On the Effective Lifetime of Reversible Bonds in Transient Networks

Abstract

The renormalized bond lifetime model (RBLM) is a popular scaling theory for the effective lifetime of reversible bonds in transient networks. It recognizes that stickers connected by a reversible bond undergo many (J) cycles of dissociation and reassociation. After finally separating, one of these stickers finds a new open partner in time 𝝉 open via a subdiffusive process whose mean-squared displacement is proportional to t 𝜶 , where t is the time elapsed, and 𝜶 is the subdiffusion exponent. The RBLM makes convenient mathematical approximations to obtain analytical expressions for J and 𝝉 open . The consequences of relaxing these approximations is investigated by performing fractional Brownian motion (FBM) simulations. It is found that the scaling relations developed in the RBLM hold surprisingly well. However, RBLM overestimates both 𝝉 open and J, especially at lower values of 𝜶. For 𝜶 = 0.5, corresponding to the Rouse limit, it is found that 𝝉 open is overestimated by a factor of approximately 4x, while the approximation for J is nearly exact. The degree of overestimation worsens as 𝜶 decreases, and increases to 1-2 orders of magnitude at 𝜶 = 0.25, corresponding to the reptation limit. This has important ramifications for experimental studies that use RBLM to interpret rheology and dielectric spectroscopy observations.

Time–Temperature Superposition of Polybutadiene Vitrimers

“…As this behavior occurs at frequencies above ω cross , we propose that subdiffusive segmental motions facilitate cross-link exchange events. This concept is consistent with previous theoretical arguments regarding reactive polymer systems. − …”

“…Although we introduced the σ parameter in our previous work, the relationship between cross-link mobility and τ XL has already been studied through the lens of microscopic theories, albeit mostly for networks with dissociative cross-links that rearrange via an elimination/addition pathway. ,, For the Renormalized Bond Lifetime Model (RBLM) of Rubinstein and Semenov, τ XL accounts for both the subdiffusive motions of backbone repeat units and cross-link reassociations that do not contribute to stress relaxation. , We derive explicit expressions for the cross-link mobility and long-time dynamics activation energy (σ RBLM and E a RBLM , respectively) using the RBLM framework and assuming the cross-links undergo Rouse-like subdiffusive motion σ normalR normalB normalL normalM = Z ( 1 + Z 3 exp true( prefix− E normala normals normalm R T true) ) E normala normalR normalB normalL normalM = E normala normalW normalL normalF + E normala normals normalm 1 + Z 3 0.25em exp ( − E a s m R T ) where Z is the number of times a cross-linkable group reassociates with an old partner (see the Supporting Information for derivation). Another transient network theory is the model of Ghosh and Schweizer (GS), in which local frictional resistance of the backbone segments attenuates the overall temperature dependence of...…”

“…Although we introduced the σ parameter in our previous work, 58 the relationship between cross-link mobility and τ XL has already been studied through the lens of microscopic theories, albeit mostly for networks with dissociative cross-links that rearrange via an elimination/addition pathway. 3,69,74 For the Renormalized Bond Lifetime Model (RBLM) of Rubinstein and Semenov, τ XL accounts for both the subdiffusive motions of backbone repeat units and cross-link reassociations that do not contribute to stress relaxation. 69,73 We derive explicit expressions for the cross-link mobility and long-time dynamics activation energy (σ RBLM and E a RBLM , respectively) using the RBLM framework and assuming the cross-links undergo Rouse-like subdiffusive motion…”

Time-temperature superposition of polybutadiene vitrimers

“…This concept is consistent with previous theoretical arguments regarding reactive polymer systems. 64,65,66,67,68,69,70,71,72,73 For the long time dynamics, the interpretation is much more complex. The beginning of this regime is signaled by 𝜏 𝑋𝐿 (or, equivalently, 𝜔 𝑐𝑟𝑜𝑠𝑠 ).…”

“…Although we introduced the 𝜎 parameter in our previous work, 58 the relationship between cross-link mobility and 𝜏 𝑋𝐿 has already been studied through the lens of microscopic theories, albeit mostly for networks with dissociative cross-links that rearrange via an elimination/addition pathway. 3,68,73 For the Renormalized Bond Lifetime Model (RBLM) of Rubinstein and Semenov, 𝜏 𝑋𝐿 accounts for both the subdiffusive motions of backbone repeat units and cross-link reassociations that do not contribute to stress relaxation. 68,72 We derive explicit expressions for the cross-link mobility and long time dynamics activation energy (𝜎 𝑅𝐵𝐿𝑀 and 𝐸 𝑎 𝑅𝐵𝐿𝑀 , respectively) using the RBLM framework and assuming the cross-links undergo Rouse-like subdiffusive motion…”