Q C MQuartz Crystal Microbalance - Theory and Modeling

11/25/14

Revision 2.0

Marco MauroScientific Coordinator Novaetech S.r.l.co-founder openQCMemail mauro@novaetech.ittwitter @2m_marco

This work is licensed under Creative Commons Attribution-ShareAlike 4.0 International License

Hypothesis• film thickness less than that of the quartz• inertial load uniform, rigid and fixed to the quartz surface • no dissipation crystal oscillation takes place in air or in vacuum

Sauerbrey equation

Sauerbrey equation changes in resonant frequency of the quartz shear vibration is proportional to the mass deposited on its active surface

Additional mass causes the decrease of the resonance frequency.

Sauerbray equation: linear relationship between resonance frequency variations and deposited thin-stiff mass

NOT working• liquid is not integral to the quartz, resistance to the

free motion• viscosity loss of mechanical energy

qµ = shear modulus

qρ = density

= resonance frequency0f

A = electrode area

Sauerbrey G. Zeitschrift Für Physik, vol. 155, no. 2, pp. 206-222, (1959)

Sauerbrey equation is directly derived from the solution of the one-dimensional shear wave equation (Helmholtz equation).The proportionality factor depends on the physical-geometrical characteristics of quartz crystal.

Resonance Frequency and Vibration Amplitude as function of the distance from the center of the quartz (Mecea 2005).

sv

q

f

qq

f

Amf

Sρ

ρµ

202

2/1

202 =

=

∆

∆=

Mecea, V. M. “From Quartz Crystal Microbalance to Fundamental Principles of Mass Measurements,”Anal. Lett., vol. 38, no. 5, pp. 753-767, (2005)

QCM Sensitivity is not uniform over the surface of the quartz, it is maximum in the center and decreases near the edges of the electrode.Gaussian distribution of the QCM mass sensitivity → Distribution of the vibration amplitude.

QCM sensitivity with frequency resolution of 1 Hz

QCM measures deposited mass in the range of nanogram 10-9 g

exe

QCM SensitivityRelationship between the variation of the resonance frequency and that of the density mass deposited:

Sauerbrey equation

f 0=25 MHz A=0.1cm2 → Δm=0.07ng

μq=2.947×1011g−1cm2 s−1

ρq=2.648g cm−3

vq=√ μq/ρq=3.340×105 cms−1

QCM in liquid Kanazawa – Gordon Equation

Kanazawa - Gordon model Hypothesis• No Dissipation quartz is a perfectly elastic solid• Newtonian Liquid purely viscous liquid, linear relationship between stress - strain rate

QCM in Liquid Relationship between the variation of the resonance frequency of a quartz crystal immersed in a fluid

The vibration consists of both a stationary shear wave in the quartz and an acoustic damped wave in the liquid. (Kanazawa Gordon 1985)

Kanazawa, K. and Gordon G.J. “Frequency of a Quartz Microbalance in Contact with Liquid” Anal. Chem., vol. 57, no. 9, pp. 1770-1771, (1985).Kanazawa, K. and Gordon G.J. “The oscillation frequency of a quartz resonator in contact with liquid” Anal. Chim. Acta, 175, pp. 99-105, (1985)

Characteristic Length of the damped vibration

ηL

= liquid viscosity

ρL = liquid density

exeQCM into water at T = 20˚C → λ = 250 nm

Liquid Effective Mass

Resonance Frequency Variation

qq

LLffρπµ

ρη2/3=∆

exeQCM in water at T = 20˚C and f

0 = 5 MHz →

Δf = 6100 Hz

Electro-Mechanical ModelGeneral model predicts quantitatively the effects of a generic load on the quartz surface . This model is the base for using QMC as biosensor (Johansmann).

Electrical System Mechanical System

Resistance R Friction ξ

Inductance L Mass m

Capacitor C Elasticity k

Current i(t) Velocity v(t)

Voltage V(t) Force F(t)

Schematic of the electro-mechanical equivalence

Johansmann, D. “Modeling of QCM data” , unpublished manuscript, available on-line

Butterworth - Van Dick CircuitEquivalent circuit of the quartz crystal• Small loads • Variation close the resonance

Mechanical BranchInductance → Quartz crystal initial massCapacitor → Mechanical ElasticityResistance → Friction/Dissipation

Electrical BranchCapacitor → Capacitance between the electrodes

Load Impedance Zm

Ratio between surface stress and velocity field.

Mechanical – Electrical System Equivalence

Electro - Mechanical Model Small-Load Approximation

Resonance Frequency and Load ImpedanceLoad Impedance is the crucial physical quantity in QCM sensing applications under the most general conditions.

Conductance curve (i.e. resistance inverse) as a function of the quartz vibration frequency

Complex Resonance Frequency

Load Impedance ZL in BVD circuit is the electrical equivalent of a generic load on the quartz surface.

Small – load approximation

At the resonance the complex part of total impedance is null

= Shear stress

= Quartz surface velocity

= 8.8 x 109 gm-2s-1 Acoustic impedance of AT-cut quartz= Resonance frequency= Half Band Half Width (HBHW)

Electro - Mechanical Model Small-Load Approximation

ωr

Γ Zq

u̇

σ

Layered Systems – Applications of InterestWe use the small-load approximation model to calculate the variation of the resonance frequency for layered systems, that is planar loads uniformly distributed on the quartz crystal surface

• Viscoelastic Medium (semi-infinite)• Inertial Load (Sauerbrey equation)• Viscoelastic Film• Viscoelastic Film in Liquid

Hypothesis•Quartz crystal and the layers are laterally homogeneous and infinite•The vibration is a transverse shear wave, whose direction is perpendicular to the surface (thickness – shear mode). •The stress tensor is proportional to the deformation, which means that linear viscoelasticity is verified•the contribution due to the piezoelectric stiffness is neglected

Johansmann, D. “Modeling of QCM data” , unpublished manuscript, available on-line“Quartz Crystal Microbalance”, Wikipedia, http://en.wikipedia.org/wiki/Quartz_crystal_microbalance

Electro - Mechanical Model Small-Load Approximation

Viscoelastic Medium (Semi-Infinite)Shear wave inside the crystal and a propagation wave inside the medium, that travels away from the crystal surface.

Quartz Crystal in contact with a semi-infinite viscoelastic medium

Inertial Load - Sauerbrey Equation

Quartz Crystal in contact with a purely inertial load.

Newtonian liquid → η’ = cost. e η’’ = 0 not a function of frequency. The variation of the resonance frequency is equal and opposite to that of the bandwidth of the resonance curve → Kanazawa – Gordon equation

Inertial Load there is only the shear wave propagating inside both the quartz crystal and the layer. The surface stress is caused only by the inertia of the deposited film.

The variation of the complex resonance frequency is given by:

Electro - Mechanical Model Small-Load Approximation

Viscoelastic Film More general formulation respect to Sauerbrey hypothesis, it is considered viscoelastic film of arbitrary thickness . The vibration consists of a shear wave inside the quartz crystal and a wave transmitted – reflected inside the viscoelastic film.

Quartz Crystal in contact with a viscoelastic film of arbitrary thickness

The relation can be written in function of the physical quantity related to the viscoelastic film

If the film thickness is very small, we can derive an approximated relation that explicitly contains the film viscoelastic constants (by expanding the tangent function in Taylor’s series to the second order):

Sauerbrey equation film thickness is negligible

The variation of the complex resonance frequency is given by:

Viscoelastic Correction film thickness is not negligible

Electro - Mechanical Model Small-Load Approximation

Viscoelastic Film in LiquidOn the quartz crystal surface is deposited a viscoelastic film of arbitrary thickness and they are immersed in a liquid. Vibrations consists of a shear-wave inside the quartz, a transmitted wave propagating through the film and reflected at the boundary quartz-film and film-liquid, and a propagation wave in the liquid, that travels away from the film surface.

Quartz Crystal in contact with a viscoelastic film and immersed in a fluid

The variation of the complex resonance frequency is given by:

If we consider small thickness film, by expanding in Taylor’s series df ≈ 0

Kanazawa-Gordon Missing Mass

Johannsmann, D. “Viscoelastic analysis of organic thin films on quartz resonators”, Macromol. Chem. Phys. 200, 501 (1999)

Voinova, M.V. , Jonson, M.B. and Kasemo, B., “‘Missing mass’ effect in biosensor's QCM applications” Biosens. Bioelectr. 17, 835 (2002)

Missing mass effect viscoelastic correction reduces the mass measured for soft-film (deviation from Sauerbrey equation)

Resonance frequency variation is measured respect to reference state, the quartz is immersed in the fluid (KG + Sauerbrey)

Sauerbrey

Electro - Mechanical Model Small-Load Approximation

QCM-D Quartz Crystal Microbalance with Dissipation Monitoring

Dixon, M. C. “Quartz crystal microbalance with dissipation monitoring: enabling real-time characterization of biological materials and their interactions.,” Journal of biomolecular techniques, vol. 19, no. 3, pp. 151-8, (2008).

Resonance frequency variation Δf → Mass deposited on the quartz surface Measurement of energy losses due to dissipation D → Viscoelastic properties of liquid in contact with quartzLife sciences applications DNA, proteins, lipids and cells and so on...

Principle of MeasurementMeasurement of the characteristics of quartz damped oscillations, caused by a rapid excitation of the quartz.- Protein = rigid layer, frequency variation (mass)- Biomacromolecules = frequency and dissipation variation (viscolastic mass)

Adsorption of the protein serum albumin step a) and his antibody step c). Step b) and d) rinse with buffer solution.a) Frequency variation - no change in dissipation → serum albumin is a rigid proteinc) Frequency and dissipation variation → antibody causes an increase in mass and viscoelasticity (water)d) Dissipation variation → change in antibodies conformation

D =1

π f τ=

1Q

=E dissipated

2π Estored

QCM – Spotlighting Research Surface-Specific Interactions

Kanazawa K. and Cho C., “Quartz Crystal Microbalance to characterize Macromolecular Assembly dynamics”, Journ. Sens. (2009)

Adsorption Kinetics of vesicles on solid layersa) Lipids double-layer on SiO2

b) Intact vescicles on Au layer c) Two-steps kinetics of vescicles

unilamellar → bilayerd) Adsorption Kinetics of intact vescicles –

frequency and dissipation variation

a) b)

d)c)

Marco MauroScientific Coordinator Novaetech S.r.l.co-founder openQCMemail mauro@novaetech.ittwitter @2m_marco

This work is licensed under Creative Commons Attribution-ShareAlike 4.0 International License

ADDITIONAL RESOURCES AND REFERENCES

Resources

References

● openQCM Is the unique open source quartz crystal microbalance. The brand new open source project to build your own QCM or buy a low-cost easy to mount kit.

● quartzcrystalmicrobalance.org the most detailed and updated resource on quartz crystal microbalance● Stanford Research System “Quartz Crystal Microbalance References”, available on-line (last revision 2005)

Review Papers on the most recent research works on QCM sensing● Cooper M. A. and Singleton V. T. “A survey of the 2001 to 2005 quartz crystal microbalance biosensor literature :  

applications of acoustic physics to the analysis of biomolecular interactions,” J. Mol. Recognit. , pp. 154-184, (2007)● Vashist S.K. and Vashist P. “Recent advances in Quartz Crystal Microbalance-Based Sensors”,

Journal of Sensors, Volume 11, Article ID 571405, (2011)● Becker B. and Cooper M.A. “A survey of the 2006-2009 quartz crystal microbalance biosensor literature”,

J. Mol. Recognit. 24(5):754-87 (2011)● Speight R.E. and Cooper M.A. “A survey of the 2010 quartz crystal microbalance literature”

J. Mol. Recognit. 25(9):451-73 (2012)