Evolution of time constants during sol-gel transition

Friedrich, Chr.; Heymann, Lutz; Berger, H.‐R.

doi:10.1007/bf01332925

Cited by 17 publications

(9 citation statements)

References 9 publications

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“…This longest relaxation time increases with growing cluster size (increasing extent of reaction) and, at the gel point, diverges to infinity. The relaxation exponent n can also be a function of p . , When modeling the relaxation spectrum of critical gels of high molecular weight model polybutadiene precursors with a combination of BSW and CW spectra, De Rosa and Winter 4 found it necessary to decrease the slope of the entanglement power law, n e , which is a possibility also considered in the present work by introducing a modified slope, An e , in this region.…”

Section: Relaxation Patterns and Model Spectrummentioning

confidence: 79%

“…Several models for the relaxation behavior of nearly critical and critical gels have been presented. ,− None of these models was able to describe all experimental observations in the vicinity of the gel point. Either the models were restricted to the material at the gel point, or they were developed for small-molecule precursors only, or they simply did not yield good agreement with experimental data.…”

Section: Introductionmentioning

confidence: 99%

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Relaxation Patterns of Nearly Critical Gels

Mours¹,

Winter²

1996

Macromolecules

142

116

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We measure the complex rheological behavior of nearly critical gels and analyze the data by searching for characteristic patterns and abstracting those patterns into a self-consistent model. The sample is a linear, flexible, nearly monodisperse polybutadiene which gets cross-linked on its vinyl side groups. The dynamic mechanical storage and loss moduli of cross-linking polymers change smoothly during the liquid-solid transition, while equilibrium rheological properties (e.g., zero-shear viscosity and equilibrium compliance) diverge. During gelation, the relaxation occurs in a distinct pattern which can be described in a quantitative way with a minimum number of parameters. The pattern can be understood as a combination of the BSW spectrum (representing the precursor relaxation behavior) and the selfsimilar Chambon-Winter gel spectrum (modeling the terminal relaxation due to growing clusters). The spectrum is cut off at the material's longest relaxation time, λmax. Our model parameters, λmax and Ge (equilibrium modulus), exhibit characteristic scaling behavior with respect to the distance from the gel point, |p -pc|. The relaxation exponent, n, in the terminal zone is a function of the extent of reaction. Hence, dynamic scaling (requires constant n values) is not valid for our system. The proposed model passes the self-consistency test by predicting the mechanical behavior (at different frequencies) as a function of the extent of reaction and other rheological observations during the sol-gel transition.

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Section: Relaxation Patterns and Model Spectrummentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Relaxation Patterns of Nearly Critical Gels

Mours¹,

Winter²

1996

Macromolecules

142

116

View full text Add to dashboard Cite

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“…Theoretically, the application of the Boltzmann's superposition principle to the complex modulus G * gives the expression of η*(ω) by the stress relaxation modulus G ( t ): 18 To fit η*(ω) data to this equation, the form of G ( t ) has to be known in advance. Friedrich et al proposed to represent the relaxation function prior to the sol−gel transition. Here S is the quality related to the stiffness of a pregel, k the parameter, and τ 0 the relaxation time, all of which are determined by the physical nature of a gelling material.…”

Section: Resultsmentioning

confidence: 99%

Rheological Images of Poly(vinyl chloride) Gels. 2. Divergence of Viscosity and the Scaling Law before the Sol−Gel Transition

Uchida

Aoki

et al. 1997

Macromolecules

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The zero shear viscosity η0 of poly(vinyl chloride) (PVC)/bis(2-ethylhexyl) phthalate (DOP) pregels has been measured as a function of polymer concentration c as well as molecular weight. It was observed that the zero shear viscosity diverged as the gelling system approached the gel point. The establishment of the scaling law, η0 ∝ ε-γ, was examined where γ is the scaling exponent and ε the relative distance defined as (c g − c)/c g. Here c g is the critical concentration for the sol−gel transition. Two methods were used to determine the scaling exponent γ. One is the direct determination of γ using the c g obtained by means of the frequency independence of loss tengent. The other is called the Takigawa method that determines simultaneously γ and c g by mathematically transforming the scaling law into −η0 -1 (dη0 -1/dc)-1 = (c g − c)/γ. Good agreement was obtained between the two methods. The scaling law, η0 ∝ ε-γ, was found to hold well for the PVC pregels, and the scaling exponent γ was a constant (=1.5 ± 0.1) and was independent of the PVC molecular weight. The results suggest that the gelation rate defined as −dη0/dε (=γη0/ε) is related to PVC molecular weight, and at the same ε increases with decreasing molecular weight. The errors in determining the scaling exponent γ were also discussed.

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“…When applied to oscillatory shear flow, the intermediate model leads to simple power-law dependences on frequency for both dynamic moduli with the same exponent a [443], and, hence, eqn. Since the early 80's, the fractional derivative approach expanded in several directions, to define more sophisticated models or to find a link between molecular theories and the empirical fractional calculus approach to viscoelasticity, also examining the models in the light of their consistency with thermodynamics principles [444][445][446][447][452][453][454][455][456][457][458][459]. Actually, the efforts of generalization are addressed in two directions: Friedrich's approach is based on the standard solid model, the Zener model, and can be designated as the fractional differential standard solid model (FDM) [459], whereas Nonnenmacher 's model starts from the integrated version of the standard solid model (FIM) [458].…”

Section: Viscoelastic Modelsfor Physical Gelsmentioning

confidence: 99%

Rheology of polysaccharide systems

Lapasin

Pricl

1995

Rheology of Industrial Polysaccharides: Theory and Applications

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The transport properties and, specifically, the rheological behavior of real and complex materials such as polysaccharide systems can be significantly affected by several factors, mainly related to molecular and supermolecular features. Most of these factors are common to all polymeric systems, as they have universal character; others are peculiar to carbohydrate polymers. If we bear in mind the concepts discussed in §1.4, we can easily imagine for polysaccharides a variety of structural conditions much wider than that generally observed for synthetic polymers. On both molecular and supermolecular scales, polysaccharides possess special characteristics that reflect specific behavior so that different classes of materials can be identified.Let us consider the molecular scale first: at this level, the polymer backbone and its related features, such as its length in solution, shape, stitfuess and the eventual presence of ionizable groups, playa major role in determining the macroscopic properties of the system, including its rheology. Many of these molecular parameters correlate among themselves (e.g. chain stitfuess and length); others must be combined with external factors such as, for instance, the characteristics of the solvent medium. These latter may, in turn, exert an influence on the conformation adopted by the macromolecule in the medium itself, as well as on the possibility of forming supermolecular structures. In other words, changes in ionic strength, the nature of counterions, temperature or other solvent parameters may induce a conformational transition, thus modifying the hydrodynamic resistance of the macromolecule to flow. Also, if the polymer concentration is high enough, the variations described above may promote structural changes at a supermolecular level and, accordingly, modifications in the rheological behavior. Before leaving this brief comment on solvent characteristics we must mention the viscosity although, R. Lapasin et al. (eds.), Rheology of Industrial Polysaccharides: Theory and Applications

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Evolution of time constants during sol-gel transition

Cited by 17 publications

References 9 publications

Relaxation Patterns of Nearly Critical Gels

Relaxation Patterns of Nearly Critical Gels

Rheological Images of Poly(vinyl chloride) Gels. 2. Divergence of Viscosity and the Scaling Law before the Sol−Gel Transition

Rheology of polysaccharide systems

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