Primary molecule

Generalization of Flory s Theory for Vinyl/Divinyl Copolvmerization Using the Crosslinkinq Density Distribution. Flory s theory of network formation (1,11) consists of the consideration of the most probable combination of the chains, namely, it assumes an equilibrium system. For kinetically controlled systems such as free radical polymerization, modifications to Flory s theory are necessary in order for it to apply to a real system. Using the crosslinking density distribution as a function of the birth conversion of the primary molecule, it is possible to generalize Flory s theory for free radical polymerization. 

The parenthesis in Equation 19 is the probability that a randomly selected monomer unit bound in the polymer chain belongs to the sol fraction, and therefore, the meaning of the above equation is obvious. At conversion 4 the weight fraction of the primary molecules which were formed at x=0 is given by ... 

Since the primary molecules are linear chains, Wj (0) is given by ... 

Number- and Weight-Average Chain Length of the Primary Molecules which Belong to the Sol Fraction. 

Wj. weight chain length distribution of the primary molecules,... 

If the primary molecules are large, the proportion of cross-linkages required for gelation becomes remarkably small a ratio of one cross-linkage for two primary molecules is sufficient for onset of formation of infinite structures (gel) according to Eq. (11). 

The relation (13) should hold regardless of the primary molecular weight distribution, provided only that the cross-linking proceeds between units at random. The number of cross-linked units per primary molecule, which has been called the cross-linking indexequals pyn-For a homogeneous primary polymer — 2/, and the critical value... 

At the gel point, (3 —l) = l/p, which with the foregoing expression gives Eq. (14), thus establishing equivalence of the two procedures. The primary molecules in a condensation polymer must almost invariably conform to a most probable distribution (see Chap. VIII). The random cross-linking of primary molecules otherwise distributed in size has no counterpart in polyfunctional condensation, therefore. 

The two conditions stated above do not assure the occurrence of gelation. The final and sufficient condition may be expressed in several ways not unrelated to one another. First, let structural elements be defined in an appropriate manner. These elements may consist of primary molecules or of chains as defined above or they may consist of the structural units themselves. The necessary and sufficient condition for infinite network formation may then be stated as follows The expected number of elements united to a given element selected at random must exceed two. Stated alternatively in a manner which recalls the method used in deriving the critical conditions expressed by Eqs. (7) and (11), the expected number of additional connections for an element known to be joined to a previously established sequence of elements must exceed unity. However the condition is stated, the issue is decided by the frequency of occurrence and functionality of branching units (i.e., units which are joined to more than two other units) in the system, on the one hand, as against terminal chain units (joined to only one unit), on the other. 

If the units of primary molecules of uniform length are subjected to cross-linking at random, the weight fraction of polymer molecules comprised of z primary molecules is ... 

The probability that a primary molecule composed of y units possesses i cross-linked units is... 

The total number of molecules in the sol fraction, neglecting intramolecular cross-linking, will equal the number of primary molecules minus the number of cross-linkages in the sol. Expressing these as numbers of moles per equivalent of structural units, we have N = 1/% primary molecules and p /2 cross-linkages in the sol. The number average degree of polymerization in the sol is then... 

The commonly observed, and technically troublesome, formation of gel in the polymerization of dienes testifies to the occurrence of cross-linking processes during their polymerization. According to Eq. (13), gel should begin to form when p reaches the critical value pc = l/y , where is the weight average degree of polymerization of the primary molecules. Thus... 

Under these circumstances the probability that a given structural unit is cross-linked is not entirely independent of the status of other units in the same primary molecule. If an abnormally large fraction of some of the units of a given primary molecule are found to be cross-linked, the likelihood that it was formed toward the end of the polymerization process is enhanced hence the probability that one of its other units is cross-linked will be greater than the over-all p for the system. Calculations indicate that the magnitude of the non-randomness is not excessive below about 70 percent conversion. For most purposes its effect probably may be ignored without serious error, thus obviating a more elaborate theory which would take into account non-randomness of this nature. 

Fig. 91.—(a) Schematic representation of cross-linking as in rubber vulcanization. (b) Incidence of cross-linked units within a given primary molecule. 

The two quantities p and pt are satisfactory for the quantitative description of any random network structure. Alternative quantities sometimes are used to advantage, however. Instead of pt, one may prefer to specify the number N of primary molecules... 

Any real network must contain terminal chains bound at one end to a cross-linkage and terminated at the other by the end ( free end O of a primary molecule. One of these is indicated by chain AB in Fig. 92, a. Terminal chains, unlike the internal chains discussed above, are subject to no permanent restraint by deformation their configurations may be temporarily altered during the deformation process, but rearrangements proceeding from the unattached chain end will in time re-... 

Fig. 92.—Effects of flaws consisting of ends of primary molecules on the network structure. indicates a cross-linkage, O the terminus of a primary molecule, and signifies continuation of the network structure. ...

The total number of chains, both internal and terminal, is v+Nj as is evident from Fig. 91,b. For every primary molecule there will be two terminal chains, i.e., a total of 2N. The number of internal chains must therefore be v — N. Hence a fraction of the chains given by... 

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