# unexpected magnitude of frequency shift

I am using the openQCM-Q to measure the viscosity of liquids. For some reason, the frequency shift is consistently about 39% larger than my calculations anticipated. I have double checked that I was calculating it correctly and my plots of frequency shift vs viscosity are good, they show a linear correlation that passes through the origin, they are just not the anticipated magnitudes by the 39%. I was wondering if anyone else has had a similar problem and might have found a solution?

any insight would be much appreciated!

thanks,

Rick

## Comments

it is a good result that the frequency vs dissipation experimental data shows linear correlation that passes through the origin

I'd need more information to figure out your results and check the calculation. particularly what kind of liquid is in contact with the surface of the qcm.

For example, if the liquid in contact with the quartz crystal surface can be modelled as a semi - infinite viscoelastic layer the following relations works:

Frequency variation

Dissipation variation

where

is the fundamental resonant frequency

are the piezoelectric shear strength and mass density of the quartz crystal

are the density and viscosity of the liquid

In particular, by combining the equations above we have:

no matter what kind of liquid is in contact with quartz crystal surface.

if the above hypothesis is applicable to your experiment, we can check your results together on the basis of these calculations

Sorry for the delay. I was hoping to figure it out, but no dice. The equation you give that should hold for any liquid in contact with the crystal doesn't appear to work for me, even for something as straightforward as DI water.

Here is a screenshot of the GUI after DI water addition:

https://gyazo.com/110fc6c84ee0bb7ad4ce0ae985a4388a

Attached below is the file containing the data for this run (Jul 11th), csv isn't supported on this forum so I converted to xlsx.

The results of a MATLAB program I made to read the QCM data suggests the delta_f should be about 2025Hz but that the measurements show a delta_f of roughly 2456Hz (over many runs, using different crystals, similar numbers occur).

If you would like to continue this conversation outside of the forum feel free to email: rsugden@uwo.ca.

Any help is much appreciated!

thanks,

Rick

we can continue the discussion here, it will help others to learn more about this subject.

Indeed, in the openQCM device calibration test, we measure frequency and dissipation variation caused by the passage from air to water, when one face of a 10 MHz quartz sensors are in contact with liquid. Here below a typical openQCM calibration result

The frequency shift caused by the passage from air to water is roughly

and the dissipation increased by roughly:

Our results coincide with those you have measured. Having used different sensors and openQCM devices, the results show that the measurements are reproducible and reliable, which is a fundamental feature for a device for scientific application.

Let us come back to the equation derived by Kanazawa and Gordon Anal. Chem. 1985, 57, 1771-1772

indeed they predict a frequency and dissipation variation for the passage from air to pure water, when one face of a 10 MHz quartz sensors is in contact with liquid, is given by:

where the density and viscosity of pure water at room temperature is given by:

Experimental results of frequency and dissipation variation caused the passage from air to pure water can be found in literature in Rodahl et al "Quartz crystal microbalance setup for frequency and Q‐factor measurements in gaseous and liquid environments"

This is one of the first, pioneering and most cited work about the use of the decay method to measure simultaneously frequency and dissipation in gaseous and liquid. They use AT-cut quartz crystal with a resonant frequency of 10 MHz with only one face in contact with liquid and the other covered with a lid. The experimental results obtained by the researchers and reported in this paper are the following

as you can verify also in this paper the experimentally measured values differ from the theoretical prediction, but despite this, the technique of decay is widely used in the field of QCM. In conclusion, the QCM setup is not straightforward and many factors must be taken into account for the prediction of experimental results observed, for example very often the same electronic interface connected to the sensor must be properly modelled.

What is a positive value is that the openQCM device provides consistent measurements, which can be considered near to the theoretical prediction.

Of course, discussion is still open, in order to improve together with the community help.

marco

We have been working lately on calibration experiments with different liquids (water: density=997 kg/m3 and viscosity=8.9e-4 Pa s; Heptane: density=679.6 kg/m3 and viscosity=3.91e-4 Pa s; and tetradecane: density=764 kg/m3 and viscosity=2.09e-3 Pa s).

For the frequency shift part, we have collected measurements of all the liquids in the fundamental frequency, and in the 3rd and 5th overtones, using different sensors (all of them of 10 MHz).

We have compared our results with the theoretical predictions using Kelvin-Voigt and Kanazawa-Gordon, ith the following parameters: Density of quartz=2648 kg/m3, Shear modulus of quartz = 2.947e10 kg/(cm s2) and thickness of crystal = 1.6e-4 m. I'm attaching a table showing the theoretical predictions and the experimental values (comparison of models)

As you can see, both models are in very good agreement, however, the experimental values are consistently 50-80% higher than the prediction. For the first model, I'm attaching a graph plotting the square root term of the Kelvin-Voigt model vs the shift in frequency. For the second model, I did the same of for the Kanazawa-Gordon. (see pdf attached "linear plots")

For both models, we can observe a very good linear dependence, regardless of the liquid or the overtone number

According to the equation, the slope of the KV model is the reciprocal of (2*pi*quartz density*crystal thickness). Using the reported values, that constant should be 0.3756. The experimental value of the slope is 0.6426

The slope of the KG model is the reciprocal of the square root of (pi*modulus*density of quarts). It should be 6.3867e-8, but the experimental value is 1.1359e-7

If I use those adjusted values for the, the predictions are in very good agreement with the experimental values. See the second attached table (adjusted models)

My question here is: Is this procedure valid? In some other techniques (such as spectroscopy) the instruments are calibrated and a "constant for the instrument" is calculated, rather than using reported extinction coefficients

Here I would like to ask my first question: It is usually assumed that the dissipation values for the crystal in air are negligible. However, the equipment records dissipation values in air of about 3e-4 for the fundamental frequency, 1e-4 for the 3rd overtone and 6e-5 for the fifth overtone. When we fill the chamber with the fluid the dissipation values increase (but the increase is of an order of magnitude around 1e-4). For the values of dissipation in fluid, should we use the difference between the dissipation in fluid and the dissipation in air, or the whole value of dissipation in fluid? The reason of my question is that it appears that by recording the whole value of dissipation we may be closer the the above value of 2.0 rather than using the Delta(D), as I explain further ahead.

I'm attaching 2 graphs showing those values for different experiments, different fluids and different overtones (see attached pdf "dissipation vs freq shift". Water-Blue dots, heptane-Green dots, tetradecane-red dots). The first graph was obtained using the value of Delta(D) (dissipation with fluid-dissipation in air), while the second graph was obtained using the value of dissipation with fluid.

For both cases, we can see that there is a huge dependency of that value with the overtone number and some dependency with the type of fluid.

In the first graph, only the heptane measured in the fundamental frequency is close to 2.0, and for the rest of the fluids and overtones, the value is much less

In the second graph, there are big discrepancies in the fundamental frequency, especially with the heptane, but almost all the values for any liquid in the 3rd and 5th overtone are close to 2 (and it appears to have an asymptotical behavior).

I understand that many researchers avoid measurements in the lower overtones (because they tend to be too sensitive to factors such as the mounting stress of the o-rings). Is this a possible explanation for this behavior? Is the use of the whole value of dissipation in liquid the correct way? If not, what could be the explanation of the big differences vs the value of 2.0 for the case where we used Delta(D)? As you can see, our experiments are very consistent when we repeat.

Thank you for sharing your results. This is an analytical, detailed, and clear examination of the openQCM Q-1 measurement performances. This will help definitely people working with openQCM.

## Discussion part 1 frequency:

the first observation is that you confirm the repeatability of measurement with openQCM Q-1, using both different bulk liquids (water, Heptane, and tetradecane) and different harmonics (up the 5th overtone using 10 MHz quartz resonators). This is the basic requirement to go further.The linearity of the experimental frequency data compared to both KV model square root (2*pi*quartz density*crystal thickness) and KG model square root (pi*shear modulus*density of quartz) makes your “calibration” procedure reasonable. It makes absolutely sense to me introducing a “constant of the instruments” to fit the experimental data to the theoretical expectations. This procedure is supported by the fact that the same calibration constant is valid for different samples and it is valid on three different overtones.

## Discussion part 2 dissipation:

it is my care to introduce first how we have defined dissipation in this scheme of measurement. This is crucial for further data interpretation. openQCM scheme of measurement is based on the passive interrogation of the quartz resonator around the resonance. The amplitude of the resonance curve here is related to the gain – loss between the input actuation signal and the signal passing through the resonator, using the AD8302 Analog Devices RF/IF Gain and Phase DetectorThe dissipation is measured as the inverse of the quality factor, which is defined as the ratio between the gain curve bandwidth and the resonance frequency

D = 1/Q = frequency bandwidth / resonance frequency

The bandwidth is defined at -3dB which corresponds roughly to 70% of the maximum amplitude. It is a way to define the quality factor, which is consistent between different harmonics (see figure quartz_crystal_microbalance_peak_bandwidth).

The theoretical prediction (resonance frequency)*(dissipation)/(shift in frequency) = 2.0 can be slightly off for the fundamental mode, as you noted and I agree. But the lower value you measured can be explained by the fact that we are underestimating the dissipation. It is clear that if choose to define the frequency bandwidth at lower level, for example 50% of the amplitude maximum, the dissipation value will increase.

It is an interesting observation that by using the whole value of dissipation in the liquid you are able to get a value close to 2.0. Although in this case it would be necessary, as done for the frequency, measure the dissipation shift for air-liquid Delta(D). If we accept the hypothesis to exclude the fundamental mode, the value fRn * Dn/Δfn is close to 1.2 for any liquid in the 3rd and 5th overtone. The value actually is lower than the theoretical value of 2.0, and this can be explained by the fact that Delta(D) is underestimated.

These results are very inspiring, however, it should be noted that air and liquid ambient condition are really different. It would be interesting to measure the ratio (resonance frequency)*(dissipation)/(shift in frequency) using a liquid bulk mixture at a different percentage of solute, for example: water – sucrose or water-glycerol mixtures just for example. We can measure the ratio (resonance frequency)*(shift in dissipation)/(shift in frequency) using this kind of sample, and referring the shift to the bulk solution without solute.

anyway, interesting discussion. Let’s go further!

In the first attached pdf (linear plot frequency.pdf), the graph plots the square root term of the Kelvin-Voigt model vs. the shift in frequency. The continuous line is the theoretical line using the reported values for density and thickness of the quartz crystal (Density = 2648 kg/m3 and thickness = 0.14 mm). The theoretical value of the slope is the reciprocal of (2*pi*quartz density*crystal thickness), and should be 0.3756. The dashed line is fitted to the experimental points, and the value is 0.5138. This is similar to our previous results, and since we are using liquids that range between 1 cP (water) to 55 cP (75% glycerol) in three different overtones, we can think of this as the first part of the “calibration”

I’m using the term “first part of the calibration”, because then there is the dissipation part. I’m also attaching another pdf (linear plot dissipation.pdf), showing the KV model for dissipation. In the first graph, I’m plotting the square root term of the KV model vs dissipation for all the samples and all the overtones. The theoretical value of the slope is the reciprocal of (quartz density*crystal thickness), and should be 2.2727 (shown in the continuous line). The fitted value is 1.9867. From the graph, it is obvious that there are some points that are very far from the linear behavior. Those points all correspond to the fundamental frequency, which we have already established to be very inconsistent. In the second graph, I’m plotting just the values obtained for the 3rd and 5th overtones, and all the results behave linearly. Using just the higher overtones, the fitted value of the slope is 2.5613.

The next pdf attached (Dissipation vs frequency shift relative to air.pdf) shows the value of (resonance frequency)*(dissipation)/(shift in frequency) vs. overtone number. The theoretical value should be 2.0. We can see that for the fundamental frequency, there is higher dispersion, and that for the subsequent overtones, it approaches an asymptotic value around 1.6. Another observation, is that for water (the less viscous fluid), the value reduces with overtone number, and with increasing viscosity, the value increases with overtone number.

Now, according to your last post, we could redefine the level at which we measured the bandwidth, in such a way that we match the theoretical prediction of (resonance frequency)*(dissipation)/(shift in frequency) of 2.0. Then this would be the “second part of the calibration”. Then my questions are: 1) How do we change this level? Is it something that can be controlled in the software?, 2) Is there any way to get more overtones (i.e. the 7th for the 10 GHz crystal) to verify the approach to an asymptote, and 3) Would this be a valid protocol?

Finally, in your response you mention the differences between air and liquid conditions, and suggest to measure the (resonance frequency)*(dissipation)/(shift in frequency) of different mixtures with the shift in frequency and dissipation referred to water. In the last attached pdf (Dissipation vs frequency shift relative to water.pdf) I’m showing the results for the glycerol solutions with respect to water. As you can see, there is no difference of that behavior vs. the results referred to air. We can also observe an asymptotic behavior approaching about 1.6