Gels with multiple junctions

Let us indicate the multiplicity by as in the convention. We then replace the notations as nig jk and Pf Pf, Pg Pk in the molecular distribution (3.68) while the notation // is kept as it is. Here, pk gives the probability for a chosen A group to belong to the junction of multiplicity k. The average functionality of B monomers becomes = J kpk, and we write this as p., where shows the average multiplicity of the cross-links. 

There are two fundamental geometrical relations which hold for clusters of tree type with multiple junctions. For the total number of monomers in the cluster, the relation 

The combinatorial factor in themolecular weight distribution function (3.68)simplifles as the factor p l-p)q l-q) disappears, and the limit 1 of complete reaction can be taken. As a result, the molecular weight distribution is transformed to 

The branching coefficient of this system when regarded as an A/B mixture is found to be O = (gw - 1)P7 by generalizing (3.52) to polydisperse systems. Fixing = 1 in 

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Most physical gels have multiple cross-links. They are formed by the association of the particular segments on the polymer chains. Therefore, gels with multiple junctions may be understood more profoundly when they are treated by thermodynamic theory rather than reaction theory. In Section 7.1, we shall present some of the equilibrium thermodynamics of physical gels with multiple junctions. 

Solution properties of thermoreversible gels with multiple junctions... 

Figure 7.6(a) plots the reduced concentration k T)(p /n at the gel point as a function of the junction multiplicity. The functionality is changed from curve to curve. For bifunctional molecules /=2, at least multiplicity 3 is necessary for gelling. The gelation concentration monotonically decreases with multiplicity. For functionalities higher than 2, however, there is an optimal multiplicity for which gelation is easiest. In such cases, network growth becomes difficult due to an increase in the number of branches at the junctions. 

Most physical gels have complex multiple junctions. In Section 7.4, we studied ther-moreversible gelation with junctions of variable multiplicity. In this section, we consider a new method to find the local structure of the networks, i.e., the structure of the network junctions. 

We now examine the opposite case where polymers carry only two functional groups / = 2, but form multiple junctions with k>2>. We find o =- 1.( 1 = 1 — xi, and 2 = 0. The last relation = 0 is obvious because an unreacted functional group on a chain can only be connected to the gel through the chain carrying it (i.e / = 1) in the special case of / = 2. The number of effective chains now becomes... 

To study the network structure, we first specify the type of junctions in more detail. A junction of multiplicity k that is connected to the gel network through i paths is referred to as an (/, /c)-junction. Let be the number of junctions specified by the type (i,k) for k = 1,2,3,4,. .. and for 0 < < 2k. The total number of junctions with multiplicity... 

Emulsions and microfluidic structures have been used for many purposes, including fusion of reactants present in two droplets, preparation of gel beads, preparation of multiple emulsions, etc. for a comprehensive overview, please consult the review paper of Vladisavljevic and colleagues [1]. Besides this, the microfluidic systems discussed in this entry can be used as analytical tools in various ways. To illustrate this, the use of Y-shaped junctions for dynamic interfacial tension measurement [14] and the use of T-junctions in combination with a coalescence chamber for emulsion stability research [15] are discussed next. 

Our estimate from simulation of a in the range 0.1-0.2, when interpreted by this equation, yields 6-11 for the multiplicity of the junctions. This number is comparable with the number of insoluble blocks in the micelles that provide the junction points in the network that is formed at < gei in the simulations. Of course, it is this number of insoluble blocks that determines the stability of the micelle, which is the junction in our gel. 

Fig. 8.2 Modified Eldridge-Ferry analysis for aqueous poly(vinyl alcohol) solutions. The gel melting concentration c measured at a constant temperature for different molecular weight polymers plotted against the molecular weight finds the junction multiplicity from the slope (broken lines), while those measured at constant molecular weight by changing the temperature find the number of the repeat units per chain in the junction (solid lines). (A) 91°C ( ) 87°C (O) 83 C ( ) 78°C (A) 74°C (o) 71 (Reprinted with permission from Ref. [6].)...

Later, Tanaka and Nishinari [7] estimated the structure of the junction zone using the experimental results rqrorted by Domszy et al. [3]. They proposed the model shown in Fig. 1 for the junction zone formed by the aggregation of polymer chains. They also derived parameters that correspond to the number of aggregating chains and the parameter C that corresponds to the number of chemical repeat units forming the crosslink [8, 9]. The parameter s is called junction multiplicity. In this model, gels are formed with crystallites in the junction zone. 

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