Article Cite This: Macromolecules XXXX, XXX, XXX−XXX
pubs.acs.org/Macromolecules
Revisiting the Elasticity Theory for Real Gaussian Phantom Networks Tzyy-Shyang Lin,† Rui Wang,† Jeremiah A. Johnson,‡ and Bradley D. Olsen*,† †
Department of Chemical Engineering and ‡Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States
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S Supporting Information *
ABSTRACT: In the classical phantom network theory, the shear modulus of a polymer network is derived assuming the underlying network has a treelike topology made up of identical strands. However, in real networks, defects such as dangling ends, cyclic defects, and polydispersity in strand sizes exist. Moreover, studies have shown that cyclic defects, or loops, are intrinsic to polymer networks. In this study, we illustrate a general framework for calculating the rubber elasticity of phantom networks with arbitrary defects. Closed form solutions for the elastic effectiveness of strands near isolated loops and dangling ends are obtained, and it was found that under classical assumptions of phantom network theory loops with order ≥3 have zero net impact on the overall elasticity. However, when a simple approximation for strand prestrain is considered, the modified network theory agrees well with experimentally measured moduli of PEG gels.
I. INTRODUCTION Polymer networks are used in a myriad of applications, ranging from everyday commodity materials1−4 to advanced functional systems.5−9 A quantitative understanding of how molecular structure impacts the mechanical properties of polymer networks is crucial to the design of new materials. However, even after decades of research, linear elasticity, one of the most important and fundamental attributes of a polymer network, is not fully understood. While models have been proposed to describe the elastic modulus of polymer networks, many experimental observations are still unexplained. One example is the phantom to affine transition recently observed in tetraPEG gels.10 Furthermore, recent developments in experimental techniques have provided new information about the microscopic molecular connectivity of networks, such as the local cyclic topologies, which was previously unavailable.11−15 These cyclic defects are intrinsic to polymer networks.16−18 These studies motivate revisiting the molecular theory of rubber elasticity for polymer networks and re-examining the assumptions and applicability of different phantom models, which will serve as a tool for the quantitative design of materials as well as a foundation for the accurate quantification of the topological contributions of a gel or elastomer’s mechanical properties. Theories describing rubber elasticity have advanced significantly over the past few decades. For example, the elasticity © XXXX American Chemical Society
model developed by Rubinstein and Panyukov effectively describes the strain softening behavior of elastomers.19 In their model, network strands are considered to be constrained by effective phantom chains to the nonfluctuating background, which provides a conceptually simple description of the elastic behavior of network strands. More recently, a physics-based model with the minimal possible number of parameters proposed by Davidson and Goulbourne further captures the strain hardening behavior of elastomeric networks at very large strain.20 However, the elastic moduli for individual networks are left as adjustable model parameters, and the quantitative predictions of the elastic modulus require the development of more elaborate network theories. For unentangled networks, the most fundamental network theory is the phantom network theory. In classical phantom network theory, every strand in a perfect network is assumed to bear the same stress when a macroscopic strain is applied, leading to the direct proportionality between the shear modulus G and the density of polymer strands ν in a network, G = CνkBT, where kB is the Boltzmann constant and T is the temperature. The proportionality constant C is the elastic effectiveness of each Received: August 3, 2018 Revised: October 22, 2018
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DOI: 10.1021/acs.macromol.8b01676 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
conversion). These observations suggest that the first-order effects of defects dominate over the higher-order effects for a considerably large range. Therefore, a quantitative theory that accurately captures the linear behavior is both theoretically important and useful in practical applications for quantitative mapping between the elastic moduli of polymer networks and the number of defects.
strand in the network, denoting the contribution of a single strand to the shear modulus in units of thermal energy per unit volume kBT/V. For an ideal treelike network, the phantom network theory elastic effectiveness is Cph = (f − 2)/f, where f is the functionality of the junctions in the network.21 However, recent advances in network theory found that loops of finite size have significant impacts on elasticity. The real elastic network theory (RENT)22 demonstrated that the impact of loops of finite size must be accounted for to explain the deviation of the modulus from the phantom network value. It was found that the elastic effectiveness of each strand becomes dependent on the exact local connectivity near the strand (even if the strand is not in a loop), and the macroscopically observed overall elastic effectiveness C of the network manifests the ensemble average of the elastic effectiveness Cη of different strands in the network, C = ⟨Cη⟩. In RENT, while the calculations for loops with one or two junctions are exact, the treatment for larger loops is only approximate. Later, RENT was extended to account for dangling ends, and the predictions were compared to the moduli of tetra-PEG gels that are not fully converted.23 More recently, Lang has proposed an equivalent resistance method to exactly solve for the impact of finite loops with arbitrary order on elasticity.24 Lang’s theory predicts that finite sized loops with order ≥3 have greater negative impacts on elasticity than that predicted by RENT. Lang also pointed out that the difference in the mean end-to-end distance of a strand after gelation, and its free chain counterpart is an important factor not considered so far in the network theories. The extent of prestrain experienced by strands may differ significantly for strands in loops of different sizes,25,26 and these effects can contribute significantly to the elasticity of the network. However, while Lang developed an ingenious approach to exactly solve for the junction fluctuation using a recursive method, the procedure is complicated and difficult to track analytically even for symmetric cases. Furthermore, while the extent of junction fluctuation is exactly captured, his calculation of the elastic effectiveness of loop strands is inexact. This calls for a more general network theory framework for the treatment of defects. This work re-examines the results of the aforementioned network theories with a mathematically exact framework and investigates how different assumptions in the phantom network theory affect the calculated elasticity of a polymer network. Three different types of defects, including cyclic defects, dangling ends, and dispersity of network strand sizes, are treated exactly. The quantitative predictions of shear moduli provided in this work complement models that focused on describing the features of stress−strain curves, such as the models proposed by Rubinstein and Panyukov19 and Davidson and Goulbourne.20 This eliminates the need of treating the shear modulus as a fitting parameter. However, it should be noted that all the discussion in this article is carried out under the ideal defect gas approximation,23 where defects are assumed to be sparsely distributed in an otherwise ideal network, and that the impact of each defect can be calculated separately, with their overall impact being simply the algebraic sum of contributions from individual defects. While this assumption is strictly only applicable in the limit of infinite dilution of the defects, experimental studies on the impact of loops22 and dangling ends27 showed that the impacts on shear moduli of gels are linear functions of the concentrations of defects (fraction of primary loop and the distance from full conversion) over quite large ranges (up to 20% primary loop fraction and 30% from full
II. GENERAL FORMULATION Following the approaches of the classical phantom network theory,28−30 we assume the mean end-to-end distances of network strands deform affinely with macroscopic strain. The fluctuations of the end-to-end distances are also assumed to remain unchanged under deformation, making this approach applicable in the small deformation regime. The free energy increase under linear deformation can be calculated by summing the contributions of individual strands, and the resulting elastic effectiveness of a single network strand η with degree of polymerization Nη can be expressed as the difference between its normalized mean end-to-end distances and the normalized extent of fluctuation of its end-to-end distances (see the Supporting Information for detailed derivation): Cη = αη − βη
(1)
The coefficient αη = ⟨rη 2⟩0 /Nηb2 = 2k η⟨rη 2⟩0 /3 denotes the ratio between the mean distance of network strands at undeformed state and the size of the corresponding free Gaussian chain Nηb2, or equivalently the product between the spring constant kη of the strand and its mean end-to-end distance, whereas βη = ⟨Δrη 2⟩0 /Nηb2 = 2k η⟨Δrη 2⟩0 /3 denotes the ratio between the extent of fluctuation in the separation between cross-links and the chain size. In previous studies,31−33 the end-to-end distribution of network strands is often assumed to be similar to that of in the pregelation melt; hence, the ratio α is usually assumed to be unity. When this condition is met, the problem of calculating the elastic modulus of a phantom network can be effectively reduced to the calculation of the extent of junction fluctuations in the network. Summing the contribution from each strand, the shear modulus of a network can be expressed as G = kBT /V
∑ Cη = ∑ [αnm − βnm] η
n1.
(3)
Because the connections are not directed, the matrix Γ is symmetric. The diagonal entries describe the connections to or from an individual cross-link, which is described by the sum of the row (or equivalently the column) of off-diagonal elements Γnn =
∑ Cη/Cph
III. ELASTIC EFFECTIVENESS OF NONLOOP STRANDS For a strand that is connected to the fixed background through two uncorrelated subnetworks, as illustrated in Figure 1a, the
∑ knm + kn,eff m≠n
(4)
where the additional term kn,eff reflects the connection of a junction to the exterior of the network. If all the connections to the exterior of the network are directly linked to nonfluctuating parts, then kn,eff is simply the spring constant of the strand that connects cross-link n to the nonfluctuating background. However, for networks that are subnetworks embedded within a larger network consisting of other cross-links that can also fluctuate, the magnitude of kn,eff is given by an appropriate value that retains the same dynamics. In the case where no cross-links in the subnetwork are correlated through its exterior, i.e., no loop that consists of both cross-links in the subnetwork and junctions outside the subnetwork exists, the magnitude of kn,ext can be determined by computing the electrical conductance of an analogous electrical network (cf. section III in the Supporting Information for details). It should be noted that this approach is equivalent to the virtual chain constraint concept presented by Rubinstein and Panyukov in their theory.19 The Kirchhoff matrix embodies all connectivity information about a Gaussian network, and its inverse describes the extents of the correlations between the fluctuations of different pairs of junctions. The off-diagonal terms of the inverse of the Kirchhoff matrix describe the cross-correlation between pairs of crosslinks, whereas the diagonal terms capture the autocorrelation of the fluctuation of individual junctions. If the weighted Kirchhoff matrix Γ is nondimensionalized by a reference spring constant kref by Γ = krefB, then the fluctuation term for cross-links n and m in eq 2 reads [(B−1)nn + (B−1)mm − (B−1)nm − (B−1)mn ] βnm = 1/|Bnm|
Figure 1. Schematic illustration of transforming complex networks into an effective network for (a) nonloop strands and (b) loop strands. The solid straight lines indicate nonfluctuating background.
network can be simplified into a three-strand equivalent network that consists the original strand of interest and two effective
(5)
phantom chains that connect the central strand to the
where the inverse of the Kirchhoff matrix is related to the fluctuation by (B−1)ij = 2k ref ⟨ΔR iΔR j⟩/3. Substituting eq 5 into eq 2 and summing over all pairs of junctions, the shear modulus of the network can be more compactly expressed as the trace of the product of inverse B and L
nonfluctuating background. This simplification procedure is
∑ βnm = tr(B−1L) = tr[B−1(B − Bext)] = tr(I − B−1Bext) n
