Revisiting the Elasticity Theory for Real Gaussian Phantom Networks

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.

Theory of Flexible Polymer Networks: Elasticity and Heterogeneities

“…In polydisperse networks, the distribution function of such virtual chains is determined by Equation 27, in which f − 1 has the meaning of the number of branches on the first generation of this tree. In the ideal defect gas approximation, the contribution of the structural defects to the elastic modulus of a perfect network was calculated in works [19][20][21][22]. Note that typical network loops are ignored in this approximation, which takes into account only explicitly treated loops of small concentration.…”

Polymer Networks: From Plastics and Gels to Porous Frameworks

“…Recently, Lang proposed an equivalent resistance method to exactly solve for the impact of finite loops on elasticity . Inspired by this approach, Olsen and co‐workers modified RENT and found that under classical assumptions of phantom network theory, loops with order three or above have zero net impact on the overall elasticity …”

“…[125] Inspired by this approach, Olsen and co-workers modified RENT and found that under classical assumptions of phantom network theory,loops with order three or above have zero net impact on the overall elasticity. [126] Deliberately designed model polymer networks have demonstrated both affine and phantom network behaviors. In 2013, by varying the polymer concentration in A 4 + B 4 Te tra-PEG gels,S akai and co-workers observed at ransition between phantom and affine behaviors near the overlapping concentration of the network precursors.…”

Polymernetzwerke: Von Kunststoffen und Gelen zu porösen Gerüsten

“…Kürzlich schlug Lang eine Methode der äquivalenten Widerstandskraft vor, um die Auswirkung begrenzter Loops auf die Elastizität exakt aufzulösen . Inspiriert von diesem Ansatz modifizierten Olsen und Mitarbeiter die REN‐Theorie und stellten fest, dass unter den klassischen Annahmen des Modells des Phantomnetzwerks die Nettoauswirkung der Loops der Ordnung 3 oder höher auf die Gesamtelastizität null ist …”

“…[125] Inspiriert von diesem Ansatz modifizierten Olsen und Mitarbeiter die REN-Theorie und stellten fest, dass unter den klassischen Annahmen des Modells des Phantomnetzwerks die Nettoauswirkung der Loops der Ordnung 3o der hçher auf die Gesamtelastizitätn ull ist. [126] Bei gezielt aufgebauten Modellpolymernetzwerken wurde sowohl affines Netzwerkverhalten als auch Phantomnetzwerkverhalten gefunden. 2013 beobachteten Sakai und Mitarbeiter beim Va riieren der Polymerkonzentration in Te tra-PEG-Gelen (A 4 + B 4 )e inen Übergang zwischen Phantomnetzwerkverhalten und affinem Netzwerkverhalten in der Nähe der sich überschneidenden Konzentration der Netzwerkvorstufen.…”