The effect of a surrounding fluid medium on the breathing mode vibration of an isotropic
elastic sphere are discussed. Both damping of the oscillations and an downward shift of the center
frequency result.
Consider an isotropic elastic sphere with radius R, density ρa
and elastic constants C11 and C44. The dynamical
equation for an isotropic elastic medium is:
C11 ∇(div u) -
C44 ∇ × ∇ × u = ρa d2u/dt2
where u(r,t) is the displacement field.
Here we are only concerned with the "breathing" mode of vibration, in which the
displacement is purely radial. Since curl u = 0, u is
necessarily the gradient of a scalar field Φ,
u = ∇ Φ
and
C11 ∇2 Φ = ρa d2 Φ /dt2
The desired (radial breathing mode) solution of this equation is
Φ = j(0,s) exp(- i ω t)
where s = kL r, kL = ω / CL, CL = sqrt(C11/ρa),
and j(0,s) is the spherical Bessel function of the first kind of order 0:
j(0,s) = sin(s) / s
Taking the gradient, u expressed in spherical coordinates
has uθ = uφ = 0 and
ur exp(iωt) = (cos(s)/s - sin(s)/s2) (ω/CL)
The boundary conditions for the solution will involve the stress at the surface of the
sphere, r = R. The xx component of the stress tensor, which equals the negative of the external
pressure P, is given by:
-P = σxx = λ ddiv u + 2 C44 dux/dx
where the first Lamé constant λ = C11 - 2 C44. First,
we find the divergence:
where now r = R, so s = ω R / CL.
At this point we consider the fluid medium surrounding the sphere. It exerts pressure P
on the surface of the sphere. The outward velocity of the surface of the sphere is vr
= d ur/ dt.
P / vr = zb
where zb is the specific acoustic impedance at the surface of the fluid. For a plane
interface, specific acoustic impedance z is density times the speed of sound. However, for a
spherical interface [Kinsler, Frey, Coppins, Sanders 1982 eq. 5.54], specific acoustic impedance
is a complex number:
zb = ρb Cb ( (kbR)2 - i (kbR) ) / (1+(kbR)2)
where
i = sqrt(-1)
ρb is the density of the surrounding fluid
Cb is the speed of sound in the surrounding fluid
kb = 2 π / λ = ω / Cb in the fluid
R is the radius of the sphere
The outward velocity of the surface of the sphere is:
It is convenient now to note that CL = sqrt(C11/ρa)
and CT = sqrt(C44/ρa), where ρa
is the density of the elastic sphere.
(sin(s)/s) +
4 (CT/CL)2 (
cos(s)/s2
- sin(s)/s3)
=
- i (zb/(ρaCCL)) (cos(s)/s - sin(s)/s2)
and the final result:
s2 tan(s)
=
[4 (CTa/CLa)2
+ i s (zb/(ρaCLa))] (tan(s) - s)
which is used together with the formula for zb:
zb = ρb Cb ( (kbR)2 - i (kbR) ) / (1+(kbR)2)
and the formula kbR = s CLa / Cb.
A short C++ program finds.cpp is used to find the complex root s.
As a numerical example, we consider an isotropic elastic sphere of diameter 55 Angstroms in which
CTa = 2640 m/s
CLa = 3830 m/s
ρa = 5680 kg/m3
It is surrounded by a fluid medium for which
Cb = 2350 m/s
ρb = 1110 kg/m3
Using the program, the resulting value of s is 2.104 - 0.204 i.
Next, s is multiplied by CLa/R to get complex angular frequency ω.
The imaginary part of ω is minus the reciprocal of the damping time.
The damping time τD = 3.5 ps.
To see how the presence of the fluid affects the vibrations, the equation is solved with
ρb=0, getting s = 2.167.
Thus, the vibration frequency is decreased by 3% by the added inertia of the surrounding fluid.
References:
The same thing was done earlier for a solid elastic (glass) matrix by:
A. M. Alcalde, G. E. Marques, G. Weber. T. L. Reinecke
"Electron acoustic phonon scattering rates in II-IV quantum dots:
contribution of the macroscopic deformation potential"
Solid State Communications 116 (2000) 247-252
pdf::
Another earlier reference is:
N. N. Ovsyuk, V. N. Novikov, Phys. Rev. B 53 (1996) 3113.
Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca