Membrane-enhanced surface acoustic wave analysis of grafted polymer brushes

An analysis is developed for the frequency response of a quartz crystal resonator (often referred to as a quartz crystal microbalance) that is modified with a grafted solvent-swollen polymer brush and placed in contact with a membrane capping layer. The shear wave generated at the resonator surface couples into the membrane layer with an efficiency that is strongly dependent on the thickness of the swollen brush layer. As a result, the resonant frequency changes by a maximum amount that is closely approximated by the Sauerbrey shift for the capping layer. The calculated shift substantially decreases for increases in the brush thickness of approximately 10 nm, which gives a net frequency response that is extremely sensitive to the degree of swelling of the polymer brush. An optimum capping layer thickness is determined by balancing the Sauerbrey shift against dissipative effects that weaken the crystal resonance. This optimum membrane thickness depends only weakly on the properties of the membrane material and is in the micron range. Detailed multilayer calculations are presented for the specific case of a poly(ethylene glycol) brush swollen with water and brought into contact with an elastomeric water-permeable membrane. These calculations confirm that the method is sensitive to the properties of the brush layer in the experimentally relevant thickness regime. Connections are also made to conceptually simpler two and three layer models of the acoustic impedance of the material systems that are brought into contact with the resonator.