Critical gels

Righetti and CagUo [315] have also found that Bis reacts faster than acrylamide and that this leads to the formation of nonhomogeneous regions in the gel. At the critical gel point they found 50% of the acrylamide monomer and 80% of the Bis monomer had reacted. The reaction was found to continue beyond the gel point, with an eventual 99%... 

The polymer at the gel point is in a critical state [3], and the name critical gel [4] is appropriate for distinguishing polymers at the gel point from the various materials which commonly are called gels. The critical gel has no intrinsic size scale except for the size of its oligomeric building block, and molecular motions are correlated over large distances. The combination of liquid and solid... 

Material properties at a critical point were believed to be independent of the structural details of the materials. Such universality has yet to be confirmed for gelation. In fact, experiments show that the dynamic mechanical properties of a polymer are intimately related to its structural characteristics and forming conditions. A direct relation between structure and relaxation behavior of critical gels is still unknown since their structure has yet evaded detailed investigation. Most structural information relies on extrapolation onto the LST. 

The self-similar spectrum is not valid at short times, X < X0, where the details of chemical structure become important (glass transition, entanglements, etc.). The cross-over to the glass transition at short times is typical for all polymeric materials, for both liquids and solids. The critical gel is no exception in that respect. X0 could be used as a characteristic time in the CW spectrum since it somehow characterizes the molecular building block of the critical gel however, it has no direct relation to the LST. At times shorter than X0, the LST has no immediate effect on the rheology. Indirect effects might be seen as a shift in the glass transition, for instance, but these will not be studied here. 

For the relaxation of the solid near the gel point, the critical gel may serve as a reference state. The long time asymptote of G(t) of the nearly critical gel, the equilibrium modulus Ge, intersects the G(t) = St n of the critical gel at a characteristic time (Fig. 6) which we will define as the longest relaxation time of the nearly critical gel [18]... 

Fig. 6. Evaluation of the longest relaxation time for a sample beyond the gel point, p > pc intersect of horizontal line for Ge with the power law of the critical gel, St n...

Only two of the exponents (a and n, for instance) are sufficient to describe the rheology of nearly critical gels. The front factor is more difficult to estimate, but it most likely differs on both sides. 

Based on the fractal behavior of the critical gel, which expresses itself in the self-similar relaxation, several different relationships between the critical exponent n and the fractal dimension df have been proposed recently. The fractal dimension ds of the polymer cluster is commonly defined by [16,42]... 

We expect that the classical framework of linear viscoelasticity also applies at the gel point. The relaxation spectrum for the critical gel is known and can be inserted into Eq. 3-3. The resulting constitutive equation will be explored in a separate section (Sect. 4). Here we are mostly concerned about the material parameters which govern the wide variety of critical gels. 

Our first two critical gels had an exponent value of n as 0.5, which made us believe initially that this would be the only possible value... 

Fig. 11. Schematic of relaxation time spectrum of the critical gel of PBD 44 (Mw = 44 000). The entanglement and glass transition is governed by the precursor s BSW-spectrum, while the CW spectrum describes the longer modes due to the crosslinking [60]. denotes the longest relaxation time of PBD44 before crosslinking...

Fig. 12. Dynamic moduli master curves of PBD 44 precursor (p = 0) and PBD 44 critical gel [60]. The entanglement and glass transition regime is hardly affected by the crosslinking. Open symbols correspond to G, filled ones to G"...

The simplest expression incorporating the basic features of self-similarity and cut-off for nearly critical gels has the spectrum of the critical gel altered by a cut-off at the longest time [19] ... 

Predictions using the observed relaxation time spectrum at the gel point are consistent with further experimental observations. Such predictions require a constitutive equation, which now is available. Insertion of the CW spectrum, Eq. 1-5, into the equation for the stress, Eq. 3-1, results in the linear viscoelastic constitutive equation of critical gels, called the critical gel equation ... 

The critical gel equation is expected to predict material functions in any small-strain viscoelastic experiment. The definition of small varies from material to material. Venkataraman and Winter [71] explored the strain limit for crosslinking polydimethylsiloxanes and found an upper shear strain of about 2, beyond which the gel started to rupture. For percolating suspensions and physical gels which form a stiff skeleton structure, this strain limit would be orders of magnitude smaller. 

The transient viscosity f] = T2i(t)/y0 diverges gradually without ever reaching steady shear flow conditions. This clarifies the type of singularity which the viscosity exhibits at the LST The steady shear viscosity is undefined at LST, since the infinitely long relaxation time of the critical gel would require an infinitely long start-up time. 

The gelation transition is observable for Ng > 10. Otherwise, the material behaves as a liquid (Ng < 1). Little is known about materials near Ng = 1. For the following, we consider only materials with iVg 1 and treat them just like chemical gels. The expression T(n + 1, (t — t )/Xpg)/T(n + 1) in Eq. 5-2 approaches a value of one in this case of Ng g> 1, and the critical gel equation, Eq. 4-1, is recovered. However, much work is needed to understand the role of non-permanent physical clusters on network formation and rheological properties. 

The above equations are generally valid for any isotropic material, including critical gels, as long as the strain amplitude y0 is sufficiently small. The material is completely characterized by the relaxation function G(t) and, in case of a solid, an additional equilibrium modulus Ge. 

Time-resolved measurements on the changing sample have the advantages that the critical gel properties can be obtained from a single experiment and that a value for the rate of evolution of properties comes with the data. 

Adolf and Martin [15] postulated, since the near critical gels are self-similar, that a change in the extent of cure results in a mere change in scale, but the functional form of the relaxation modulus remains the same. They accounted for this change in scale by redefinition of time and by a suitable redefinition of the equilibrium modulus. The data were rescaled as G /Ge(p) and G"/Ge(p) over (oimax(p). The result is a set of master curves, one for the sol (Fig. 23a) and one for the gel (Fig. 23 b). 

Dynamic mechanical data near the gel point allow easy determination of the parameters of the critical gel, Eq. 1-1. Tan 8, as shown in Fig. 26, gives the relaxation exponent n... 

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