# QCM Calculator > Source page: https://openqcm.com/qcm-calculator > Markdown version for AI agents and readers of plain text. Site guide: https://openqcm.com/llms.txt · Everything in one file: https://openqcm.com/llms-full.txt ## Turn Δf into Δm. Four QCM models in one tool — the Sauerbrey equation for rigid films, Kanazawa-Gordon for Newtonian liquids, Voigt-Voinova for viscoelastic layers, and a multi-overtone Voigt inverse fit. Math runs locally in your browser; nothing is sent anywhere. Jump to calculator ↓[See the source ↗](https://github.com/openQCM) 01 / Which model ### Match the model to your regime. Each model assumes a different physical picture of what’s on the crystal. Pick by how your film behaves — rigid, liquid-loaded, or soft and solvated. ##### Sauerbrey — thin, rigid films Thin (< 1% of f₀ shift), rigid coatings firmly coupled to the crystal: thin metals, evaporated films, dry polymers, monolayers. Δf maps straight to Δm; a single overtone is enough. ##### Kanazawa-Gordon — Newtonian liquid A bare crystal in contact with a Newtonian liquid. The frequency shift comes from the liquid’s density-viscosity product (ρ·η), not adsorbed mass — the baseline for any in-liquid measurement. ##### Voigt-Voinova — soft, solvated layers For swollen, soft, viscoelastic films where ΔD grows significant and different overtones give different Δm. Run the forward model to predict Δf and ΔD from a known layer (thickness, shear modulus, viscosity). ##### Voigt inverse — multi-harmonic fit When one overtone can’t pin down a viscoelastic layer, feed Δf/n and ΔD measured at several harmonics (n = 1, 3, 5, 7, 9) into a least-squares fit that recovers film thickness, shear modulus and viscosity together. 02 / Calculator ### Pick a model. Drop in your numbers. ### Researcher’s Guide & Quick Reference ▼ Overview How to use this calculator This calculator provides **real-time analysis** for Quartz Crystal Microbalance (QCM) experiments using four 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; the **Voigt Inverse** tab fits the same model to multi-overtone data. All four models share the same AT-cut quartz constants (ρ_Q = 2.648 g/cm³, μ_Q = 2.947 × 10¹¹ g/(cm·s²), Z_Q ≈ 8.834 × 10⁵ g/(cm²·s)), displayed at the bottom of each panel for reference. 1 — Sauerbrey Model Rigid thin film in air 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 ρ_f. The calculator returns: - **Mass sensitivity C_f** — 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). 2 — Kanazawa–Gordon Model Newtonian liquid loading 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 ρ_L and viscosity η_L, predict Δf, ΔD, ΔΓ and the viscous penetration depth δ. - **Inverse:** given a measured Δf, extract the ρ_L·η_L product. If ρ_L is known, extract η_L 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** = √(η_L/(π·f₀·ρ_L)) defines the liquid layer thickness sensed by the shear wave. In water at 10 MHz, δ ≈ 178 nm. 3 — Voigt–Voinova Model Viscoelastic film ± liquid Use this tab for **viscoelastic films** — hydrogels, polymer brushes, protein adlayers, lipid bilayers, cell monolayers. Provide the film’s density ρ_f, thickness h_f, storage modulus G′, and a loss parameter (either film viscosity η_f 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 - **Δf_S** — Sauerbrey rigid-film prediction for comparison A **regime assessment** automatically classifies the film behaviour: Rigid Soft-Rigid Viscoelastic ΔD/|Δf| < 0.05 0.05 – 0.2 > 0.2 **Limiting cases:** G′ → ∞ recovers the Sauerbrey equation; h_f → 0 with liquid contact recovers Kanazawa. Use this to verify continuity across models. 4 — Voigt Inverse Fitting Multi-harmonic parameter extraction 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:** 1. **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. 2. **Set fixed parameters.** Enter the fundamental frequency f₀ (typically 5 MHz), the film density ρ_f (g/cm³), and select Air or Liquid medium. For liquid, provide ρ_L and η_L. 3. **Initial guesses (optional).** Expand the section to provide starting values for h_f, G′, G″. If left at zero, the calculator auto-estimates h_f from the Sauerbrey equation and uses sensible defaults for the moduli. 4. **Click ▶ Run Fit.** The Levenberg–Marquardt optimizer runs 4 restarts with different initial conditions and selects the best solution. The optimizer extracts: - **h_f** — film thickness (nm), typically larger than the Sauerbrey estimate for viscoelastic films - **G′** — storage (elastic) modulus (Pa) - **G″** — loss (viscous) modulus (Pa) - **η_f** — film viscosity (Pa·s), computed as G″/(2πf₀) - **|G*|** — complex modulus magnitude, **tan δ** — loss tangent - **χ²_r** — 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. Practical Tips For QCM-D and multi-harmonic users - 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** (Δf_S), so you can quantify how much viscoelasticity shifts the result. - Use the **loss parameter toggle** (η_f ↔ G″) in the Voigt panel for convenience. The calculator auto-converts between the two using G″ = ω·η_f 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 ΔD_n 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. Measuring Film Swelling — Recommended Workflow Dry → Wet → Swell ratio for polymer, hydrogel and resist films 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:** 1. **Bare crystal in air** — record f₀ on each overtone you intend to use (n = 3, 5, 7, …). This is the absolute reference. 2. **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 ρ_L and η_L 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. 3. **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 ρ_f to obtain h_dry. If Δf/n varies with n, the dry film is already viscoelastic — use the **Voigt Inverse Fitting** tab in *Air* mode instead. 4. **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 T_g and solvent compatibility). Switch to the **Voigt Inverse Fitting** tab, set Medium = *Liquid* , enter the solvent’s ρ_L and η_L (use the *Solvent preset* dropdown for common values), and run the fit. The output h_f is the swollen thickness h_wet. 5. **Compute swelling quantities** — use the *Swelling Analysis* helper in the Voigt Inverse panel. With h_dry and h_wet it returns the swell ratio SR = h_wet/h_dry, the volume change ΔV/V = SR − 1, and the solvent volume fraction φ_s = 1 − h_dry/h_wet (assuming negligible lateral spreading). **Why multiple overtones are non-negotiable in step 4:** the Voigt model has three film unknowns (h_f, 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. Key References Foundational literature for the four models - **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). - **QCM-D instrument:** M. Rodahl, F. Höök, A. Krozer, P. Brzezinski, B. Kasemo, “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). Input Parameters A = 0.1963 cm² Resonant Frequency (f₀) Hz Valid range: 1 kHz – 1 GHz Sensitive Area Diameter (d) cm Valid range: 0.01 – 10 cm Frequency Change (Δf) Hz Negative → mass deposition | Positive → mass removal Sample Density (ρ_f) g/cm³ Valid range: 0.01 – 30 g/cm³ **Validity:** Rigid, elastically uniform film in air. Keep |Δf/f₀| < 1/1000 for reliable results. Theoretical Sauerbrey limit ≈ 1/100. Results Mass Sensitivity C_f ? 4.417 ng/(Hz·cm²) Mass per Hz over active area ? 0.867 ng/Hz Mass Saturation (at Δf/f₀ = 1%) ? 8.6726e-5 g Maximum Sample Thickness (at saturation) ? 4417 nm Film Thickness ? 4.42 nm Mass Change Δm ? 8.6726e-8 g Thickness — alternate units 0.0044 μm | 44.2 Å Mass per Unit Area Δm/A ? 4.41692e-7 g/cm² Quartz Constants (AT-cut) Density ρ_Q2.648 g/cm³ Shear modulus μ_Q2.947 × 10¹¹ g/(cm·s²) Acoustic impedance Z_Q8.834 × 10⁵ g/(cm²·s) Sauerbrey Equation \Delta m = -\,\Delta f \cdot A \cdot \dfrac{\sqrt{\rho_Q\,\mu_Q}}{2 f_0^{2}} C_f = \dfrac{\sqrt{\rho_Q\,\mu_Q}}{2 f_0^{2}} \quad [\mathrm{g/(Hz\cdot cm^2)}] t = -\,\Delta f \cdot \dfrac{C_f}{\rho_f} **Limits:** For a 10 MHz crystal, Cᶠ ≈ 4.42 ng/(Hz·cm²). Mass resolution is sub-nanogram per cm². Deviations from linearity (Sauerbrey breakdown) begin when the film acoustic thickness approaches that of the quartz. Monitor ΔD to verify the rigid-film assumption holds. Input Parameters Resonant Frequency f₀ Hz Liquid Density ρ_L g/cm³ Liquid Viscosity η_L mPa·s Water 20°C ≈ 1.002 mPa·s | Glycerol ≈ 1412 mPa·s Measured Δf Hz Enter the measured frequency shift (negative = liquid loading) Liquid Density ρ_L g/cm³ (optional) Provide ρ_L to extract η_L individually **Kanazawa model:** QCM in contact with a semi-infinite Newtonian liquid. Characteristic signature: |Δf| = ΔΓ, i.e. ΔD = 2|Δf|/f₀. Higher ΔD/Δf ratios indicate non-Newtonian behaviour. Results Forward Frequency Shift Δf ? -2021.67 Hz Dissipation Change ΔD (× 10⁻⁶) ? 404.33 Half-Bandwidth Change ΔΓ ? 2021.67 Hz ρ_L · η_L product ? 0.00100 (g/cm³)·(Pa·s) Viscous Penetration Depth δ ? 178.6 nm Extracted ρ_L · η_L product ? — Extracted η_L (using given ρ_L) ? — Predicted ΔD (× 10⁻⁶) — Viscous Penetration Depth δ — Quartz Constants (AT-cut) ρ_Q2.648 g/cm³ μ_Q2.947 × 10¹¹ g/(cm·s²) Z_Q = √(ρ_Q·μ_Q)8.834 × 10⁵ g/(cm²·s) Kanazawa – Gordon Equation \Delta f = -\,f_0^{3/2}\,\sqrt{\dfrac{\rho_L\,\eta_L}{\pi\,\rho_Q\,\mu_Q}} \Delta\Gamma = |\Delta f| \implies \Delta D = \dfrac{2|\Delta f|}{f_0} \delta = \sqrt{\dfrac{\eta_L}{\pi\,f_0\,\rho_L}} **Inverse:** \rho_L\,\eta_L = \dfrac{\Delta f^{2}\,\pi\,\rho_Q\,\mu_Q}{f_0^{3}} **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. Input Parameters Resonant Frequency f₀ Hz --- Film Properties Film Density ρ_f g/cm³ Polymer ~1.0–1.5 | Protein layer ~1.3–1.4 Film Thickness h_f nm Storage Modulus G’ Pa Soft gel ~10³–10⁴ Pa | Stiff polymer ~10⁶–10⁸ Pa Loss Parameter Film Viscosity η_f Pa·s Soft gel ~ 0.001–0.1 Pa·s --- Contact Medium → Switch to **Liquid** for swelling, biosensing or any in-solvent measurement. ρ_L and η_L fields plus a solvent preset will appear below. Solvent preset 25 °C unless stated Liquid Density ρ_L g/cm³ Liquid Viscosity η_L mPa·s Results In Air Frequency Shift Δf ? -2434.83 Hz Dissipation Change ΔD (× 10⁻⁶) ? 29.81 Half-Bandwidth Change ΔΓ 149.04 Hz --- ΔD / |Δf| ratio (× 10⁻⁶ / Hz) ? 0.0122 Complex Modulus |G*| ? 118100.981 Pa Loss Tangent tan δ = G” / G’ ? 0.6283 Sauerbrey rigid-film limit Δf_S ? -2716.83 Hz Rigid regime: Sauerbrey equation applicable. Very low viscoelastic losses. Quartz Constants (AT-cut) ρ_Q2.648 g/cm³ μ_Q2.947 × 10¹¹ g/(cm·s²) Z_Q8.834 × 10⁵ g/(cm²·s) ReferenceVoinova et al., 1999 Voigt – Voinova Model \widetilde{\Delta f} = \Delta f + i\,\Delta\Gamma = -\dfrac{f_0}{\pi Z_Q}\,\tilde{Z}_{\text{load}} \tilde{Z}_{\text{load}} = \tilde{Z}_f\,\dfrac{\tilde{Z}_L + \tilde{Z}_f\tanh\tilde{\xi}_f}{\tilde{Z}_f + \tilde{Z}_L\tanh\tilde{\xi}_f} \tilde{Z}_f = \sqrt{\rho_f\,\tilde{G}_f} \qquad \tilde{G}_f = G^{\prime} + iG^{\prime\prime} = G^{\prime} + i\omega\eta_f \tilde{Z}_L = \sqrt{i\omega\,\rho_L\,\eta_L} \quad [0\ \text{in air}] \tilde{\xi}_f = \omega\,h_f\,\sqrt{\dfrac{\rho_f}{\tilde{G}_f}} **Limits:** G^{\prime}\!\to\!\infty \implies \text{Sauerbrey} | h_f\!\to\!0 \implies \text{Kanazawa} \Delta D = \dfrac{2\Delta\Gamma}{f_0} High ΔD/|Δf| ratio indicates strong viscoelastic losses. Typical threshold: ΔD/|Δf| > 0.2×10⁻⁶/Hz signals Sauerbrey breakdown. Experimental Data (multi-harmonic) Loss input: | Use | n | Δf/n (Hz) | ΔD (× 10⁻⁶) | | --- | --- | --- | --- | | | 1 | | | | | 3 | | | | | 5 | | | | | 7 | | | | | 9 | | | | | 11 | | | | | 13 | | | Fundamental Frequency f₀ Hz Film Density ρ_f g/cm³ Medium Solvent preset 25 °C unless stated ρ_L g/cm³ η_L mPa·s ▸ Initial Guesses (optional — auto-estimated if left at 0) h_f initial nm G′ initial Pa G″ initial Pa Fitted Parameters Enter Δf/n and ΔD (or ΔΓ) data at ≥ 2 overtones, then click **▶ Run Fit** . The optimizer will simultaneously fit the Voigt–Voinova model to all harmonics and extract h_f, G′, G″. Film Thickness h_f —nm Storage Modulus G′ —Pa Loss Modulus G″ —Pa Film Viscosity η_f —Pa·s |G*| Complex Modulus —Pa tan δ = G″/G′ — Sauerbrey Thickness (rigid limit) —nm χ² (reduced) — Measured vs. Fitted | n | Δf/n meas(Hz) | Δf/n fit(Hz) | ΔD meas(×10⁻⁶) | ΔD fit(×10⁻⁶) | Δf err(%) | | --- | --- | --- | --- | --- | --- | **Interpretation:** Results will appear here after fitting. Swelling Analysis — differential dry / wet metrics 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). h_dry nm h_wet nm Swell ratio SR = h_wet/h_dry ? — Volume change ΔV/V — Solvent volume fraction φ_s ? — Quartz Constants (AT-cut) ρ_Q2.648 g/cm³ μ_Q2.947 × 10¹¹ g/(cm·s²) Z_Q8.834 × 10⁵ g/(cm²·s) ReferenceVoinova et al., 1999 Voigt Inverse — Minimisation Problem \min_{\{h_f,\,G^{\prime},\,G^{\prime\prime}\}} \chi^{2} = \sum_n \dfrac{\big[(\Delta f/n)_{\text{calc}}-(\Delta f/n)_{\text{exp}}\big]^{2}}{\sigma_f^{2}} + \dfrac{\big[\Delta D_{\text{calc}}-\Delta D_{\text{exp}}\big]^{2}}{\sigma_D^{2}} where Δf_calc, ΔD_calc come from the full Voigt–Voinova forward model evaluated at each overtone frequency f_n = n\,f_0 **Method:** Levenberg–Marquardt with log-transformed parameters for guaranteed positivity and improved conditioning. 4 random restarts for robustness. **Requirements:** ≥ 2 overtones (4 data points) for 3 free parameters. 3+ overtones (6 data points) recommended. --- openQCM by Novaetech S.r.l., Pompeii, Italy · Buy: https://store.openqcm.com/ · Contact: info@openqcm.com · https://openqcm.com/contacts/