Biomedical Engineering Reference
In-Depth Information
a function of concentration,
B
can be obtained from the slope and
M
w
from
the intercept.
2.2.2 Large particles
The above discussion is restricted to particles that are typically about 250 Å.
This is the size of many globular proteins. However, particles used in bioma-
terials research are often much larger. In such cases, the interference of light
from scattering centers within a molecule which are more or less fi xed rela-
tive to each other, needs to be considered. The effect of the large size can be
described by a function
P
(
θ
), defi ned as the ratio of the scattered intensity
at an angle
θ
to that which would be observed if there was no interference,
that is, Rayleigh scattering. The expression for
P
(
θ
) as derived by Debye is
sin
N
N
1
qr
∑
∑
ij
r
(
θ
=
[2.5]
P
2
N
qr
i
1
j
1
ij
r
=
=
where
N
is the number of scattering centers within the molecule, atoms for
instance,
r
ij
is the distance between any pair of centers
i
and
j
, and
Q
is the
scattering vector given by
in( /2
4
[2.6]
πθ
λ
q
=
To obtain the size of the particle from Equation [2.5], the term under the
summation sign is expanded as a power series, and only the fi rst two terms
are kept. Next, a term radius of gyration (
R
g
) in introduced to capture the
summation of
r
ij
within the particle using the relation
N
N
∑
=
=
2
r
∫
2
d
ij
r
rv
i
1
j
1
2
[2.7]
R
=
=
∫
2
g
2
N
d
v
By recalling that
P
(
θ
) is the ratio of the scattering from a real particle to that
of the same particle if it showed Rayleigh scattering (
R
θ
), one can combine
Equations [2.4] and [2.5] to obtain
⎡
16
3
22
R
π
2
λ
⎤
KC
R
⎡
⎢
⎤
⎥
θ
1
π
R
2
Bc
[2.8]
⎡
⎢
g
⎤
⎥
1
sin
2
=+
⎡
⎣
+
⎤
⎦
M
⎢
λ
2
⎥
⎣
⎣
⎦
⎦
θ
In the case of heterogeneous particles,
M
is replaced by
M
w
. To obtain the
molecular weight from Equation [2.8], it is necessary to extrapolate
KC
/
R
θ
graphed as a function of (sin
2
θ
/2 +
kc
), where
k
is an arbitrary constant to
