Loops in Polymer Networks | Macromolecules - ACS Publications

Article Cite This: Macromolecules 2019, 52, 4145−4153

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Loops in Polymer Networks Sergey Panyukov*

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P. N. Lebedev Physics Institute, Russian Academy of Sciences, Moscow 117924, Russia ABSTRACT: The classical theory suggests that phantom polymer networks have a treelike structure. In this study, we investigate the actual polymer networks, in which the polymer loops have a finite size and strongly overlap with each other. We show that the elastic modulus of such networks has two contributions: of elastically effective strands and of finite size loops shunting the network strands. The latter contribution substantially depends on the interaction of the network strand monomers at network preparation conditions. The result of calculations performed in the framework of the replica theory of polymer networks is interpreted using a generalized combined chain model. This model allows us to describe quantitatively the elasticity of the polymer network with finite size loops and the deformation of its individual strands. We also calculate the impact of primary loops and cyclic defects of arbitrary concentration on the elasticity of such a network.

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INTRODUCTION

Numerous applications of polymer networks are associated with their exceptional mechanical properties.1−3 The design of new materials requires a deep understanding of how the elasticity of a network is related to its molecular structure. The importance of the theoretical description of the microscopic molecular connectivity of networks increases significantly because of the development of new experimental methods for determining the local cyclic topologies.4−6 However, so far, this understanding is very far from complete even after decades of research. The reason for this is the complexity and randomness of the molecular structure of polymer networks: the cross-linking of the precursor polymer chains is a random process, leading to numerous topological defects. Usually, the defects are considered in the ideal defect gas approximation.7−10 They are assumed to be sparsely distributed in an otherwise perfect network. The contribution of each defect is calculated separately, and their overall impact is taken as the algebraic sum of the contributions from all individual defects. In the classical model, the “perfect polymer network” has a treelike structure, see Figure 1a. Because polymeric trees cannot bear shear stress, even the perfect network must have loops connecting different infinitely distant generations of these trees. In actual networks, the loops have finite sizes, see Figure 1b. Such networks behave as polymer liquids on small scales and acquire solid-like elasticity only on scales exceeding the size of typical loops.11 Specifically, the network deforms affinely on scales exceeding the loop size and nonaffinely on smaller scales. The network loops are strongly overlapped and interact with each other. The impact of such typical loops, which are responsible for the solid-state elasticity of the network, cannot be described in the ideal defect gas approximation. There is another important conceptual restriction on the ideal defect gas approximation, even if we consider defects that are located far from each other. An imaginary perfect network does not consist of a single tree but of an infinite number of © 2019 American Chemical Society

Figure 1. (a) Model of perfect network with treelike connection of network strands (solid curves) and infinite size loops (dotted lines). A network strand with N monomers can be considered as a part of the combined chain, connected through two effective chains with n1 monomers (dashed green curves) to the nonfluctuating background. (b) Actual network with finite size loops. The combined chain consists of three effective chains. The filled circles and crosses indicate cross-links and nonfluctuating background, respectively. The end-toend vector of the combined chain X deforms affinely with network deformation.

polymer trees (or “layers”) overlapping with each other. In the ideal gas approximation, any structural defect is connected with the network through the root trees, see Figure 1a. These trees can be modeled by the effective (virtual) chain, attached at the other end to the affinely deformed nonfluctuating background12 (see the bottom image in Figure 1a). In an actual network, the number of overlapping trees is small enough, and different root trees should be interconnected by loops already at first generations of the trees, see Figure 1b. To describe the contribution of such loops of finite size to the elasticity of polymer networks, the ideal defect gas approximation must be replaced by a theory that takes into account all main effects inherent to actual polymer networks: the polydispersity of network strands, a complex molecular Received: April 16, 2019 Revised: May 6, 2019 Published: May 22, 2019 4145

DOI: 10.1021/acs.macromol.9b00782 Macromolecules 2019, 52, 4145−4153

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Macromolecules

exceed 1 order of magnitude. The probability distribution of the number l of network strands in a loop can be roughly approximated by eq 2 with a cutoff at l = lmax ≃ l̅. This simple expression qualitatively correctly describes the main features of the result of numerical simulations in ref 15. The width of this distribution is small compared with the average size l̅, so most of the loops in the network are of the size l̅ (see also discussion on excluded volume effects, eq 11). A numerical algorithm for finding this distribution was proposed in ref 16. The shear stress is transmitted by loops in the network. This idea can be illustrated by treating the network as P(0) overlapping yet independent “elementary” networks, the strands of which consist of l̅N ≃ N monomers, see Figure 2.

structure of networks, the interaction of strand monomers at preparation conditions, and so on. In our paper, we take into consideration these effects using the replica model of the network. This approach was proposed long time ago and used to calculate the correlation functions of density fluctuations in polymer networks.13 In Appendix, we generalize these replica calculations to bear in mind finite size loops in the network. In this paper, we will consider only well-developed networks obtained far beyond the gel point. Of course, the properties of polymer networks depend significantly on the experimental conditions. Therefore, most of the works devoted to the study of this particular dependence, but that is not the purpose of this paper. The molecular structure of the network depends only on the conditions of its preparation but not on the conditions of the experiment. In this work, we will be interested only in how the conditions of the network preparation affect the network elasticity.

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RANDOM STRUCTURE APPROXIMATION In this section, we review the main results of the traditional approach, which assumes that the formation of a loop in the network is a random process, independent of the presence of other loops. A typical polymer network has both chemical cross-links and topological entanglements. In this paper, we consider only unentangled networks, in which the number, N, of Kuhn monomers of the network strand is small with respect to the number of monomers, Ne, in the entangled strand, N < Ne (Ne ≃ 20 for the melt). In a typical network, there are many network strands overlapping within the volume pervaded by a network strand. The overlap parameter is defined as the number of network strands within the volume R3 pervaded by one network strand and for the solution of chains with monomer density ρ(0), P(0) ≃ ρ(0)R3/N. In Gaussian networks, the size of the network strand with Kuhn length a is R ≃ aN1/2, and P(0) ≃ ρ(0)a3N1/2

Figure 2. (a−d) Set of P(0) = 4 elementary networks with nonoverlapping strands. (e) Actual network can be assembled by cross-linking (shown by brown circles) P(0) overlapped elementary networks. The loops of this network are strongly interconnected with each other.

There is about one network strand per volume R3 in the “elementary” network, and its elastic modulus is estimated as Gelem ≃ kT/R3, where kT is the thermal energy. Because the actual network consists of P(0) elementary networks, its elastic modulus is P(0) times larger14

(1)

G = P(0)Gelem ≃ kTρ(0) /N

To understand how this strong overlap of network strands, P ≫ 1, affects the topological structure of the network, consider the network formed by end-linking monodisperse precursor linear polymers.14 Each cross-link joins f ends of such polymers. On small scales, the network has a treelike structure, and the number of network strands at the lth generation of this tree is (f − 1)l−1. Because the network strands heavily overlap with each other, they form loops that contain many generations of the tree per loop. The number of possible ways to form a loop of l network strands is (f − 1)l−1, and the probability that two ends of such a tree are inside the contact volume a3 is (0)

φ(l) ≃ (f − 1)l − 1

a3 1 ≃ (f − 1)l − 1 1/2 3 (Rl ) (Nl)3/2

Combined Chain Model, Figure 1a. To describe the elasticity of polymer networks, several models were developed. In the affine network model, it is assumed that the ends of each network strand are pinned to a “nonfluctuating elastic background”, which deforms affinely with the whole network.12 In the perfect network model, the strand ends are not fixed in spacethey are connected to the root trees of network strands, as shown in Figure 1a. The elasticity of these root trees is described by an effective (virtual) chain with the number of monomers n1 attached by the other end to the nonfluctuating elastic background. In each generation, k = 1, 2, ... , of the root tree (left and right in Figure 1a), (f − 1)k network strands are effectively connected in parallel. Summarizing the contributions of all generations of the tree, we find the number of monomers of the effective chain17

(2)

1/2 3

where (Rl ) is the volume pervaded by a linear chain of l network strands. An average number, l̅, of network strands in a loop is determined by the condition φ(l̅) ≃ 1, which gives at N ≫1 l̅ ≃

1 3/2 ln(N3/2 l ̅ ) f−1

(4)

n1 =

∑ k>0

N N = f−2 (f − 1)k

(5)

The network strand with N monomers together with the two effective chains with n1 monomers each can be considered as one combined chain. The ends of this combined chain are not fluctuating and can be considered as attached to an elastic nonfluctuating background. The vector X between the ends of this chain is deformed affinely with macroscopic network

(3)

The iterative solution of this equation quickly converges. Notice that because for unentangled networks N < Ne, the average number of generations of the tree cannot significantly 4146


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Macromolecules deformation, and the average vector R between the ends of the network strand is determined from the force balance condition: R=

X 1 + 2n1/N

(iii) Equation 9 cannot describe the cyclic defects, as each of them is characterized by its “elastic effectiveness”, which depends on the type of the defect. Here, we will not go into detail because all these effects are “automatically” taken into account in the framework of the replica approach, see eq 18. Excluded Volume Effects. In actual networks, the loops are strongly interconnected with each other (see Figure 2e). As at the preparation conditions the set of all overlapping loops is “packed” with a given monomeric density ρ(0), the chains of these loops are prestrained. This effect is not taken into account in the framework of the random structure approximation described in this section, which assumes unperturbed Gaussian conformations of polymer strands in the network. An attempt to evaluate the effect of the excluded volume effect at preparation conditions on the number l̅ of polymer chains in a loop (loop length) was made in ref 15. Here, we generalize this estimate, taking into account the dependence of the loop length on the monomeric density ρ(0). The number of chains connected to the origin at a distance of i chains along the network structure is ∼f(f − 1)i−1, and the volume of the tree up to generation l is on order of the volume of a sphere with radius a(lN)1/2

(6)

where n1 is the number of monomers of the effective chain. This model predicts the free energy of an actual network with monomer density ρ(0) F = V (0)

G ∑ λα 2 2 α

(7)

Here, V(0) is the network volume at preparation conditions and the elastic modulus is G=

ρ(0) ijj kTρ(0) /N 2 yz = kT jjj1 − zzzz, 1 + 2n1/N N k f{

perfect network (8)

−1

Here, (1 + 2n1/N) is the strand fraction of the combined chain. While obtaining the second equality, we substituted n1 from eq 5. Equation 8 can be recast in a more general form with the network modulus proportional to the difference of the number densities of network strands ν and cross-links μ = 2ν/f, as there are f/2 strands per cross-link G = kT(ν − μ)

l

N

∑ f (f − 1)i− 1 ≈ a3(lN )3/2 (0)

ρ

i=1

(10)

By calculating the sum of the geometric progression, for N ≫ 1, this equation can be converted to the form of eq 3

(9)

It is usually assumed that this expression can also be applied to networks with defects, if we include in the densities ν and μ only elastically effective strands and cross-links.17 In this case, (ν − μ)V(0) has the meaning of the cycle rank of the networkthe number of its independent loops. Note that expression 9 was derived for networks with loops of infinite size, see Figure 1a. Below we show that it is violated for cyclic defects of finite size in the network, see eq 19. The perfect network model is usually called the phantom network model, although neglecting the entanglement effects for phantom chains is not equivalent to the approximation of infinite loops size in the perfect network model. The results of numerical simulation of the network with Gaussian phantom strands are presented in ref 18. It is shown that the elastic modulus of this network is different from that of the perfect network model and coincides with the elastic modulus of the classical nonfluctuating grid, in which each strand is replaced by a corresponding elastic thread with the same elastic stiffness coefficient. In Appendix, we prove this coincidence and show that it is due to affine deformation of the average distances between the cross-links in perfect Gaussian networks. In ref 9, the combined chain model is generalized to take into account the contribution of cyclic fragments to the elasticity of polymer networks in the ideal defect gas approximation. The impact of finite size cycles is determined by several effects: (i) Because part of the network strands is “spent” on creating the cyclic fragment itself, the length of the effective chain linking this fragment to the elastic nonfluctuating background is different from eq 5. (ii) At network preparation conditions, the strands of a cyclic fragment are contracted and the surrounding strands are stretched (prestrained) with respect to Gaussian sizes of these chains.10

l≈

1 ln(P(0)l 3/2) f−1

(11)

The overlap parameter P(0) is defined in eq 1. The iteration method for solving this equation quickly converges. Note that it also predicts a weak logarithmic dependence of the average loop length on N, although the authors15 argue that the solution of this equation has the form of a power dependence, l ≈ N1.1/( f − 1). The case P(0) ≈ 1 with the cycle size l ≈ 1 describes the “elementary networks” shown in Figure 2a−d. In accordance with the results of numerical simulations,15 the loop length l decreases with dilution and with increasing functionality f of cross-links. Comparing expressions 3 and 11, we find that steric repulsion in the network leads to a decrease in the length of its typical loops by the value of Δl ≈

1 N ln f − 1 ρ(0)a3

(12)

The statistics of the minimal size loops in the network does not directly determine its elastic properties, as each elastically effective chain is simultaneously part of a large number of loops. The cumulative effect of these loops is self-averaged and can be described in the effective mean field (one-loop) approximation.

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FINITE SIZE LOOPS The excluded volume effects at preparation conditions have a deep impact on the molecular structure of a network. Unlike perfect networks, the monomers of actual networks mutually repel each other, “crowding out” the network loops on small length scales, see Figure 1 and eq 12. Therefore, the elasticity of a polymer network should explicitly depend on the excluded volume parameter v(0), characterizing the monomer interaction 4147


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Macromolecules at the preparation conditions. The classical theory considered in the previous section leads to a paradoxical result about the independence of the elasticity of the network on an interaction of its monomers at the conditions of the network synthesis. In Appendix, we find the elastic modulus of an actual network with the polydisperse (exponential) distribution of strand lengths and the number x1 of primary loops per cross-link G(x1) = kTρf(0)

x1 =

f /2 − 1 + (f − 4)x1/2 Q (0) − kTv(0)ρ(0) 1 + x1 2 (13)

where is the density of cross-links, and the loop formation probability density is

∫0

∞

dNPN (0) ≃

∑ PN(0) N

(14)

PN(0) is the probability density of forming a closed loop of N monomers. Equation 13 generalizes classical expression 8 for the perfect network model. The number x1 of primary loops per cross-link is proportional to the number of ways, f(f − 1), to form a loop on the cross-link x1 = f (f − 1)

8ρ

1/2

(18)

where a is the monomer size. Excluded volume leads to swollen conformations of elastic strands (see, for instance, Figure 5 in ref 22). This effect is well studied, and below we consider only the case z ≪ 1 of Gaussian strand conformations. In the opposite limit, z ≫ 1, the strands take self-excluding conformations, and such a network is described by the scaling theory of elasticity.13,23 (ii) The second “channel” is described by the second term in expression 13, which is proportional to the excluded volume parameter v(0). Generally speaking, this new term is not small even in the limit of a small perturbation parameter z ≪ 1. It is determined by the contribution of loops to the elasticity of the polymer networks. At the conditions of network preparation, the loops can be of two types: due to cross-links and temporary loops that are formed when two monomers of different strands touch each other in the process of thermodynamic fluctuations, see Figure 4a,b. Note that expression 13 was obtained in the one-loop approximation for the replica theory. It can be shown that the multiloop contributions produce only higher-order terms of the series expansion in the parameters z (eq 18) and 1/P(0)

(15)

The denominator of the first term in expression 13 describes an increase in the effective number of monomers between cross-links due to primary loops, see Figure 3a. N̅ → N̅ (1 + x1)

(17)

1 ijj 2v(0) yzz jj zz z= 8πa3 jk ρ(0) z{

Q (0) (0)

2ρ(0)

In such vulcanized networks, the most wasted loops consist of a rather small number of segments, while other more complex loop fragments are not so important (see Figures 10 and 12 in ref 21). For general f > 2, the first term in eq 13 can be interpreted as a contribution from cross-links of two types: with functionality f and density ρ(0) and with functionality f − 2 and density f x1ρ(0) f , and also with a renormalized number of monomers of network strands, see eq 16. There are two “channels” through which the excluded volume effect contributes to the elasticity of the polymer network: (i) The first “channel” is described by the perturbation theory parameter z, which determines the difference between the conformations of network strands and Gaussian chain conformations

ρ(0) f

Q (0) =

Q (0)

(16)

Figure 3. Two main contributions to the network elastic modulus: (a,b) renormalization of the effective length of elastically active chains (red) due to pending loops (f = 3) and primary loops (f > 3) (green) and (c) shunting the elastically active chains by a set of typical network loops (two of them are shown in green).

Although primary loops cannot bear shear stress in the network, they renormalize the length of the elastically effective chains, which deform and store elastic energy upon network deformation.19,20 For f = 3, expression 13 describes a decrease in the number of elastically effective chains in the network because of the presence of pending loops, see Figure 3a. For f = 4, the primary loops do not contribute to the numerator of expression 13, as the formation of an additional primary loop does not lead to any new loops consisting of elastically effective chains, see Figure 3a. This case, f = 4, also describes a polymer network obtained by cross-linking polymer chains with the number of primary (wasted) loops per cross-link13

Figure 4. (a) Primary loop permanently fixed in the topological structure of the network and (b) temporary loop, which appears as a result of the collisions of fluctuating in space monomers of the polymer strand. (c) Ring fragment with i = 4 internal cross-links and the functionality f4 = 8. (d) Loops taken into account in the one-loop approximation in eq 13 and (e) loops not captured whose contribution is taken into account in eq 19 (with i = f 2 = 2). 4148


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Combined Chain Model, Figure 1b. To get a deeper understanding of the origination of both terms in eqs 13 and 19, one may study the influence of elastic environment of a network strand on its conformations. In a perfect network, these conformations can be described by the combined chain model, see Figure 1a. In an actual network, the combined chain includes an additional effective chain of n2 monomers forming a closed loop with the network strand of N monomer, see Figure 1b. This virtual chain represents effective elasticity of all loops of finite size in the network, shunting the real strand of N monomers. The average end-to-end vector R of the network strand is related to the end-to-end vector X of the combined chain via the force balance condition

(see eq 1), which are assumed to be small for the networks under consideration. The selection of accounted molecular structures of the polymer network is based on the topology of the loops, and not on their concentration, which can be arbitrary. For example, the loops shown in Figure 4d are taken into account and the loops shown in Figure 4e are not captured. The actual polymer networks have numerous topological defects.24 The primary loops taken into account in eq 13 are not elastically effective. Below we also consider the contribution of elastically effective cyclic defects in model networks obtained essentially above the overlap concentration of their strands. Expressions 15 and 17 for the fraction of primary loops x1 are calculated for the network obtained by random end-linking/cross-linking the solution of polydisperse chains. In general, the fraction xi of cyclic defects (small fragments of the network with i ≥ 1 “internal” cross-links) depends on the type of reactions leading to the formation of the network, and it can be controlled experimentally.4,5 On the other hand, the properties of typical (large) cycles in the network, which determine its elasticity, are universal and do not depend on the kinetics of the reactions. The approach developed in the Appendix combines both of these ideas: we calculated the (universal, independent of the type of the reactions) elastic modulus of the network obtained by random end-linking polydisperse chains by cross-links with functionality f and concentration ρ(0) and cyclic fragments with f functionality {f i} and (arbitrary and nonuniversal) concentrations {xiρ(0) f } G{xi} = kTρf(0)

R=

G=

1 + ∑i ixi Q (0) 2

kTρ(0) /N 1 , 1 + 2n(1/N + 1/n2) 1 + N /n2

actual network (22)

−1

An additional factor (1 + N/n2) in this expression (compare with eq 8) is the fraction of energy related to the network strand when it is connected in parallel with the effective chain of n2 monomers, see Figure 1b. We can rewrite this result in the form

(19)

The functionality f i defined as the number of “external” network strands joined to the cyclic fragment (the number of its chemically active sites). For the ring fragment with i “internal” strands and cross-links, f i = i(f − 2), as two of the functional groups of each cross-link are involved in creating the ring, see Figure 4c,e. For all xi = 0, except x1, this expression turns into eq 13. Some important generalizations of eqs 13 and 19, which are valid in the leading order in the small parameters 1/P(0) ≪ 1 (see eq 1) and z ≪ 1 (see eq 18), are as follows (i) The excluded volume parameter v(0) describes the interaction of the monomers of the network in a good solvent. In the case of networks obtained in conditions close to the θpoint, the energy density of steric repulsion is v(0)(ρ(0))2/2 + w(0)(ρ(0))3/3, and the parameter v(0) is replaced by the sum of the contributions of the second (v(0)) and third (w(0)) virial coefficients at the network preparation conditions v(0) → v(0) + 2w(0)ρ(0)

(21)

Here, n is the number of monomers of the effective chains, connecting the strand ends with the nonfluctuating background. A part of the root tree strands, connected in parallel in the perfect network (see eq 5), switches to the formation of loops of finite sizes. Therefore, the effective chains have a greater number of monomers, n > n1, compared to the perfect network, see Figure 1b. The factor [1 + 2n(1/N + 1/n2)]−1 is the fraction of the combined chain which belongs to the parallel connection of the network strand with the effective chain of n2 monomers. The vector X deforms affinely with the applied deformation. Neglecting the primary loop contribution (which can be taken into account by the renormalization of the strand length, eq 16), the elastic modulus of this model is

f /2 − 1 + ∑i (fi /2 − 1)xi

− kTv(0)ρ(0)

X 1 + 2n(1/N + 1/n2)

G=

kTρ(0) kTρ(0) − N + 2n1 N + n2

(23)

The first term in this expression has the same meaning as for the perfect network shown in Figure 1a, describing the two effective chains connecting the network strand to the nonfluctuating background. They have renormalized total number of monomers 2n1 = 2nn2/(2n + n2), corresponding to parallel connection of two effective chains: with 2n and n2 monomers (see eq 8 and Figure 1b with two crosses, strapped into one). The novel second contribution in eq 23 is negative. In agreement with our prediction for the elastic modulus in eq 13, the generalized combined chain model shown in Figure 1b reproduces the negative contribution to the elastic modulus because of the finite size of typical loops in the network. A perfect network discussed in the previous section has a treelike structure on all spatial scales. Because of the excluded volume effect, trees cannot “grow” to infinity (the Malthus effect), as such infinite trees cannot fit in a real threedimensional (3D) space. Therefore, in actual networks, the size of the typical tree is finite, and the network consists of a large number of mutually overlapping loops of finite length, see Figure 2 and eq 11. The factor Q(0) in the second term of expression 13 determines the probability of formation of such

(20)

(ii) Experimentally, it is pretty hard to reach total conversion at the network preparation conditions. In the case of incomplete conversion, p < 1 (but still far from the gelation threshold, |1 − p| ≪ 1), in expression 19, instead of functionalities f and f i, their average values should be substituted, pf and pf i. Such dependence of the network elastic modulus on the conversion is in agreement with numerical simulations of Gaussian networks, performed in ref 17. 4149


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at x1 = 0 of the experimental elastic modulus is well below the prediction of the classical theory of phantom network, eq 8. The difference between the experimental and theoretical values is due to the novel contribution to the elastic modulus. This second contribution is always negative and gives an impact of a large number of typical loops of the network. It describes the effect of shunting of the elastically effective chains by finite size polymer loops and depends on the interaction of monomers at the preparation conditions of the network. We also demonstrated that both contributions to the network elastic modulus can be represented by the generalized combined chain model with an additional effective chain (see Figure 1b) that accounts the cumulative effect of finite size loops in actual networks. At the end, we briefly list some possible generalizations of our approach. Although in this paper we studied only Gaussian networks, the theory can be generalized for the case of strong non-Gaussian deformations. Only phantom networks were considered in this paper. The approach proposed in ref 25 allows to introduce an entanglement tube into the replica space and describe the case of topologically entangled network strands. This approach leads to an additional term in the free energy of the network, which takes into account the entanglements. One of the most intriguing applications of the proposed approach is the description of spatially inhomogeneous networks, for example, near nanoparticles or other inclusions in composite networks. In this case, the order parameter φ(0) is not a constant but becomes a function of the coordinates describing the change in the molecular structure of the network near such inclusions.

closed loops. These loops shunt elastically effective chains of the network, an effect that is taken into account by an additional effective chain in the model of the combined chain shown in Figure 1b. Comparing eq 22 with 23, we estimate the total number of monomers of the effective closed loop as n2 + N = 2/[v(0)Q (0)]

(24)

The stronger the excluded volume effect, v(0), and the greater the loop formation probability Q(0), the shorter the effective shunt loop. In Figure 3a,b, we show the network configurations contributing to the corresponding terms in eq 22.

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DISCUSSION Investigation of the structure of loops in polymer networks is of fundamental significance for the understanding of their elastic properties. Most researchers consider the loops as defects and study their contribution to the elasticity of polymer networks in the ideal defect gas approximation. In this approximation, only explicitly treated loops of small concentration can be taken into account. In actual networks, there are both topological defects and typical loops, which are not sparsely distributed. In this paper, we study the cumulative effect of a large number of strongly overlapping typical loops of finite size using an effective mean field approximation. This approximation works for polymer networks because of the large overlap parameter of network strands, P(0) ≫ 1, see eq 1. As shown in Appendix, the properties of polymer networks are described by the liquid−solid (sol−gel) order parameter φ1(ς), which is actually not a parameter, but a function of the variable ς (see eq 30 in Appendix), and is determined by a complex integrodifferential equation. The only, but very important exception is the elastic modulus, which is expressed through the value of the order parameter at ς = 0, and the corresponding equations for φ1(0) become algebraic. This greatly simplifies the calculations of the elastic modulus of the network with an arbitrary number of cyclic fragments of finite size. In contrast to the classical theory of phantom networks, the mean field of loops, φ1(0) (see Appendix), explicitly depends on the excluded volume parameter v(0) and the density of monomers ρ(0) at network preparation conditions. This dependence takes into account the limitations imposed by the packing of highly overlapping typical loops in a real 3D space on the molecular structure of the network being formed. We show that the resulting elastic modulus of actual networks has two main contributions, see eqs 13 and 19. The first of them can be interpreted as the contribution of elastically effective network strands. Only this contribution is predicted in the classical model of phantom networks. We calculated the renormalization of this classical expression because of the presence of topological defects in the networkprimary loops and cyclic fragments of finite size. Our prediction, eq 19, generalizes the elastic modulus calculated in the approximation of an ideal defect gas (the so-called “network theory with strand prestrain”9) to the case of arbitrary concentration of primary loops. Its expansion in a series in the parameters xi (the number of cyclic fragments per cross-link) with accuracy up to first-order terms reproduces the result of this work.9 Comparison with experimental data at small x1 and all other xi = 0 has already been made, see Figure 5 of the discussed work. Note that the extrapolated limit value

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APPENDIX

Replica Model of the Network

In this Appendix, we present the main ideas of the replica model of the network prepared by random end-linking linear polydisperse chains by f-functional cross-links. Replica Space

The average free energy F of the final system (a deformed network) is expressed through the analytic continuation to m = 0 of the free energy Fm of the replica system F = lim

m→0

dFm dm

(25)

The replica system consists of the initial system (the network at preparation conditions) and m identical replicas of the final system,19 see Figure 5. The network has an identical molecular structure both at initial and final systems with network strands adopting different conformations. One can combine the coordinates x(0) in the initial system and x(k) (k = 1, ... , m) in m replicas of final systems into single vector x̂ = (x(0), x(1), ... , x(m)) in a 3(1 + m) dimensional replica space, see Figure 5. The monomer densities in the initial, ρ(0)(x(0)), and in the final, ρ(k)(x(k)), k = 1, ... , m, systems are expressed through the monomer density in the replica space, ρ(x̂) ρ(k)(x(k)) =

∏ ∫ dx(l)ρ(x̂) l≠k

(26)

Hamiltonian

In the initial system, it is extremely important to take into account the interaction of the network monomers, as in neglecting the excluded volume effects, the polymer network 4150


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Macromolecules

point. In addition to make the consideration strict, one has to perform analytical continuation of the field u(0) to imaginary values. We do not use this trick, since fluctuation modes of the field φ(x̂) in the maximum direction of the Hamiltonian H[φ,u(0)] (fluctuations of the monomer density ρ(0)(x(0)) in initial system), “compensate” the corresponding fluctuations of the field u(0)(x(0)). The field φ(x̂) has the meaning of the order parameter of the liquid−solid (sol−gel) phase transition. The steepest descent solution φ1(ς) > 0 depends on a single variable ÄÅ ÉÑ ÑÑ 1 ÅÅÅÅ 2 2Ñ ς = ÅÅx̂ − ∑ (eα̂ x̂) ÑÑÑ ÑÑ 2 ÅÅÅ ÑÖ (30) α Ç

Figure 5. Replica system consists of the initial system (k = 0)the polymer network at preparation conditions (blue cube) and m identical replicas of the final system (k = 1, ... , m)the network, deformed by factors λα along the principal axes α = x, y, z of deformation (yellow cuboids). The polymer network in the replica space of dimension 3(1 + m) is mainly localized inside the (green) cylinder with the diameter R ≃ aN̅ 1/2 directed along unit vectors {êα}, corresponding to affine deformation of the polymer. The networks (and all their monomers and strands) in the initial system and any of the m replicas of the final system are the projections of the network in the replica space onto the corresponding subspaces k = 0, ... , m.

where êα is a unit vector in the replica space along the direction (0) x(k) α = λαxα of the affine deformation and λα are deformation ratios along the principal axes α = x, y, z of deformation, see Figure 5. This function falls exponentially in the replica space far from the cylinder of diameter R ≃ aN̅ 1/2 directed along the axes {êα} of the affine deformation of the network, see Figure 5. In contrast to usual solids, the “order parameter” φ1(ς) is really a function, describing the distribution of elastic strands to ground. This function is determined by the differential equation. However, the free energy of the network depends only on its value at ς = 0. At ς = 0, the differential equation becomes an algebraic one with the solution ÄÅ É ÅÅ (f − 1)! ÑÑÑ1/(f − 2) ÅÅ ÑÑ ÑÑ φ1(0) = ÅÅ ÅÅ Nz ÑÑ ̅ f (31) ÅÇ ÑÖ

collapses to the size of a single strand. Following the idea of φ4 formulation of the excluded volume problem,26 we introduce n → 0 component field, φ(x̂) in the replica space, which is related to the monomer density as n

ρ (x ̂ ) =

1 2 1 φ (x̂) = ∑ φi 2(x̂) 2 2 i=1

(27)

where N̅ = (μ + u(0))−1 is the average number of monomers between the neighboring cross-links. Substituting eq 29 into 25, we find the free energy of the network13 ÄÅ ÉÑ 2 ÅÅ ÑÑÑ λ (0) Å 1/2 α + ln(aN )ÑÑÑ F = V GÅÅÅ∑ ÅÅ ÑÑ ÅÇ α 2 ÑÖ (32)

The free energy Fm of the network in replica system is described by Hamiltonian ÄÅ É ÅÅ 1 2 z f f ÑÑÑÑ a2 ̂ 2 Å H[φ] = dx̂ÅÅÅ μφ + (∇ φ) − φ1 ÑÑÑ ÅÅÇ 2 f ! ÑÑÖ 2 Ä É2 Å Ñ Å ÑÑ Ñ v(0) ÅÅÅ ÅÅ∏ dx(k)φ 2 ÑÑÑ + dx(0) Å ÑÑ 8 ÅÅ k ≠ 0 ÑÑÖ (28) ÅÇ Here, ∇̂ is the gradient of the variables x̂, μ is the monomer chemical potential, and zf is the fugacity of f-functional crosslinks. The last term in eq 28 describes the two-body monomer interaction in the initial system characterized by the excluded volume parameter v(0). As this term “spoils” the symmetry of the Hamiltonian in the replica space, we transform it by introducing a new field u(0) (x(0)), conjugated to the monomer density in the initial system, ρ(0)(x(0)), and find the Hamiltonian depending on two fields φ(x̂) and u(0)(x(0)) É ÅÄÅ z f f ÑÑÑÑ ÅÅ μ + u(0) 2 a2 ̂ 2 (0) Å φ + (∇ φ) − φ1 ÑÑÑ H[φ , u ] = dx̂ÅÅ ÅÅ 2 2 f ! ÑÑÑÖ ÅÇ (u(0))2 − dx(0) (0) (29) 2v

∫

∫

if y G = ρf(0)jjj − 1zzz kT k2 {

with the elastic modulus

∫

where

ρ(0) f

(33)

is the cross-link concentration.

Effective Hamiltonian

Fluctuations of the fields φ and u(0) can be accounted by the effective action method,27 replacing the bare Hamiltonian H[φ,u(0)] by the effective Hamiltonian Heff[φ,u(0)], taking into account the contribution from all loops. As in the mean field approximation, the extremum of the effective Hamiltonian in the fields φ(x̂) and u(0)(x(0)) determines a new ground state, which includes exactly the loop effect on network elasticity. The extreme value of the field u(0) = v(0)ρ(0) has a meaning of the average field acting on monomers because of their interaction at preparation conditions. The fluctuations of this field due to the temporary contacts of the network monomers in the process of thermodynamic fluctuations in the initial system are described by corresponding loop diagrams. Specifically, the loops can be either permanent, quenched into the topological structure of the network, or temporary, see Figure 4. The extremum value of the field φ1(ς) depends only on one variable ς, eq 30, and varies in the interval (0, φ1(0)). In the limit m → 0, the difference of the values of the effective

∫

∫

Before studying the properties of the loops, we reproduce the results of the mean field approximation, which implies that the network has a treelike structure with infinite loops (Figure 1a), see ref 13 for all details. In this approximation, the steepest descent values of the fields φ1 and u(0) are found from the extremum of the Hamiltonian in eq 29. The steepest descent solution does not correspond to a minimum, but the saddle 4151


Article

Macromolecules In the limit m → 0, eq 37 takes the form

Hamiltonian at the boundaries of this interval determines the elastic modulus of the network (see eq 32) G=

Heff [φ , u(0)]

kT

lim {Heff [φ1(0), u(0)] − Heff [0, u(0)]} V (0) m → 0

V (0)

(34)

In the mean field approximation, Heff = H, and this expression turns to eq 33. The field φ1(ς) describing the distribution of elastic strands to the ground is determined by a complex integrodifferential equation. However, the elastic modulus of the network depends only on the integral characteristic of this distribution, φ1(0). The condition ς = 0 corresponds to affine deformation of average end-to-end distances of all strands, and eq 34 determines the elastic modulus of a random (nonfluctuating) grid in which each strand is replaced by a corresponding elastic thread with the same elastic stiffness coefficient. The effective Hamiltonian also determines the free energy of the initial system (0)

+

ρf(0)

1

∂F (0)(μ , z f )

kTV (0)

∂μ

=−

Λ q [φ1(0)] L

Λ⊥q

(41)

L

First, we calculate the contribution of small loops of the first order, l = 1 in network elasticity. Expanding the logarithm of the eigenvalues in eq 41 in the limit a2qL2 ≫ τ⊥, τ∥, we find ÄÅ ÉÑ Λ q [φ(0)] ÑÑ zf 1 ÅÅÅ f −2 L ÑÑ Å ln (0) ≃ τ − τ − φ Å ÑÑ ⊥ ÑÑÖ (f − 2)! 1 a 2qL 2 ÅÅÅÇ Λ⊥q L

The fugacity zf of cross-links and the chemical potential μ of monomers can be obtained from the conditions on the density of cross-links and monomer density ρ(0) =

dq

∫ (2πL)3 ln

Primary Loops

(35)

m→0

1 2

When substituting this expression into eq 34 for the elastic modulus, τ⊥ and τ∥ should be considered as independent parameters, the value of which is determined from eq 36 for the chemical potential μ and the fugacity zf. Different interpretations of the φ1(0) dependence of τ∥ in eqs 34 and 35 are related to different functions φ1(0) and φ1(ς) in these expressions before performing the limit m → 0 in eq 34. After this limit, only the explicit dependence on φ1(0) should be taken into account in eq 34.

(0)

F (μ , z f ) = kT lim Heff [φ1 , u ]

zf (u(0))2 μ + u(0) 2 φ1 (0) − φ1 f (0) − f! 2 2v(0)

=

(42)

,

1

∂F (0)(μ , z f )

kTV (0)

∂ln z f

The divergence of the integral with respect to qL in eq 41 is associated with the inadequacy of the Gaussian chain model on small spatial scales 0 and turn to zero at qL = 0 only at the crosslink saturation threshold.13

where Q(0) is the probability of a primary loop, eq 14. Using this expression in eqs 35 and 36, we find

L

(f − 2)!

ρ(0) = ρf(0) 4152

1 2 φ (0), 2 1 zf Q (0) zcφ f − 2(0) = φ1 f (0) + 4(f − 2)! 1 f!

(46)


Article

Macromolecules The fields u(0) and φ1(0) are determined by the maximum condition of the functional (eq 45), in which τ⊥ is the function of φ1(0), as defined in eq 39

(6) Gu, Y.; Zhao, J.; Johnson, J. A. A (Macro)Molecular-Level Understanding of Polymer Network Topology. Trends Chem. 2019, 1, 1−17. (7) Zhong, M.; Wang, R.; Kawamoto, K.; Olsen, B. D.; Johnson, J. A. Quantifying the impact of molecular defects on polymer network elasticity. Science 2016, 353, 1264−1268. (8) Lin, T.-S.; Wang, R.; Johnson, J. A.; Olsen, B. D. Topological Structure of Networks Formed from Symmetric Four-Arm Precursors. Macromolecules 2018, 51, 1224−1231. (9) Lin, T.-S.; Wang, R.; Johnson, J. A.; Olsen, B. D. Revisiting the Elasticity Theory for Real Gaussian Phantom Networks. Macromolecules 2019, 52, 1685−1694. (10) Lang, M. Elasticity of Phantom Model Networks with Cyclic Defects. ACS Macro Lett. 2018, 7, 536−539. (11) Panyukov, S.; Rabin, Y.; Feigel, A. Solid Elasticity and LiquidLike Behaviour in Randomly Crosslinked Polymer Networks. Europhys. Lett. 1994, 28, 149−154. (12) Rubinstein, M.; Panyukov, S. Elasticity of Polymer Networks. Macromolecules 2002, 35, 6670−6686. (13) Panyukov, S.; Rabin, Y. Statistical Physics of Polymer Gels. Phys. Rep. 1996, 269, 1−131. (14) Cai, L.-H.; Panyukov, S.; Rubinstein, M. Hopping Diffusion of Nanoparticles in Polymer Matrices. Macromolecules 2015, 48, 847− 862. (15) Lang, M.; Kreitmeier, S.; Göritz, D. Trapped Entanglements in Polymer Networks. Rubber Chem. Technol. 2007, 80, 873−894. (16) Lang, M.; Michalke, W.; Kreitmeier, S. A statistical model for the length distribution of meshes in a polymer network. J. Chem. Phys. 2001, 114, 7627−7632. (17) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: Oxford; New York, 2003. (18) Gusev, A. A. Numerical Estimates of the Topological Effects in the Elasticity of Gaussian Polymer Networks and Their Exact Theoretical Description. Macromolecules 2019, 52, 3244. (19) Deam, R. T.; Edwards, S. F. The Theory of Rubber Elasticity. Philos. Trans. R. Soc. London, Ser. A 1976, 280, 317. (20) Panyukov, S. V.; Rabin, Y. Replica Field Theory Methods in Physics of Polymer Networks, in theoretical and mathematical models in polymer research. In Modern Methods in Polymer Research and Technology; Grosberg, A. Y., Ed.; Acad. Press, 1989; pp 83−185. (21) Lang, M.; Göritz, D.; Kreitmeier, S. Length of Subchains and Chain Ends in Cross-Linked Polymer Networks. Macromolecules 2003, 36, 4646−4658. (22) Lang, M. Monomer Fluctuations and the Distribution of Residual Bond Orientations in Polymer Networks. Macromolecules 2013, 46, 9782−9797. (23) Panyukov, S. V. Scaling theory of high elasticity. Sov. Phys. JETP 1990, 71, 372−379. http://jetp.ac.ru/cgi-bin/dn/e_071_02_0372. pdf. (24) Wang, R.; Alexander-Katz, A.; Johnson, J. A.; Olsen, B. D. Universal cyclic topology in polymer networks. Phys. Rev. Lett. 2016, 116, 188302. (25) Panyukov, S. V. Topological Interactions in the Statistical Theory of Polymers. Sov. Phys. JETP 1988, 67, 2274−2284. http:// www.jetp.ac.ru/cgi-bin/dn/e_067_11_2274.pdf. (26) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, New York, 1979. (27) Zinn-Justin, J. Quantum Field Theory and Critical Phenomena; Clarendon Press: Oxford, 1989.

u(0) = v(0)ρ(0) , zf

μ + u(0) − −

zf (f − 3)!

(f − 1)!

φ1 f − 2(0) + v(0)

φ1 f − 4(0)

Q (0) =0 4

Q (0) 2 (47)

Substituting the solution of eqs 46 and 47 into eqs 45 and 34, we find eq 13. Cyclic Fragments

The Hamiltonian in eq 28 of the network with ring fragments has additional contributions i

−∑ θi ∏ k=1

i

zf (f − 2)!

∫ dxî φ1 f −2(xî )G(xî − xî +1)

(48)

To simplify the notation, we adopted x̂i+1 = x̂0 and the function G depends on the polydispersity of the internal chains of the ring. The factors θi depend on the method of network synthesis (θi ∼ 1 for instant cross-linking) and are determined by the known fractions xi of the ring fragments per cross-link using equations (see also eq 36) xiρf(0) = −

1

∂F (0)(μ , z f , {θi})

kTV (0)

∂ln θi

(49)

In the m → 0 limit, the effective Hamilton in eq 45 gains additional contributions i

−

∑ θi ∏ i

k=1

zf (f − 2)!

φ1 f − 2(0)

∫ dxiG(xi − xi+1)

(50)

Repeating the calculations of the previous section, we find expression 19 for the elastic modulus of such a network.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Sergey Panyukov: 0000-0003-0514-3310 Notes

The author declares no competing financial interest.

■

REFERENCES

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