Polymer Chain in a Dilute Solution

A long flexible polymer chain is a typical object addressed by meso thermodynamics. In dilute solutions a polymer chain can exhibit either a random walk or a self-avoiding walk. These two regimes are separated by the theta point, the point of polymer-solvent phase separation in the limit of infinite degree of polymerization. The random walk occurs when the polymer chain and the solvent form either ideal or quasi-ideal solutions. The ideal polymer chain exhibits Gaussian fluctuations of the distance R between the two ends of the 

The Gaussian probability distribution function, as discussed in Section 7.2.1, for a 3-dimensional random walk is 

As follows from eq 7.29, the configurational entropy change per polymer chain is 

Stretching the ideal chain, therefore, increases the Gibbs energy by 

Real polymer solutions are not ideal. In the mean-field Flory-Huggins model of a polymer solution the virial expansion of the osmotic pressure reads  

---

The volumes of polymer chains in a dilute solution are representable through... 

The above calculations assume that the gross chain conformations are those of a random walk, which is the case in the melt. However, for an isolated polymer molecule in a dilute solution, the average conformation is affected by excluded-volume interactions between one part of the chain and another. Because the chain must avoid self-intersection, the conformation of the chain will be that of a self-avoiding walk, rather than a random walk, if the solution is athermal—that is, if all interactions are negligible except excluded volume. Self-avoiding walks lead, on average, to more expanded coil dimensions, since expanded configurations are less likely than contracted ones to lead to self-intersection of the chain. Thus, in an athermal solution, the mean-square end-to-end dimension of a polymer molecule scales as... 

In dilute solutions, hydrodynamic interactions between the monomers in the polymer chain are strong. These hydrodynamic interactions also are strong between the monomers and the solvent within the pervaded volume of the chain. When the polymer moves, it effectively drags the solvent within its pervaded volume with it. For this reason, the best model of polymer dynamics in a dilute solution is the Zimm model, which effectively treats the pervaded volume of the chain as a solid object moving through the surrounding solvent. 

It is not well understood how torsional motion of a polymer side group is affected by the rest of the polymer chain in highly dilute solutions. In the case of intramolecular exciplex formation of l-(l-pyrenyl)-3-(4-N,N-dimethylaminophenyl) propane bonded to polystyrene [14], the rate of exciplex formation in dilute polymer solution is much slower than that of a reference small molecule system. 

The exponent v characterizes the swelling of a long polymer chain in very dilute solutions. In theory, it could be measured in several ways. However, can we trust results obtained by the simplest technique which consists in measuring the intrinsic viscosity These measurements produce values for the exponent v which are always lower than those obtained by light scattering measurements of the radius of gyration. It was necessary to explain this discrepancy in order to make a proper comparison between experimental and theoretical values of the exponent v. 

This estimation of c is rather arbitrary. In the following section, the value of c will be estimated in a more realistic manner and a very different result will be found. To do this, we shall study how the top of the demixtion curve varies with temperature. Subsequently, using the value of c thus determined, we shall interpret experiments concerning the collapse of polymer chains in very dilute solutions. 

Polymeric solutions are inherently viscoelastic due to their long molecular chains. Dilute polymeric solutions will be of interests for microfluidic and nanofluidic applications, due to their ease of implementation. Since the polymer coils in a dilute solution are isolated and independent in their molecular movements, their viscoelastic behavior can be explained by the deformation of the individual coil or molecule in the stream [2,3]. As a result, the viscoelasticity of polymer solutions is generally attributed to the deformation of polymer chains and the consequent generation of unequal normal stresses. 

Crystalhzation analysis fractionation (Crystaf) is a recently developed characterization technique that fractionates polymer chains according to their crystaUizabUities in a dilute solution [1,2]. This techiuque is based on the continuous noiusothermal crystalhzation of polymer chains from a dilute solution. During crystalhzation, the concentration of polymer in solution is measured as a function of crystalhzation temperature, generating a cumida-tive concentration profile such as the one shown in Fig. 1. The derivative of this cumulative concentration profile is proportional to the fraction of polymer crystalhzed at each temperature interval and represents the distribution of chain crystallizabihties in the sample. 

A different type of microgels can be obtained by solution polymerization. Since an increase of dilution during crosslinking increases the probability of intramolecular crosslinking, the growing polymer chains in a highly dilute solution become intramolecularly crosslinked and their structure approaches that of the microgels formed within the micelles. 

In a dilute solution, when the polymer is in a coil state (Fig. 6a), the diffusion of hydrophobic particles into the coil is normally faster than the chemical reaction [53]. In this case, the local concentration of particles H inside the coil is practically the same as in the bulk. Therefore, we expect that at the initial stage, the reaction will lead to a random copolymer some of the P monomeric units will attach to H reagent and thereby they will acquire amphiphilic (A) properties P + H —A (Fig. 6b). As long as the number of modified A units is not too large, the chain remains in a swollen coillike conformation (Fig. 6b). However, when this number becomes sufficiently large, the hydrophobically modified polymer segments would tend to form... 

However, viscometric measurements of dilute polymer solutions in a steady flow are inadequate for this purpose although, as already indicated, viscosity is related to molecular rotation. This has been demonstrated by Zimm s theory ). Zimm considered the kinetics of the motion and deformation of a kinetically flexible polymer chain in a weak mechanical field with harmonic velocity gradient g at frequency v. It has been found that under steady and weak flow conditions... 

The main difficulty in the experimental investigations of EB in flexible-chain polymer solutions is due to the low value of the observed effect. Specific Kerr constants for a flexible-chain polymer bearing no charge are K 10 cm g (V/300) even for polar macromolecules and, hence, the difference between birefringence in a dilute solution and the Kerr effect in the solvent alone is very slight. 

Consider a polymer containing N Kuhn monomers (of length b) in a dilute solution at the 0-temperature, where ideal chain statistics apply. 

When a polymer molecule moves in a dilute solution it undergoes frictional interactions with solvent molecules. The nature and effect of these frictional interactions depend upon the size and shape of the polymer molecule. Thus, the chain dimensions of polymer molecules can be evaluated from measurements of their frictional properties [25]. 

---