The structural and thermodynamical properties of weak polyelectrolytes end-tethered to surfaces of arbitrary geometry are studied using a molecular theory. The theory is based on writing down the free energy functional of the system including all the basic interactions and the explicit acid-base equilibrium for the chargeable groups of the polymer. The theory explicitly includes the size, shape, conformations, and charge distribution of all the molecular species. The electrostatic interactions include a density-dependent dielectric function, modeled with the Maxwell-Garnett mixing formula, to account for the composition-dependent permittivity. The minimization of the free energy leads to the distribution of all molecular species and their dependence on bulk pH and salt concentration. We apply the theory to polymer chains end-tethered to planar, cylindrical, and spherical surfaces. The radius of the curved surfaces is small to enhance the curvature effect. We find that when the grafting surfaces are uncharged, the approximation of a constant dielectric function works very well for both structural and thermodynamic properties. The structure of weak polyelectrolytes tethered on cylindrical and spherical surfaces is different from that of polymers tethered on planar surfaces due to the available volume as a function of the distance from the surface. Specifically, the degree of dissociation increases with increasing curvature of the surface. This is a manifestation of the coupling between the local density of protons, counterions, and polymer segments. The results can be interpreted in terms of the local Le Chatelier principle for the acid-base equilibrium, with proper account of the three local contributions: counterions, protons, and chargeable groups. We find that one can achieve local changes of pH between one to two units within 1-2 nm. The thickness of the tethered layers as a function of bulk pH shows a large increase when the pH is equal to the bulk pK. However, the variation with salt concentration is different for the different geometries. The largest swelling is found for cylindrical surfaces. The predictions from scaling theories of a maximum in the thickness of the film as a function of salt concentration is found for planar films, but not for curved surfaces. Finally, the interactions between cylinders with tethered polyelectrolytes is very different from the equivalent planar surfaces. These results are important for the interpretation of force measurements with nanoscale AFM tips. The implications of the results for the rational design of responsive tethered polymer layers is discussed together with the limitations of the theoretical approach. Figure 4. (a) The percentile difference between the polymer density obtained with the Maxwell-Garnett and the constant dielectric function. (b)The percentile difference for the density of the positive ions. The polymer surface coverage is σ = 0.2 nm −2 ; the polymer chain length is n = 50, the bulk pH = 6, and the pK a = 5.

We describe experiments pertaining to the formation of surface-anchored poly(acrylic acid) (PAA) brushes with a gradual variation of the PAA grafting densities on flat surfaces and provide detailed analysis of their properties. The PAA brush gradients are generated by first covering the substrate with a molecular gradient of the polymerization initiator, followed by the “grafting from” polymerization of tert-butyl acrylate (tBA) from these substrate-bound initiator centers, and finally converting the PtBA into PAA. We use spectroscopic ellipsometry to measure the wet thickness of the grafted PAA chains in aqueous solutions at three different pH values (4, 5.8, and 10) and a series of ionic strengths (IS). Our measurements reveal that at low grafting densities, σ, the wet thickness of the PAA brush (H) remains relatively constant, the polymers are in the mushroom regime. Beyond a certain value of σ, the macromolecules enter the brush regime, where H increases with increasing σ. For a given σ, H exhibits a nonmonotonic behavior as a function of the IS. At large IS, the H is small because the charges along PAA are completely screened by the excess of the external salt. As IS decreases, the PAA enters the so-called salt brush (SB) regime, where H increases. At a certain value of IS, H reaches a maximum and then decreases again. The latter is a typical brush behavior in so-called osmotic brush (OB) regime. We provide detailed discussion of the behavior of the grafted PAA chains in the SB and OB regimes.

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