---
title: "When a slower quartz crystal can measure added mass"
description: "A rigid film adds shear inertia, but liquid drag also lowers frequency. Follow current resonance to learn when a quartz microbalance can infer mass."
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  zh-CN: "https://sandboxphysics.com/zh/stories/instruments/quartz-microbalance/"
last_modified: "2026-11-03"
---

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Inside the instrument

# When a slower quartz crystal can measure added mass

A thin rigid coating moves with a quartz surface, adds inertia and lowers its shear resonance. A quartz crystal microbalance can use that change to measure very small deposits. Water and compliant attachments also change the motion, so a lower frequency needs a suitable blank and a measure of loss before it supports a mass estimate.

Sandbox Physics · 2026-11-03

## A coating must turn around with the surface

Place electrodes on opposite faces of a quartz plate and apply an alternating voltage. Its piezoelectric response repeatedly deforms the crystal. With a suitable cut, the two faces move sideways in opposite directions. This thickness-shear motion resembles a thin stack of paper whose top and bottom slide opposite ways. A common sensor repeats the motion about five million times each second.

A uniform, rigidly attached coating must accelerate whenever the surface reverses direction. The elastic restoring action remains, but the moving inertia increases, so the resonance becomes slower. The coating must be thin and stiff enough to follow the quartz. A loosely attached particle or a compliant connection can have relative motion of its own.

Sources: [1]

## Find the mechanical resonance in the current

A voltage drives a larger mechanical response near resonance. An instrument can sweep a narrow frequency interval and save the current at each point. It also records whether the current peak arrives before or after the voltage peak. This position within a cycle is its phase. Current size and phase together preserve the shape of the response.

The electrodes also charge capacitively. That current can flow without driving the same mechanical motion, and it contributes to the total reading. An equivalent circuit separates a motional path representing inertia, elasticity and loss from a parallel capacitance representing electrodes and wiring. Fitting the two current components avoids identifying the largest total current with the mechanical series resonance.

The peak position identifies a preferred frequency. Its width reflects loss: more rapid decay generally gives a wider resonance. Counting frequency precisely is therefore only part of a mass measurement.

Sources: [1]

## One microgram per square centimetre gives about 56.6 hertz

For a five-megahertz crystal, adding a uniform rigid film of one microgram per square centimetre lowers frequency by about 56.6 hertz under the Sauerbrey relation. A microgram is a millionth of a gram. This is mass per area; converting it to total mass also needs the effective coated area.

Write the added areal mass as $m_A$, the positive sensitivity as $C_f$, and the frequency change as $\Delta f$. The relation below expresses the downward shift caused by added rigid inertia.

A thicker film can deform in shear itself. Uneven coverage can also weight some surface regions more strongly than others. Even for a small frequency shift, the conversion requires a thin, stiff coating that follows the surface.

$$
\Delta f=-C_f m_A
$$

Frequency change uses hertz and areal mass uses micrograms per square centimetre. The sensitivity is about 56.6 hertz per unit areal mass for this five-megahertz AT-cut quartz. The linear conversion assumes a uniform thin, rigidly attached load.

Sources: [1] [2]

## Water slows a bare crystal too

Let one face of a bare crystal touch water. The moving surface drags nearby liquid sideways. More distant liquid follows less closely, and viscosity dissipates energy within that shear layer. The crystal is now moving against more than a solid deposited film.

At twenty degrees Celsius, ideal one-face water loading lowers a five-megahertz resonance by about 715 hertz. Applying the dry-film sensitivity would suggest about 12.6 micrograms per square centimetre of added material even though no film was deposited. This liquid model depends on the product of density and viscosity, so one response cannot determine them separately.

The resonance also broadens, providing another clue. Compliant attachments can change both coupled inertia and loss. Measuring position and width helps select a load model. Separating mass from mechanical properties requires further frequencies or independent information about the material.

Sources: [1] [2]

## The blank must describe comparable conditions

The crystal frequency itself changes with temperature. Clean-crystal scans at several temperatures provide a baseline for the sample temperature and can remove calibrated static drift. The current receiver needs known references for scale and phase, and the frequency clock needs its own scale reference. Saving those standards with the raw sweep gives the measured shift a clear origin.

Mass measurement in liquid remains possible. If the liquid properties and temperature stay fixed and an added layer remains sufficiently rigid, a matched liquid blank and a suitable load model can support a mass increment. Changing liquid, swelling films or relative motion require another interpretation. The instrument measures how a surface drags its load; the mass estimate rests on that physical picture.

Sources: [1]

## [Compare a rigid film with a liquid-loaded shear crystal](https://sandboxphysics.com/labs/quartz-crystal-microbalance/)

Drag the drive dial, save complex-current sweeps, then compare the frequency and width changes from a rigid film and bare-water contact.

Fundamental BVD sweeps use independent standards and dry temperature blanks. This dry-mass estimator reports areal mass for compatible rigid gas-phase films.

## Sources and further reading

1. [SRS QCM200 / QCM25 Operation and Service Manual · Revision 2.5 (6/2018)](https://www.thinksrs.com/downloads/pdfs/manuals/QCM200m.pdf)

   Printed pp.15–16 introduce rigid-film sensitivity and the BVD circuit; Eq.12 on p.27 gives Sauerbrey sensitivity, Eq.15 on p.29 gives liquid loading and p.30 gives the approximately 715 Hz water example. Pages35–37 discuss rigid increments and viscoelastic changes in solution; pp.42 and49–55 describe dissipation, thickness shear and temperature. These adopted sections were checked on 28 October 2026.

2. [openQCM Advanced Calculator · Newtonian liquid model and model-selection guide](https://openqcm.com/shared/QCM-Calculator/QCM_Advanced_Calculator.html)

   The Sauerbrey and Kanazawa–Gordon guide sections distinguish rigid gas-phase films from one-face Newtonian liquid loading. The liquid section states that frequency shift and half-bandwidth increment have equal magnitudes and identifies density times viscosity. Consulted 28 October 2026; no stable page revision is exposed. Its automatic multi-parameter inversion claims are not adopted here.

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