Interpenetration free energy

We consider two polymers A and B with the same radius Ra = Rb-Rj (o = RaIRb = 1) When the two polymers occupy the same volume, their interaction free energy is proportional to their number of contact points. In a Flory-like model, the interpenetration free energy per chain is thus given by ... 

If the interpenetration free energy Fint is much smaller than kT, the two polymers interpenetrate almost freely. They are diaphanous to each other and a mean-field theory (Eq. 4) may be used to calculate the mutual virial coefficient Gab ... 

Comparison of Theory and Experiment. The expression for the free energy of interpenetration of sterically stabilized particles may be obtained by combining Equations 2, 3 and 6. Using these expressions can be calculated as a function of both... 

Figure A The total free energy of interpenetration of two particles is shown plotted as a function of temperature for atmospheric and 180 bars pressure.

Repulsive forces between Fe oxide particles can be established by adsorption of suitable polymers such as proteins (Johnson and Matijevic, 1992), starches, non-ionic detergents and polyelectrolytes. Adsorption of such polymers stabilizes the particles at electrolyte concentrations otherwise high enough for coagulation to occur. Such stabilization is termed protective action or steric stabilization. It arises when particles approach each other closely enough for repulsive forces to develop. This repulsion has two sources. 1) The volume restriction effect where the ends of the polymer chains interpenetrate as the particles approach and lose some of their available conformations. This leads to a decrease in the free energy of the system which may be sufficient to produce a large repulsive force between particles. 2) The osmotic effect where the polymer chains from two particles overlap and produce a repulsion which prevents closer approach of the particles. 

There has been considerable recent interest in the determination by XPS of the surface composition of co-polymers whose components differ considerably in their surface free energy (51). When the individual components of block co-polymers or interpenetrating networks can be detected and individually identified by XPS, a fairly rapid co-polymer composition determination is possible as well as the determination of the composition of... 

The free energy change which takes place when polymer chains interpenetrate is influenced by factors such as temperature, pressure and solvent composition. The point at which this free energy change is equal to zero is known as the 0 (theta)-point and such a solvent is called a 0-solvent. More formally, a 0-point is defined as one where the second virial coefficient of the polymer chains is equal to zero. It can be determined by light scattering and by osmometry. 

Napper (14) has derived analytical expressions for the free energy of interpenetration and compression of the adsorbed polymer chains as a function of the distance of separation between two particles. The region of dose approach was divided into two domains ... 

Here A is the surface to surface separation between the bare particles and 6 is the thickness of the adsorbed layer. For distances of separation A > 26, the free energy of mixing of the chains is zero. Assuming constant segment density distribution in the adsorbed layers, Evans and Napper (15) derived the following expression for the free energy in the interpenetration domain, which is due only to the mixing of the chains ... 

In order to apply the above procedure to determine the conditions of phase separation, we have chosen the system of polyisobutene-stabilized silica particles with polystyrene as the free polymer dissolved in cyclohexane. The system temperature is chosen to be the 8 temperature for the polystyrene-cyclohexane system (34.5°C), corresponding to the experimental conditions of deHek and Vrij (1). The pertinent parameters required for the calculation of the contribution of the adsorbed layers to the total interaction potential are a = 48 nm, u, =0.18 nm3, 5 = 5 nm, Xi = 0.47(32), X2 = 0.10(32), v = 0.10, and up = 2.36 nm3. It can be seen from Fig. 2 that these forces are repulsive, with very large positive values for the potential energy at small distances of separation and falling off to zero at separation distances of the order of 25, where 6 is the thickness of the adsorbed layer. At the distance of separation 5, the expressions for the interpenetration domain and the interpenetration plus compression domain give the same value for the free energy, indicating a continuous transition from one domain to the other. 

When the concentration of the free polymer is set equal to zero, the situation corresponds to pure steric stabilization. The free energy of interaction due to the interpenetration of the adsorbed polymer chains has a range of 26, where 6 is the thickness of the adsorbed layer. This free energy is proportional to the quantity (0.5 - x), where x is the Flory interaction parameter for the polymer-solvent system. Thus, a repulsive potential is expected between two particles when x < 0.5 and this repulsion is absent when x = 0.5. For this reason, it was suggested [25] that instabilities in sterically stabilized dispersions occur for x > 0.5, hence for theta or worse-than-theta conditions. However, the correlation with the theta point only holds when the molecular weight of the added polymer is sufficiently high... 

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). 

According to the free volume theory, the total free energy of close approach (or flocculation), AGr, in the interpenetrational domain is the sum of three free energy contributions... 

The combinatorial contribution The combinatorial free energy of interpenetration is readily understood for it is purely entropic in origin. It arises on interpenetration of the stabilizing... 

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