Frequency Shifts of a Quartz Crystal Microbalance Calculated with the Frequency-Domain Lattice-Boltzmann Method: Application to Coupled Liquid Mass.

In recent years the quartz crystal microbalance (QCM) has seen an impressive evolution from a film-thickness monitor to a surface-analytical instrument with capabilities much beyond gravimetry. In particular, the instrument has often been applied to adsorbates from a liquid phase and, also, to samples with structure in the surface plane. In order to quantitatively predict frequency shifts induced by such samples from a model, one needs to compute the in-phase component of the area-averaged periodic tangential stress at the resonator surface. A method is described which performs this task, making use of a variant of the Lattice-Boltzmann (LB) method. The algorithm differs from the conventional LB method in that it deals with oscillatory flows and only covers linear hydrodynamics. The adsorption of small particles (mimicking proteins) was chosen as an example to test the performance. These samples are acoustically thin, which simplifies the calculations. Also, the material's finite compliance can be neglected in this limit. The simulations predict the amount of solvent trapped between neighboring particles, which contributes to the adsorbate's apparent mass. The unknown amount of hydrodynamically coupled liquid is a serious problem in the interpretation of QCM experiments. On an experimental level, the amount of trapped solvent can be estimated from the comparison of the optical layer thickness (determined with ellipsometry) and the acoustic layer thickness (determined with a QCM). Since the amount of trapped liquid decreases when neighboring particles aggregate into clusters, this analysis can lead to a statement on the degree of clustering. The LB-based simulations show, though, that the relation between the cluster geometry and the amount of trapped solvent is highly nontrivial. The details of the geometry do matter. The LB-based algorithm can calculate the amount of trapped solvent for user-specified particle shapes, orientations, interparticle distances, and also distributions thereof. It is an essential step in the quantitative interpretation of QCM results obtained on thin samples with in-plane structure.

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