IN16 · A thin film slows a shear wave

Quartz Crystal Microbalance

When does a crystal frequency drop really measure added mass? A thin rigid coating moves with the quartz and adds inertia, lowering its shear resonance. A compatible blank removes baseline drift before converting that shift to mass per area. Liquid drag and compliant attachment also change frequency and loss, so frequency alone cannot identify dry mass.

Coat a quartz crystal and compare dry, liquid and compliant loads. Fit saved current sweeps before converting a compatible frequency shift to areal mass.

Drag the gold drive dial to preview ideal crystal motion and current. Save a sweep to compare calibrated ADC observations with clean temperature blanks.

Surface mediumUniform film areal mass

2

Equilibrated temperature

20

Drive offset from 5 MHz

-110

Three dry blanks and a coated sweep are loaded.

Liquid, attachment, receiver and sweep

Film attachment modelLiquid density / kg per cubic metre

998.2

Liquid viscosity / mPa s

1.002

Net parallel capacitance / pF

20

Sweep half span / Hz

2500

Current noise / microamp RMS per component

0.02

Bipolar ADC current range / mA

1

Random seed

189

Blank acquisition prepares a clean dry crystal at 15, 25 and 35 degrees Celsius, then restores the sample setup. It assumes repeatable cleaning and unchanged electrodes. No thermal transient or deposition kinetics is simulated.

Model assumptions

Fundamental-only 5 MHz thickness-shear teaching resonator. Complex-current ADC sweeps use a BVD circuit with a prescribed motional capacitance; fitted frequency, half-width and shunt capacitance come from raw samples. Independent current, frequency and temperature standards and three dry temperature blanks are saved. Uniform thin rigid gas-phase films use the small-load Sauerbrey relation. One-face semi-infinite Newtonian liquid loading and a passive lumped compliant attachment are counterexamples; this dry-mass estimator withholds mass in those conditions. Suitable liquid QCM experiments can still measure mass with additional assumptions and matched references. No multimode, finite-thickness viscoelastic inversion, distributed sensitivity, thermal transients or commercial accuracy is claimed. Shear displacement, quartz thickness and film thickness are explicitly enlarged; plots remain outside the apparatus.

Simulated complex-current ADC observations

Fit the peak before assigning a mass

Saved sweep

Shift from dry temperature blank

-113\,\mathrm{Hz}

Fitted half-width

25.9\,\mathrm{Hz}

Dissipation factor

1.04\times10^{-5}

Conditional dry areal mass

2.00\,\mathrm{\mu g/cm^2}

Blank, receiver calibration and fit quality

The ADC stores two RMS-current components at each indicated frequency. Independent current standards set the complex scale and offset; known frequency and temperature references set their scales. Three clean dry sweeps remove the assumed quadratic temperature drift only between their measured temperatures. New controls or standards do not replace those inside an earlier scan. The active uniform-weight area is 0.20 square centimetres; total mass additionally uses this assumed area. No statistical confidence interval or commercial uncertainty is claimed.

Y(\omega)=i\omega C_0+\frac{1}{R_m+i(\omega L_m-1/(\omega C_m))}

Mechanical model, visual scales and limits

Fundamental-only small loading of a 5 MHz AT-cut thickness-shear plate. A uniform rigid gas-phase coating uses Sauerbrey; a one-face Newtonian liquid adds frequency shift and damping. This dry-mass estimator withholds a mass for liquid or compliant loading. Other liquid-QCM protocols can measure rigid increments with a matched liquid reference and additional assumptions. The compliant case is a passive lumped mass connected by a spring and dashpot, not a finite-thickness viscoelastic-film inversion. No film modulus or liquid density and viscosity pair is uniquely inferred.

Shear displacement is enlarged one million times, quartz thickness 12 times and film thickness 50,000 times. Motion is replayed at 0.65 cycles/s. Fluid tracer height is enlarged 5,000 times to reveal the model Stokes penetration; the bulk liquid dome is schematic. Gold electrode thickness and mounting hardware are schematic. Actual thin-film and seal deformations, spatial sensitivity weighting, nonlinear piezoelectric response, liquid conductivity, surface roughness and thermal transients are omitted. Temperature affects the chosen quartz baseline only; liquid viscosity and density are independent inputs at that temperature.

\Delta f=-\frac{2f_0^2m_A}{\sqrt{\rho_q\mu_q}},\qquad \Gamma=\frac{R_m}{4\pi L_m}

SRS QCM200 manual Rev. 2.5 (6/2018), pp. 15–16, 27–30, 35–37, 42, 49–55

Start with the essentials

Focus question: When does a crystal frequency drop really measure added mass?
One-sentence intuition: A mass estimate needs a suitable load model, a compatible blank and a resonance fitted from observations.

Core mathematical model

Thin rigid film at the fundamental

\Delta f=-\frac{2f_0^2}{\sqrt{\rho_q\mu_q}}m_A

A uniform, acoustically thin and rigidly coupled gas-phase coating gives a frequency shift proportional to mass per area. Quartz density and shear modulus set the sensitivity.

The electrical resonance has two paths

Y(\omega)=i\omega C_0+\frac{1}{R_m+i(\omega L_m-1/(\omega C_m))}

Current can charge the parallel capacitance or drive the motional branch. Fitting both complex components separates the shunt from the series resonance.

Half-width and dissipation convention

\Gamma=\frac{R_m}{4\pi L_m},\qquad D=\frac{2\Gamma}{f_s}

Gamma is half the full conductance half-power bandwidth. Dissipation is the inverse resonant quality factor in this underdamped BVD model.

Common difficulties

A lower frequency means more mass

Typical misconceptionAny downward frequency shift can be converted with the same sensitivity.

Better mental modelLiquid drag, compliant motion and temperature also shift resonance. The sample model and blank must support a mass conversion.

The highest total current locates the mechanical resonance

Typical misconceptionPick the maximum magnitude of total admittance.

Better mental modelParallel capacitance changes that magnitude. Fit complex current with its shunt path to locate the series resonance.

Run the experiment

01
Drag the actual frequency dial
Drag the source knob across the resonance.
What to observe: The drive frequency changes immediately. Shear amplitude and current respond to the same BVD circuit.
02
Save a coated scan
Compare Rigid film and More rigid film.
What to observe: A stronger uniform inertial load shifts the fitted center downward while keeping the small-loss width nearly unchanged.
03
Add liquid without adding a film
Choose Bare water, then More viscous liquid.
What to observe: Both center and linewidth change. This dry-mass estimator retains those observations and withholds a mass value.
04
Check a compliant connection
Choose A slipping attachment.
What to observe: A spring-dashpot attachment changes how much of the film follows the quartz and dissipates energy. One frequency/width pair does not determine all film properties.
05
Keep the temperature reference
Acquire a known-standard set and clean dry temperature blanks, then save the sample.
What to observe: Each scan retains its original standards. The three blank temperatures remove the declared quadratic drift within their covered interval.

Quartz Crystal Microbalance

Fit the peak before assigning a mass

The surface drags its load sideways

Core mathematical model

Thin rigid film at the fundamental

The electrical resonance has two paths

Half-width and dissipation convention

Common difficulties

A lower frequency means more mass

The highest total current locates the mechanical resonance

Run the experiment

Drag the actual frequency dial

Save a coated scan

Add liquid without adding a film

Check a compliant connection

Keep the temperature reference

Related experiments and articles

How physics instruments measure