. After model development, the model is tested with the data of refs 15−17 that were obtained using a different simulation model. For this purpose, it is more convenient to provide cross-link fluctuations in Table 1 in the form of virtual chains. Cross-link

(2.2)

Here, b is the root-mean-square size of a segment. Thus n=

⎞ 1 ⎛⎜ Δr 4 Δr 2 4 4 + N 2 + 2 2 − N ⎟⎟ ⎜ 2⎝ b b ⎠

(2.3)

which becomes n ≈ Δr2/b2 for N ≫ n. For a better distinction from the entanglement degree of polymerization, Ne, we use a “confinement degree of polymerization”, Nc ≈ (dT/b)2, as in ref 6 to relate the tube diameter dT to a polymer strand of Nc with the same extension, since the ratio Ne/Nc (or Neb2/dT2) depends on the particular version of the tube model3,21 and is a constant only in the limit of N → ∞.22,23 The approximate symbol in this relation reflects additional corrections to chain conformations in dense systems24,25 like melts or networks. Note that an entangled chain in a tube is stretched by an entropic force in order to prevent the collapse of the chain to ideal size along the primitive path of the chains.4 This entropic force leads to a tension along the chains with tension blobs of the same size as the tube diameter. Corrections to tension show up as a correction to the tube diameter, which is expressed in the present work by a nonconstant Nc(N). In order to determine an estimate for Nc(N), we fit the data of Table 1 to the proposed qualitative approximation6 (see Appendix for more details) Nc(N ) ≈ Nc, ∞ − cN −1/2

(2.4)

The Nc data were obtained for the residual bond orientations of chains between junctions with four active connections each. A fit of eq 2.4 to the data yields c = 37 ± 2 and Nc,∞ = 15.7 ± 0.3. The above approximation was proposed6 to account for extra tension along the chains that arises from attaching network chains with a fluctuating number of entangled sections (or entanglements per chain) to network junctions. A quantitative model to explain this correction is currently not at hand. The B

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virtual chains for which fluctuations deform affinely (as above) lead to the same affine deformation dependence as the affine and phantom network model.3 We can apply the results of the constrained junction model to slip-tube or slip-link models of rubber elasticity directly, since the longitudinal motion of the network strands along the confining tubes is not restricted in these models. Therefore, the coupling of the motion of connected junctions and the constraints acting on these junctions must be the same as for the constrained junction model. For our considerations, we require an estimate for the virtual chains n that are attached to a given chain (the thick black line in Figure 1) to model cross-link fluctuations as part of a combined chain of N + 2n segments. In the spirit of the approach of ref 26 we can ask: what is the expected average spring constant when looking out from a chain end into the network? We find that f − 1 other real polymer strands of N segments are attached to the same virtual chains n on their other end in parallel to a virtual strand of ne segments that models the effective entanglement confinement acting on the junction. This observation is written as a recursive equation3

goal of the present work is to point out where this correction affects simulations and experimental data in order to provide a first basis for a quantitative understanding. Nevertheless, for networks where the main structural difference lies in the degree of polymerization of the precursor polymers (weight fraction of active material, gel point, etc., are rather similar for end-linked model networks of same f close to full conversion), we expect that the fluctuation correction for the number of entangled sections per chains dominates the correction to tension along the chains. This way, we arrive at the qualitative approximation of eq 2.4, where the coefficient c may absorb additional dependencies on network structure as, for instance, the crosslink functionality f. A quick view on the data of the virtual chains in Table 1 shows that n in all samples is clearly smaller than the phantom model prediction4 n < n ph = N /(f − 2)

(2.5)

for a perfect network containing only f functional junctions and no finite loops or network defects. Note that any network defect or finite loop leads to larger virtual chains. Thus, an additional constraint on junction fluctuations must explain this discrepancy. A self-consistent approach to deal with additional constraints on network junctions was elaborated in the framework of the constraint junction model,10,11 which is sketched in Figure 1.

f−1 1 1 = + n ne N+n

(2.7)

where the positive root ⎛ ⎞2 ⎛N ⎞⎞⎟ ne ⎜ ⎛ N N − ⎜ + f − 2⎟⎟ n = ⎜ ⎜ + f − 2⎟ + 4 2 ⎝ ne ne ⎠ ⎝ ne ⎠⎠ ⎝

(2.8)

provides the virtual chains that describe the cross-link fluctuations under this additional confinement (as part of a combined chain of N + 2n segments). Note that eq 2.7, without the entanglement confinement ne, leads directly to the phantom result, eq 2.5. The open question is which expression one should use for ne, since the locally anisotropic confinement of two tubes with different orientation needs to be coupled. The simplest approximation is to ignore local tube orientations and to assume that a superposition of f/2 tubes leads to an isotropic confinement that is stronger by a factor of f/2 as the lateral tube confinement, which was proposed first in refs 12 and 13. In order to learn about this point, we test eq 2.7 with the simulation data. We set ne = 2Nc(N)/f to describe the monomer fluctuations perpendicular to the tube axis and put this hypothesis to a test by plotting eq 2.8 against the virtual chain data for f = 4 without any adjustable parameter. The result is shown by the black continuous line in Figure 2. We find a quantitative match within error bars of the data except for the smallest N where the approximation of eq 2.4 must break down, since the size of the network strands becomes comparable to the tube diameter. This quantitative agreement is a remarkable result and demonstrates that the data of crosslink fluctuations and bond orientations (this is where a nonconstant Nc(N) was first observed) in ref 6 are fully consistent. As a consequence, eq 2.8 should allow for a quantitative description of the cross-link fluctuations, if Nc(N) is known or vice versa. Let us now revert the analysis and check whether one can estimate Nc(N) through eq 2.8, since analyzing cross-link fluctuations is a simpler task than analyzing residual bond orientations. We obtain Nc,∞ = 15.0 ± 0.2 and c = 30 ± 1 for the f = 4 data. The difference to the estimate of Nc,∞ from the

Figure 1. Entanglement constraints act on all junctions of the network simultaneously. The circles represent cross-links, the thick black line is the N-mer under observation, and the thin black lines show the next two generations of attached network strands. The dashed lines are the additional virtual chains ne that are attached to the nonfluctuating elastic background.

Here, additional virtual chains of ne segments are attached simultaneously to all junctions in order to model the effect of entanglement constraints on junction fluctuations. These constraints deform affinely with the sample deformation λα = Lα/Lα,0 in direction α, such that the directional components of the additional virtual chains deform as 1 ne, α = neλα 2 (2.6) 3 Here, ne is the virtual “degree of polymerization” of the isotropic (nondeformed) virtual chain and Lα and Lα,0 are the macroscopic extension of the sample in the deformed and nondeformed state, respectively. Note that only additional C

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cross-link fluctuations in the dense sample with N = 50 of about 5.15σ2, which is added to Figure 3. Svaneborg et al.16 use only

Figure 2. Virtual chain data of Table 1 analyzed with eq 2.8. The continuous line is a plot of eq 2.8 assuming ne = 2Nc(N)/f without adjustable parameter. The black dashed line represent a fit of Nc(N) through eq 2.8. The red dashed line is a fit of Nc(N) through eq 2.8 for f = 3, assuming the same Nc,∞ as for f = 4 and a variable c = 19 ± 2.

Figure 3. Cross-link fluctuation data of the bead−spring simulations of Svaneborg et al.16 and Chen et al.15 converted to virtual chain data. The continuous line is a fit of the data of ref 16 to Nc(N) through eq 2.8. The dashed line is a plot of eq 2.8 using a constant Nc,∞ = 17.4.

residual bond orientation data in ref 6 (see above) is less than 5%, while the parameter c for the correction at small N is off by 20%. This agreement shows that eq 2.8 allows for an excellent fit of Nc,∞ by using the cross-link fluctuation data and that even the correction at small N is available in a first-order approximation. The data for f = 3 allow for an explicit test of the proposed dependence on f. In Figure 2, the data are fit by using the very same Nc,∞ for f = 3 while only c of eq 2.4 was used as adjustable parameter. A smaller c = 19 ± 2 is obtained for f = 3 as compared to f = 4. Note that the data fit in the N → ∞ limit within error bars, which supports that the dependence on functionality may be correctly estimated by eq 2.8. Based upon these observations, it is concluded that in the limit of N → ∞ one observes n = 2Nc,∞/f. Thus, the fluctuations of cross-links with f ≥ 3 active connections to the network remain below tube diameter (according to the available simulation data), whereby the tube diameter dT can be estimated from extrapolating the long time limit of the mean-square displacements ⟨Δr2⟩ of network junctions with f ≥ 3 active connections to the network for N → ∞ using

half the mean-square displacements for their Figure 1a from which these authors extract limiting cross-link fluctuations.27 After correcting the different normalization, the data of ref 16 are almost in quantitative agreement with the data of Chen et al.15 (see Figure 3). In Figure 3, only the data of Svaneborg et al.16 are fitted, since the analysis of Chen et al.15 is not restricted to 4functional junctions. Recall that we were able to extract a reasonable estimate for Nc(N) from the simulation data. Thus, we attempt the same for the data of the bead−spring model and obtain Nc,∞ = 17.4 ± 0.2 and c = 23 ± 2. Therefore, also the bead−spring simulations are in accordance with an extra tension on entangled chains that increases for smaller N. In order to show the quantitative difference between a constant Nc,∞ and the correction by assuming an N-dependent Nc(N), a plot of eq 2.8 with a constant Nc(N) = Nc,∞ = 17.4 is added to Figure 3. A nonconstant Nc(N) leads to a significant change in the slope of the fit function. As obvious from Figure 3, using a constant Nc,∞ does not lead to a good qualitative fit of any of the available data sets, since the slope of the data at high conversion is always somewhat larger than the maximum slope ∝ f − 1 possible for the N-dependent term of the phantom model. Even though only end-linked model networks are investigated in the present paper, recent results on the fluctuation corrections for cross-linked brushes28 allow for a qualitative prediction of this correction in the case of random cross-linking of long precursor polymers (vulcanization). The main difference between cross-linking and end-linking is that cross-linking leads to a rather quick incorporation of most polymer into the elastically active material, while later reactions increase mainly cross-link density and thus reduce the average strand length between junctions.29,30 Thus, we ignore the change in tube diameter of the elastically active polymers as a function of conversion in a first qualitative approach. Let q denote the number fraction of cross-linked sites per chain and N0 the average degree of polymerization of the precursor polymer melt. Tube length fluctuations ∝ N01/2 freeze in at low conversion. At high conversion, qN0 ≫ 1, additional

fb2n f ⟨Δr 2⟩ ≈ (2.9) 2 2 Our analysis above can only be applied to data of networks with mixed active junction functionality (as for the present set of data), if the confinement due to entanglements acting on a junction is clearly stronger than the confinement resulting from connecting to f − 1 other chains. In this case, the entanglement confinement damps out quickly enough the phantom contributions of higher generations of attached strands such that the weight-average functionality of the network junctions further apart does not significantly affect cross-link fluctuations. This is checked by comparing 2Nc(N)/f ≲ (N + n)/( f − 1), which results in N ≳ 2( f − 1)Nc(N)/f − n that is satisfied for all data of the present study. Now let us apply eq 2.4 to the data of other publications15,16 that use the bead−spring model of Grest and Kremer20 at a bead density of 0.85σ−3. Equation 2.3 is used here to convert the cross-link fluctuation data of refs 15 and 16 to virtual chain data. Chen et al.15 report in their Figure 4b average square d T 2 ≈ b2Nc, ∞ ≈

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Macromolecules fluctuations freeze in proportional to the longitudinal fluctuations of the monomers of the network strands along their confining tubes. These fluctuations are roughly proportional to the square root of the average degree of polymerization of the network strands, N, with N1/2 ≈ q−1/2. Integration over the whole reaction process as a function of q similar to ref 28 provides a fluctuation correction ∝ q1/2 ∝ N−1/2. Thus, a fluctuation correction ∝ N−1/2 to tension along the chains seems to be universal for both end-linked and cross-linked networks. However, since only ≈1/3 of N0 is active30 when being incorporated into the active network, a quantitatively smaller correction is expected as compared to end-linked networks. The results of this section can be summarized as follows: the available data on cross-link fluctuations are in quantitative agreement with the exact solution of the constrained junction model and a confinement degree of polymerization Nc(N) that contains a correction ∝ N−1/2 to the tension along the chains. The limiting behavior for N → ∞ is not affected by this correction and in accord with the hypothesis that the additional constraint on junction fluctuations results from the superposition of f/2 entanglement tubes.

Note the qualitative difference of the results of refs 15 and 17 where the deviation from the theoretically expected behavior is either attributed to a correction in the cross-link contribution15 or the entanglement contribution17 by making assumptions for the complementary contribution to modulus. Actually, the results of the preceding section provide quantitative predictions for these corrections, either via preloaded entanglements toward small N as given by Nc(N) or via an increased cross-link contribution as given by the virtual chains n. Thus, we can actually test, which approach is in quantitative agreement with the simulation data. In order to see whether extra tension ∝ Nc,∞/Nc(N) in preloaded entanglements could explain the trend of the data observed for G − Gph, we first compute G − Gph for the data of Chen et al.15 in order to make this result comparable to the analysis of Svaneborg et al.17 Assuming that this estimate provides the entanglement contribution to modulus, Ge = G − Gph, we extrapolate the resulting Ge to perfect networks, Ge(1), by considering that inactive material reduces the plateau modulus like a theta solvent Ge(1) ≈ Ge(wa)wa−7/3.4 Here, wa is the number fraction of the active material with respect to all beads at preparation conditions. Note that there is wa ≈ 0.85 and wa = 0.925 in refs 17 and 15, respectively. Furthermore, we consider bead density ϕ = 0.85σ−3 by relating Ne(1) = ϵϕ/ Ge(1) for each measured Ge. Here, ϵ is the basic energy unit of the bead−spring model. The estimates for Nc (either constant Nc,∞ or Nc(N) = Nc,∞ − cN−1/2 with Nc,∞ = 17.4 ± 0.2 and c = 23 ± 2 as determined in the previous section) are both extrapolated toward perfect networks via Nc(1,N) = Nc(N)wa4/3 and Nc,∞(1) = Nc,∞wa4/3 assuming the same dependence on polymer volume fraction as for the entanglement degree of polymerization, Ne.4 In Figure 4, it is shown that the apparent N dependence of the estimated Ge (by computing G − Gph), and thus the

3. ELASTICITY The above results are quite unexpected and indeed should have a significant impact on the entanglement contribution to modulus or residual bond orientations via a modified Nc(N). Recall that the original estimate of Nc(N) data of ref 6 was determined from exactly these residual bond orientations, such that there is perfect agreement from this side. The remaining questions are (a) whether an enhanced entanglement contribution for small N is in accordance with what has been observed for network elasticity or not and (b) whether there is an enhanced cross-link contribution due to extra constraints on junction fluctuations or not. Svaneborg et al.17 determine the cross-link contribution independent of entanglements by switching off excluded volume interactions. The entanglement contribution to modulus is then determined by subtracting this presumed cross-link part. By this procedure, any possible cross-coupling of entanglement and cross-link contribution or an enhanced cross-link contribution as proposed, for instance, in refs 12 and 13, is here identified as an additionally increased entanglement modulus Ge. Indeed, a clear increase of Ge toward small N is observed in Figure 4b of ref 17, where the so-obtained entanglement contribution for end-linked networks almost doubles for the smallest N. Interestingly, the observed increase of the entanglement contribution of cross-linked networks in Figure 4b of ref 17 lies below the data of end-linked networks, in agreement with the discussion at the end of the preceding section. Chen et al.15 report uniaxial deformations that were analyzed by a simultaneous fit of cross-link and entanglement contributions using the slip-tube model of Rubinstein and Panyukov.3 This leads to an estimate of Gc ≈ 0.0115 and Ge ≈ 0.015 for N = 50. The authors compare this result with an estimate of the affine modulus Gaf ≈ 0.0145 that was a bit overestimated by considering the number density of effective (instead of active) chains. A ratio of Gc/Gaf ≈ 0.79 is a clear indication of an enhanced Gc as compared to the phantom case with Gph/Gaf < 1/2 here because of an excess amount of four functional cross-links for network formation.

Figure 4. Ratio between the confinement degree of polymerization, Nc, and entanglement degree of polymerization, Ne, for the data of refs 15 and 17. All data are extrapolated to perfect networks (or melts of infinitely long chains) assuming a phantom modulus for the cross-link contribution, whereby either a constant Nc,∞(1) (open symbols) or an N-dependent Nc(1,N) (full symbols) is considered.

apparaent N dependence of Nc,∞(1)/Ne(1), disappears for large N upon considering the modified tension along the chains when plotting Nc(1,N)/Ne(1) instead. One obtains that Nc(1,N)/Ne(1) ≈ 0.35 ± 0.01 for the four samples with N > 20. This ratio is significantly smaller than a coefficient of 4/7 E

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which is close to the Gc/Gaf ≈ 0.79 of Chen et al.15 Usually, the cross-link density (or efficiency) is extracted from the Gc contribution by assuming a constant plateau modulus Ge as above and assuming a phantom, affine, or constrained junction term for Gc. The analysis in Figure 5 shows that such a procedure is not quantitative, when assuming a = 1 or a = 1 − 2/f for the phantom or affine model, respectively. The Heinrich−Straube model12,13 provides qualitatively a good fit to the data, but it is also not quantitative, since the presumed cross-link fluctuations differ from the true cross-link fluctuations (virtual chains) for finite N. Thus, a determination of the cross-link density based upon elasticity data alone is likely to fail. A way out of this dilemma is possible, if a separate estimate for the phantom modulus is used, for instance, by determining the weight fraction of the elastically active material using NMR.36,37 Relating these results to the presumed network structure, Gph can be estimated. Together with data on the plateau modulus, Ge, one could identify Nc(N) by computing (G − Gph)/Ge similar to Figure 4. As a consequence of this procedure, one might obtain a different quantitative explanation for the mismatched slope of ≠1 that is typically observed when plotting modulus vs residual bond orientations (see ref 38 for illustration). The results of the present work are also relevant to better understand the dynamics and viscoelasticity of entangled melts: the Ge = 0.0145ϵσ−3 used by ref 17 at wa ≈ 0.85 extrapolates to Ge(1) ≈ 0.0212ϵσ−3 in perfect networks, which is equivalent to a limiting Ne(1) ≈ 40 in melts of infinitely long chains at a bead density of 0.85σ−3. This result is in quantitative disagreement with results of the primitive path analysis39 that proposes Ne = 65 ± 7. In contrast to this discrepancy, a consistent set of data are obtained, when comparing cross-link fluctuations (present paper), residual bond orientations,6 monomer mean-square displacements in networks,5,6 or equilibrium swelling data.40 Also, dynamical data of melts near terminal relaxation time41−43 are in agreement, if tube length fluctuations are taken into account as discussed in refs 5 and 6. This agreement does not extend to data of monomer mean-square displacements in melts near the entanglement relaxation time τe. Note that the qualitative form of the mean-square displacement data (for the simulation methods discussed in the present paper) does not agree with the motion of an ideal chain in a curved-tube-like potential: there should be an “overshoot” of mean-square displacements above the t1/4 regime near τe, since mean-square displacements at large times t ≫ τe are dominated by the motion along the tube axis (which is only one out of three directions). Since the shape of the data is different, a determination of a crossover time τe by taking the intersection point of the extrapolated regimes is therefore likely to be not quantitative. This hypothesis is supported by recent work on viscoelastic hydrodynamic interactions in polymer liquids, which enhance the monomer displacements even in the case of Langevin friction.44 This mechanism is expected be effective up to at least τe in entangled polymer melts and provides a possible explanation for the qualitative differences of monomer mean-square displacements near τe. Finally, we have to stress that throughout this work we omitted a discussion of the nonideal chain conformations in melts24,25 that we expect in similar form in polymer networks. As reason for this simplification, we take the observation of ref 6 that this effect seems to be compensated almost exactly by the above-mentioned “overshoot” of mean-square displacements

that was proposed for the slip tube model.3 It agrees perfectly with the estimate of ≈0.35 in refs 31 and 32. Note that Svaneborg et al.17 mention that the sample at N = 20 already shows significant impact of finite extensibility, which is the reason why this data was excluded from the present discussion. Now let us perform the complementary test by assuming that the entanglement contribution is independent of N. Instead, we consider an increased cross-link contribution due to restricted junction fluctuations according to eq 1.2. For this test, we assume that a constant plateau modulus Ge = 0.0145ϵσ−3, as extrapolated by Svaneborg et al.,17 provides the correct estimate for the entanglement contribution. Thus, we are interested in Gc ≈ G − Ge. As above, we extrapolate the resulting cross-link contribution toward perfect networks Gc(ϕ,wa) = Gc(1)ϕwa using the polymer volume fraction dependence of the phantom modulus4 and the statement made in ref 17 that for their data the ratio of Gc and the phantom modulus of an ideal network differ by a factor of roughly wa. We plot the normalized modulus Gc(1)/ϵ = 1/Nel in terms of an average elastic chain Nel of the affine model vs 1/N in Figure 5. The data of Chen et al.15 are also corrected analogously with respect to ϕ and wa. For comparison, we show Gc/ϵ as computed from eq 1.2 using either Nc(1,N) or a constant Nc,∞(1).

Figure 5. Effective elastic chain length Nel for the cross-link contribution (assuming constant entanglement contribution) in the framework of the affine model for the data of refs 15 and 17. The solid line shows an estimate of Gc(1)/ϵ = 1/Nel using the cross-link fluctuations as a function of Nc(1,N). For the dashed line, Gc(1) is computed assuming a constant Nc,∞(1) similar33 to the Heinrich− Straube model.12,13 The red dotted line is a linear fit to Gc(1)/ϵ = a/N that leads to a coefficient of a = 0.852 ± 0.005 for the extrapolated data of ref 17.

Figure 5 shows that the simulation data may be well described by assuming a constant33 Nc,∞ as, for instance, in the Heinrich−Straube model,12,13 but the data do not fit to the correct estimate based upon the true cross-link fluctuations using Nc(1,N). Thus, one has to consider a preloaded entanglement contribution ∝ Nc(N) as the correct physical explanation for the deviations between simulation data and the theoretical estimate for modulus. A second important conclusion from Figure 5 is that a simple linear approximation to the modified Gc data of ref 17 (assuming constant entanglement contribution) provides a rather good qualitative fit. One obtains a coefficient of a = Gc/ Gaf = 0.852 ± 0.005 in between the affine and phantom limit, F

DOI: 10.1021/acs.macromol.6b02690 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules for the bond fluctuation model. Thus, for considering fluctuations, we simply assumed ideal chain conformations to convert these into virtual chains and vice versa. Note also that nonideal chain conformations lead to a renormalization of chain size and thus do not show up in the leading term of elasticity, since chain size is canceled out when computing the change in entropy (see for instance the derivation of the affine modulus in ref 4). However, the redistribution of segments in different directions upon network deformation should become nonideal such that a larger number of segments is being pulled in the deformation direction as compared to the ideal chain case. This would cause a coefficient