Scaling model of gelation

An important difference between randomly branched and linear polymers is that the fractal dimension of branched polymers is larger than the dimension of space (d—3). This severely limits the applicability of the mean-field theory to the crosslinking of long linear chains, called vulcanization. Long chains in the melt have a fractal dimension of P = 2, which leaves lots of room inside the pervaded volume of the chain (i.e., filled by other chains in a polymer melt). The extra room created by the linear sections between crosslinks allows the fractal dimension of P = 4 to exist in three-dimensional space on a certain range of length scales (see Section 6.5.4). 

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Diffusion-limited aggregation of particles results in a fractal object. Growth processes that are apparendy disordered also form fractal objects (30). Sol—gel particle growth has also been modeled using fractal concepts (3,20). The nature of fractals requires that they be invariant with scale, ie, the fractal must look similar regardless of the level of detail chosen. The second requirement for mass fractals is that their density decreases with size. Thus, the fractal model overcomes the problem of increasing density of the classical models of gelation, yet retains many of its desirable features. The mass of a fractal, Af, is related to the fractal dimension and its size or radius, R, by equationS ... 

As discussed above, the large length- and time-scales involved in most experimental realizations of gelation and other aspects of aerogel preparatimi are not directly accessible by molecular simulations. An alternative approach is to use coarse-grained models, which alleviate these scale problems at the expense of the loss of atomistic detail. The construction of coarse-grained models affords considerable freedom in choice of the primary objects simulated, their interaction, and the associated dynamics so much so, in fact, that much work has focussed on the simplest possible models, both to expose the most fundamental physics involved and to avoid laborious and possibly underdetermined parametrization problems. 

Fig. 4 The (scaled) gelation concentration (solid lines) of telechelic polymers and cmc of surfactants (dotted lines) plotted against the (scaled) concentration of surfactants for fixed multiplicity model. The multiplicity is varied frpm curve to curve. For s larger than 4, the gel concentration shows a minimum...

As for the description of gelation, the framework of scaling is frequently used to account for both experimental and theoretical research results. As is well known, pwwer law appears for many physical parameters to capture polymer pa-operties in the scaling law. Also in the model of Bethe lattice, quite a few power law dependences can be seen for gel fraction, degree of polymerisation etc. In Section 2.1.2, those relationships are shown to discuss the analogues between properties of gel and other physical phenomena... 

Another trend towards the model of a satisfactory interpretation of the experimental data comes from physics of phase transition which regards the gelation as an example of critical phenomenon. This trend is developing with the viewpoint of scaling approaches for the static and dynamic properties of gels. The scaling concepts introduced into the theory of polymer solutions were subsequently extended to the description of swollen networks. 

Both the Flory-Stockmayer mean-field theory and the percolation model provide scaling relations for the divergence of static properties of the polymer species at the gelation threshold. 

In these expressions, tg is the reaction time at gel point, s and t are the static scaling exponents which describe the divergence of the static viscosity, nO 6 , at trstatic elastic modulus. Go at tggelation mechanism has been discussed on the basis of several models based on the percolation theory (for review, see ref 16), that provide power laws for the divergence of the static viscosity and the elastic moduli. Characteristic values for the s, t and A exponents are predicted by each of these models (Table I). 

Modulus-frequency master curves have been constructed by applying appropriate time dependent renormalisation factors to the frequency and modulus individual data. From the scaling of these factors with reaction time, the static scaling exponents t and s have been calculated and observed to be independent of the chemical nature of the midblock, suggesting a unique gelation mechanism. For all the samples, 1.84scalar elasticity percolation model. 

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