Polymers hard chain predictions

We now turn to the actual polymerization process and we will try to present a series of pictures that clarifies how chain-end control can be used to obtain either syndiotactic or isotactic polymers. Subsequently we will see how a chiral site can influence the production of syndiotactic or isotactic polymers. Finally, after the separate stories of chain-end control and site control, the reader will be confused by introducing the following elements (1) pure chain-end control can truly occur when the catalyst site does not contain chirality (2) but since we are making chiral chain ends in all instances, pure site control does not exist. In a polymerization governed by site control there will potentially always be the influence of chain-end control. This does not change our story fundamentally all we want to show is that stereoregular polymers can indeed be made, and which factors play a role but their relative importance remains hard to predict. 

We now can easily visualize polymer molecules — they are long chains tangled up into coils. However, even knowing the molecular structure of a substance, it may still be hard to predict for sure all its properties. For example, water, consisting always of the same well-known molecules H2O, depending on the conditions can be a liquid, a solid (ice), or a gas (steam). So what about pol3mier substances How do they look, and what states can they exist in ... 

Figure 8 Comparison of PRISM predictions (solid lines) to Monte Carlo simulation data for the pair correlation functions in blends of hard-chain branched (open circles) and linear (filled circles) polymers...

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

One may wonder to what extent our predictions for hard spheres apply to a system of soft particles in a polymer solution. A definite answer to this question cannot be given at the moment since numerical data for the depletion of free polymer chains in the neighbourhood of a surface with terminally attached chains are not yet available. Some qualitative features for such a system have been discussed using scaling arguments (24). We may expect that the depleted amount of polymer is, at least in some cases, less than near a hard surface, giving rise to weaker attraction. Both the destabilization concentration (J) and the restabilisation concentration (<(> ) could be much lower. Experimental observations support this qualitative conclusion (1-5). 

The solution properties of dendrigraft polybutadienes are, as in the previous cases discussed, consistent with a hard sphere morphology. The intrinsic viscosity of arborescent-poly(butadienes) levels off for the G1 and G2 polymers. Additionally, the ratio of the radius of gyration in solution (Rg) to the hydrodynamic radius (Rb) of the molecules decreases from RJRb = 1.4 to 0.8 from G1 to G2. For linear polymer chains with a coiled conformation in solution, a ratio RJRb = 1.48-1.50 is expected. For rigid spheres, in comparison, a limiting value RJRb = 0.775 is predicted. 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

To illustrate some commonly encountered classification methods, a data set obtained from a series of polyurethane rigid foams will be used.55 In this example, a series of 26 polyurethane foam samples were analyzed by NIR diffuse reflectance spectroscopy. The spectra of these foams are shown in Figure 8.25. Each of these foam samples belongs to one of four known classes, where each class is distinguished by different chemistry in the hard block parts of the polymer chain. Of the 26 samples, 24 are selected as calibration samples and 2 samples are selected as prediction samples. Prediction sample A is known to belong to class number 2, and prediction sample B is known to belong to class number 4. Table 8.8 provides a summary of the samples used to produce this data set. 

Complications. In practice, precise calculation of the interaction free energy is not always easy. Eq. (12.12) applies to a pair of identical hard spheres, and even in that case the result may differ from the prediction. The relation <5 = rg is not exact, even if the polymer is monodisperse, because some chains will protrude beyond rg. If the particles are somewhat deformable, because they are very soft or because they have a deformable adsorption layer, the depletion interaction forcing them together may cause local flattening, by which Fdepi becomes even larger. Particle shape has a large effect, and for platelets the depletion interaction is far stronger than for spheres of equal volume (can you explain this ). 

Simultaneous polymerization of two monomers by chain initiation usually results in a copolymer whose composition is different from that of the feed. This shows that different monomers have different tendencies to undergo copolymerization. These tendencies often have little or no resemblance to their behavior in homopolymerization. For example, vinyl acetate polymerizes about twenty times as fast as styrene in a free-radical reaction, but the product obtained by free-radical polymerization of a mixture of vinyl acetate and styrene is found to be almost pure polystyrene with hardly any content of vinyl acetate. By contrast, maleic anhydride, which has very little or no tendency to undergo homopolymerization with radical initiation, readily copolymerizes with styrene forming one-to-one copolymers. The composition of a copolymeir thus cannot be predicted simply from a knowledge of the polymerization rates of the different monomers individually. The simple copolymer model described below accounts for the copolymerization behavior of monomer pairs. It enables one to calculate the distribution of sequences of each monomer in the macromolecule and the drift of copolymer composition with the extent of conversion of monomers to polymer. 

The theory, which is really only qualitative in character, was found to predict correctly that Vj shoiild decrease with increasing molecular weight of the free polymer, increasing particle size and increasing particle concentration. Indeed, by appropriate adjustment of the arbitrary parameters, semi-quantitative agreement could be achieved between the pr ictions of theory and the results of experiments. Of course, any theory that treats polymer chains as hard spheres is foredoomed to quantitative inaccuracy. 

For the calculations, different EoS have been used the lattice fluid (LF) model developed by Sanchez and Lacombet , as well as two recently developed equations of state - the statistical-associating-fluid theory (SAFT)f l and the perturbed-hard-spheres-chain (PHSC) theoryt ° . Such models have been considered due to their solid physical background and to their ability to represent the equilibrium properties of pure substances and fluid mixfures. As will be shown, fhey are also able to describe, if not to predict completely, the solubility isotherms of gases and vapors in polymeric phases, by using their original equilibrium version for rubbery mixtures, and their respective extensions to non-equilibrium phases (NELF, NE-SAFT, NE-PHSC) for glassy polymers. 

De Gennes (1971) postulated that polymer molecules were constrained to move along a tube formed by neighbouring molecules. In a deformed melt, the ends of the molecules could escape from the tube by a reciprocating motion (reptation), whereas the centre of the molecule was trapped in the tube. When the chain end advanced, it chose from a number of different paths in the melt. This theory predicts that the zero-shear rate viscosity depends on the cube of the molecular weight. However, in the absence of techniques to image the motion of single polymer molecules in a melt, it is hard to confirm the theory. 

Neither the uniform strain model nor the uniform stress model is appropriate for this microstructure. Consequently, the elastic moduli of polyurethanes lie between the limits set by Eqs (4.11) and (4.12). For a network chain of Me = 6000, the rubber elasticity theory of Eq. (3.20) predicts a shear modulus of about 0.4 MPa. The hard blocks will have the typical 3GPa Young s modulus of glassy polymers. Increases in the hard block content cause the Young s modulus to increase from 30 to 500 MPa (Fig. 7.13). For automobile panel applications it is usual to have a high per cent of hard blocks so that the room temperature flexural modulus is 500 MPa. 

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