Polymers adsorption kinetics, model

Most theories for polymer adsorption kinetics start from (combinations of) the models discussed earlier. Other theories, often proposed for (bio)polymer adsorption, are based on the random sequential adsorption (RSA) model. According to this model, the adsorbate molecules arrive randomly at the interface and they stick where they hit. It implies that both desorption and tangential motion of the adsorbate at the interface are absent. Because the center of a newly arriving spherical molecule cannot be accommodated within the shaded areas enclosed by the dashed circles shown in Figure 15.6, only the unshaded fraction < ) of the surface is available for adsorption. It is obvious that ( ) is a function of 0, the fraction of the surface that is covered by the adsorbate. For sphere-like molecules, 0 = niUi (R being the radius of the molecule). The following expressions for the available surface function < )(0) can be derived from the RSA theory ... 

Most spraying processes work under dynamic conditions and improvement of their efficiency requires the use of surfactants that lower the liquid surface tension yLv under these dynamic conditions. The interfaces involved (e.g. droplets formed in a spray or impacting on a surface) are freshly formed and have only a small effective age of some seconds or even less than a millisecond. The most frequently used parameter to characterize the dynamic properties of liquid adsorption layers is the dynamic surface tension (that is a time dependent quantity). Techniques should be available to measure yLv as a function of time (ranging firom a fraction of a millisecond to minutes and hours or days). To optimize the use of surfactants, polymers and mixtures of them specific knowledge of their dynamic adsorption behavior rather than equilibrium properties is of great interest [28]. It is, therefore, necessary to describe the dynamics of surfeictant adsorption at a fundamental level. The first physically sound model for adsorption kinetics was derived by Ward and Tordai [29]. It is based on the assumption that the time dependence of surface or interfacial tension, which is directly proportional to the surface excess F (moles m ), is caused by diffusion and transport of surfeictant molecules to the interface. This is referred to as the diffusion controlled adsorption kinetics model . This diffusion controlled model assumes transport by diffusion of the surface active molecules to be the rate controlled step. The so called kinetic controlled model is based on the transfer mechanism of molecules from solution to the adsorbed state and vice versa [28]. 

The more complex adsorption kinetics for PPPS in Fig. 2 suggests that it cannot be described by a simple diffusion model. The sigmoidal shape also indicates that cooperative processes have to be taken into account, and these may be rate-limiting. In the case of a rigid polymer like PPPS one may assume a nematic ordering of the rod-like polymers in two-dimensional sheets and the pressure change - may reflect the amount U (t) of polymers in... 

Another interesting but very complex model was discussed by Douillard Lefebvre (1990). Their final linear differential equation system is able to describe many experimental data, but it contains eight independent rate constants and it is very time consuming obtaining such a large number of parameters fi-om adsorption kinetics experiments. A more suitable kinetic relation was proposed by de Feijter et al. (1987) which can be easily applied to experimental adsorption data. Other adsorption kinetics and relaxation models for polymer solutions based on diffusion transport and kinetic equations are presented by Miller (1991). 

Qualitative and quantitative models of adsorption kinetics of surfactants and polymers are described in this chapter. A comprehensive presentation of the most developed physical model, the difRision-controlled adsorption and the desorption model, is given and different methods of solving the resulting differential equations are discussed (Miller Kretzschmar 1991). A direct numerical integration enables us to consider any type of adsorption isotherm relating the surfactant bulk concentration with the adsorbed amount at the interface. 

As mentioned above, beside the diffusion-controlled models, others exist to describe the adsorption kinetics and exchange of matter. De Feijter et al. (1987) have developed a relation taking into consideration simultaneous adsorption of proteins and surfactants at an interface. As a special case a relation results which describes the equilibrium state of adsorption of polymer molecules at a liquid interface. 

The whole discussion of polymer adsorption so far makes the fundamental assumption that the layer is at thermodynamic equilibrium. The relaxation times measured experimentally for polymer adsorption are very long and this equilibrium hypothesis is in many cases not satisfied [29]. The most striking example is the study of desorption if an adsorbed polymer layer is placed in contact with pure solvent, even after very long times (days) only a small fraction of the chains desorb (roughly 10%) polymer adsorption is thus mostly irreversible. A kinetic theory of polymer adsorption would thus be necessary. A few attempts have been made in this direction but the existing models remain rather rough [30,31]. 

To describe the kinetics of olefin polymerization with heterogeneous catalysts, kinetic models based on adsorption isotherm theories have been proposed [7-10], The most accepted two-step mechanism of ZN polymerization, proposed by Cossee [10-12], includes olefin coordination and migratory insertion of coordinated monomer into a metal-carbon bond of the growing polymer chain. 

Einarson and Berg (1993) have attempted to explain the data on flocculation kinetics of latex particles with a block copolymer adsorbed on them. The polymer was polyethylene oxide (PEO)/polypropylene oxide (PPO). PPO is water insoluble and forms the part that adsorbs on the latex PEO forms streaming tails into water. Some charge effects remain after the polymer adsorption. The total potential is DLVO plus elastic plus osmotic effects. After fitting the model to the experimental data, they were able to calculate the value of 6, which they called the adlayer thickness. Their data on the stability ratio of latex with and without the polymer and as a fimction of NaCl concentration are shown in Figure 3.23. Note that the polymer stabilizes the colloid by almost one order of magnimde in NaQ concentration. That is, polymers may be necessary to maintain stability in aqueous media containing substantial electrolyte. 

Douglas, J.F., Johnson, H.E., and Granick, S., A simple kinetic-model of polymer adsorption and desorption. Science, 262, 2010, 1993. 

However, most explicit kinetic equations proposed in the literature are based on rather simple adsorption models which have no relevance for polymer adsorption. In this paper we will therefore first consider mass transfer-limited polymer adsorption and desorption rates from a theoretical point of view. We will then turn our attention to measurements which were designed in such a way as to enable accurate control over the mass transfer rate, so that data can be meaningfully analyzed. We used two different methods. The first is reflectometry combined with impinging-jet flow in order to measure adsorbed mass as a function of time.This... 

It was noted from Figure 13.5 where the MB adsorption is plotted as a function of time that the free gel was saturated with the dye within 54 h and more than 70% of the dye was removed within 240 min. The sharp increase in adsorption rate, points to the fact that the adsorption takes place at the polymer surface (Hasaine et al., 2003). The adsorption rate was evaluated by two kinetic models, pseudo-first order and pseudo second order kinetic models, explained inequations 13.3 and 13.4... 

Recently, several other groups have tried to model the kinetics of polymer adsorption (and exchange) by taking into account diffusion and some aspects of reconformation of the adsorbed polymer at the surface [22-24]. 

The kinetics of polymer adsorption on porous substrates is much more difficult to tackle. Besides adsorption, desorption, and exchange, size exelusion has to be taken into account. Also, most in situ methods are not applicable to porous substrates. A major difficulty is that with all available methods smeared-out properties are measured while it is likely that strong gradients in the axial direetion of the cylindrical pore are present. The process of axial equilibration is poorly understood and in many cases extremely slow. Most studies were performed with porous substrates with broad pore size and shape distributions. Controlled-pore glasses, zeolites, or porous membranes could be used as model systems with pores of molecular size. Application of glass capillaries is interesting for controlling the hydrodynamics in a curved system. 

It is not easy to perform measurements of kinetics of polymer adsorption or exchange in porous systems. Pore geometries, even in model systems like con-trolled-pore glass or Stober silicas, are usually poorly denned. In situ measurements are difficult to perform. Usually, indirect measurements are performed in which one measures batchwise the time dependence of the concentration of the polymer in intensively stirred bulk solutions. As discussed above, depending on the size of the polymer and the pore radius, adsorption can be extremely slow. Of course, one must realize that even in model porous systems nonideality of the pore geometry, such as the presence of tortuous pore channels with junctions and branches, a pore size distribution, and a non-uniform pore diameter, may thwart the interpretation of experimental results. 

Figure 1.2 gives another example of the kinetics of polymer adsorption - this time the adsorption of PEI on pulp fibers [17]. The curve for the lowest PEI concentration is fitted to Equation 1.1, with 7to = 3.5, kads = 0.2 min and kdes = 0. The curves for higher PEI concentrations were modeled by assuming that a fraction of the poly disperse PEI is small enough to penetrate the pores or the lumen. At higher PEI concentrations, more low-molecular weight PEI is available for pore penetration [18]. 

---