Effect of polyelectrolyte multilayers on the response of a quartz crystal microbalance

The effect of a polyelectrolyte (PE) multilayers made by a layer-by-layer technique on the response of a quartz crystal microbalance (QCM) is studied by using novel mathematical methods based on the M?bius transformations and their matrix representations in the complex plane. In the first method, the basic properties of the M?bius transformation are used for obtaining the PE bilayer matrix from the QCM impedance measurements taken at four different numbers of layers. In the second method, nonlinear fitting with concomitant error estimation is used for obtaining the elements of the bilayer matrix. The methods are applied to a multilayer composed of 150 bilayers of poly(sodium 4-styrenesulfonate) and poly(diallyldimethylammonium) chloride on a quartz crystal resonator. The structure of the system is discussed, and the bulk acoustic impedance and areal mass density of the bilayer are calculated from the layer matrix.