Crosslinking Index, Molecular Weight Distribution and Rubber Equilibrium Shear Modulus During Polyfunctional Crosslinking of Existing Polymer, 6. Primary Polymer with a Schulz–Zimm Distribution of Molecular Weights

Abstract

A network model for the crosslinking of already existing polymer molecules with a so‐called Schulz–Zimm distribution of their molecular weights is presented. It is an extension of previously developed statistical network models applied to the crosslinking of primary polymers with several other molecular weight distributions and with crosslinks of any functionality. The model results in the possibility to obtain more insight into the structure of polymers, especially those with narrow distributions of the molecular weight. In more detail, the model can give a perspective on structural network parameters such as the weight fractions of ideal network, of dangling polymer ends, and of those molecules not connected to the network, i. e., the sol fraction, the number of crosslinks in which a polymer molecule is bound, the functionality of the crosslinks, or the average molar mass of the polymer molecules in between the crosslinks M̄c. Results of calculations are shown for a hypothetical crosslinking process of polymers with various molecular weight distributions. Moreover, the dependency of the network parameters on the polydispersity index and the type of molecular weight distribution is shown. Finally the increase of the functionality of the crosslinks during the ageing process of a 9.9% poly(vinyl chloride) gel as a function of the polydispersity index of the molecular weight distribution is presented.

Cross-Linking Behavior of Diskotic Side-Chain Polymers in Solution

“…Much later, te Nijenhuis generalized the model to comply with cross-links of any functionality and a polydisperse distribution of the molecular weights. [16][17][18] It has to be mentioned that there is much agreement between the results of the cross-linking process of high molecular weight polymer calculated with this model and those more specific presented in the literature (e.g., Charlesby et al, 19,20 Langley et al, [21][22][23] Graessley et al, 24,25 S ˇomva ´rsky et al, 26 and Peppas et al 27,28 ). For a so-called accumulated Schulz-Flory distribution with M h w /M h n g 2 the relationship between the equilibrium shear modulus, G e (determined with oscillatory rheological measurements), and the sol fraction, w s , is found to be where and * To whom correspondence should be addressed.…”

“…Nevertheless, Flory demonstrated the possibility of modeling tetrafunctional cross-linking of polymer molecules with a monodisperse distribution of the molecular weights at and beyond the gel point. Much later, te Nijenhuis generalized the model to comply with cross-links of any functionality and a polydisperse distribution of the molecular weights. − It has to be mentioned that there is much agreement between the results of the cross-linking process of high molecular weight polymer calculated with this model and those more specific presented in the literature (e.g., Charlesby et al, , Langley et al, − Graessley et al., , Šomvársky et al, and Peppas et al , ). For a so-called accumulated Schulz−Flory distribution with M̄ w / M̄ n ≥ 2 the relationship between the equilibrium shear modulus, G e (determined with oscillatory rheological measurements), and the sol fraction, w s , is found to be where and and where f is the functionality of the cross-links (i.e., the number of polymers leaving the cross-links), c is the mass concentration of polymer (kg/m 3 ), R is the gas constant (J/(mol K)), T is the absolute temperature (K), M̄ w is the weight-average molecular weight of the polymer molecules before cross-linking (kg/mol), and α is the monomer conversion during preparation of the polymer, which can be calculated from the polydispersity index ( D = M̄ w / M̄ n , where M̄ n is the number-average molecular weight (kg/mol)): …”

Unidirectional Dimerization and Stacking of Ureidopyrimidinone End Groups in Polycaprolactone Supramolecular Polymers

“…We have analyzed this network with the Flory-Stockmayer theory for tetrafunctional crosslinking of monodisperse high molecular weight polymers, extended by one of the authors for cross-links of any functionality and for polydisperse polymers. [74][75][76][77][78][79][80][81] Physical networks are in general far from ideal ( Figure 7). 82 Many polymer molecules are present which are not bound to the network (sol fraction w s ).…”

“…In the case of compound 2b , the physical network persists at a temperature of 130 °C, more than 70 °C above the melting point. We have analyzed this network with the Flory−Stockmayer theory for tetrafunctional cross-linking of monodisperse high molecular weight polymers, extended by one of the authors for cross-links of any functionality and for polydisperse polymers. − Physical networks are in general far from ideal (Figure ) 7 Schematic representation of a nonideal gel network. …”

Photochemistry of 2,6‐di(4′‐azidobenzylidene)‐methylcyclohexanone in polymer matrices

“…The average number of crosslinks per primary molecule, noted γ c and named crosslinking index, is in a relationship with the gel fraction g , given by the eq. . The crosslinking density, ρ, is derived form eq.…”