Tools to Probe Nanoscale Surface Phenomena in Cellulose Thin Films: Applications in the Area of Adsorption and Friction - The Nanoscience and Technology of Renewable Biomaterials - page 106

where E dissipated is the energy dissipated during one period of oscillation and E stored is the
energy stored in the oscillating system. The resonance frequency is measured when the
oscillator is on and the amplitude A of the oscillation is monitored when the oscillator
is turned off. A can be determined in its decay as an exponentially damped sinusoidal
function:

A(t) = A 0 e − t/τ sin (ωt + ϕ) + c

(4.7)

where τ is the decay time, ω is the angular frequency at resonance, φ is the phase angle
and the constant, c , is the offset.

The dissipation factor is related to the decay time

through Equation (4.8).

1
πf τ

D = (4.8)

Combining Equations (4.5) and (4.8) the dissipation changes can be expressed as
Equation (4.9). This equation shows that dissipation changes depend not only on the
properties of the adsorbed layer but also the density and viscosity of the solution
(Rodahl and Kasemo 1996a):

η f ρ f
2 πf

D = √ n

1
ρ q t q

(4.9)

Generally, soft adlayers dissipate more energy and thus are of higher dissipation value.
From this point of view, dissipation can be used as an indicator of the conformation of
the adlayer.
A typical QCM-D system records the signals of fundamental frequency (5 MHz) and
overtones (e.g. 15, 25 and 35 MHz and even high frequencies for newly developed
systems). Each overtone has its own detection range in thickness. Theoretical work by
Voinova and coworkers (Voinova, Rodahl et al . 1999) advanced a general equation to
describe the dynamics of two-layer viscoelastic polymer materials of arbitrary thickness
deposited on solid (quartz) surfaces in a fluid environment as follows:

2 h j η 3
δ 3

2

η j ω 2
µ j + ω 2 η j

1
πρ 0 h 0

η 3
δ 3 +

f ≈− h j ρ j ω − (4.10)

j = 1 , 2

2 h j η 3
δ 3

2

µ j ω
µ j + ω 2 η j

1
2 πfρ 0 h 0

η 3
δ 3 +

D ≈ (4.11)

j = 1 , 2

where ρ stands for density; h is the thickness; η is the viscosity and δ stands for
the viscous penetration depth ( δ =

2 η
ρω ) . The subscript 0, 1, 2 and 3 denote quartz
crystal layer 1, layer 2 and bulk solution respectively. From this model, the shift of
the quartz resonance frequency and dissipation factor strongly depend on the viscous
loading of the adsorbed layers and on the shear storage and loss moduli of the over-
layers. These results can readily be applied to quartz crystal acoustical measurements
of polymer viscoelasticity which conserve their shape under the shear deformations and
do not flow as well as layered structures such as protein films adsorbed from solution