2. Background
2.1. Polymers in solution
The word (poly)-(mer) means (many)-(parts) and refers to molecules consisting of many elementary units, called monomers.33 Monomers are structural repeating units of a polymer that are connected to each other by covalent bonds. Polymer solutions can be obtained by dissolving a polymer in a solvent. The dissolution of a polymer into a thermodynamically compatible solvent is a stepwise process, which include solvent diffusion and chain disentanglement:34 First, the solvent molecules penetrate and swell the polymer. Secondly, the solvated polymer molecules create a gel. Finally, the gel breaks up, and the molecules are dispersed into a true solution. Not all polymers can form a true solution in solvent.
The general phase behavior of the polymer-solvent mixtures has been described by Flory-Huggins theory, based on the principles of thermodynamics.35,36 The behavior of a polymer in solution depends both on the polymer and solvent properties. Flory-Huggins’ mathematical model implies disregard small thermal composition fluctuations as in the mean field theory.
The result is an equation for the change of the Gibbs free energy of mixing, ∆𝐺𝑚, which polymer and solvent molecules. ∆𝑆𝑚 is the entropy of mixing, T is the absolute temperature,
and ln )
ln N ( R
Sm 1 12 2
. Thus, the right-hand side is a function of the number of moles and volume fraction of solvent, 𝜑1, and polymer, 𝜑2.( 1 2 1). Thus, the Gibbs
2.1. Polymers in solution
6
Mixing takes place when the change of the Gibbs free energy is negative. The entropy of mixing is given by the number of possible configurations of the polymer chains; its change is very small for polymer-solvent systems, especially at low concentrations of a high molecular weight polymer. Thus, the sign of the change of the Gibbs free energy determines the enthalpy term, which characterizes the interaction between two molecules, e.g. polymer-polymer, polymer-solvent or solvent-solvent interactions.
𝛿𝑖 is the solubility parameter, 𝛷𝑖 is the volume fraction of ith constituent in the mixture, and V is the volume of the mixture. The difference in solubility parameters must be small to
The change in the interactions upon mixing (or enthalpy of mixing) governs the miscibility.
Empirically, the temperature dependence of the Flory interaction parameter is often written as the sum of two terms, referred to the ‘entropic part’, A, and the ‘enthalpic’ part, B/T:
T A B
12
. (2.5)
In practice, there are a number of confounding factors to the Flory-Huggins theory. When the entropy of mixing for polymer-solvent systems is not small enough, the simple Flory-Huggins approach would be not sufficient. There are several aspects, which this simple form of the interaction parameter proposed by Flory-Huggins, does not take into account, e.g.
molecular weight polydispersity or compressibility and thermal expansions effects.
The polymer solubility depends on the solvent quality, the polymer concentrations, its molecular weight, polydispersity and perhaps other values, including the solution temperature. The solvent quality is related to the balance between the enthalpy and entropy of mixing.38 When they are in balance, the energetic part of two-body interactions exactly
7 cancels the entropic part and the interaction parameter χ = ½. In this case, the solution is in so-called θ-conditions and at the θ–temperature and chains behave as “ideal”. As the parameter χ is lowered, the polymer “likes” the solvent more and the chains are expanded. As the parameter χ is increased, the polymer “likes” the solvent less, and phase separation occurs. The solubility properties of a polymer can be changed either by changing the temperature or by changing the mixing ratio of a good solvent to a poor solvent.
The dimension of the real chain is different from that of the ideal chain of the same contour length, for instance, due to the excluded volume, which corresponds to the interaction between segments.13,39 Excluded volume interaction cause a polymer coil expansion and mixing with solvent molecules in a good solvent, and the polymer coil shrinkage with formation of a very compact structure that excludes all solvent in a poor solvent.40,41 The polymer chains extension is limited by the C-C covalent bonds and the entropy of the coil.
The excluded volume effect can be described in terms of the total internal energy of the segmental interactions, U, expanded as a power series of the segment density ρ:
...) A A
( VkT
U 2 2 3 3 , (2.6)
where A2 is the second virial coefficient of the expansion, which is a measure of the solvent-mediated polymer-polymer interactions. A2 accounts for the pair interaction between the repeating units of the chain and solvent molecules, it is the characteristic of the interaction potential between the segments and in general depends on the temperature.13 A2 =0 in the Θ-state of a polymer solution, T = TΘ, when there is no interaction between the repeating units and the polymer chains are in an ideal Gaussian coil conformation.
The mean-field theory predicts the collapse of a single linear polymer chain from expanded coil to a rather dense globule, the coil-to-globule transition, in organic solvents.42 The solubility of a polymer characterized by the short-range van der Waals interactions: the stronger the interactions, the more thermodynamically poor the solvent and the lower the polymer solubility (A2 < 0). An ideal polymer chain of infinite molecular weight undergoes the transition at TΘ, whether, a real polymer chain of finite molecular weight at lower temperature, T < TΘ.43