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24
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
the particle is much smaller than the wavelength of the incident
light lo (in practice, smaller than lo
the incident light
EE0 i pnt −j)]
(1.26)
= exp[ (2
is homogeneous within the particle; it induces in it a dipole
maE
n and so does
=
=
0
′
E
(1.27)
maE =4pe a
here a′ is the polarizability volume [46] of the particle and
e
Figure 1.8 Three-dimensional coordinates where origin O represents a
scaer and P is the observaon point at the xOy plane. The vercally polarized
incident beam causes polarizaon in the scaer, which radiates in dierent
direcons.
According to Maxwell's electromagnetic equations, the electric
p with distance
r from origin o and angle of q from the incident light is
2
/
2
2 2
2
dp dt
d ( m
0
+
m)/dt
4pa
′
E
= = =−
E
(1.28)
s
2 2
2
4pe rc 4pe rc r l
0 0
0
where c is the velocity of light in vacuum and a′ is the polarizability
volume. p is the total dipole of the particle, i.e., the summation
The Powerful Tool for Study of Nano Physical Pharmaceutics
25
of the permanent and the induced dipole . Since at
2
d
t
i (kJ
m
0
electro-
in
Equation
–2
m
«
–1
m
) of
room temperature
1.28. Thus,
the
m
0
and
m
0
time-average
2
pd /
d
scattered
intensity
t2= d
m
frequency
2
m/
the particle at point p is
i = e
where
the
scattered o intensity
weight
a
is
proportional
i
/I
o
a sh
orter wavelength.
2
c E
o
s
is
the
I
since for particles made from a given isotropic material,
−4
means that the scattering is much stronger for
l
0
the ∝scattering
4 2
16
pa′ pa′
=
42
r
l
0
intensity
of
is
to the other hand,
their molecular weight. On
This
is stronger
2
eo cE
(
)
primary light.
proportional
explains
toward
4 2
16
=
42
r
l
0
I
Equation o1.29 shows that
to the square of
why
the sky is blue because
the
short-wavelength
(1.29)
molecular
light of
end of
the visible spectrum.
1.4.1.2
When
the
simple summation of the scattered intensity of
Scat
there
total
tering by many small-part
are
N-independen
scattered
intensity per
t same
small
unit
icle systems
particles in volume
scattering
volume is the
N V
/ particles:
V
I
,
4
2
′
Ii= NV
(/)
R Ir
pa
16
=
l
Vr
2
/
Io (known as the Rayleigh ratio named after the
N
42
0
I
o
“father” of the theory of light scattering), we have
4′2
pa
R
where 16(dimension
N
=
R
4
l
V
0
of L) is dependent on the concentration,
the size and the nature of the particles.
1.4.1.3
For a
One
small, dimension > /20, that light scattered
Scat
re
al system,
is
the
tering by real p
two kinds
intraparticle
l
olymer solut
of interference
interference
ion
must be considered.
(i.e., the
from
particle
is not so
two scattering
(1.31)
26
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
is
convenient is composed
each
with equivalent
each scattering
p can 1.28. Thus, the scattered
still
to
assume
unit, its
be
expressed
N
volume
units:
that it
and
scattered
polarizability
intensity
of
N
scattering
volume
at
observation
by Equation
a
o
V
units
′
.
For
point
= −
4p
l r
2
N′
′
a
E
exp[ ( )]
∑
0
2
0
l=1
N
′
E
=
E
∑
s s l o
,
1
l=
Further, the time-average scattered
becomes
i = e =
where
and at point p.
l m
Df
s
o
=
fl– f
lm
2
c E
4 ′ 2
16pa
42
l r
0
m
N′ N
o
I
∑
∑
o
l=
m=
1
Df
=⋅qr
=
r
−
m l
lm
r
q
=
r
i r f
−
lm
where
beam
r
lm
( ) and the one along the
r
f
vector module
qq=
4p
=
l
q
n
sin
2
o
i 2pnt
′
1
exp[
i( qr ⋅
scattered
− f
l
intensity
)
m
l
beam
(
at point p
(1.33)
(1.34)
),
and the
r
f
where
q has
, n q increases and q
with which
static LLS is
possible orientations (
refractiv
–1
index
of
e
been
the medium.
used
as a spatial
With
an increasing
resolution
able to probe the size of colloidal particles
r
lm
ruler
The Powerful Tool for Study of Nano Physical Pharmaceutics
27
4
p
16
()
q
i = I a
where
2
4
l
r
0
q is the inclined
beam to that of line OP, as is shown in Fig. 1.8. When
(p i q
Note
16
→0) =
r
N′ao ′ ′
= a so that Equation 1.37 is the same
R q
()
q =
R()()0 =N1 ′
After developing sin(
2
′
∑∑
oo
l=1 m=
4
I
2 4
l
0
2
′ N ′
N
′
a
oo
N′
∑
l=1
sin(qr
1
angle
)
lm
qr
lm
from the direction of
the primary
q →
2 2
N′
PP(q) as
N ′
∑
m=1
qr
sin(qr
lm
)
lm
qr
lm
) into a Taylor series (sin x = x – x3/
0,
as Equation
6 +…) and retaining the two leading terms, the result reads
2
N′ N ′
()q 1
P =−
where
1
2
N
of gyration
2
′
R
6
′
N
N
∑
∑
l 1
= m=1
2
. Thus
g
q
2
′
N
′
2
r
lm
∑∑
l=1 m=
2
+
r
⋅⋅⋅
lm
1
P()
=−11(/3 ) q
which
is q related
that is
form
volume
why it
factor
(
q 〉
, Equation 1.36 can be rewritten as
V
p N
16
()
=
V
It R is q interesting
22
Rg + ⋅⋅⋅
to
the conformation of the larger particles and
also
2
g
a′ Pl() q
called
> 1).
For
is
〈
4
4
0
R
to note that when the above discussion is
scattered intensity is zero in
the
structure factor
N-independent
all
directions
q 〈R 〉
larger (particles gin
except
in the
direction
28
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
of the incident beam. This is because for any selected small
of
the
former Then
will
exactly
paired
be up
by
cancel
in this
interference.
each
other.
way.
All the volume
However,
exist, even for pure gas and liquid of small
because the
the time
by individual
thus
they
the
many
l
will
also
a
′ = a
l
o
scattering of pure gas
we
divide scattering
substitute
a
′
o
the
interference of all units so that only the
molecules
and
the
elements
will
not be canceled
properties,
more
or
′ + da
′, and
o
a ′ a ′ da ′
in Equation
o
are
undergoing
properties
of individual
will not
have
totally
polarization
less
deviate its
it
is
the
existence
and
liquid.
volume
1.32
nothing to
volume
Following
V
into
with
scattered
the scattered waves
elements
light
scattering
molecules.
random
volume
movement
elements
identical
by
interference.
most aprobable
of da ′
o
amplitudes
′ of a scattering
l
value
leads to
the above
N
o
. We know
o
intensity
term da
because
o
be considered. Equation 1.32 then reads
can
does
It is
all
will
and
As one
of
unit
a
′ as:
o
the
light
treatment,
units and
the term
of
′
needs to
N
E
=
E
∑
s
s ,
l=
Similar 1 to
l
p
4
=
−
2
l
0
deriving
2
N
da′exp[(i 2pnt
E
∑
0
l=1
l
r
− f
)] (1.42)
l
Equation 1.33, taking time-averaged
we can rewrite Equation 1.42 as
4
N N
16
′
l
to
V
further
is
p
∑
∑
4
l
l=1
m=1
0
da
m
average
convenient
da
still random
R() q =
where and
da
We need
purpose, it
sum into terms for which l
4
16p
R =
()
q (
V
l
N
∑N∑
4
l= m == lm ≠ m=
0
1 1
da
′ ′
da iq r m−
l
to j separate
= and ∑.
2
′
)
l
exp[ ( )]
m
functions
Equation
l ≠ j
N N
+
∑
da
1
r
l
of time and space.
1.43
over time.
the terms
′
′
da
exp[iqrm − rl )]
l
m
in the
For
double
this
The Powerful Tool for Study of Nano Physical Pharmaceutics
29
and are independent of each
′
other
and 〈da〉 = 0. Therefore,
〈
da
l m
′
da
′
da
〉 = 〈
l dam m
da
′
l
〉 〈
′
〉 = 0 and
da
Equation 1.44 turns to be
4
N
N
R()q =
4
V l
0
small scattering unit,
(da′ )
l
2
as . Thus,
turns to be
4
16p
R =
where
we
macromolecule
concentration
C
and
r are independent of each other, we have
da′=
( )
2
N
4
l
V
o
have used
solution
C
∂′a
2
∂C
⋅ (da
(da
∑
∑
l = 1=m
(da
N N
∑
∑
l =1 m= 1
2
′
)
o
the relation = da
)
2
( )
C
d +
16p
′
)
l
(da
=
or a
2
2
′
)
l
2
2
′
)
l
p
16
4
V l
o
colloid
r.
da′ = (∂
∂r
2
′
(da
)
l
2
′
. Now
o
(
〈 da
a′/ +
C)
∂C
∂ (∂
2
()dr
a
2
〉
is
a′/a ∂
the
.
Consider
function
r)∂r. Since
4
(da )
⋅
da N
∂′a
2
dispersion,
2
(1.45)
(1.46)
remained
a
of
Thus, for a dilute solution, Equation 1.46 becomes
2
where
R =
solution
16
l
p
4
o
16p
V l
4
excess Rayleigh
solvent (
of
of
e
r
R
solvent
the solute
the solution.
– 1 = 4
pa
/V and er = n2, we have
by subtracting
4
∂′a
2
∂C
(∂C
()∂C
2
)
the solution
4
o
∂a′
∂C
ratio of
) respectively.
According
2
R is
excess
the
to Clausius–Mossotti
4
4
o
16p
l
∂′
∂r
4
4
o
(
R
the
a
∂a
∂
excess
net
+
16p
V l
intensity of the solvent
2
(∂rr
2
′
( )
∂r
r
) and that
scattering
2
)
2
equation
of the
intensity
from
that
[46],
30
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
(1.49)
2
∂′
a
∂
C
=
∂′
a
∂
e
r
∂e
r
∂n
∂n
∂C
=
2
nd n
p
dc
On the
and , where and are
partial
change
other hand, we know
2
∂
A
2
∂C
volume
1
= −
VVC
,
T
m
∂C
and
in concentration
from
thermodynamics
∂m
T ,V
chemical
V
m
m
potential of the
can cause a change in osmotic
(dC
solvent.
namely
∂m
∂C
In dilute macromolecular solution,
A
2
∂p
∂
C
TV,
Now
of
R
excess
, we have
∂m
=
,
∂p
,
TV
RT
= + AC +
(12
M
substitute ( ) and Equation 1.49 into
∂p
TV∂C
M
2
TV
∂m
=−
∂P
,
)
∂p
∂CC
,
TV
p RT
C M
dC =
( )
,
TV
= (1 + A CM +
2
NA V(1
=−V
〈 dC2〉 the expression
that
kBT
2
)
=
(∂2 A
/∂C
respectively
And the
pressure,
p
∂
m
∂C
TV,
where
2
)
CM
+2A2 CM +)
the
2
)
the
T ,V
dC
2
(12
CM
+ AC
2
M
+)
K
2n2
. For larger
excess
(dn/dC
22
p n dn
excess
4
=
4
l N
o
+2A2 C +
A
(
N
l
A
R
4
), we get
o
KC =1
R M
where we have omitted the footnote “excess” in R
macromolecules, a construction factor must be introduced, thus
2
/
)
The Powerful Tool for Study of Nano Physical Pharmaceutics
31
(1.52)
It shows that with ( ) measured at a series of and , we are
K(C
R q
The
is the
P
know (
concentration
(
q) = KCMP
Considering
1
=
)
M
last
+ AC
(
P q )
question
2
polydispersity
q) = 1 q
– (1/3)
C →
0,
(q) =
the additive
2
deriving
in the basic equation of static LLS
in 2real
2
cases.
R
⋅⋅⋅
. Thus,
g
KCM[1 – (1/3)q
nature of →excess Rayleigh ratio,
for a polydispersed polymer solution at
() q =
R q
()
C =
∑
i
If R we
divide
, we get, after a slight rearrangement,
C
∑
i
i
∑
R() q
=
KC
= ∑ KC
i
Equation
CiM
i
i
C
∑
i
i
M
[ 11
− (
i
i
1.54
by
2
q∑CiMR
−
1
i gi
i
CiM
3
∑
i
i
/) 3
the
2
,
+⋅⋅⋅
From
Equation 1.40 we
2
R
in the
2
+
g
limit of vanishing
⋅⋅⋅] (1.53)
the
C
2
2
+⋅⋅⋅]
q
R
gii ,
total
polymer
concentration
(1.54)
or
R() q
KC
= M
11−(/3 )
w
Now, come back to
2
2
q R + ⋅⋅⋅
g
z
Equation
When q
(1.56)
2
2
R
g
the higher order terms in series, we get
KC
1
= + qR
q M
()
R
This
is
the
equation, , is
M
square radius of gyration, , is z-average.
1
1
w
2 2
3
basic
equation of static LLS which
=
C
M
/
∑
w i
i
i
R
q C
∑
i
g
z
C
i
R
+ AC
2
2
weight-average;
2
= CMR
∑
g
z
i i
i i g
is frequently
and
the qmean
2
CM
/
∑
, i
i
i
32
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
able to determine from the slope of [ / ( )]
0
versus
; from the slope of [ / ( )] versus ; and from
2
R
〈
2
q
A
2
[
/
( )]
KC R q
C → K
→
with k being
extrapolations
to
It should be mentioned
restriction
structures,
that the polymer solution exhibits no adsorption,
the
Berry
adequately used
observed
structure
is not
ln[
KC/R(q vs q
in
the angular
is
expected
linearize
)]
d
〉
z
g
KC R q
. The Zimm plot, i.e., 0 / ( ) versus (
0
an
adjustable
be made on a
q →
constant,
single coordinate
that Equation 1.57 is valid
[48]
plot
([KC
/R(q)]
because
it
often
removes
dependence
to be
large and
but
still shows
2
+ kC that removes the upturn
an upturn.
KC R
C M
KC
R q
allows
both
plane
1/2
much
of 2the
of the
Zimm
globular,
the
In these cases, it is
In
practice,
method;
standard
namely,
such
the
Rayleigh
by
measuring
as benzene
ratio determined
is by a relative
the
or toluene,
we
scattering
can calculate
intensity
ratio of a given solution by
q
C →
w
2
q
q
[42, 47].
under
kC) is more
curvature
plot.
Berry plot
even
the Rayleigh
If
of a
C
C
the
the
Rv = R
v
vv
( q )
I q I q
( )
o
(
q)
where the subscript “
scattered
polarized;
light a intensity
is
constant
of
the light
the same from the
standard. If
scattered
to
have
a slit
we
have already
(vertical).
light
are
I and n are, respectively,
and
refraction
betwe
correction
en 1 and
scattering
scattering volume solution
we
take
light
as
a linear
is used
correction
to determine
seen
On the
− n
I q
)
standa” r
(
solutio(n
vv
vertically
the refractive
for the
2, depending
instrument,
the
incident light
the
y-dire
the
all
the scattered
other
hand,
)
solvent
nn
d
means
(z-axis
index.
scattering
because
(i.e.,
ction
of
the refraction
scattering
a
if
g
solvenrt
standa
d
both
the incident and the
direction in Fig. 1.8)
the
time-averaged
The
term (
solvent
volume nand
on the
detection
we
should
and the
the x-direction
as
q = 90°), we only
in the x-direction if
volume,
lights in
g = 1 because
i.e.,
the a z-direction
pinhole with
scattered
standard
)
/n
g
geometry
compare
reference
and
the
need
size much
g
The Powerful Tool for Study of Nano Physical Pharmaceutics
33
smaller than the diameter of the incident beam at the center of
the scattering cell, we have to correct the refraction in both the
x- and z-directions, i.e., g = 2. However, if the pinhole size is
comparable to the beam diameter, 1 < g < 2. In practice, we should
avoid this situation by choosing either a slit or a smaller pinhole
[41, 51].
1.4.2 Dynamic Laser Light Scaering (DLLS)
Motions (translational, rotational or internal motion) of
macromolecules or colloidal particles in solution can be
conveniently studied by using dynamic LLS. Measurement at a
dimension of macromolecules or colloidal particles in the solution
with reasonable accuracy. Unlike the static LLS version, dynamic
LLS does not rely on the excess scattering intensity between
the pure solvent and a dilute solution. The signal from the slowly
moving polymer is unambiguously separated from the signal
that originates from the rest of the solution. The principle of
dynamic LLS has been utilized in some commercial particlesizing systems for many years. The measurement and data analysis
are automated. Users only need to prepare clean solutions by
online detector in size exclusion chromatography (SEC) [51].
Nowadays, the most commonly used method in QELS is the digital
technique of photon correlation spectroscopy (or optical mixing)
time domain. Practically there are two basic forms of optical
mixing: heterodyne and homodyne (self-beating). By heterodyne
mixing we refer to the mix of the scattered light with a reference
beam (local oscillator) unshifted or shifted in frequency from the
incident light beam. In self-beating optical mixing, the scattered
wave is not mixed with a reference signal but directly detected.
Here we only consider the self-beating intensity-intensity time
correlation spectroscopy.
1.4.2.1 Power spectrum of scattered light
Now we consider again an N particle (macromolecule or colloidal
particles) scattering system with scattering volume V. We view