sub-stance differ substantially. The adsorption isotherms obtained for low molecu-lar mass systems cannot be applied to polymers. However, for very dilute solution, adsorption can be described by the Langmuir isotherm
θ= Ap/Aps= bc/(1+bc) [1.2]
whereθis the degree of surface coverage, Apis the amount of polymer adsorbed at concentration c, Apsis the adsorption at saturation, and b is a constant. Eq 1.2 is derived for solutions in which adsorbing molecules are spherical, do not inter-act with each other, and do not change their shape on adsorption. None of these conditions are valid for polymer solutions and therefore experimental isotherms coincide with those calculated using the Langmuir equation only at very low
concentrations. For a wider concentration range, the polymer adsorption can be described by the empirical Freundlich isotherm:
Ap=βcµ [1.3]
whereβandµare constants. Eq 1.3, however, is not applicable at low concentra-tions. The applicability of the Freundlich equation over a wide concentration range can be explained by the mechanism of aggregative adsorption (see 1.8), in which aggregates of macromolecules, having independent kinetics or structural units, interact with the adsorbent surface together with individual molecules.10 In this instance the adsorption mechanism is not as specific as for dilute solu-tions because the conformational effect is less important. Polymer adsorption isotherms for dilute solutions have been derived theoretically by Simha, Frisch and Eirich.15,16The polymer solution is assumed to be infinitely dilute, whereas a polymer molecule is regarded as a gaussian coil. Active centers are located reg-ularly on the surface with the area of the active center corresponding to the sur-face of adsorbing segment, and each active center can bind only with one segment. In this instance, only monomolecular adsorption is taken into account.
For this case, the adsorption isotherm assumes the form:
( )
θe2K1θ / 1−θ v Kc
= [1.4]
whereθis the degree of the surface coverage, K1is a constant characterizing the interaction of polymer molecules with each other, K is a complex function of the molecular mass of the segment, solvent, temperature and other variables, <ν> is the mean number of bound segments of each molecule consisting of t segments;
<ν> = pt. The isotherm equation for <ν> = 1, i.e., for full adsorption of all seg-ments, becomes the Langmuir adsorption isotherm, as K1θ<< 1. Although the derivation of the isotherm equation is based on simplified assumptions, the es-sential point is that at the stage of determination of values involved in the equa-tion, one considers the interaction of polymer segments with each other, i.e., the concept of a reflecting barrier is introduced, due to which the already adsorbed segments hinder further adsorption. The magnitude of the barrier is character-ized by a number of loops restricting access to adsorption centers and a function
of the degree of surface coverage. Experimental data on the adsorption of poly-mers from dilute solutions, however, show that the value of equilibrium adsorp-tion, as a rule, exceeds the values calculated for monomolecular adsorption.
Therefore, Frisch and Simha further assumed that one adsorption center can bind S layers of segments if the adsorption is carried out through individual loops; <ν> equals the mean number of molecular bonds of each molecule in all layers. The equation of the adsorption isotherm can be derived in the following way. Let the surface contain Nsadsorption centers capable of binding one seg-ment each. In the polymer solution, there are N molecules from t segseg-ments, of whichνsegments are bound with the surface and Nosolvent molecules. The frac-tion of the surface occupied by polymer segments,θ, and molecules of solvent,θo, can be found from the equation:
θ = νN′/Ns; θo=N′o/Ns [1.5]
where N′is the number of adsorbed macromolecules and N′ois the number of ad-sorbed solvent molecules. The number of polymer molecules remaining in the solution of concentration, c, is (N - N′), whereas the number of the solvent mole-cules is (No- N′o) at the solvent concentration co. Let us consider the following equilibriums:17
a) macromolecule in the solution +νfree sites give N′adsorbed
macromolecules bound by n segments (equilibrium constant Kν= k1/k2) b) solvent molecules + 1 free site give N1o of adsorbed molecules of the solvent (equilibrium constant K = k ko 1o / ).2o
The concentration of free sites on the surface is (1 -θ − θo). Applying the law of mass action, we find an expression for the rate of the adsorption of the polymer:
r1= k1c(1 -θ − θo)ν [1.6]
and the rate of desorption:
r2= k1oco(1 -θ − θo) [1.7]
Accordingly, the rate of the solvent adsorption is:
r = k c (11o 1o o − −θ θo) [1.8]
and the rate of desorption is:
r = k2o 2oθo [1.9]
In Equations 1.8 and 1.9, we equate r1and r2and divide both sides of the equation by (1 -θ)n:
(θ/ν)/(1 − θ)n= Kνc[1 -θo/(1 -θ)]ν [1.10]
where Kν= k1/k2. Taking K = k ko 1o / , we obtain:2o
θ(1 − θ)= Koco/(1 + Koco) =βo<< 1 [1.11]
Let us divide equation by (1 -θ). After a number of conversions, we find:
θ/ν(1 − θ)n= ( 1 -βνo)Kν c = Kc [1.12]
At Ko →1, the equation reduces to the Simha and Frisch equation, and at Kν →0(i.e.,θ →0) to the Langmuir equation. The same equation may be ob-tained on the basis of statistical concept of the behavior of the flexible molecule in space. There are other approaches to derive the isotherms of adsorption based on the statistical physics of polymer. The equations obtained cannot be proven by experiment, due to a great number of unknown parameters.
A more perfect form of the adsorption isotherm was derived by Silberberg,11 based on concepts of conformation of adsorbed chains and the structure of the adsorption layer formed by the sequences of bound segments and loops extending into the solution. According to Silberberg, the shape of the chain is determined by the adsorption energy and surface structure, i.e., by the character of the arrangements of active centers in it. Real isotherms of polymer
adsorption strongly depend on the polymer polydispersity, due to various adsorbance of low and high molecular mass fractions.11,18
1.3 THERMODYNAMIC INTERACTION BETWEEN POLYMER AND SURFACE