thermody-namic quality of solvent and the interaction energy between polymer and sur-face. All theories of adsorption include the thermodynamic parameter of interaction of the Flory-Huggins theoryχ12. The thermodynamic interaction be-tween polymer and solvent determines the conformation of macromolecules in solutions and thus the conditions of its interaction with the surface. The inter-action between polymer and surface is characterized by the parameter of ther-modynamic interaction, which was introduced by Silberberg,19,20χs, using the model of quasi-crystalline lattice of the surface layer, describing the properties of polymer solutions. This parameter may be determined as follows:
∆Us= −χskT [1.13]
where∆Usis the change in enthalpy by adsorption, i.e., the difference between energies of contacts segment-surface and segment-solvent. The physical mean-ing of parameter is discussed below.4If a molecule of a solvent, adsorbed by the surface, is displaced by the segment of macromolecule, the interaction between segment and solvent is changed. It is assumed that the total number of contacts, z, of a given segment or molecule of solvent produces z′contacts on the surface, and (z - z′) contacts are between neighboring solvent molecules and other seg-ments in the bulk of solution. Parameterχscharacterizes the total change in enthalpy (in kT units) in the course of exchange of the segment having 1/2(z \ z′) contacts with solvent molecules and 1/2(z - z′) contacts with other segments. The solvent molecules in solution have 1/2z contacts with other solvent molecules and 1/2z contacts with segments of macromolecules. It is supposed that adsorp-tion sites on the surface have an equal number of contacts between segments and solvent molecules.
As a result of such an exchange, segment substitutes z′ contacts seg-ment-surface (S-2) on 1/2 z contacts polymer-polymer (2-2) and 1/2 z contacts polymer-solvent (2 -1). The molecule of solvent losses 1/2 z′contacts (1-2) and 1/2z′contacts (1-1), and gains z′ contacts (S-1). Other (z-z′) segments are
un-changed for solvent molecule and segments and do not contribute to the total enthalpy. The enthalpy of exchange will then be:
z′(Hs1- Hs2+ 1/2 H22+ 1/2H11) [1.14]
or
χs =z′(Hs1 - Hs2+ 1/2H22+ 1/2H11)/(kT) [1.15]
The valueχsis positive if by interaction of polymer with adsorbent, there exists a critical valueχs = (χs)crit at which adsorption is possible. This critical value is of the order some tenth per segment. Value (χs)critexists because the loss in conformational entropy, connected with segment adsorption, must be com-pensated by the decrease in free energy due to the formation of contacts seg-ment-surface. Whenχsbegins to exceed (χs)crit, the adsorption sharply increases with growth ofχs. Such behavior distinguishes adsorption of polymers from ad-sorption of small molecules, which is possible at all positive values ofχs and gradually increases with the growth of this parameter. The estimation of this value is of great importance. However, only recently was a general method of the determination ofχsproposed.21
The method is based on the following theoretical premises. The lattice model for a regular solution in contact with an adsorbing surface is used. The physical meaning of value -χs/kT allows one to conclude that if the contact sur-face-solvent has a low free energy, valueχsmay be below critical value and ad-sorption of a polymer does not proceed. It means that there are such solvents which prevent adsorption. A mixture of two solvents allows us to bring about the desorption of previously adsorbed polymers. When the composition of mixed sol-vent is changed, the excess amount of adsorbed polymer,θexp , expressed as the number of equivalent monolayers, diminishes and eventually passes through the valueθexp = 0. At the critical composition of binary solvent, the adsorbed poly-mer is fully substituted by a more strongly adsorbing component of mixed sol-vent (solsol-vent-displacer). In such a three-component system, threeχsparameters play a role, one for each pair of components. Let us label eachχby two indices, specifying the respective component pair. Thusχspo is the adsorption energy pa-rameter for adsorption of polymer from solvent,χspd for polymer from displacer,
andχsdo for displacer adsorbing from the solvent. Then there is a simple relation between the three parameters:
χspo -χspd =χsdo [1.16]
The preferential adsorption of displacer is characterized by the negative value ofχspd. In the three-component system, the adsorption may be represented as an exchange process between each of the three-component pairs: polymer-sol-vent, polymer-displacer, and solvent-displacer. The free energy change of all three processes, which may be expressed by threeχsand threeχsparameters for sixconcentrations in the surface layers and in the bulk solution, must be zero. In order to describe the equilibrium in a three-component solution, it is sufficient to calculate the free energy of two exchange processes in which all three compo-nents participate.
In present theories,22,23it is considered a multilayer model for description of the interfacial region, i.e., a lattice consisting of many layers of sites parallel to the surface. Each lattice site may be occupied by either a solvent molecule, a displacer molecule, or a polymer segment. Each site has z nearest neighbors, fractionsλoandλ1of which are in the same layer and in each of the neighboring layers, respectively. When the chain consisting r segments is adsorbed with the fraction of bound segments, p, that means that pz molecules of a solvent are dis-placed from the surface. The free energy change∆Gpoconsists of several parts:
• the entropy of mixing per polymer chain, kTln(Φ1/F*), whereΦ1is the vol-ume fraction of polymer in the adsorption layer, andΦ* is the volume frac-tion of polymer in the bulk solufrac-tion
• the mixing entropy for the pr solvent molecules, -kTpr ln(Φ Φ1o / *o), where Φ1o andΦ*o are the volume fractions of the solvent in the adsorption layer and in solution, respectively
• the change in conformational entropy per one chain. Its loss is equal to kT(pr−1)(χs)crit, where (χs)critis a critical parameter equal to ln (1 +λ)
• the adsorption energy with respect to the solvent, -kTprχspo
• the mixing energy per chain, pr∆Hpo. The last value determines the changes in the contact free energy in the case, when a polymer segment in solution is exchanged with a solvent molecule in the adsorption layer.
The final expression for the interaction parameterχspo is:24
χspo = lnΦcrit + lns + ( )χs crit −λ χ1 po + [(1− Φcrit)(1−λ1)−λ1]∆χdo
[1.17]
∆χdo =χpd −χdo −χpo
where s = exp(χsdo +λχdo) is the initial slope of the displacer isotherm.
The experimental verification was presented for the adsorption of polyvinylpyrrolidone on silica,24and adsorption of PMMA on fumed silica.25The theory developed gives the possibility of experimental evaluation of the ener-getic factors and their role in adsorption.
Another experimental method of estimation of the energetics of adsorption is based on direct calorimetric determination of heat of adsorption.26,27,28The in-tegral heat of adsorption is determined from the relation:
-∆H = n∆Hps+ n∆Hso+ n∆Hpo+∆Hpp [1.18]
where n is the number of surface sites covered by polymer segments,∆Hpsis the enthalpy of bonding per a mol of monomeric units,∆Hsois the heat of wetting per mole of active sites, ∆Hpo is the enthalpy of desolvation per mole of adsorbed monomeric unit, and∆Hppis the energy of interaction of adsorbed polymer mole-cules. If the last term is neglected, it may be written:
-∆H = n(∆Hps-∆Hpo-∆Hso) [1.19]
where the adsorption enthalpy is proportional to the number of active sites on the adsorbent surface and constant enthalpy contribution (the term in brack-ets). The differential enthalpy of adsorption is:
DA= -∂∆H/∂Ν = ∂Η/∂Ν (∆Hps-∆Hpo-∆Hso) [1.20]
where N = A/Mmis the number of monomeric units of adsorbed macromolecules, A is the amount of adsorbed polymer, g, and Mm is the molecular mass of monomeric unit. In such a way, the value DAmay be considered as a measure of the fraction of segments bound with the surface.