Temperature Flory

The temperature TF is related to the demixtion curve of polymer solutions, since 

There are also special temperatures which are associated with a solvent-solute system of area S and which have the same limit TF when S - oo. This fact reveals that the system is in a tricritical state. 

Tf can be determined by measuring the temperature at which the two-body interaction b vanishes. We showed in Chapter 15 how values of b can be deduced from the swelling of a chain in good solvent. By letting the temperature T vary in good solvent, one obtains, in the zero concentration limit, a function b T) which is linear with respect to /T (see, for instance, Fig. 15.9). The extrapolation for b - 0 gives a value of Tr defined by  

The method which seems to give the best result consists in measuring the critical temperature TC(S) for various values of S (see, for instance, Fig. 16.2). The result Tf is obtained by extrapolation 

The most current method consists in determining the temperature TrF(S) at which the second virial coefficient vanishes. This coefficient is obtained either by radiation scattering or by osmotic pressure measurements. The temperature 

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Theta temperature (Flory temperature or ideal temperature) is the temperature at which, for a given polymer-solvent pair, the polymer exists in its unperturbed dimensions. The theta temperature, , can be determined by colligative property measurements, by determining the second virial coefficient. At theta temperature the second virial coefficient becomes zero. More rapid methods use turbidity and cloud point temperature measurements. In this method, the linearity of the reciprocal cloud point temperature (l/Tcp) against the logarithm of the polymer volume fraction (( )) is observed. Extrapolation to log ( ) = 0 gives the reciprocal theta temperature (Guner and Kara 1998). 

We discuss this more fully below, but one thing to note immediately is that T = 9 describes the same state as described by x = 1/2, namely, the condition of B = 0. It is apparent that 9 is a temperature —variously known as the theta temperature or the Flory temperature. Introducing this parameter indicates why the B = 0 situation is called the 9 condition. In order to justify the equivalence of the two sides of Equation (81), consider the following steps ... 

What is theta temperature (or, the Flory temperature) What are the relative magnitudes of the excluded-volume interactions and the energetic interactions in a dilute polymer solution at its theta temperature ... 

The solubility limits of a given polymer are closely related to the Flory-temperatures of the polymer in various solvents. 

The Flory-temperature or theta-temperature (0F) is defined as the temperature where the partial molar free energy due to polymer-solvent interactions is zero, i.e. when y = 0, so that the polymer-solvent systems show ideal solution behaviour. If T = 0F, the molecules can interpenetrate one another freely with no net interactions. For systems with an upper critical solution temperature (UCST) the polymer molecules attract one another at temperatures T < 0F. If the temperature is much below 0F precipitation occurs. On the other hand for systems with a lower critical solution temperature (LOST) the polymer molecules attract one another at temperatures T > F. If the temperature is much above 0F precipitation occurs. Aqueous polymer solutions show this behaviour. Systems with both UCST and LCST are also known (see, e.g. Napper, 1983). 

It is clear that the Flory-temperature is the critical miscibility temperature in the limit of infinite molar weight. Fox (1962) succeeded in correlating 0F-temperatures of polymer-solvent systems with the solubility parameter 5S of the solvent. Plots of 8S as a function of 0F are shown in Fig. 7.8. 

At a given temperature, a solvent for the polymer should have a (5-value approximately between the limits, indicated by the two straight lines in the figure. An even better correlation of Flory-temperatures with solubility parameters can be given in a <5h-<5v-diagram. This is shown in Fig. 7.9 for polystyrene. The circle drawn in Fig. 7.9 corresponds again with Eq. (7.18). 

It is sometimes convenient to define the temperature at which a polymer of infinite molecular weight just becomes insoluble in a given solvent this temperature is the Flory temperature or theta temperature, 6, which may also be defined by... 

Theta conditions occur at a particular temperature T = 6, known as the theta (or Flory) temperature, for which (jj,i — = 0. It means... 

Equation (3.104) provides an alternative definition of the theta temperature, as being the proportionality constant relating AiJj to ASj. According to Eq. (3.105), the theta (or Flory) temperature can be defined as the temperature at which the condition k — ijj prevails. Substituting Eqs. (3.99) and (3.100) into Eq. (3.104) gives... 

Theta (6) solvents are solvents in which, at a given temperature, a polymer molecule is in the so-called theta-state. The temperature is known as the theta-temperature or the Flory temperature. (Since P. J. Flory was the first to show the importance of the theta-state for a better understanding of molecular and technological properties of polymers, theta temperatures are also called Flory temperatures. ) In the theta-state, as explained above, the solution behaves thermodynamically ideal at low concentrations. 

A better definition of the Flory temperature Te is given in Chapter 14. 

On the contrary, with Daoud and Jannink,49 we can consider the length of the polymers as fixed and assume that the temperature and consequently the solubility of the solution may vary. In this case, a polymer solution is represented by a point in the (C, T) plane. The good-solubility domain corresponds to temperatures T > TF where Tr in the Flory temperature. By definition TrF(N) is the temperature at which the second virial coefficient vanishes and TF = lim Trf (N - qo ). Anyway, if N is large TrF(N) is close to TF. 

The subject was renewed by application of renormalization theory to this problem. P.G. de Gennes pointed out in 1975 that, at the Flory temperature Tf, the polymer solution can be considered as a tricritical system. Since that time,. this fact has been used by various physicists, among them, P.G. de Gennes, M.J. Stephen, and B. Duplantier (1981). However, these new approaches have to be developed the theoretician or the experimentalist who studies the behaviour of polymer solutions in the vicinity of Tf encounters difficult problems in fact, the question remains very complex. 

Fig. 14.1. The reduced osmotic pressure tlMjRTp is plotted against the mass concentration p, for various temperatures (according to Strazielle 1 polystyrene M = 130000 solvent cyclohexane temperatures 40 °C, 35 °C, 30 °C). The temperature T corresponding to 35 °C is close to Tf (the Flory temperature).

This result is important. It shows that a more sophisticated theory is needed to study excluded volume effects in the vicinity of the Flory temperature. Therefore, the properties of polymer solutions, above and below the Flory temperature, are not directly related, and this is an important fact which has not always been recognized.5,6... 

It is convenient to consider c as a constant and b as a linear function of temperature. Thus, a question arises which values of z and y correspond to the Flory temperature TFT. The simplest and most physical definition of TF is as follows it is the temperature at which the second virial coefficient vanishes. This condition can be written in the form... 

The properties of a polymer solution become extremely complex in the vicinity of the Flory temperature TF, when the size of the chains increases. In fact when N -+ oo, Tc - Tf. Then, the system becomes critical in a double manner. First, a system made of very long chains is equivalent to a Landau-Ginzburg model, in which the order parameter is a zero component vector (n = 0). Secondly, the top of the demixtion curve, even for chains of finite length, is a critical point, and near this point the system can be described by another Landau-Ginzburg model the order parameter of which is a one-component vector (n = 1). 

To describe the coil-globule transition in the vicinity of the Flory temperature, in 1968 Lifshitz8 proposed an original method which amounts to expressing the free energy of a polymer in terms of the local monomer concentration. This method was subsequently reexamined and developed in various articles.2 We shall describe it here but, for reasons of convenience, we shall use a slightly different formalism. 

In this context, the Flory temperature Tv is defined by the condition that the second virial coefficient vanishes, a fact which occurs when x = 1/2. Consequently, in the vicinity of TF, we have... 

It is assumed here that the temperature of the solution does not differ much from the Flory temperature Tp. Now, we recall that for d = 3, the three-body interaction is marginal, as can be seen from (14.6.2). Thus, for g 1 and d = 3, the situation of the system reminds us of the situation of the standard continuous model for d = 4, and these marginal systems will be treated in very similar manners (see Chapter 12, Section 3.3.4.2). This means that in practice, for g 4 1, the chains are nearly Brownian but that logarithmic corrections must be calculated. However, the fact that demixtion may occur when g is weakly negative introduces additional difficulties. 

In order to calculate the swelling of a chain in the vicinity of the Flory temperature, we need c, — H) to first order in k2. In particular, this quantity... 

The experiments described above indicate the existence of a Brownian behaviour for atactic polystyrenes in the two following physical situations polymers in the melt or in dilute solutions at the Flory temperature Tv. 

The distinction between poor and good solvent was introduced in the 1950s by Fox and Flory after experimental studies of the intrinsic viscosity of polymer solutions. These authors recognized that the viscosity varies in relation to the dependence of the chain sizes on temperature the poor solvent state is the state of a solution in which the chains have quasi-Brownian configurations. Systematic experiments have been made in this domain, for instance to determine the Flory temperature, but they have never given very precise results. Physicists are just now beginning to overcome the experimental and theoretical difficulties. Experiments have been made to show the existence of a collapse of the polymer chain, and certain authors have been prone to compare it with the coil-globule transition in proteins. 

Let 7y be the Flory temperature. The interval TF — Te decreases when the equivalent Brownian area S of the polymers increases. For the polystyrene-cyclohexane system, we found a relation [see (16.2.7)] which can be written in... 

Equation (3.93) indicates that the size of a polymer coil increases with increasing molecular weight of the polymer. According to this equation, at a temperature equal to the Flory temperature (i.e., T = 0), - a = 0, i.e., a = 1. Thus,... 

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