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.

• 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, run the Voigt model at each overtone frequency (e.g. 5, 15, 25, 35 MHz for n = 1, 3, 5, 7) and compare Δf/n and ΔD_n 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.