On the topic of QCM, the different resonance harmonics, i.e., the fundamental and the overtone harmonics, are often subject to discussion. So, what is the deal with the overtones? Are they really needed, and if they are, when and why are they important?

Like a guitar string, a QCM crystal can be excited to resonate at several different harmonics, labelled with a number ‘*n’*. The available harmonics are the fundamental, n = 1, and a set of overtones to the fundamental, n > 1.

The fundamental is the harmonic with the lowest resonance frequency that is possible to excite, and the overtones resonate with a frequency that is higher than the fundamental. For AT-cut QCM crystals, which oscillate in the thickness shear mode, only the odd harmonics, n = 1, 3, 5,…. are possible to excite electrically, Fig. 1. For example, if the fundamental mode is 5MHz, then the overtones resonate at odd multiples of the fundamental, i.e. 15 MHz, 25 MHz, 35 MHz etc.

*Figure 1. Schematic illustration of the excitation of two harmonics, n = 1 and n = 3, of an AT-cut crystal resonating in the thickness shear mode. To the left, the fundamental mode, n = 1 and to the right, the overtone n = 3.*

In a multi-harmonic QCM measurement, each harmonic will probe how the system under study responds to be shaken at that particular frequency (i.e., the harmonic’s resonance frequency). We can use this information, that the system responds the same or differently at one harmonic compared to another, to say something about the material properties of the system. This added set of information (compared to only using one harmonic) is useful when interpreting QCM data.

An analogy to measure at just one harmonic to measuring at multiple harmonics is photographing in black and white or in color. A color photograph will reveal much more information about the object under study than a black and white one. Even if the object is only black and white, one cannot be sure from just looking at a black and white picture. Only from a from a color photograph is it possible to say that the object does not have any other colors than black and white.

When it comes to QCM data, not only will the extra information from multiple harmonics provide relevant qualitative information, but it is also key to be able to perform viscoelastic analysis. To be able to perform viscoelastic modeling, one needs to fit several unknown parameters (such as thickness, viscosity, shear modulus and the frequency dependence of the viscosity and shear modulus) to data. To fit the unknown parameters, at least the same number of measured variables are needed to feed into the model. In addition to the frequency input, the mandatory information about the energy loss, *D*, will offer one parameter. By capturing *f *and *D* for three harmonics we will have six measured variables and 5 unknown which is theoretically enough if we can assume a perfect measurement and a perfect model. However, since there is always noise in the data and reality rarely is perfectly described by a mathematical model, it is wise to use as many input variables as can be measured to see how well the model fits reality. As a comparison, it is always possible to draw a straight line through two measured points, but to be confident that the measured system behaves according a linear model one need to measure more data points and see how they fall along the line.

To maximize the information extraction from collected QCM data, and enable viscoelastic film analysis, data from multiple harmonics is needed. In viscoelastic modeling, there are multiple unknowns. To solve for these, single input from the resonance frequency, *f*, or even both the resonance frequency and the energy loss, *D*, is not sufficient. For the modeling of a single layer, information on *f* and *D* from at least two harmonics is needed, which means that overtone measurements are necessary for adequate data analysis.

Cover photo by Dominik Scythe on Unsplash

Topics:

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