Segmented chain

Fig. XI-4. Schematic diagram of the structure of an adsorbed polymer chain. Segments are distributed into trains directly attached to the surface and loops and tails extending into solution.

This relation also applies to any portion of the chain segments as long as the number of segments in the portion is sufficient. Therefore, if one proceeds segmental steps, starting from a point in the interior of the chain, the resulting average displacement is of the order of. Conversely, the number of monomers contained in a sphere of... 

In otlier words, tire micelle surface is not densely packed witli headgroups, but also comprises intennediate and end of chain segments of tire tailgroups. Such segments reasonably interact witli water, consistent witli dynamical measurements. Given tliat tire lifetime of individual surfactants in micelles is of tire order of microseconds and tliat of micelles is of tire order of milliseconds, it is clear tliat tire dynamical equilibria associated witli micellar stmctures is one tliat brings most segments of surfactant into contact witli water. The core of nonnal micelles probably remains fairly dry , however. 

For a carbon-carbon bond located along a polymer backbone, the preceding molecular representation must be modified to Fig. 1.8c. The chain segments on either side of the bond of interest are substituents for which the amount of steric hindrance follows a slightly different pattern than for the unsubstituted ethane. Using the same convention for [Pg.58]

At the beginning of this section we enumerated four ways in which actual polymer molecules deviate from the model for perfectly flexible chains. The three sources of deviation which we have discussed so far all lead to the prediction of larger coil dimensions than would be the case for perfect flexibility. The fourth source of discrepancy, solvent interaction, can have either an expansion or a contraction effect on the coil dimensions. To see how this comes about, we consider enclosing the spherical domain occupied by the polymer molecule by a hypothetical boundary as indicated by the broken line in Fig. 1.9. Only a portion of this domain is actually occupied by chain segments, and the remaining sites are occupied by solvent molecules which we have assumed to be totally indifferent as far as coil dimensions are concerned. The region enclosed by this hypothetical boundary may be viewed as a solution, an we next consider the tendency of solvent molecules to cross in or out of the domain of the polymer molecule. 

This last factor overcounts the number of couplings, since the random placement of chain segments makes it improbable that each entanglement will involve a new molecule. Thus an entanglement may be redundant the chain might already be coupled to the original molecule. 

Stretching a polymer sample tends to orient chain segments and thereby facilitate crystallization. The incorporation of different polymer chains into small patches of crystallinity is equivalent to additional crosslinking and changes the modulus accordingly. Likewise, the presence of finely subdivided solid particles, such as carbon black in rubber, reinforces the polymer in a way that imitates the effect of crystallites. Spontaneous crystal formation and reinforcement... 

The attitude we adopt in this discussion is that only those chain segments in the middle of the chain possess sufficient regularity to crystallize. Hence we picture crystallization occurring from a mixture in which the concentration of crystallizable units is Xj and the concentration of solute or diluent is Xg. The effect of solute on the freezing (melting) point of a solvent is a well-known result T j, is lowered. Standard thermodynamic analysis yields the relationship... 

The probability that a chain segment is capped at both ends by a branch unit is described by the branching coefficient a. The branching coefficient is central to the discussion of gelation, since whether gelation occurs or not depends on what happens after capping a section of chain with a potential branch point. 

Our approach to the problem of gelation proceeds through two stages First we consider the probability that AA and BB polymerize until all chain segments are capped by an Aj- monomer then we consider the probability that these are connected together to form a network. The actual molecular processes occur at random and not in this sequence, but mathematical analysis is feasible if we consider the process in stages. As long as the same sort of structure results from both the random and the subdivided processes, the analysis is valid. 

Since the branching coefficient gives the probability of a chain segment being capped by potential branch points, the above development describes this situation ... 

The number of iso triads in a sequence of nj iso repeat units is nj - l,and the number of syndio triads in a sequence of n syndio repeat units is n - 1. We can verify these relationships by examining a specific chain segment [XVIII] ... 

We saw in Sec. 1.11 that coil dimensions are affected by interactions between chain segments and solvent. Both the coil expansion factor a defined by Eq. (1.63) and the interaction parameter x are pertinent to describing this situation. 

Only a fraction of the chain segments will be present in this spherical shell, but whatever their number is, it will increase with the degree of polymerization n. Therefore, in the volume element associated with the expansion of the coil, the volume fraction of chain segments 0 is proportional to n/dV, or 0 n/a ro dro ... 

Noncrystalline domains in fibers are not stmctureless, but the stmctural organization of the polymer chains or chain segments is difficult to evaluate, just as it is difficult to evaluate the stmcture of Hquids. No direct methods are available, but various combinations of physicochemical methods such as x-ray diffraction, birefringence, density, mechanical response, and thermal behavior, have been used to deduce physical quantities that can be used to describe the stmcture of the noncrystalline domains. Among these quantities are the amorphous orientation function and the amorphous density, which can be related to some of the important physical properties of fibers. 

A variety of experimental techniques have been used to prepare and characterize polymer blends some of the mote important ones for estabHshing the equiHbtium-phase behavior and the energetic interactions between chain segments ate described here (3,5,28,29). 

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

Fig. 4. Simple model of an IgG molecule showing light- and heavy-chain segments where a line ( ) between the chains represents a disulfide bond. General Methodology.

There are two principal forces that govern the abdity of a polymer to crystallise the interchain attractive forces, which are a function of the chain stmcture, and the countervailing kinetic energy of the chain segments, which is a function of the temperature. The fact that polymers consist of long-chain molecules also iatroduces a third parameter, ie, the imposition of a mechanical force, eg, stretching, which can also enhance interchain orientation and favor crystallisation. 

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