# Viscoelastic modeling

### How does the Sauerbrey equation read?

The Sauerbrey model describes the linear relationship between frequency shifts and mass changes for thin films. For a detailed description of the Sauerbrey model, see here.

### When do I need to use the viscoelastic (Voigt) model?

The simple answer is: as soon as D is larger than zero. Theoretically, as soon as there is any viscoelastic behavior in the adsorbed layers, the mass will not couple 100% to the oscillatory motion of the crystal, and the true mass will be under-estimated if only the Sauerbrey equation is being used. Putting it the opposite way: as soon as you are able to obtain a stable fit to your data set, it is never wrong to model it… Practically though, it is obvious that small values of D still hold for as a rigid film, especially if the D/f ratio still is very small (for instance, 1:30 or above).

### What is the difference between Voigt and Maxwell? What are their respective applications?

Viscoelastic behavior of polymers and other materials are often simulated using elastic springs and viscous dashpots. The Maxwell model is the simplest model for a more liquid-like viscoelastic soft matter material and it consists of a spring and dashpot in series. The Voigt model is the simplest model for a more rigid viscoelastic soft matter material and it consists of a spring and dashpot in parallel.

A detailed description, including the equations, of the implemented models can be found in "M.V.Voinova, M.Rodahl, M.Jonson and B.Kasemo "Viscoelastic Acoustic Response of Layered Polymer Films at Fluid-Solid Interfaces: Continuum Mechanics Approach", Physica Scripta 59 (1999) 391-396

### Can I model a Newtonian fluid?

Yes, you can model either the density OR the viscosity of a Newtonian fluid. It is not possible through modeling to separate the density and viscosity of a Newtonian fluid. You need to know either one or the other. The frequency and dissipation response going from vacuum to such a fluid depends only on the density x viscosity product. So you need to fix one of the parameters in order to get the other.

### Can I get both viscosity and density of a Newtonian fluid through modeling?

It is not possible by modeling to separate viscosity and density of a Newtonian fluid. The frequency and dissipation response going from vacuum to such a fluid depends only on the viscosity*density product. You need one of the parameters in order to get the other.

### How do I work with fixed versus fitted parameters?

Under the tab 'Parameters' you can choose to fit the parameters, or to use them as fixed values in order to help determining other fitted parameters. By double-clicking on each parameter, you move them between the two alternative categories.

If you want to model the thickness of a layer, it is recommended to have the density of that layer fixed in the model, since the two parameters are connected to each other (high density/small thickness is equal to low density/large thickness, for a certain amount of mass) and the software will have a problem to find one single solution.

Normally, the parameters "thickness", "viscosity" and "shear modulus" are chosen as parameters to fit. The density is then assumed to be the same during the whole measurement. The input values for the parameters to fit are set as an interval, with 'maximum guess' and 'minimum guess' values inserted in the table.

The values for the fixed parameters need to be rather accurate - the easiest way to find out how exact they need to be, is to play around with the values and observe the fluctuations of the model output.

### When do I use the extended modeling with overtone dependence?

The extended model has been introduced since there was a demand from many customers to be able to test the frequency dependence of the viscoelastic parameters. In the extended model, possible frequency dependence has been introduced but only in the simplest way, i.e. a linear dependence. So strictly speaking, this model is only valid when the film you are modeling has linear frequency dependence from the viscoelastic parameters. The more the frequency dependence deviates from a linear relationship, the less the extended viscoelastic model will apply for that particular system.

### How do I work with L1 and L2 modeling?

Checking “L1” when modeling a two-layer system (or any multilayer), means that QTools will treat the film as one homogeneous film, finding an average value for each parameter (viscosity, shear modulus and density) for the whole film. The modeled thickness will be the total thickness for the whole film.

Checking both “L1” and “L2” means that QTools will try to find a solution with two regions having different properties, with “L1” being the innermost layer. The more distinct the interface is (i.e. the larger the difference is between the two layers), the better is the chance to find a good fit.

A good way to investigate a two-layer system is to build it sequentially during an on-going measurement, so that you can model the first layer (first part of the measurement) separately, using only “L1”. When this is done, set all modeled “L1” parameters as “fixed parameters”, then include “L2” and model the properties of the second layer. This approach assumes that the properties of the first layer do not change when introducing the second one.

### If I choose to model system with two layers, would I get the same mass and thickness of L1 and L2 as when I choose a one-layer model?

The thickness of a L1 modeling would be the same as the total thickness of a L1+L2 modeling, if the viscoelastic properties of the two layers are close to each other. In case of e.g. a rigid layer and a soft layer and different densities the L1 modeling will find viscoelastic output parameters that are an average between the two layers. In such case the total thickness might be slightly different from the one obtained using a two layer model.

### My system includes more than 2 layers – can I still model them one by one?

You cannot model them all at one time. The approach to model each layer separately would be to group the “L1” and “L2” as one layer – forming a new “L1” layer – and treat layer 3 as the new “L2”, while disregarding the averaged viscoelastic results for the “L1” layer (same idea as describe above, under “How do I work with L1 and L2 modeling?”). The approach requires that the build-up of the multilayer is done sequentially within one measurement data set).

### Can I apply the modeling to a film made outside the QCM-D instrument?

Yes, you can: measure the non-coated crystal first, so that you get a QCM-D baseline before the treatment of the sensor crystal; then, after the external treatment you measure the processes on the treated sensor crystal, as usual. Remember to use the same medium above the sensor crystal in both measurements to be able to compare data. With QSoft401, you then merge the two files by using the “Stitch files” function and model the data in QTools the standard way. With QSoft 301 it is a bit more work: you need to un-offset the data in the first measurement file, which means when you open the QSoft file, you do not accept the “File transcriber” suggestions to offsets and scaling; then copy the baseline from the first measurement and insert it in the beginning of the second, un-offseted, measurement file, to get the right baseline reference in QTools.

### How to perform degradation experiments? What is there to consider compared to adsorption experiments?

Generally experiments are made on a bare sensor and then something is adsorbed to it during the measurement. In these cases you can say that if D is larger than zero and if the overtones are spreading, the film is soft and requires viscoelastic modeling.

In some cases a film is coated on the sensor before the measurement and then something is adsorbed onto that during the measurement. If the initial film is rigid and is not affected (for example degraded or swollen) during the measurement, the statement above is also valid.

But, if you coat the sensor with a film before the measurement which is not rigid and/or which is affected during the measurement (degradation, swelling etc) then the statement above is not valid. The reason that we cannot judge if the film is soft or not from just looking at Δf and ΔD is that f and D are not shown as absolute values but only in relation to the base line (therefore called Δf and ΔD). And since the baseline is the sensor at the beginning of the measurement, and if the properties of this change, we are moving the baseline and then relations will change. It is a bit tricky to explain.

But if we can add the baseline of the bare sensor before coating, to the beginning of the degradation measurement then this would be a solid ground to stand on and in relation to this we can judge if the film is soft or not based on the above statement.

So how do we do this?

Measure the sensor in the instrument with the same buffer as you will use in your experiment. Then coat the sensor and do your measurement. The first measurement could thereafter be stitched to the beginning of the second measurement, so that they are in the same file. Now the measurement will show the bare sensor in fluid, thereafter there will be a jump in f and maybe in D to the coated sensor in fluid and thereafter the degradation.

Note that this is not a standard procedure! There might be some stress in the sensor from mounting so the absolute f and D can differ from time to time. One should try mounting the sensor several times to see if the difference in absolute f and D varies a lot by looking at the absolute values of f and D in the data sheet in QTools. Hopefully it doesn’t differ too much and then you can use this procedure to get the baseline in order to model.

### Is there a way to estimate the quantity of an adsorbed protein, or similar, with water excluded, even though it is probed in hydrated form?

The QCM-D will give the hydrated mass, i.e. the mass of the molecules adsorbed to the surface and the solvent that is trapped in between. There is no means to extract the mass of only the molecules with the QCM-D technique. However, the dissipation may give a hint on the hydration of the formed layer. For example if you have a really, really high dissipation with spreading of overtones you may conclude that the molecules have arranged themselves in an extended and sparse fashion which allows plenty of solvent to be trapped. If the dissipation is low and the spreading of overtones is not so significant the molecules have probably arranged themselves in a more dense and ordered fashion not allowing so much solvent to be trapped. In this case the QCM-D mass will be a closer estimate to the mass of the molecules than the QCM-D mass for the soft layer.

### If I increase/decrease the density, the thickness values also change. So, how to "guess" the "best" value for the density?

Yes your thickness will vary depending on L1 density value, since they are correlated. You need to state in your experimental section “an estimated L1 density of YYY kg/m^{3} was used”. You can change the L1 density, the fit should be equally good, but your thickness will change. If you know that your film is 50% hydrated, you can calculate the L1 density: 0.5 x density of the dry layer + 0.5 x density of the bulk solution. But this is something that is almost impossible to know. For hydrated biomaterials the density is close to, however slightly larger than, that for water.

### Would the viscosity and shear modulus be zero or infinite for a rigid material?

If it is a solid, the elasticity is very high, and the viscosity is very low. Think of the energy losses when sound is transmitted through a material: a good material for sound transfer has low viscosity, and so has quartz almost no viscosity but high elasticity. Maple syrup has high viscosity which means it is not a good material for sound transfer - it dissipates a lot of energy.

### I have difficulties finding a good fit to my data.

There may be several reasons why it is difficult to find a good fit:

- The *D*-factor is very low (<1), especially in comparison to the frequency shifts. The modeling requires that there is viscoelastic behavior of the adsorbed film. If there is not, the modeling is not applicable and the Sauerbrey function is enough to extract mass and thickness data. Viscoelastic data is not possible to extract, simply because the film is too rigid.

- You are trying to model both density and thickness at the same time. When modeling a viscoelastic film, it is generally not possible to separate between these two parameters in much the same way as it is not possible to separate density and viscosity for a Newtonian fluid. You will usually get an equally good fit if you at the same time increase the density by 10% and lower the thickness by 10% (so that the product stays the same).

- The adsorbed films do not fulfill the assumptions made in QTools. A good fit requires that the adsorbed films have a homogeneous density, viscosity and/or shear modulus. If you for instance have fitted your data using a fixed density, it could be worth considering if this parameter is changing during different parts of the measurement.

- Your minimum and maximum guesses for the initial search grid are not optimal. Changing the extreme values will give a new set of coordinates, even within the grid, and new solutions to the fitting may be found.

- The data set does not include all coating/measurement steps. QTools assumes that the beginning of the measurement reflects the non-coated rigid sensor surface material. If you for instance coat the crystal ex situ with a film and then insert it in the chamber for measurements, you need to have taken a short baseline on the non-coated surface before the coating step as reference, and then combine the two data sets before using QTools.

- The fundamental tone is included in the fitting. We do not recommend using the fundamental frequency (5 MHz) when doing liquid measurements since this resonance is much more susceptible to mounting mediated stresses which are aggravated by liquid loading. If the first harmonic is unchecked in the modeling, you will get a better fit.

### I get much larger noise in QTools, than in QSoft, for the same data.

If you zoom in your QSoft data, they will appear the same way. It is the averaging function in QSoft that does this: if every single data point would be plotted in QSoft, the software would be too slow, i.e. too much processor time would be spent on updating the plots, which would slow down the acquisition speed drastically. So, instead of plotting all points, only one average of them is plotted for each x pixel.

### My QTools spreadsheets are blacked out.

Some of the Windows settings for data table cell color do not show cell contents properly in QTools. The settings should be accessible under Control Panel/ Display/Appearance. Select any of the other "Color Schemes" and you should be able to see the data in the cells.

### I can’t get a reliable modeled viscosity “baseline”.

A reason for unexpected values of modeled viscosity or other parameters when there is nothing adsorbed onto the sensor surface, could be that the measured f and D values are very close to zero. If so, the model may have a problem finding good solutions when fitting. This typically happens in the beginning of an adsorption process, or more precisely, right before anything has been adsorbed. Because of this, the default modeling settings start the fit at the end of the measurement and proceed backwards (this is indicated by the quick-button illustrated by an arrow in the Modeling Center).

### I have experienced that thick coatings and/or non-even coatings sometimes results in high overtones being noisy. Are higher overtones less sensitive because they are dampened too much?

The electrical signal strength (voltage) that reaches the instrument decreases as the overtone number goes up. Dampening will decrease the signal strength and when it is low enough, QSoft will lose the overtone.

### What is a good ChiSqr?

You cannot really predict Chi-square, since it will be different for different data sets. It is not divided by the number of data points in the data set, which means that if a data set is twice the size as another data set, it will also give twice as large Chi-square for an equally good fit.