Theoretical background of experimental electrochemistry (KÉM/041E)
THEORY AND PRACTICE OF THE
ELECTROCHEMICAL QUARTZ CRYSTAL
MICROBALANCE
Balázs B. Berkes
2011
1
Introduction
The quartz crystal microbalance (QCM) is a piezoelectric device capable of
extremely sensitive mass measurements. It oscillates in a mechanically
resonant shear mode by application of an alternating, high frequency electric
field using electrodes which are usually deposited on both sides of the disk.
Sauerbrey was the first to recognize that these devices could be used to
measure mass changes at the crystal surface. [1]
1.Electroanalytical Chemistry ed. by Allen J. Bard, Volume 17, Marcel Dekker, INC., New York, Basel, Hong Kong
2
History
1918 – Quartz crystal oscillator (Walter Gouyton Cady)
(control the frequency of radio broadcasting stations)
1928 – Warren Marrison (Bell Telephone Laboratories), first quartz crystal clock
World War II. – radios, radars (Paul Langevin)
1950s – QCM as thickness monitor, gas phase deposition
(physics was well established, presence of a contacting liquid would prevent
the oscillation)
1980s – EQCM
3
Quartz
SiO2
2nd most abundant mineral
constituent of sandstone and many rocks (e.g. granite)
crystalline forms: α-quartz (crystallizes below 573 °C),
β-quartz, tridymite, cristobalite, quartz glass
Colored varieties: citrine, rose quartz, amethyst, smoky quartz and milky quartz
Enantiomorphic forms of α-quartz
4
Quartz
5
Piezoelectricity - History
1824 – Brewster, pyroelectricity
Haüy (the ”father of crystallography”), Becquerel
1880
”We have found a new method
for the development of polar
electricity in these same
crystals, consisting in
subjecting them to variations
in pressure along their
hemihedral axes.”
Jacques Curie
29 October 1855 – 19 February 1941
Pierre Curie
15 May 1859 – 19 April 1906
6
Piezoelectricity - History
The piezoelectric formulation was carried out most fully and rigorously by
Woldemar Voigt in 1894. (Lehrbuch der Kristallphysik, 1910)
Piezoelectricity
”Electricity or electric polarity due to
pressure, especially in a crystallized
substance, as quartz.”
(Webster’s New International Dictionary, 1939)
”Electricity, or electric polarity, resulting
from the application of mechanical
pressure on a dielectric crystal.”
(H. Granicher, "Piezoelectricity," in AccessScience,
©McGraw-Hill Companies, 2008)
Woldemar Voigt
(2 September 1850 – 13 December 1919)
7
Piezoelectricity
Materials exhibiting piezoelectricity:
Naturally occuring crystals / materials:
Berlinite (AlPO4), sucrose, quartz, rochelle salt, topaz, tourmaline
Bone, apatite, collagen, silk, dentin
Man-made crsytals / ceramics:
Gallium orthophosphate (GaPO4), langasite (La3Ga5SiO14)
Barium titanate (BaTiO3), PbTiO3, potassium niobate (KNbO3), LiNbO3 etc.
Polyvinylidene fluoride (PVDF)!
8
Piezoelectricity
Necessary condition for the piezoelectric effect: absence of a center of
symmetry in the crystal structure.
7 crystal systems – 32 crystal classes
T Th O Td Oh
Cubic
C4 S4 C4h D4 C4v D2d D4h
Tetragonal
D2 C2v D2h
Orthorhombic
C2 Cs C2h
Monoclinic
C1 Ci
Triclinic
C3 S6 D3 C3v D3d
Trigonal
C6 C3h C6h D6 C6v D3h D6h
Hexagonal
Piezoelectric (20): C1, C2, Cs, D2, C2v, C4, S4, D4, C4v, D2d, C3, C3h, D3, C3v, C6,
D6, D3h, C6v, T, Td
The presence of the effect is dependent upon the crystal class and not upon
the material, although the degree of the effect is material specific.
9
Piezoelectricity – The Thermodynamic Potentials
Green* in 1837 introduced the „strain-energy function” in the treatment of
problems in elasticity: the function, when applied to a reversible system is
called free energy.
Expressed in terms of strains – first thermodynamic potential (ξ)
Expressed in terms of stresses – second thermodynamic potential (ζ)
1. These potentials can be expanded in powers and products of the
components of strain or stress, thus becoming the sum of homogeneous
functions of various degrees.
2. Since for an unstrained body the potential energy is a true minimum, the
first-degree term vanishes.
3. Insofar the strains are small, as is usually the case, only quadratic terms
need be retained (Hook’s law).
*George Green (14 July 1793 – 31 May 1841)
yields
10
Piezoelectricity – The Thermodynamic Potentials
𝜉
= −
1
2
6
+ 𝜗
𝑒𝑙𝑎𝑠𝑡𝑖𝑐
6 6
𝑐ℎ𝑖 𝑥ℎ 𝑥𝑖 +
ℎ
𝑖
𝜁
=
1
2
− 𝜗
′′
𝜂𝑘𝑚
𝐸𝑘 𝐸𝑚 +
𝑘
𝑚
𝑒𝑚ℎ 𝐸𝑚 𝑥ℎ +
𝑚
ℎ
𝑡ℎ𝑒𝑟𝑚𝑎𝑙
1 𝐶𝜗 2
2 𝑇
𝑚
𝑝𝑦𝑟𝑜𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐
𝑠ℎ𝑖 𝑋ℎ 𝑋𝑖 +
𝑖
3
𝑎ℎ 𝑋ℎ + 𝜗
ℎ
𝑡ℎ𝑒𝑟𝑚𝑜𝑒𝑙𝑎𝑠𝑡𝑖𝑐
𝑝𝑖𝑒𝑧𝑜𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐
3 6
𝑝𝑚 𝐸𝑚
𝑒𝑙𝑎𝑠𝑡𝑖𝑐
6 6
ℎ
6
1
2
3
𝑞ℎ 𝑥ℎ + 𝜗
ℎ
𝑡ℎ𝑒𝑟𝑚𝑜𝑒𝑙𝑎𝑠𝑡𝑖𝑐
𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐
3 3
1
2
𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐
3 3
𝑝𝑖𝑒𝑧𝑜𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐
3 6
′
𝜂𝑘𝑚
𝐸𝑘 𝐸𝑚 +
𝑘
𝑚
𝑑𝑚ℎ 𝐸𝑚 𝑋ℎ +
𝑚
ℎ
𝑡ℎ𝑒𝑟𝑚𝑎𝑙
1 𝐶𝜗 2
2 𝑇
𝑝𝑚 𝐸𝑚
𝑚
𝑝𝑦𝑟𝑜𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐
11
Piezoelectricity – The Thermodynamic Potentials
𝑥ℎ , 𝑥𝑖 : components of the total strain due to all causes
𝑋ℎ , 𝑋𝑖 : components of externally applied mechanical stress
𝐸𝑘 , 𝐸𝑚 : components of field stength in the crystal
𝑐ℎ𝑖 : elastic constants (more appropriate: stiffness coefficients) (𝑐ℎ𝑖 = 𝑐𝑖ℎ )
𝑠ℎ𝑖 : compliance coefficients (m. a.: elastic susceptibility) (𝑠ℎ𝑖 = 𝑠𝑖ℎ )
′′
′′
′′
𝜂𝑘𝑚
: susceptibility at constant mechanical strain (𝜂𝑘𝑚
= 𝜂𝑚𝑘
)
′
′
′
𝜂𝑘𝑚
: susceptibility at constant mechanical stress (𝜂𝑘𝑚
= 𝜂𝑚𝑘
)
𝑒𝑚ℎ : piezoelectric stress coefficients (𝑒𝑚ℎ ≠ 𝑒ℎ𝑚 )
𝑑𝑚ℎ : piezoelectric strain coefficients (𝑑𝑚ℎ ≠ 𝑑ℎ𝑚 )
𝐶: specific heat capacity
𝜗 = ∆𝑇: temperature differing from some standard temperature
𝑞ℎ : coefficients of thermal stress
𝑎ℎ : coefficinets of expansion
𝑝𝑚 : pyroelectric constant
12
Piezoelectricity – The Thermodynamic Potentials
21 elastic terms, 6 dielectric, 18 piezoelectric, 6 thermal coefficients and 3
pyroelectric
In the elementary theory of elasticity the three elastic constants of an isotropic
solid are Young’s modulus (Y), the rigidity or shear modulus (n) and the bulk
modulus or voulme elasticity (κ).
Stresses and Their Components:
A stress is defined as the force per unit area exerted by the portion of the body
on one side of a surface element within it upon the portion on the other side.
(tensorial nature of stress)
In general such a force can be resolved into a normal component, which is a
simple pressure (positive or negative) and a tangential component, which is
one of the pair of forces producing a shearing stress.
13
Piezoelectricity – The Thermodynamic Potentials
Strains and Their Components:
𝜕𝜁
=
𝜕𝑋ℎ
𝑥𝑥 = 𝑠11 𝑋𝑥 + 𝑠12 𝑌𝑦 + 𝑠13 𝑍𝑧 + 𝑠14 𝑌𝑧 + 𝑠15 𝑍𝑥 + 𝑠16 𝑋𝑦
𝑦𝑦 = 𝑠21 𝑋𝑥 + 𝑠22 𝑌𝑦 + 𝑠23 𝑍𝑧 + 𝑠24 𝑌𝑧 + 𝑠25 𝑍𝑥 + 𝑠26 𝑋𝑦
𝑧𝑧 = 𝑠31 𝑋𝑥 + 𝑠32 𝑌𝑦 + 𝑠33 𝑍𝑧 + 𝑠34 𝑌𝑧 + 𝑠35 𝑍𝑥 + 𝑠36 𝑋𝑦
𝑦𝑧 = 𝑠41 𝑋𝑥 + 𝑠42 𝑌𝑦 + 𝑠43 𝑍𝑧 + 𝑠44 𝑌𝑧 + 𝑠45 𝑍𝑥 + 𝑠46 𝑋𝑦
𝑧𝑥 = 𝑠51 𝑋𝑥 + 𝑠52 𝑌𝑦 + 𝑠53 𝑍𝑧 + 𝑠54 𝑌𝑧 + 𝑠55 𝑍𝑥 + 𝑠56 𝑋𝑦
𝑥𝑦 = 𝑠61 𝑋𝑥 + 𝑠62 𝑌𝑦 + 𝑠63 𝑍𝑧 + 𝑠64 𝑌𝑧 + 𝑠65 𝑍𝑥 + 𝑠66 𝑋𝑦
𝑋𝑥 = 𝑐11 𝑥𝑥 + 𝑐12 𝑦𝑦 + 𝑐13 𝑧𝑧 + 𝑐14 𝑦𝑧 + 𝑐15 𝑧𝑥 + 𝑐16 𝑥𝑦
𝑌𝑦 = 𝑐21 𝑥𝑥 + 𝑐22 𝑦𝑦 + 𝑐23 𝑧𝑧 + 𝑐24 𝑦𝑧 + 𝑐25 𝑧𝑥 + 𝑐26 𝑥𝑦
𝑍𝑧 = 𝑐31 𝑥𝑥 + 𝑐32 𝑦𝑦 + 𝑐33 𝑧𝑧 + 𝑐34 𝑦𝑧 + 𝑐35 𝑧𝑥 + 𝑐36 𝑥𝑦
𝑌𝑧 = 𝑐41 𝑥𝑥 + 𝑐42 𝑦𝑦 + 𝑐43 𝑧𝑧 + 𝑐44 𝑦𝑧 + 𝑐45 𝑧𝑥 + 𝑐46 𝑥𝑦
𝑍𝑥 = 𝑐51 𝑥𝑥 + 𝑐52 𝑦𝑦 + 𝑐53 𝑧𝑧 + 𝑐54 𝑦𝑧 + 𝑐55 𝑧𝑥 + 𝑐56 𝑥𝑦
𝑋𝑦 = 𝑐61 𝑥𝑥 + 𝑐62 𝑦𝑦 + 𝑐63 𝑧𝑧 + 𝑐64 𝑦𝑧 + 𝑐65 𝑧𝑥 + 𝑐66 𝑥𝑦
These are the generalized form of Hook’s law.
6
𝑠ℎ𝑖 𝑋𝑖 = 𝑥ℎ
𝑖
6
𝑐𝑖ℎ 𝑠𝑖ℎ = 1
ℎ
6
𝑐𝑖ℎ 𝑠𝑘ℎ = 0 (𝑖 ≠ 𝑘)
ℎ
𝑖, ℎ, 𝑘 = 1, 2, … , 6
14
Piezoelectricity – Fundamental Equations
A piezoelectric crystal is placed to an electric field E, and at the same
time subjected to a mechanical stress X. The total change in its
intermal energy maybe exoressed as:
𝑑𝑈 = 𝑃𝑑𝐸 − 𝑥𝑑𝑋
Assuming the process to be reversible, we can write:
𝜕𝑃
𝜕𝑋
𝐸
𝜕𝑥
=
𝜕𝐸
≔𝛿
𝑥
(Lippmann’s theory. He predicted the converse effect.)
15
Piezoelectricity – Fundamental Equations
In terms of strains (at constant T):
𝜕𝜉
=
𝜕𝑥ℎ
𝜕𝜉
=
𝜕𝐸𝑚
6
3
𝐸
𝑐ℎ𝑖
𝑥𝑖 −
𝑖
3
𝑒𝑚ℎ 𝐸𝑚 = 𝑋ℎ
(converse effect)
𝑚
6
′′
𝜂𝑘𝑚
𝐸𝑘 +
𝑘
𝑒𝑚ℎ 𝑥ℎ = 𝑃𝑚
(direct effect)
ℎ
In terms of external stresses:
𝜕𝜁
=
𝜕𝑋ℎ
𝜕𝜁
=
𝜕𝐸𝑚
6
3
𝐸
𝑠ℎ𝑖
𝑋𝑖 +
𝑖
3
𝑑𝑚ℎ 𝐸𝑚 = 𝑥ℎ
𝑚
6
′
𝜂𝑘𝑚
𝐸𝑘 +
𝑘
(converse effect)
𝑑𝑚ℎ 𝑋ℎ = 𝑃𝑚
(direct effect)
ℎ
16
Piezoelectricity – Fundamental Equations
Simple example (T = 0, E = E, single stress, single strain):
1
1
𝜉 = 𝑐 𝐸 𝑥 2 + 𝜂′′ 𝐸 2 + 𝑒𝐸𝑥
2
2
1 𝐸 2 1 ′ 2
𝜁 = 𝑠 𝑋 + 𝜂 𝐸 + 𝑑𝐸𝑋
2
2
The piezoelectric strain and stress coefficients are related:
6
𝐸
𝑑𝑚𝑖 𝑐𝑖ℎ
𝑒𝑚ℎ =
𝑖
6
𝐸
𝑒𝑚𝑖 𝑠𝑖ℎ
𝑑𝑚ℎ =
𝑖
17
Electrical Equivalent Description
Butterworth-van Dyke model of Quartz Crystal Resonator
Quartz
crystal
Au-film (excitation electrode)
Au-film (excitation electrode)
~
Rm (resistor) corresponds to the dissipation of the oscillation energy from mounting structures and
from the medium in contact with the crystal (i.e. losses induced by a viscous solution). Rm, can also
provide important information about a process since soft films and viscous liquids will increase
motional losses and increase the value of Rm
Cm (capacitor) corresponds to the stored energy in the oscillation and is related to
the elasticity of the quartz and the surrounding medium
Lm (inductor) corresponds to the inertial component of the oscillation, which is related to the mass
displaced during the vibration. Lm, is increased when mass is added to the crystal electrode
Co, represents the sum of the static capacitances of the crystal’s electrodes, holder, and connector
capacitance
18
Electrical Equivalent Description
The impedance of the network can be written as:
𝑍 𝑠 = 𝑠𝐿𝑚 +
1
1
+ 𝑅𝑚 ||
𝑠𝐶𝑚
𝑠𝐶0
where 𝑠 = 𝑖𝜔, or
𝑍 𝑠 =
𝑅
𝑠 2 + 𝑠 𝐿 𝑚 + 𝜔𝑠2
𝑚
𝑠𝐶0
where 𝜔𝑠 =
𝑅
𝑠 2 + 𝑠 𝐿 𝑚 + 𝜔𝑝2
𝑚
1
𝐿𝑚 𝐶𝑚
and 𝜔𝑝 =
1
𝐿𝑚 𝐶𝑚
1
+𝐿 𝐶
𝑚 0
+
2
𝑅𝑚
𝐿2𝑚
= 𝜔𝑠 1 +
𝐶𝑚
𝐶0
≈ 𝜔𝑠 1 +
19
Electrical Equivalent Description
(The resonant frequency of the circuit is defined as the frequency at which the
admittance has zero imaginary part.)
Quartz crystal resonators applied for microweighing purposes typically operate
in series resonance mode.
An increase or decrease in the mass of a resonator is equivalent to a
corresponding change in the value of L.
Impedance
The amount of energy lost during
oscillation at the resonant frequency is
at a minimum.
The resonance
frequency
decreases if
something is
deposited at
the surface of
the crystal
Picture from http://www.gamry.com/App_Notes/Basics_of_QCM.pdf
20
Electrical Equivalent Description
Influence of the parameter values
21
Electrical Equivalent Description
Common impedance responses of an EQCM during metal or polymer film
deposition
Deposition of a metal significantly
changes the resonance frequency
Deposition of a viscous polymer influences both the
resonance frequency and resonance resistance
Pictures from http://www.gamry.com/App_Notes/Basics_of_QCM.pdf
Many of commercial EQCM devices can detect the
resonance frequency and resistance automatically
22
Electrical Equivalent Description
H.L. Bandey et al. J. Electroanal. Chem. 410 (1996) 219
23
Frequency – Mass Relationships
What are possible contributions to the measured values of Δf?
∆𝑓 = ∆𝑓𝑚 + ∆𝑓η + ∆𝑓𝑝 + ∆𝑓𝑅 + ∆𝑓𝑠𝑙 + ∆𝑓𝑇
Effects of:
∆𝑓𝑚 - mass loading,
∆𝑓η - viscosity and density of the medium in contact with the vibrating crystal,
∆𝑓𝑝 - the hydrostatic pressure,
∆𝑓𝑅 - the surface roughness,
∆𝑓𝑠𝑙 – “slippage” effect etc.,
Electroanalytical Chemistry ed. by Allen J. Bard and I. Rubinstein,
Volume 17, 2003 (V. Tsionsky)
∆𝑓𝑇 - the temperature,
The different contributions can be interdependent. Assuming that a sufficiently
accurate frequency measurement technique is applied and the effect of
extraneous factors are insignificant, then the absolute accuracy of the QCM is
predicted by the accuracy of the formula used to convert the frequency
measurement to mass change.
24
Frequency – Mass Relationships
A linear frequency-to-Mass equation for small mass loads
Sauerbrey equation describes the relationship between the resonant
frequency shift (Δf) and the added mass (Δm):
∆𝑓 = ∆𝑓𝑚 + ∆𝑓η + ∆𝑓𝑝 + ∆𝑓𝑅 + ∆𝑓𝑠𝑙 + ∆𝑓𝑇
∆𝑓 = 𝑓𝑐 − 𝑓0 = −
2𝑓02 ∆𝑚
𝐴 𝜌𝑞 𝜇𝑞
1
2
= −𝐶𝑓 ∆𝑚
fc is the measured resonant frequency,
f0 is the resonant frequency of the unloaded crystal
A is the acoustically active surface area,
ρq = 2.648 g cm-3 and μq = 2.947 × 1010 N m−2 are the density and the shear modulus of quartz,
respectively,
m is the change of the surface mass density,
Cf is the integral mass sensitivity (56.6 Hz μg-1 cm2 for a 5 MHz AT-cut quartz crystal at room
temperature)
25
Frequency – Mass Relationships
A linear frequency-to-Mass equation for small mass loads
Sauerbrey equation describes the relationship between the resonant
frequency shift (Δf) and the added mass (Δm):
∆𝑓 = ∆𝑓𝑚 + ∆𝑓η + ∆𝑓𝑝 + ∆𝑓𝑅 + ∆𝑓𝑠𝑙 + ∆𝑓𝑇
∆𝑓 = 𝑓𝑐 − 𝑓0 = −
2𝑓02 ∆𝑚
𝐴 𝜌𝑞 𝜇𝑞
1
2
= −𝐶𝑓 ∆𝑚
Valid for small mass changes (Δm<10% of the total mass of the quartz).
Valid for purely elastic material as quartz or equivalent.
Valid if the film is rigidly attached to the electrode; for non-rigid films, calculated
mass changes are lower than the "true" values.
26
Frequency – Mass Relationships
A general formula for mass determination
x
tf
film
µf, ρf
quartz
crystal
µq, ρq
0
u
-tq
One-dimensional composite resonator model
𝑢𝑞 𝑥, 𝑡 = 𝐴 exp 𝑖𝜔
𝑢𝑓 𝑥, 𝑡 = 𝐶 exp 𝑖𝜔
𝜌𝑞
𝜌𝑓
1
𝜇𝑞
𝜇𝑓
1
2
2
𝑥 + 𝐵 exp −𝑖𝜔
𝑥 + 𝐷 exp −𝑖𝜔
𝜌𝑞
𝜌𝑓
1
𝜇𝑞
𝜇𝑓
1
2
2
𝑥
𝑥
27
Frequency – Mass Relationships
The continuity of particle displacement and stress at the film/quartz interface
requires the following conditions to be satisfied:
𝑢𝑞 0, 𝑡 = 𝑢𝑓 0, 𝑡
𝜇𝑞
𝑑𝑢𝑓
𝑑𝑢𝑞
= 𝜇𝑓
𝑑𝑥
𝑑𝑥
(at 𝑥 = 0)
Since the two free surfaces are antinodes, we also have
𝑑𝑢𝑞
= 0 (at 𝑥 = −𝑡𝑞 )
𝑑𝑥
𝑑𝑢𝑓
= 0 (at 𝑥 = 𝑡𝑓 )
𝑑𝑥
28
Frequency – Mass Relationships
Substituting the wave equations into the boundary conditions:
𝜇𝑓 𝜌𝑓
𝐴−𝐵−
𝜇𝑞 𝜌𝑞
1/2
𝜇𝑓 𝜌𝑓
𝐶+
𝜇𝑞 𝜌𝑞
1/2
𝐷=0
𝐴+𝐵−𝐶−𝐷 =0
𝐴 − 𝐵 exp 2𝑖𝜔 𝜌𝑞 /𝜇𝑞
𝐶 − 𝐷 exp −2𝑖𝜔 𝜌𝑓 /𝜇𝑓
1/2
𝑡𝑞 = 0
1/2
𝑡𝑓 = 0
These equations have nonzero solutions for the coefficients only if
1
1
1
1
1
0

 ( f  f /  q  q )
 exp 2 i    q /  q 
1/ 2
0
tq

1/ 2
( f  f /  q  q )
1
1
0
0
1

1/ 2
 exp  2 i    f /  f
0
1 / 2 t f 
29
Frequency – Mass Relationships
exp 2𝑖𝜔 𝜌𝑞 /𝜇𝑞
exp 2𝑖𝜔 𝜌𝑞 /𝜇𝑞
1/2
1/2
𝑡𝑞 − 1
𝑡𝑞 + 1
=
𝜌𝑓 𝜇𝑓
𝜌𝑞 𝜇𝑞
1/2
exp −2𝑖𝜔 𝜌𝑓 /𝜇𝑓
exp −2𝑖𝜔 𝜌𝑓 /𝜇𝑓
1/2
1/2
𝑡𝑓 − 1
𝑡𝑓 + 1
In terms of trigonometric functions:
tan 𝜔
𝜌𝑞 /𝜇𝑞
1/2
𝑡𝑞
1/2
𝜌𝑓 𝜇𝑓
=−
𝜌𝑞 𝜇𝑞
1/2
tan 𝜔
𝜌𝑓 /𝜇𝑓
1/2
𝑡𝑓
1/2
Substituting 𝜔 = 2𝜋𝑓𝑐 , 𝜇𝑞 /𝜌𝑞
= 𝑣𝑞 , 𝜇𝑓 /𝜌𝑓
= 𝑣𝑓 and introducing a
convenient parameter to characterize the film-quartz combination, the acoustic
impedance ratio: 𝑍 = 𝑍𝑞 /𝑍𝑓 , where 𝑍𝑞 = 𝜌𝑞 𝜇𝑞
obtains
1/2
, and 𝑍𝑓 = 𝜌𝑓 𝜇𝑓
1/2
, one
tan 𝜋𝑓𝑐 /𝑓𝑞 = −1/𝑍 tan 𝜋𝑓𝑐 /𝑓𝑓
30
Frequency – Mass Relationships
v
Using the definition of 𝑓𝑓 (𝑓𝑓 = 2tf ) and the trigonometric identity tan 𝜋 𝑓𝑞 − 𝑓𝑐 /
f
31
Possible Failures of EQCM
32
Applications of EQCM
The quartz crystal microbalance is a piezoelectric sensing device that consists
of an oscillator circuit and a crystal, which is incorporated into the feedback
loop of the circuit.
QCM is normally a stand-alone instrument with a built-in frequency counter and
resistance meter.
Crystal oscillator
(controller)
Series resonance frequency and resistance are measured and displayed, and
there is an analog output proportional to frequency which can
be used to interface e.g. with a potentiostat
33
Applications of EQCM
Electrosorption
Underpotential deposition of metals
Adsorption / Desorption of Surfactant Molecules
Multilaye deposition / dissolution
Polymer films / conducting polymers
Integration with other techniques:
EQCM +
EQCM +
FTIR
ellipsometry
PBD
SPM
cyclic voltammetry
chronoamperometry
chronopotentiometry
electrochemical impedance
spectroscopy
34
Compendium
The quartz crystal microbalance is a piezoelectric sensing device that consists
of an oscillator circuit and a crystal, which is incorporated into the feedback
loop of the circuit. The frequency response is sensitive to the nature of the
contacting environment.
Sensitivity of the EQCM is enough to detect sub-monolayer amounts of
adsorbates at the electrode surfaces.
Selectivity is an obvious drawback of EQCM.
Combination of the EQCM with other electrochemical techniques is particularly
powerful to elucidate mechanisms of the reactions, as well as to monitor and
control many electrochemical processes.
Interpretation of EQCM data, however, requires careful account for possible
small changes of the electrolyte temperature, viscosity of the electrolyte close
to the interface and other effects.
35
References
1. Electroanalytical Chemistry ed. by A.J. Bard, Volume 17, Marcel Dekker,
INC., New York, Basel, Hong Kong (D.A. Buttry)
2. Piezoelectricity, Walter Guyton Cady, Volume 1, Dover Publications INC.,
New York
3. Electroanalytical Chemistry ed. by A.J. Bard and I. Rubinstein, Volume 17,
Marcel Dekker, INC., New York, Basel, Hong Kong (V. Tsionsky)
4. Methods and Phenomena ed. by C. Lu, Volume 7, Elsevier, Amsterdam,
Oxford, New York, Tokyo (A.W. Czanderna)
5. Hillman, R.: The EQCM: electrogravimetry with a light touch. J. Solid State
Electrochem. 15 (2011) 1647-1660
36
Electrical Equivalent Description
An impedance spectrum of real quartz crystals with two Au-electrodes
High frequency
oscillator
Quartz
crystal
Au-film (excitation electrode)
Au-film (excitation electrode)
~
37