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

journal contribution

posted on 21.07.2015 by Diethelm Johannsmann, Gunther Brenner#### journal contribution

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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.