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34
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
N particles as N
scattered intensity at point p in Fig. 1.8 can still be expressed
using Equations 1.32 and 1.33 and they are,
N
4p
2
N
′
Et
() =
Et
() =−
2
a
E
exp[ i 2p
vt
− f ()]
t
(1.59)
∑
i
l
r
o 0
∑
i
i=1
0
i=1
and
N N
It ∝
∑∑
exp iqr i t − rj ( )] t
(1.60)
() [()
i=1 j=1
Figure 1.9 The power spectrum S(w) of scaered light: a Lorentzian opcal
frequency distribuon centered by the angular frequency of the incident light
w
0
with line width of G.
Note fi(t) represents the phase term of the ith particle and
is now a function of time due to the motion of the particles. The
same situation should be applied to I(t) and Dfij(t) because
Dfij(t) =
⋅ ()
t
and
t
qr
r
()
ij
ij
t.
When the scatters are undergoing Brownian motion,
Et
( )
has
a randomly modulated phase. The scattered light is broadened
in frequency with an optical frequency distribution, or, power
spectrum S(w) as illustrated in Fig. 1.9. Since the motion of the
particles has no preferred direction, the optical spectrum of
scattered light contains a continuous distribution of frequencies,
i.e., Lorentzian distribution centered by wo, the angular frequency of
the incident light:
The Powerful Tool for Study of Nano Physical Pharmaceutics
35
(1.61)
-
w
S() =
It
can
be
w = w
,
S(w
) =
o
when the
scattered
density function
of
its peak value. For this reason,
to measure
G
« w
. It is known from
o
G
2
G
+
( 2ww
−
seen
2/G; and
of power
G S w
(or (
2
)
o
Fig.
from
when
frequency
spectrum
)) directly
mathematics
1.9
and
w
o
shifts
in frequency
〈E(0)E ∗(t)
±
a
distance
S
G is called
S
transform and inverse Fourier transform:
∞
∗
() t =
E () 0 E
w =
1
2
S()
These
Thus,
S(w) twand
the frequency
p
o equations
Sx()w
∫
−∞
∞
E () 0Et
∫
−∞
〈E(0)E
domain
exp( −it w )dw
∗
()exp( it w )dt
are
know
∗
(
t)〉,
and
n as Wiener–Khintchine
functions
two
time
domain respectively, are now
connected with each other.
Equation
G
, S(w) = 1/
1.61 that when
of
(
w
) is
half the valu
domain
that
(w
〉 are
a
pair
originally located
G
.
That means
from
G
half-w
because
of Fourier
, the
w
o
e that
idth
theory.
at
in
1.4.2.2
Another
Without
light
Siegert relat
important
a local oscillator
reaching
autocorrelation
relation:
(2)
( , t q
) = 1 + |
q, t)〉/〈
(2)
q
t
E((q, , E
functions.
where qg
E∗(
relation
function
ion [41]
equation
the detector
function,
(1)
( t)|
≡〈I
(
q, 0)I
∗
(q
, (0)
〉
Thus,
in dynamic LLS is the Siegert relation.
(i.e.,
a constant
from
(2)
g
q
,
t
) based
( on the Sieget
2
,
(q,
t) /
〉 〈
I q
( , 0)〉2) and
the
intensity-intensity
fraction
various
in essence
of the incident
intentional
sources,
(1)
g
(q, t)
(
≡〈E(q
time
correlation
, 0)
36
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
(2)
( , ) = ( , 0) ( , ) = ( , 0)
2 (2)
( )
G q t
= 〈I(q, 0)
the fact
In practice,
small is. Therefore,
it the scattered
coherent and
(2)
that
G
the
an instrument parameter, b is in
〈I q 〉
(q, tt
)
detection
I
q t 〉
2
[1 +
〈I(q, 0) be
Equation 1.65:
(2)
G
(q
, t) = A g
(1+ b (qI, t)
A
where
parameter
t
and I(t) is
time
, including tcontributions from
Therefore,
I
q
(G, )] and Equation 1.66 becomes
solute
()2
q
( ( , 0)
depending
the detected
(2)
G
(q, t) = [I
(,
q t ) = A + b
(1)
2
) is the
on
the
scattered
solvent
I I
solvent
I
solutio
1
〈I
(1)
area
〉
g
q, t
2
(q)
]
〉 can
(2)
q G q
g t
( , ) and
measured
(2)
( , )
experimentally.
cannot be zero no matter
light detected
cannot
introduced
2
) (1.66)
baseline,
coherence
(q, 0) + I
| g
n
intensity
()1
solven
t
t is
the
of
the
or photon
the solvent
q, 0)] [I
( , t) +
solute
(, q t )|+
I
s
solution
delay
detection
and the
solute
| g
time,
solvent
()1
solut
e
(1.65)
be purely
b is a
optics,
counts
solute.
(q
2
(q,)t |
(1.67)
at
where
the light
correlated.
than
faster than larger
all
the cross terms have been dropped by assuming that
by
ered
scatt
It
sh
ould
| g
(1)
solut
(
q,
e
t
|
)
be noted
2
particles.
Equation 1.67 becomes
( 2)
G
(
q, t ) ≅ A
where = (
b
app
1 + b
A
1
=
+ b
app olut
solute
/
b I
intensity from
solvent
I
solute
I
solution
(
1)
g
s
e
I
)2. For a dilute
solution
molecules
molecules
( )
1
| g
that
solut
(q, t )
e
a
Thus, after
2
(1)
| g
solut
2
e
(q, t )|
(q,t)
could
and
2
|
very short
2
solution,
become
particles
delay
the
appreciable
is
not
faster
time,
scattered
The Powerful Tool for Study of Nano Physical Pharmaceutics
37
(i.e.,
solute < solution
) and thus the apparent coherence
app
would
be lower, i.e., ( ,0) appears to have a very low value than
expected. IWe Ishould
weakly scattered dilute
For
example,
for each
In fact,
particular
I
solute
distributed
.
(2)
G
q
be
if
I
solute
I
optical
be
estimated
aware
and
solvent, app
of this situation, especially bfor
low-molar-mass
b
b
geometry
of the
from
latex
standard
whose
scattering
b
app
polymer
that is
scattering
intensity
solutions.
constant
b
instrument.
of at
b
is much
[52]
1.4.2.3
Now will
the
function
the
Translat
we
particles
(2)
G
see
fro
m
q
, t
( ).
ional dif
how
the
Generally,
particles.
For monodispe
fusions
to get
the information about the motion of
measure
d intensity-intensity
the
relaxation of |
g
rsed spherical
(1)
scatters,
time (correlation
)| includes
q, t
(1)
g
|
(q, t)| is
theoretically represented as an exponential decay function:
(1)
q, t)| = G
(
(–Gt) (1.69)
exp
where
width,
G
respecti
representing
a polydispersed
of molar mass
()1
| g
G
where
statistic
line width
angle is related to q
G
vely,
the
rate of dynamic relaxation in self-beating. For
polymer
M
, Equation 1.69 may be generalized as
∞
(q) ,)
t |
G
(
is
weight
G
For
. a
2
(1 + kCd )( 1 + fq 2R
G
=
∫
0
called
of
the
dilute
C
and by [53]
the Gfactor
and
( G )exp( −G td) G
the
particles
of
=
proportionality
−c 1
, the
characteristic
and
decay
the
time
sample with a continuous distribution
line width
distribution and
or Gmacromolecules
lution,
so
2
) (1.71)
g
z
( )
G G
which
dG
possess
line
38
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
where
molecule at 0,
to the molecular friction factor through the Stokes–Einstein
relation
D
C → k
d
and f is a dimefnsionless parameter depending on polymer chain
good
q, CD ≈
higher
dependence
solvent,
2
G q
/
scattering
is between
of the
angle.
Figure
line with
0.1
and 0.2).
1.10
G on
Hence,
(1)
(q, t)| decays faster at a
for small
demonstrates
the gscattering
vector
the linear
( ) and
q k
and
g
q2G(
G
intensity
distribution. And, since |
(1)
g q, t
( )| approaches tunity
(1)
(q
, )|,
( ) =
G
D
as
t
*
0 )(
Eq
Eq
=
∞
G
∫
0
(,
(,
Eq
()GG
()1
g (q t
→ 0, →
|
0)| =
∞
D =
∫
0
D) D dD
0)
d
, t
Eq*(, →0)
∞
=
GD
∫
0
〈D〉
()
0)
=1
dD
(1.73)
G(
D
f
vector q.
The Powerful Tool for Study of Nano Physical Pharmaceutics
39
= / (1.74)
elsewhere [57, 58].
D k T
where
temperature
f
phRR, wher
coil,
is replaced by its hydrodynamic radius
R
h
1.4.2.4 Analysis of the correlation funct
Equation
(2)
( , ) through
q and
G
computed from the
In the tlast three
developed.
very important factor in the development of the programs.
constraint
computer
many
[56]
is still one of
this
computatio
is an
ill-conditioned
of
photon
the
measured
data points.
uniquely.
intensity-intensity
important
it
is crucial
free”) very thoroughly
the measured and calculated baselines not exceeding
f
B
and
k
=
B
6ph
kT
B
respectively.
e T h is the visc
D
For
a hard
osity
sphere
of the
and a the
with
solvent.
, so that
R
h
absolute
radius
of R,
For a polymer
ion profile
1.70 indicates
Equation
decades, many computation
At
the earlier stage, computation qspeed
has
gradually
has
become
programs, the
the
n.
However,
correlation
time correlation
In other
Thus, in practice,
than
choosing
that
the
that
once |
1.66,
Laplace
inversion
been
faster
and
CONTIN
program
most widely
it should
problem
because
instruments,
words, the
reducing the gnoise
time correlation
a program for data
sample
before
solution
it is subjected
(1)
g
(
G G
the
removed
faster
used
be
of the
some
function,
data
is cleaned
(
q, t
)| is
)
of
then
(1)
|
g
( ,
programs
determined
( ) can
G D
t) | [45,
54,
have
from
55].
been
was
This
because
in the
last 10 years.
developed
and
accepted
noted
the personal
by Provencher
programs
that Equatio
Among
n 1.66
bandwidth limitation
unavoidable
and
a (limited
(1)
of
, ) do not
q
t
in
noises in
number
always
the measured
G(G)
function becomes more
analysis. For this reason,
(i.e.,
made “dust-
to
LLS
measurements.
0.1%.
be
for
of
The error analysis related to the above problem can be found
a
40
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
It is worth noting that there is a temptation among the users
of
dynamic
measured
literature,
It is meaningless
from experimental
distribution
evidence
that
many
past are
in retrieving the desired
average 〈G
width
∞
=
m
2
∫
0
very
helpful
),
(
G G
should
LLS to extract
intensity-intensity
three or
because
noises.
of (
G D
or
pre-experimental
of
the
Laplace
useless.
m 〈G〉
(
(G2−
On
line
width and the relative
2
/
)
of the line width distribution ( ( )) with
2
G ()
GGd
G
)
method
used
be
too much
time
correlation
even four
It
peaks
they
has
to
G D
in
were
actually
be warned
( ) were
knowledge.
∞
∫
0
programs
they
G
G(G)
have
especially
d G
inversion developed in the
the contrary,
information,
〉
≡
(
. Therefore, the Laplace inversion as a
in the
analysis
with
of the
a clear
understanding
conditioned nature and its limitations.
In practice,
〈
G〉
and m
by
Koppel
2
/
〈G〉2,
[59]
if one is only
a fast
but
mor
can
be
used,
interested
e
limited cumulants
wherein
[G
(2)
in
(q, t
as
2
m
t
2
2
!
ln
()2
qt −
G
(,
A
)
A
(
=+
1
ln)
b − Gt +
information
function.
from
In
often reported.
extracting
that
even
a
“data”
bimodal
This does not mean
been quite successful
in terms
of the
)
G G
line width distribution
of its ill-
the determination of
analysis
)
– A
]/A is expanded
3
m
t
3
−
+⋅⋅⋅
3
!
adopted
the
the
where
m
m
is the
m
cumulant
∞
=
(G − G )
∫
0
th
moment
G
, sometimes
m
G () GGd
of the line width distribution ( ). A th
G
also called
the
initial
slope,
is an important
G
m
m
t
quantity, since it can be calculated for many physical systems
and situations
the
width of
slightly polydispersity polymers in
[60,
61].
the
distribution.
The second
For
cumulant
unimodal
m
is a measure
2
distributions
of
of
solution, the following relation
has been derived [61]:
The Powerful Tool for Study of Nano Physical Pharmaceutics
41
/ ( / 1)/4 (1.78)
(1.79)
2
G
2
m
z
used mfor 〈Grelatively
For
〉
/
2
2
m 〈G〉
/
, higher
2
meaningful results
very tedious.
reliable
〈G〉
the measured
statistics, e.g., the baseline
it warns that the line width
Laplace inversion
the
limitations,
methods can provide useful
unimodal
when
the peak
2
order
On
and
time
and
multimodal
positions are
Mw –
narrow
because
the other
m
/
2
is only an
yet must
characteristic line width distributions.
m
expansions
〈
G〉
correlation
2
/〈G
〉
2
using
hand, the
2
(A) has
estimate.
realize
information
line
separated
the range
should
be used.
too
many terms
use
of CONTIN
function
was
a total count
distribution
One
that
the Laplace
and distinguish between
width
distributions,
by
obtained
over 106. However,
obtained
should
factor of
in the
2. It is pointed out that function 1.76 was a always
analysis of the intensity-intensity
study of semid
analyzing
by
changing
ilute solutions.
the autocorrelation
exponents
of
that is,
bf = 1 and 0 <
2
()
q, t)= b
g
(
b
b
(=
S
coupling
A f exp
apparent
{
time
Because
function
b
f
correlation
in
most previous
is so arbitrary.
b
function
f
function
as follow
) < 1, as:
b
−( t /
f
+ A exp [−(t /
]
)
f S
~0.2–0.3,
Howeve
cumulants
could
within a
from the
be
aware of
inversion
especially,
mor
used
literature,
However,
the
it
r,
yield
e than
for the
in our
s,
and
b
S
)
S
is
22
}
42
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
nanoformulations post administration.
1.4.3 Typical Application of Laser Light Scattering in
NPP Accomplished by Our Group
The light scattering
of
static light scattering
a wealth
average
of information about
molecular
dynamics,
In
so on. addition
suitable for
vesicles,
nanoparticles
distribution,
size
studying the
technique
and
dynamic
the polymer,
weight, z-average
molecular
to polymers,
light scattering
behavior
and
so on
is one
of
light scattering
such as the
root-mean-square radius
chain conformation
technology is also
of
biomacromolecules,
in
solutions. Several
applications of light scattering are described below.
1.4.3.1
The special
in-house designed powerful tool for
NPP study
Figure
system by a special
generates
solution. This
indicate
1.11 shows the
the
in vitro
equipment is special
and evaluate
in vivo blood conditions mimic
designed equipment
the real
for the
state,
study
structure and proper
of nanomedicines
the
important
can
obtain
weight-
and
colloids,
typical
testing
with
in vivo
ties
of
barriers mimic panel conditions tuning panel
Figure 1.11 The special in-house built equipment designed for the NPP study.
In-suit D/SLLS detecting system
with flowing cell
The Powerful Tool for Study of Nano Physical Pharmaceutics
43
1.4.3.2 Characterization of polymer chains dynamics in
solution
Figure 1.12 shows the variation of the polymer concentration
with evaporation time as measured by light scattering. As time
goes on, the solvent volatilizes and polymer chain entanglement
increases, resulting in the increase of polymer concentration.
The polymer solution gradually changes from dilute to subconcentrated solution. This in-suit time-dependence of polymer
errors, which is a big trouble for the LLS study.
Based on the smart LLS experimental design, the semidiluted
solutions of polystyrene in toluene are successfully obtained. It is
easy to know that the polymer chains will entangle each other as
the entangled points, that is the chain blob, is thus recorded and
plotted. And a so-called scalding law between the average scattering
intensity (〈I〉) and the polymer concentration (C) is obtained in
tell us that the LLS is a powerful tool for studying the polymers
in solutions.
Figure 1.12 The variaon of the polymer concentraon with evaporaon
me. Reproduced with permission from [62].