Effective Crosslinking Density

In order to compare theories with experiments, the number of effective crosslinks should be determined. Actually, this parameter cannot easily be analyzed, because in any one elastomer, the ratio of linkages that do not contribute to an infinite network is difficult to evaluate. In addition, mesogenic groups induce sterical hindrance, which should modify the chemical reactivity of the netpoints as well as the theories referring to ideal rubbers that are used for the determination of the effective crosslinking density. 

IV Behavior and Properties of Side Group Thermotropic Liquid Crystal Polymers 

Polymer concentration in the reaction bath Consistency of the final product Weight swelling ratio in toluene 0.3 

Swelling experiments and stress-strain measurements are widely used to estimate the crosslinking density in liquid-crystalline networks [101,106-108], For ideal rubbers with tetrafunctional crosslinks, these techniques allow the evaluation of the number average molecular weight between crosslinks [104, 105]. 

This theory refers to ideal rubbers. The additional entropic component, introduced by the presence of the mesogenic cores [91], is not taken into account, and so the absolute values of are not relevant. 

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The pre-gel model calculates the weight average molecular weight of the reaction mixture, while the post-gel model calculates the weight of the sol fraction and the effective crosslink density. A simple computer program using the derived expressions has been written in BASIC and runs on IBM-PC compatible computers. The importance of secondary reactions on cure in typical coatings is discussed. 

Once P(F ° ) and P(Fg° ) have been calculated, it is possible to calculate a number of network structure parameters including the weight fraction of sol, wg, and the "effective" crosslink density. A given polymer or crosslinker will be part of the sol only if all of its groups are attached to finite chains. Thus, the weight of the sol is given by... 

It is possible to calculate a number of different kinds of "effective" crosslink densities. Bauer et al have used a quantity they termed the "elastically effective crosslink density " (Cel) correlate cure with solvent resistance and other physical properties of coatings (7-10). The correlation was basically empirical. Formally, the is a calculation of the number of functional groups attached to the infinite network for which there are at least two other paths out to the network on the given polymer or crosslinker. Thus, chains with only one or two paths to the infinite network are excluded. The following expression can be written for... 

The effective crosslink density is used to predict state of cure in thermoset coatings. Side reactions have been found to play an important role in cure of typical coating mixtures. 

PRINT "THE OLD ELASTICALLY EFFECTIVE CROSSLINK DENSITY EXPRESSION OF BAUER" 150 PRINT "AND BUDDE. THE SECOND CALCULATES A CROSSLINK DENSITY WHICH IN" THEORY SHOULD BE PROPORTIONAL TO THE RUBBERY ELASTIC MODULUS."... 

PRINT "THE OLD ELASTICALLY EFFECTIVE CROSSLINK DENSITY - CEL 3330 PRINT... 

In this case, the crosslinking density is the sum of the effective crosslinking density and the secondary cyclization, since both crosslinkages are elastically effective. The calculation conditions are the same as Figure 1 except that the number of secondary cycles formed per effective crosslinkage T) is 20. 

DC Harsh, SH Gehrke. Characterization of ionic water absorbent polymers Determination of ionic content and effective crosslink density. In L Brannon-Peppas, RS Harland, eds. Absorbent Polymer Technology. Amsterdam Elsevier, 1990, pp 103-124. 

Figure 2. Elastically effective crosslink density versus bake temperature for a 17 minute bake. Low solids 0 experimental values, model values High solids 1 ----- ...

Differences in Network Structure. Network formation depends on the kinetics of the various crosslinking reactions and on the number of functional groups on the polymer and crosslinker (32). Polymers and crosslinkers with low functionality are less efficient at building network structure than those with high functionality. Miller and Macosko (32) have derived a network structure theory which has been adapted to calculate "elastically effective" crosslink densities (4-6.8.9). This parameter has been found to correlate well with physical measures of cure < 6.8). There is a range of crosslink densities for which acceptable physical properties are obtained. The range of bake conditions which yield crosslink densities within this range define a cure window (8. 9). 

Gel type Approximate gelation time (min) Shear modulus (gcm 2) Equilibrium degree of swelling Effective crosslinking density (mol cm-3) Crosslinking efficiency (%)... 

Figure 2 illustrates the temperature dependence of the swelling degree as a function of precursor polymer type. Methylcellulose (MC), hydroxypropyl-methylcellulose, type E (HPMC-E) and hydroxypropylmethylcellulose, type K (HPMC-K) gels have comparable effective crosslink densities of about 2 x 10 5 mol/cm3 (as determined from uniaxial compression testing), while the crosslink density of the hydroxypropylcellulose (HPC) gel is about half this [52]. The transition temperature for each gel is within several degrees of the precursor polymer lower critical solution temperature (LCST), except for the MC gel, which has a transition temperature 9 °C higher than the LCST. The sharpness of the transition was about 3%/°C, except for the HPC gel transition, which was much sharper - about 8%/°C. 

Fig. 4. Dependence of the effective crosslinking density of cured adhesive KL-3 on time for different media 1 proteinase solution, 2 extract of rabbit kidneys, 3 extract of rabbit liver, 4 extract of rabbit muscles, 5 Ringer-Lock solution, 6 saline, 7 animal organism...

It was shown that the stress-induced orientational order is larger in a filled network than in an unfilled one [78]. Two effects explain this observation first, adsorption of network chains on filler particles leads to an increase of the effective crosslink density, and secondly, the microscopic deformation ratio differs from the macroscopic one, since part of the volume is occupied by solid filler particles. An important question for understanding the elastic properties of filled elastomeric systems, is to know to what extent the adsorption layer is affected by an external stress. Tong-time elastic relaxation and/or non-linearity in the elastic behaviour (Mullins effect, Payne effect) may be related to this question [79]. Just above the melting temperature Tm, it has been shown that local chain mobility in the adsorption layer decreases under stress, which may allow some elastic energy to be dissipated, (i.e., to relax). This may provide a mechanism for the reinforcement of filled PDMS networks [78]. 

Strength increases with an increase in the effective crosslink density of the gel or in the concentration and average molecular weight of the polymer. However, a rise in temperature may increase or decrease the apparent viscosity, depending on the molecular interactions between the polymer and solvent. In addition, the direction of change in apparent viscosity may not be readily predictable when additives such as ions, non-electrolytes, solvents or non-solvents, and other compatible polymers are mixed with a gel. 

Fig. 2 The modulus of elasticity Go, the effective crosslink density Vo (a) and the volume swelling ratio 2v of PAAc hydrogels in water (b) as function of the polymer network concentration ip2, 1.2 mol % MBAAm. The inset to Fig. 2a shows a double logarithmic Go vs. 9 plot. Reprinted from Yazici and Okay (2005) with kind permission from Elsevier...

Dissociation degree Polymer-solvent interaction parameter Deformation ratio Effective crosslink density Nominal stress Compressive stress... 

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