Concentration, polymer, model chain solution conformations

The deficiencies of the Flory-Huggins theory result from the limitations both of the model and of the assumptions employed in its derivation. Thus, the use of a single type of lattice for pure solvent, pure polymer and their mixtures is clearly unrealistic since it requires that there is no volume change upon mixing. The method used in the model to calculate the total number of possible conformations of a polymer molecule in the lattice is also unrealistic since it does not exclude self-intersections of the chain. Moreover, the use of a mean-field approximation to facilitate this calculation, whereby it is assumed that the segments of the previously added polymer molecules are distributed uniformly in the lattice, is satisfactory only when the volume fraction (f>2 of polymer is high, as in relatively concentrated polymer solutions. 

Due to the screening effect in the volume exclusion of polymer chains, singlechain conformation in the concentrated solutions will exhibit the size scaling similar to the ideal-chain model, as... 

The diffo ence in the character of the nematic ordering in solutions of semiflexible macromolecules with diffaent mechanisms of flexibility is not only manifested in the thermodynamic characteristics of the phase transition itself, but also in the conformations of the polymer chains in the liquid-crystalline phase. For example, the dependences of the root-mean-square distance between chain ends (/ 2) on the concentration of polymer in the solution for semiflexible freely jointed and persistent chains calculated in [43,44] are shown in Fig. 1.4. Note that for the freely jointed model, the value of (jf) is almost independent of the concentration of the solution in the anisotropic phase (i.e., orientation of the segments but not uncoiling of the macromolecules takes place), while for a solution of persistent chains, the increase in (/ ) in the anisotropic phase with an increase in the concentration is very signiflcant (exponential). A solution of chains with the rotational-isomeric mechanism of flexibility (cf. Fig. 1.2c) behaves analogously in this case, as demonstrated in [35], in the... 

Polymers can adsorb spontaneously from solution on to surfaces if the interaction between the polymer and the surface is more favorable than that of the solvent with the surface. For example, a polymer like poly(ethylene oxide) (PEO) is soluble in water but will adsorb on various hydrophobic surfaces and on the water/air interface. This is the case of equilibrium adsorption where the concentration of the polymer monomers increases close to the surface with respect to their concentration in the bulk solution. We discuss this phenomenon at length both on the level of a single polymer chain (valid only for extremely dilute polymer solutions), see Section II, and for polymers adsorbing from (semidilute) solutions, see Section III. In Fig. 2a we schematically show the volume fraction profile (p(z) of monomers as a function of the distance z from the adsorbing substrate. In the bulk, i.e., far away from the substrate surface, the volume fraction of the monomers is (p], whereas, at the surface, the corresponding value is (p > (p]. The theoretical models address questions in relation to the polymer conformations at the interface, the local concentration of polymer in the vicinity of the surface, and the total amount of adsorbing polymer chains. In turn, the knowledge of the polymer interfacial behavior is used to calcu-... 

It is important to observe that only the first term in equation (137) depends on chain architecture and the lattice coordination number, and that both it and the second term cancel out of the entropy of mixing, leaving only the third term equal to Q/Qq. Consequently the theory would apply alike to solutions of flexible and stiff-chain polymers (except that above a certain concentration the latter would respond to packing constraints and minimize the free energy by separation of an ordered phase). The factorization of the combinatorial factors in equation (137) is the fundamental reason why the lattice calculation works at all, despite the extreme artificiality of picturing the chain as fitting a sequence of regular lattice sites with a definite coordination number. These aspects of the model simply disappear in the final result. The independence of the intermolecular factor also implies that the chain conformation should be independent of dilution Rq should be the same in pure liquid polymer as in solution. Naturally this rationale would not hold for dilute solutions, for which the intermolecular factor in equation (137) is not valid. 

A model of chain conformations in solution has been studied numerically on a computer. The polymer chains less than fully flexible were considered. Each chain was represented by relatively rigid groupings of base units called compact bundles, intercalated with sequences of base units called extended bundles. Several parameters characterizing the chains were varied. Thus, consequences of the model were found from the point of view of the exchange interaction energy, polymer concentration, number of segments in a bundle, molecular mass and temperature. Experimental evidence supporting conclusions from our model as well as the model itself is reviewed. 

Monte Carlo simulations provide a rewarding and invaluable approach to solving these systems, and computer simulations and theory can isolate the molecular factors that control polyelectrolyte conformations in solution. Therefore, they are exU cmely useful to address the optimization of colloid-polymer mixtures and guide the design of new experiments. A simple model involving one chain interacting with one particle has been described, but the same model can be extended to more concentrated systems, e.g. involving several chains (and/or colloidal particles) with explicit counter ions, co-ions and solvent molecules. 

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