Partial molar area

This means that the partial molar area may directly be determined from the change in molecular area, when an amphiphilic molecule is introduced into a host liquid crystalline pattern. Of course, this area is the change of area per molecule at the introduction of one molecule of the substance in question and may be influenced by the interaction between the host molecules and the guest molecules. Since this interaction is an essential part of the present problem, it appears obvious that the method exactly meets the requirements. 

We commence with the adsorption of nonionic surfactants, which does not require the consideration of the effect of the electrical double layer on adsorption. The equilibrium distribution of the surfactant molecules and the solvent between the bulk solution (b) and at the surface (s) is determined by the respective chemical potentials. The chemical potential /zf of each component i in the surface layer can be expressed in terms of partial molar fraction, xf, partial molar area a>i, and surface tension y by the Butler equation as [14]... 

The average To of the partial molar area for all components or all possible states at the interface is often used in conjunction with the Lucassen-Reynders dividing surface, which can be equivalently described as... 

The average partial molar area is determined by Eq. 12 as l + niFi/Fcr- ... 

Later, Alexander and Barnes ( ) showed that the constant conditions of equation 7 are not those required for the partial molar area. This is given by ... 

It should be noted that in both of these approaches it is necessary to make an assumption about the partial molar area of monolayer (equation 8 or 11). 

Molar area, adsorbed phase Parameter, cubic equations of state Partial parameter, cubic equations of state Second virial coefficient, density expansion... 

First, two-dimensional partial quantities can be introduced. For the area, = 0A/3n° ) o it represents the increase in molar area if an infinitesimal amount... 

Figure 3.16. Possible dependence of the molar area in a binary Langmuir monolayer on composition, k and Tare fixed. and A are the partial areas of si and s2 in point B.

The equations can be essentially simplified for globular protein molecules, which can exist in the surface layer in two states only. In this case, the adsorption values in the states 1 and 2 (Tj and T2, respectively), characterised by different values of the partial molar area a) (cd2 > u>i), are related via the adsorption equation... 

The mean partial molar area (averaged over the two states) can be expressed by... 

Differences in the molecular area are of obvious relevance in mixed monolayers, where larger molecules have a larger partial molar area than smaller ones. Such differences lead to a situation where the smaller molecules cire increasingly preferentially adsorbed with increasing surface pressure, even in the absence of any surface interactions [16]. In adsorption layers consisting of a single surface-active compound similar effects can occur if, due to the asymmetry in different adsorption states the molecules can occupy different areas [3, 4, 15, 19, 21, 22], The fraction of molecules which are in the state characterised by a particular partial molar area depends on the surface pressure. In a thermodynamic study by Joos and Serrien [21] it was shown that if the molecule possesses, say, the two modifications 1 and 2, with different partial molar surface areas coi and o)2 (in absence of intermolecular interactions) their ratio in the surface layer obeys the equation... 

Thus, deviations from the ideal Langmuir isotherm can be caused both by intermolecular interactions, which result in an enthalpy of mixing, and by area differences between molecules, which produce a non-ideal entropy of mixing [18]. For a simple case where the interactions are of the Frumkin type and the partial molar areas of solvent and surfactant are constant the entropic effect of area differences results in typical features of macromolecular adsorption, e.g., a steep initial increase of adsorption ( high affinity adsorption) and a very slow rise once the surface is approximately half filled [18]. 

It is seen that the additional (nonnalised) activity coefficients introduced in Eq. (2.10) to establish the consistency between the standard potentials of the pure components and those at infinite dilution, can be incorporated into the constant Kj in Eq. (2.15). Therefore, if a diluted solution with activity coefficients of unity is taken as the standard state, the form of Eqs. (2.13) and (2.14) remains unchanged. The equations (2.14) and (2.15) are the most general relationships from which meiny well-known isotherms for non-ionic surfactants can be obtained. For further derivation it is necessary to express the surface molar fractions, x-, in terms of their Gibbs adsorption values Tj. For this we introduce the degree of surface coverage, i.e. 9j = TjCOj or 0j = TjCO. Here to is the partial molar area averaged over all components or all... 

The equations which describe the reorientation of surfactant molecules in the surface layer can be derived from Eqs. (2.26) and (2.27). It is assumed that the reorientation results in a variation of the partial molar area coj. Note that for the derivation of Eq. (2.7) it was assumed that the tOj are independent of y. This requirement, however, does not contradict with the assumption of variable molar areas, because only the ratio of the surfactant adsorptions in different states, i.e., the states with different coj-values, depends on y. 

On the other hand, the number of possible states for adsorbed molecules, corresponding to different partial molar areas cOj, can be quite large. Theoretically one can assume a continuous change of co between cOmin and cOmax, and successive values varying from each other by an infinitesimal increment of the molar area Aco. The transition from a discrete to a continuous reorientation model can be performed formally, replacing the summation in Eqs. (2.78) and (2.89) by an integration. 

For ideal (with respect to the enthalpy) surface layers of a surfactant capable of adsorbing in two states (1 and 2) with different partial molar areas cOj (coi > 2) and different adsorption equilibrium constants, Eqs. (2.26) and (2.27) can be transformed into a generalised von Szyszkowski-Langmuir equation of state [25]... 

This is illustrated in Fig. 2.8, where the surface layer coverage is plotted vi the bulk concentration. Here the partial coverage FjOj was calculated for the two states of the adsorbed molecules (i = 1, 2) with isotherm parameters coi= lO m mol, C02= 2.5-1 O m mol and a = 3, which are typical for oxyethylated alcohols (see below). It is seen that the maximum coverage by the molecules in state 1 is reached for a concentration of about 2-10 mol/1, while for c = 5-10 mol/1, i.e., when the curve corresponding to this parameter set at Fig. 2.6 exhibits a sharp increase, the coverages for the two states of the adsorbed molecules become approximately equal to each other (see Fig. 2.8). It should be noted that at concentrations above 2-10 mol/1 the total adsorption increases only due to a decrease of the fraction of molecules adsorbed in the state with maximum partial molar area. 

Here a is a constant which determines the variation in surface activity of the protein molecule in the i state with respect to the state 1 characterised by a minimum partial molar area C0[ = co j , bj = b,i . The value i can be either integer or fractional and the increment is defined by Ai = Aco/coi. For a = 0 one obtains b = bj = const, while for a > 0 the bj increase with increasing coj. 

It is seen that for low values of n the adsorption achieves its maximum at = max = 40 nmVmolecule. With the increase of n the value of (Oj(rmax) decreases monotonously, while finally at n 2.0 mN/m the maximum adsorption corresponds to the state which possesses minimum partial molar area of tOmin = 2 nmVmolecule. Therefore, the protein adsorption layer is characterised by almost a complete denaturation at low surface pressure while at large surface pressures the adsorption layer is comprised of molecules in a state with minimum surface area demand. 

The theory which describes the penetration of a soluble surfactant into a monolayer formed by molecules possessing equal partial molar area (mixtures of homologues), was extended recently to include the actual process of protein penetration into 2D aggregating phospholipid monolayer [155, 157]. This extension was based on the concept of independent segments of the protein molecules, occupying an area equal to that of the phospholipid molecule. In the theoretical models, various mechanisms for the effect of the soluble surfactant on the aggregation of the insoluble component can be considered ... 

Here Fj and Fc are the partial and critical adsorption of monomers, respectively, and the average molar area co should be expressed via the equation... 

---