Scaling and mean-field theories applied to polymer brushes.

Analysis of the osmotic pressures of aqueous poly(ethylene glycol) solutions indicates a power-law behavior in the range of volume fractions from 0.08–0.12 to 0.33 or higher, for polymers with average molecular masses in the range 1500– 4000 Da. The exponent of the power law is closer to a value of 2, which is predicted by mean-field theory, than to the value of 5/4, which is predicted by scaling theory for a good solvent. The latter value is attained by long polymers, with molecular masses 8000–20,000 Da. Quantitative predictions of the repulsive pressure between grafted polymer brushes requires a further scaling factor, additional to that relating bulk osmotic pressure and polymer volume fraction, in both scaling and mean-field theories. Recently, Hansen et al. (2003) established the validity of scaling theory applied to lipid-grafted polymer brushes. The criterion used was that the effective monomer concentration of polymer in the brush should equal that at which the osmotic pressure, Pf, of the corresponding free poly(ethylene glycol) (PEG) polymer in bulk solution conforms to that for a very long polymer, PEG:20,000. With the latter, the dependence of Pf on volume fraction, ff, of free polymer is that predicted by scaling theory for a good solvent, viz., Pf ; f 9=4 f ; over a wide range of ff in the semidilute regime. This criterion is achieved in polymer brushes only for relatively long PEG polymers at relatively high grafting densities, e.g., PEG:5000 lipid at a mol fraction of X5000 ffi 0.07–0.1, or PEG:2000 lipid at a mol fraction X2000 ffi 0.23 (Hansen et al., 2003). These contents of PEG lipid exceed those at which micellization is initiated in unsupported liposomal membranes (Montesano et al., 2001), and those that are used routinely in the steric stabilization of liposomes for drug delivery (Lasic and Needham, 1995). It is therefore of considerable practical interest to inquire as to whether, with less restrictive criteria, scaling-theory behavior could still be fulfilled in polymer brushes. The requirement of convergence to universal behavior, independent of molecular weight, used by Hansen et al. (2003) ensures a fortiori that scaling theory holds. Nevertheless, it is possible that the characteristic power-law dependence of Pf on volume fraction of free polymer might be achieved with smaller polymers before strict conformity to the osmotic pressure of the very long polymers is reached. An associated consideration is the extent to which mean-field theory might apply to the polymer brush. It is likely that criteria for the latter may be less restrictive than for scaling theory, because uniform monomer density throughout the brush is not required by mean-field theory of stretched polymers (Pincus, 1991; Milner et al., 1988). In addition to scaling theory, mean-field theory also has been used extensively for interpreting data on polymer-lipid brushes (Hristova and Needham, 1994; Marsh et al., 2003; Netz and Andelman, 2003). For free polymers in bulk solution, the dependence of osmotic pressure on polymer volume fraction predicted by mean-field theory isPf ;f 2 f in the semidilute regime, which differs significantly from that for scaling theory (De Gennes, 1979). The mean-field prediction corresponds to dominance of the second-order term in a virial expansion, for the semidilute regime. In the above connection, it is worthwhile to review the predictions of scaling theory over the entire range of polymer concentration. At very low mol fractions, the osmotic pressure of the dilute polymer will obey the ideal ‘‘gas’’ law: Pfa 3 m=kBT;ff ; with exponent one. At higher volume fractions, the more general version of scaling theory is (Grosberg and Khokhlov, 1994):