QCM Advanced Calculator
Quartz Crystal Microbalance — Multi-Model Frequency Analysis
Researcher's Guide & Quick Reference
▼This calculator provides real-time analysis for Quartz Crystal Microbalance (QCM) experiments using three complementary physical models. All input fields update results instantly — no button click required. Hover over the ? icons for contextual explanations of each parameter and result.
Choose the appropriate model tab based on your experimental conditions: Sauerbrey for rigid thin films in air, Kanazawa–Gordon for Newtonian liquid loading, and Voigt–Voinova for viscoelastic films with or without liquid contact. All three models share the same AT-cut quartz constants (ρQ = 2.648 g/cm³, μQ = 2.947 × 10¹¹ g/(cm·s²), ZQ ≈ 8.828 × 10⁵ g/(cm²·s)), displayed at the bottom of each panel for reference.
Use this tab for rigid, acoustically thin films deposited in air (or vacuum). Enter the crystal's resonant frequency f₀, active electrode diameter, observed frequency shift Δf, and film density ρf.
The calculator returns:
- Mass sensitivity Cf — Sauerbrey constant [ng/(Hz·cm²)]
- Mass per Hz — total mass per 1 Hz shift over the active area
- Saturation mass — maximum mass at the 1% Δf/f₀ limit
- Film thickness — in nm, μm and Å
- Total mass change Δm and areal mass Δm/A
Validity criterion: |Δf/f₀| < 1/1000 for reliable results; theoretical Sauerbrey limit ≈ 1/100. Beyond this, the acoustic impedance mismatch between film and quartz causes nonlinear deviations.
Switch between QCM Characterization (sensor properties at a given f₀ and electrode geometry) and Film Measurement (thickness and mass from a measured Δf).
Select this tab when the crystal contacts a semi-infinite Newtonian liquid with no adsorbed film. Two operating modes are available:
- Forward: given liquid density ρL and viscosity ηL, predict Δf, ΔD, ΔΓ and the viscous penetration depth δ.
- Inverse: given a measured Δf, extract the ρL·ηL product. If ρL is known, extract ηL individually.
The Kanazawa signature is |Δf| = ΔΓ, equivalently ΔD·f₀/(2·|Δf|) = 1. Significant deviation from this ratio indicates non-Newtonian behaviour, viscoelastic film presence, or interfacial slip.
Penetration depth δ = √(ηL/(π·f₀·ρL)) defines the liquid layer thickness sensed by the shear wave. In water at 10 MHz, δ ≈ 178 nm.
Use this tab for viscoelastic films — hydrogels, polymer brushes, protein adlayers, lipid bilayers, cell monolayers. Provide the film's density ρf, thickness hf, storage modulus G′, and a loss parameter (either film viscosity ηf or loss modulus G″). Choose between Air and Liquid contact medium.
The calculator predicts:
- Δf — frequency shift (includes both mass and viscoelastic contributions)
- ΔD — dissipation change (energy lost per oscillation cycle)
- ΔD/|Δf| ratio — key diagnostic for regime classification
- |G*| — magnitude of the complex shear modulus
- tan δ — loss tangent (G″/G′), ratio of dissipated to stored energy
- ΔfS — Sauerbrey rigid-film prediction for comparison
A regime assessment automatically classifies the film behaviour:
Limiting cases: G′ → ∞ recovers the Sauerbrey equation; hf → 0 with liquid contact recovers Kanazawa. Use this to verify continuity across models.
This tab solves the inverse problem: given experimental Δf/n and ΔD values measured at multiple overtones by a QCM-D instrument, it extracts the film's physical properties by fitting the Voigt–Voinova model simultaneously to all harmonics.
How to use it — step by step:
- Enter your data. For each measured overtone, tick the checkbox and type Δf/n (Hz) and ΔD (× 10⁻⁶). If your instrument reports the half-bandwidth shift ΔΓ (Hz) instead of dissipation, use the ΔD / ΔΓ toggle above the table — the calculator converts automatically. You need at least 2 overtones (≥ 3 recommended for the 3 free parameters). Common sets: n = 3, 5, 7 or n = 3, 5, 7, 9, 11.
- Set fixed parameters. Enter the fundamental frequency f₀ (typically 5 MHz), the film density ρf (g/cm³), and select Air or Liquid medium. For liquid, provide ρL and ηL.
- Initial guesses (optional). Expand the section to provide starting values for hf, G′, G″. If left at zero, the calculator auto-estimates hf from the Sauerbrey equation and uses sensible defaults for the moduli.
- Click ▶ Run Fit. The Levenberg–Marquardt optimizer runs 4 restarts with different initial conditions and selects the best solution.
The optimizer extracts:
- hf — film thickness (nm), typically larger than the Sauerbrey estimate for viscoelastic films
- G′ — storage (elastic) modulus (Pa)
- G″ — loss (viscous) modulus (Pa)
- ηf — film viscosity (Pa·s), computed as G″/(2πf₀)
- |G*| — complex modulus magnitude, tan δ — loss tangent
- χ²r — reduced chi-squared, a goodness-of-fit metric (values near 1 = excellent)
The Measured vs. Fitted comparison table shows, for each overtone, the experimental and predicted Δf/n and ΔD with percentage error. An automatic interpretation classifies the film regime and comments on fit quality.
When to use this: If your Sauerbrey thickness differs significantly across overtones (Δf/n is not constant), the film is viscoelastic and this inverse fitting gives you the correct thickness along with the mechanical properties of the film.
- All calculations update automatically with a short debounce delay (280 ms) as you type.
- Negative Δf means mass was added to the crystal; positive Δf means mass was removed or liquid was displaced.
- For QCM-D users: ΔD = 2ΔΓ / f₀, reported in units of 10⁻⁶.
- The Voigt panel includes a Sauerbrey rigid-film comparison (ΔfS), so you can quantify how much viscoelasticity shifts the result.
- Use the loss parameter toggle (ηf ↔ G″) in the Voigt panel for convenience. The calculator auto-converts between the two using G″ = ω·ηf at the fundamental frequency.
- For multi-harmonic analysis, use the Voigt Inverse tab: enter Δf/n and ΔD at each overtone (e.g. n = 3, 5, 7) and the optimizer will extract thickness, G′ and G″ simultaneously. Alternatively, run the forward Voigt model at each overtone frequency and compare Δf/n and ΔDn manually to diagnose frequency-dependent viscoelastic behaviour.
- Unit conventions: viscosity in mPa·s (= cP), density in g/cm³, frequency in Hz, moduli in Pa. The calculator uses CGS internally (standard for QCM literature) and converts for display.
A swelling experiment is a differential measurement between two states of the same coated crystal: dry film in air (or pure solvent vapour) and the same film equilibrated in the chosen solvent. Each state needs the right model — there is no single tab that "gives swelling" because the dry and wet regimes obey different physics.
Important conceptual point: the Kanazawa–Gordon model describes a bare crystal in a Newtonian liquid — there is no film in its physical picture, so it cannot return a film thickness by construction. Use it only as a baseline / setup-validation step before the coated measurement, not for the swollen-film analysis.
Step-by-step:
- Bare crystal in air — record f₀ on each overtone you intend to use (n = 3, 5, 7, …). This is the absolute reference.
- Bare crystal in pure solvent (cell-flow setup, fully thermalised — typically 30–60 min). Use the Kanazawa–Gordon tab in Forward mode with the solvent's ρL and ηL at your working temperature. The measured Δf and ΔD must match the prediction within a few percent, and the diagnostic ratio ΔD·f₀/(2|Δf|) should sit close to 1. If it does, your setup, mounting and thermalisation are validated. If not, fix that before going further — do not proceed with a coated measurement on top of an uncalibrated baseline.
- Coated crystal, dry, in air — measure Δf and ΔD on each overtone. If ΔD/|Δf| is small (< 0.05) and Δf/n is constant across overtones, the dry film is rigid: use the Sauerbrey tab with the film density ρf to obtain hdry. If Δf/n varies with n, the dry film is already viscoelastic — use the Voigt Inverse Fitting tab in Air mode instead.
- Coated crystal in solvent — flow the same solvent over the coated crystal and acquire Δf/n and ΔD on the same overtones until the signals stop drifting (swelling can take minutes to hours depending on Tg and solvent compatibility). Switch to the Voigt Inverse Fitting tab, set Medium = Liquid, enter the solvent's ρL and ηL (use the Solvent preset dropdown for common values), and run the fit. The output hf is the swollen thickness hwet.
- Compute swelling quantities — use the Swelling Analysis helper in the Voigt Inverse panel. With hdry and hwet it returns the swell ratio SR = hwet/hdry, the volume change ΔV/V = SR − 1, and the solvent volume fraction φs = 1 − hdry/hwet (assuming negligible lateral spreading).
Why multiple overtones are non-negotiable in step 4: the Voigt model has three film unknowns (hf, G′, G″) plus the fixed liquid loading. Fitting at the fundamental alone gives an under-determined system; at least three overtones (n = 3, 5, 7 minimum, and 9, 11 if signal quality allows) are needed for a well-posed inversion. Avoid the fundamental in liquid — edge effects and lower stability degrade the data.
Diagnostic flag during swelling: when the solvent first contacts the film, watch ΔD/|Δf|. If it rises sharply, the film has crossed into the viscoelastic regime — confirmation that Sauerbrey is no longer applicable on the wet state and that the Voigt Inverse path is the correct one.
- Sauerbrey: G. Sauerbrey, "Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung", Z. Phys. 155, 206–222 (1959).
- Kanazawa–Gordon: K.K. Kanazawa & J.G. Gordon II, "Frequency of a quartz microbalance in contact with liquid", Anal. Chem. 57, 1770–1771 (1985).
- Voigt–Voinova: M.V. Voinova, M. Rodahl, M. Jonson, B. Kasemo, "Viscoelastic acoustic response of layered polymer films at fluid-solid interfaces: Continuum mechanics approach", Phys. Scr. 59, 391–396 (1999).
- Comprehensive review: M. Rodahl, F. Höök, B. Kasemo et al., "Quartz crystal microbalance setup for frequency and Q-factor measurements in gaseous and liquid environments", Rev. Sci. Instrum. 66, 3924 (1995).
- QCM-D technique: M. Rodahl & B. Kasemo, "On the measurement of thin liquid overlayers with the quartz-crystal microbalance", Sens. Actuators A 54, 448–456 (1996).
- Voigt Inverse Fitting: D. Johannsmann, "Viscoelastic, mechanical, and dielectric measurements on complex samples with the quartz crystal microbalance", Phys. Chem. Chem. Phys. 10, 4516–4534 (2008).
- Polymer film swelling by QCM-D: B.D. Vogt, E.K. Lin, W.-l. Wu, C.C. White, "Effect of film thickness on the validity of the Sauerbrey equation for hydrated polyelectrolyte films", J. Phys. Chem. B 108, 12685–12690 (2004).
- Levenberg–Marquardt: K. Levenberg, "A method for the solution of certain non-linear problems in least squares", Q. Appl. Math. 2, 164–168 (1944); D.W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431–441 (1963).
Cf = √(ρQ·μQ) / (2·f₀²) [g/(Hz·cm²)]
t = −Δf · Cf / ρf
ΔΓ = |Δf| ⟹ ΔD = 2|Δf|/f₀
δ = √(ηL / (π·f₀·ρL))
Validity: Semi-infinite, homogeneous Newtonian liquid, no-slip at interface, no film present. Combined ρ·η characterisation of a liquid from a single Δf measurement.
Z̃load = Z̃f · (Z̃L + Z̃f·tanh ξ̃f) / (Z̃f + Z̃L·tanh ξ̃f)
Z̃f = √(ρf·G̃f) G̃f = G' + iG'' = G' + iωηf
Z̃L = √(iω·ρL·ηL) [0 in air]
ξ̃f = ω·hf·√(ρf/G̃f)
ΔD = 2ΔΓ/f₀
High ΔD/|Δf| ratio indicates strong viscoelastic losses. Typical threshold: ΔD/|Δf| > 0.2×10⁻⁶/Hz signals Sauerbrey breakdown.
| Use | n | Δf/n (Hz) | ΔD (× 10⁻⁶) |
|---|---|---|---|
| 1 | |||
| 3 | |||
| 5 | |||
| 7 | |||
| 9 | |||
| 11 | |||
| 13 |
▸ Initial Guesses (optional — auto-estimated if left at 0)
The optimizer will simultaneously fit the Voigt–Voinova model to all harmonics and extract hf, G′, G″.
| n | Δf/n meas (Hz) |
Δf/n fit (Hz) |
ΔD meas (×10⁻⁶) |
ΔD fit (×10⁻⁶) |
Δf err (%) |
|---|
Enter the dry film thickness (from a Sauerbrey or Voigt-in-air analysis on the same coated crystal before solvent contact) and the wet film thickness (from the Voigt Inverse fit above with the solvent in place). The helper computes the swelling metrics assuming negligible lateral spreading (1D thickness change only).
where Δfcalc, ΔDcalc come from the full Voigt–Voinova forward model
evaluated at each overtone frequency fn = n·f₀
Requirements: ≥ 2 overtones (6 data points) for 3 free parameters. 3+ overtones recommended.