Vibration-Based Anti-Biofouling of Implants 213
D
E


 G\QHFP
Figure 9.4. (a) Schematic diagram of rotating plate system. (b) Fluorescence intensity of
different rotating plate surfaces with different shear stresses applied on one BSA molecule.
Reproduced with the permission from Yeh et al. (2007).
where η(cPoise) is the viscosity of the medium, r(cm) is the radial distance from
the center of the disk, ω(rpm) is the radial velocity of the rotating disk (which can
be measured by laser doppler vibrometer (LDV)), and υ(Stokes) is the kinematic
viscosity of the medium. Properties of buffer solution for experiment are close to
those of water at 37
C, in which, η and υ are 0.89 (cPoise) and 0.008583 (Stokes),
respectively. The shear stress is a function of radius, so the average shear stress
overthewholeplateis:
59
0.8η(ω
3
/υ)
1/2
r2πrd r/πr
2
(9.12)
The wall shear stress on an individual BSA molecule can then be estimated by
knowing the surface area of BSA, which is simpliﬁed as a sphere with radius of
3.04 nm. The shear stress on one BSA molecule at different rotating speeds is listed
in Table 9.3; for example, for a rotating speed of 40.79 Hz, the average shear stress
is 0.267*0.8*8.9*10
3
*((40.79)
3
/8.583*10
3
)
0.5
= 5.35 dyne/cm
2
on the circular area
of plate from radius 0.1 to 0.38 cm. Now, consider the shear stress from acoustic
ﬂow induced by vibration on one BSA molecule. The shear stress is deﬁned as
τ = η∂u/z,whereη is the viscosity of the medium, u is the ﬂow velocity and
SO13997_text.indd 221SO13997_text.indd 221 26/01/2011 3:50 PM26/01/2011 3:50 PM
214 P. Y. J. Yeh, J. N. Kizhakkedathu and M. Chiao
y is the distance from the surface. The ﬂow velocity is related to the vibration
amplitude of the surface as the following:
60
u =
5
4
( fAmp)
2
v
p
κ
s
α
3
α
κ
s
+
κ
s
β

1
dα
2
2
(9.13)
where u is the maximum velocity of the acoustic stream; f is the frequency of
vibration; Amp is the z-directed component of membrane vibration amplitude;
υ
p
is the phase velocity of the plate wave; κ
s
is the propagation constant, equal to
2π/λ; λ is the wavelength of FPW; α = κs(1-(υ
P
/υ
F
)
2
)
1/2
; υ
F
is the sound velocity
in PBS; β
2
= ρf/2η, which relates to the viscosity effect of PBS; ρ is the density of
PBS; η is the dynamic viscosity of PBS; and d is the membrane thickness. From
Eq. (9.13), the ﬂow velocity is proportional to the square of the vibration ampli-
tude, which is linearly related to the driving voltage. This relationship between
vibration amplitude and ﬂow velocity has also been experimentally observed in
the literature.
60,61
The velocity of acoustic streaming is calculated from Eq. (9.13), which is
8.84*10
6
m/s (10 V
pp
) and is the maximum velocity parallel to the plate surface.
The ﬂow on the wall is assumed to be no-slip, hence the velocity at the wall is zero.
The velocity of ﬂow reaches a maximum at the evanescent decay length, which is
the reciprocal of α in Eq. (9.13). For our experimental setup, 1/α equals 3.26*10
3
m. Assuming that the dependence of velocity with distance from wall is linear,
the shear rate of acoustic streaming is 8.84*10
6
/3.26*10
3
= 2.71*10
3
1/s. The
shear stress, τ = η∂u/z,is8.9*10
4
(kg/m.s) * 2.71*10
3
(1/s) = 2.41*10
6
N/m
2
= 2.41*10
5
dyne/cm
2
where η is 8.9*10
4
(kg/m.s). Parameters for calculating
shear stress from acoustic streaming are also listed in Table 9.3.
The ﬂuorescence intensities for plates rotating at different speeds for 5 min
were compared with plates without rotation (Fig. 9.4(b)) and no signiﬁcant differ-
ence occurs between different speeds. Since shear stresses induced by a rotating
Table 9. 3 Shear stress applied on a BSA molecule. Reproduced with the permission
from Yeh et al. (2007)
Applied
Voltage (v
p
p)
Rotating
Speed (Hz)
Shear Stress
(dyne/cm
2
)
Rotating
Plate
0.7 1.49 3.73 × 10
2
1 6.66 3.53 × 10
1
3 40.79 5.35 × 10
0
Vibrating Applied Max Shear Rate Shear Stress
PZT Voltage (V
p
p) Vibration
Amplitude
(nm)
(1/s) (dyne/cm
2
)
10 500 2.71 × 10
3
2.41 × 10
5
SO13997_text.indd 222SO13997_text.indd 222 26/01/2011 3:50 PM26/01/2011 3:50 PM

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