QCM Advanced Calculator

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).

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.

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:

Limiting cases: G′ → ∞ recovers the Sauerbrey equation; h_f → 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:

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.

• 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.