Chains flexible

Free-draining models were among the first to be considered [14-18]. For flexible polymer chains of sufficient length, [77] behaves as if the polymer coil occupied a spherical volume through which the solvent cannot flow. Under these conditions, 

Finally, some rather recent devdopments must be noted. Several years ago, Yamakawa and co-workers [25-27] developed the wormlike continuous cylinder model. This approach models the polymer as a continuous cylinder of hydrodynamic diameter d, contour length L, and persistence length q (or Kuhn length / ). The axis of the cylinder conforms to wormlike chain statistics. More recently, Yamakawa and co-workers [28] have developed the helical wormlike chain model. This is a more complicated and detailed model, which requires a total of five chain parameters to be evaluated as compared to only two, q and L, for the wormlike chain model and three for a wormlike cylinder. Conversely, the helical wormlike chain model allows a more rigorous description of properties, and especially of local dynamics of semi-flexible chains. In large part due to the complexity of this model, it has not yet gained widespread use among experimentalists. Yamakawa and co-workers [29-31] have interpreted experimental data for several polymers in terms of this model. 

From Eq. 1, the evaluation of the characteristic ratio requires measurement of the unperturbed dimensions of the polymer chain. Polymers exhibit their characteristic unperturbed dimensions in the bulk amorphous state, i.e., chain dimensions under these conditions reflect the influence of short range, rotational isomeric state, effects only. Prior to the availability of small-angle neutron scattering (SANS) in the mid-1970s [32-34], no method was available that allowed chain dimensions in polymer melts to be measured directly. SANS utilizes the fact that different isotopes result in different scattering amplitudes for neutrons. Thus, selective deuterium labelling of some chains, followed by dispersing these chains in a solvent of otherwise identical but non-deuterated chains, allows the conformational properties of individual chains to be probed in the melt. 

SANS allows Rg o, the unperturbed mean-square radius of gyration, to be measured. For a linear unperturbed chain, it is well-known that 

the value of R o may be computed. A detailed discussion of the SANS technique is beyond the scope of the present review, and details of the experimental protocol and data analysis have been reviewed previously [35]. Although unperturbed dimensions from SANS studies on melts are still very 

In practice, there is a growing body of experimental evidence that shows that the choice of theta solvent can have an impact on the magnitude of measured unperturbed dimensions [39, 42-44] and/or the Flory hydrodynamic parameter d under theta conditions [30, 45]. Theory anticipated [46] and has rationalized [45, 47] such effects. In fact, today, polymer chains are viewed by theory as being only quasi-ideal at the theta state [48-50]. 

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Siepmann J I and Frenkel D 1992 Configurational bias Monte Carlo—a new sampling scheme for flexible chains Moi. Phys. 75 59-70... 

Rosenbluth algorithm can also be used as the basis for a more efficient way to perform ite Carlo sampling for fully flexible chain molecules [Siepmann and Frenkel 1992], ch, as we have seen, is difficult to do as bond rotations often give rise to high energy rlaps with the rest of the system. 

Siepmann J I and D Frenkel 1992. Configurational Bias Monte Carlo A New Sampling Scheme f Flexible Chains. Molecular Physics 75 59-70. 

Next let us apply random walk statistics to three-dimensional chains. We begin by assuming isolated polymer molecules which consist of perfectly flexible chains. 

This kind of perfect flexibility means that C3 may lie anywhere on the surface of the sphere. According to the model, it is not even excluded from Cj. This model of a perfectly flexible chain is not a realistic representation of an actual polymer molecule. The latter is subject to fixed bond angles and experiences some degree of hindrance to rotation around bonds. We shall consider the effect of these constraints, as well as the effect of solvent-polymer interactions, after we explore the properties of the perfectly flexible chain. Even in this revised model, we shall not correct for the volume excluded by the polymer chain itself. 

The one-dimensional random walk of the last section is readily adapted to this problem once we recognize the following connection. As before, we imagine that one end of the chain is anchored at the origin of a three-dimensional coordinate system. Our interest is in knowing, on the average, what will be the distance of the other end of the chain from this origin. A moment s reflection will convince us that the x, y, and z directions are all equally probable as far as the perfectly flexible chain is concerned. Therefore one-third of the repeat units will be associated with each of the three perpendicular directions... 

Making these substitutions gives the probability of finding one end of a perfectly flexible chain of n units a distance r from the other end by... 

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. 

Figure 9.15 Schematic illustration of size exclusion in a cylindrical pore (a) for spherical particles of radius R and (b) for a flexible chain, showing allowed (solid) and forbidden (broken) conformations of polymer.

Figure 9,16 Comparison of theory with experiment for rg/a versus K. The solid line is drawn according to the theory for flexible chains in a cylindrical pore. Experimental points show some data, with pore dimensions determined by mercury penetration (circles, a = 21 nm) and gas adsorption (squares, a= 41 nm). [From W. W. Yau and C. P. yidXont, Polym. Prepr. 12 797 (1971), used with permission.]...

Properties. One of the characteristic properties of the polyphosphazene backbone is high chain dexibility which allows mobility of the chains even at quite low temperatures. Glass-transition temperatures down to —105° C are known with some alkoxy substituents. Symmetrically substituted alkoxy and aryloxy polymers often exhibit melting transitions if the substituents allow packing of the chains, but mixed-substituent polymers are amorphous. Thus the mixed substitution pattern is deUberately used for the synthesis of various phosphazene elastomers. On the other hand, as with many other flexible-chain polymers, glass-transition temperatures above 100°C can be obtained with bulky substituents on the phosphazene backbone. 

Polyols. Analogous to the use of linear a,C0-dibasic acids, such as adipic and sebacic, polyols with long, flexible chains between hydroxyl groups, such as 1,4-butanediol [110-63-4] 1,6-hexanediol [629-11-8J, and diethylene glycol [111-46-6] may also be used to impart greater flexibiUty ia the resia. 

Simplified models for proteins are being used to predict their stmcture and the folding process. One is the lattice model where proteins are represented as self-avoiding flexible chains on lattices, and the lattice sites are occupied by the different residues (29). When only hydrophobic interactions are considered and the residues are either hydrophobic or hydrophilic, simulations have shown that, as in proteins, the stmctures with optimum energy are compact and few in number. An additional component, hydrogen bonding, has to be invoked to obtain stmctures similar to the secondary stmctures observed in nature (30). 

Crystallinity. Generally, spider dragline and silkworm cocoon silks are considered semicrystalline materials having amorphous flexible chains reinforced by strong stiff crystals (3). The orb web fibers are composite materials (qv) in the sense that they are composed of crystalline regions immersed in less crystalline regions, which have estimates of 30—50% crystallinity (3,16). Eadier studies by x-ray diffraction analysis indicated 62—65% crystallinity in cocoon silk fibroin from the silkworm, 50—63% in wild-type silkworm cocoons, and lesser amounts in spider silk (17). 

Optical properties of cyanines can be usefiil for both chiral substituents/environments and also third-order nonlinear optical properties in polymer films. Methine-chain substituted die arbo cyanines have been prepared from a chiral dialdehyde (S)-(+)-2-j -butylmalonaldehyde [127473-57-8] (79), where the chiral properties are introduced via the chiral j -butyl group on the central methine carbon of the pentamethine (die arbo cyanine) chromophore. For a nonchiral oxadicarbocyanine, the dimeric aggregate form of the dye shows circular dichroism when trapped in y-cyclodextrin (80). Attempts to prepare polymers with carbocyanine repeat units (linked by flexible chains) gave oligomers with only two or three repeat units (81). However, these materials... 

Both polymers are linear with a flexible chain backbone and are thus both thermoplastic. Both the structures shown Figure 19.4) are regular and since there is no question of tacticity arising both polymers are capable of crystallisation. In the case of both materials polymerisation conditions may lead to structures which slightly impede crystallisation with the polyethylenes this is due to a branching mechanism, whilst with the polyacetals this may be due to copolymerisation. 

We consider a fluid of flexible chain molecules made of tangent hard sphere monomers. Each chain consists of m monomers of diameter ctq] the distance between centers of adjacent monomers in a chain is fixed and equals ctq. However, the angle formed by any three consecutive monomers in a chain is not fixed. The only restriction is that monomers belonging to a given chain do not overlap each other. Let us describe first a computer simulation procedure. 

A set of flexible chains is considered in the canonical NVT ensemble. The density of chains, pch = - ch/ (- ch is a chosen number of chains), is the parameter of simulation. The system of flexible chains in question has been equilibrated, then during the productive part of the simulation run the pair... 

FIG. 11 Adsorbed amount as a function of bulk concentration for a non-interacting (empty symbols) and adsorbing (full symbols) wall. Diamonds and triangles correspond to a system with semi-rigid chains, circles and squares for flexible chains [28]. 

FIG. 13 Average center-of-mass position of flexible chains of length / with respect to the nearest solid surfaces for different /. Diamonds denote a system of semi-rigid chains in which the opposite effect is observed [28]. 

FIG. 14 Semi-log plot of mean chain length L vs width of the open slit D at various temperatures in 3d. Full symbols denote flexible chains and empty symbols semirigid chains with activation energy a = 0.5 [61]. 

On the other hand, as the width of a closed three-dimensional gap V is decreased, L(T>)) gradually decreases for absolutely flexible chains whereas for semi-rigid chains it goes through a minimum at D = 2 monomer diameters and then grows again for D = 1 (Fig. 15). 

In some of these models (see Sec. Ill) the surfactants are still treated as flexible chains [24]. This allows one to study the role of the chain length and chain conformations. For example, the chain degrees of freedom are responsible for the internal phase transitions in monolayers and bilayers, in particular the hquid/gel transition. The chain length and chain architecture determine the efficiency of an amphiphile and thus influence the phase behavior. Moreover, they affect the shapes and size distributions of micelles. Chain models are usually fairly universal, in the sense that they can be used to study many different phenomena. 

Synthetic rubber (elastomers) are high molecular weight polymers with long flexible chains and weak intermolecular forces. They have low crystallinity (highly amorphous) in the unstressed state, segmental mobility, and high reversible elasticity. Elastomers are usually cross-linked to impart strength. 

For flexible chain copolymers based on acrylic and methacrylic acids (AA and MA) crosslinked with a polyvinyl component, the inhomogeneity of the structures formed depends on the nature of the crosslinking agent, its content in the reaction mixture and the thermodynamic quality of the solvent [13,14],... 

As a result of thermodynamic analysis it is shown that protein bonding to carboxylic CP exhibiting a local internal chain structure is determined by the entropy factor, whereas, if the arrangement of flexible chain parts on the protein globule is possible, the energetic component predominates. 

Elyashevich, G. K. Thermodynamics and Kinetics of Orientational Crystallization of Flexible-Chain Polymers. Vol. 43, pp. 207 — 246. 

Thermodynamics and Kinetics of Orientational Crystallization of Flexible-Chain Polymers... 

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