Transcript Chapter 12. Light scattering calibration) Transmission Reflection

Chapter 12. Light scattering (determination of MW without
calibration)
Electromagnetic radiation 과 물질과의 상호작용의 결과
Transmission
Reflection
Absorption
Scattering
Incident Radiation
네 가지 현상:
1. transmission: transmitted radiation passes through the medium unaltered.
2. absorption: energy from the incident beam is taken up, resulting in: (1)heating, (2) reemitting at another wavelength (fluorescence, phosphorescence), (3)supporting chemical
reactions. * In this discussion, we assume that radiation heating is negligible. Other
absorption effects are specific to the particular medium, and will also not be considered here.
3. scattering: scattering is non-specific, meaning the incident radiation is entirely re-emitted in
all direction with essentially no change in wavelength. Scattering results simply from the
optical inhomogeneity of the medium.
4. reflection: scattering at the surface of a matter (not considered here)
Now we focus on the light scattering.
Application of Light Scattering for Analysis
1.Classical Light Scattering (CLS) or Static Light Scattering (SLS)
2.Dynamic Light Scattering (DLS, QELS, PCS)
CLS
• 정의: Scattering center = small volumes of material that scatters light. 예:
individual molecule in a gas.
• Consequences of the interaction of the beam with the scattering center:
depends, among other things, on the ratio of the size of the scattering center to
the incident wavelength (λo). Our primary interest is the case where the radius
of the scattering center, a, is much smaller than the wavelength of the incident
light (a < 0.05λo, less than 5% of λo). This condition is satisfied by dissolved
polymer coils of moderate molar mass radiated by VISIBLE light. When the
oscillating electric field of the incident beam interacts with the scattering center, it
induces a synchronous oscillating dipole, which re-emits the electromagnetic
energy in all directions. Scattering under these circumstances is called
Rayleigh scattering. The light which is not scattered is transmitted: I o  I s  I t
, where Is and It are the intensity of the scattered and transmitted light,
respectively.
Elastic Scattering
Scattering
Transmission
• Oscillating electric field of incident beam interacts
with scattering center, induces a synchronous
oscillating dipole, which re-emits electromagnetic
energy in all directions.
I =Is+It
Io
Rayleigh scattering에 의한 산란광의 세기는 측정 위치에 따라 변한다: (1+cos2θ)에 비례하고, scattering
center와 observer사이의 거리(r)의 제곱에 반비례.
• 1944, Debye
I
• Rearrange:
Io
 1  cos2  

] 
2

r


I
[
Io


r


 1  cos2    [


1 + cos2
340
350 2.0
320
]
310
(2)
300
290
280
270
[
 
 2 2  dn
 no
]   4

dc
 o N A 


 RTc 
 

c T 

 
2

10
20
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
330
2
0
30
40
50
60
70
80
90
260
(3)
100
250
110
240
120
230
Constant, K
130
220
140
210
150
200
λo = 입사광파장, dn/dc = refractive index increment
no: 용매의 refractive index, π=삼투압, c=시료농도[g/mL]
190
180
170
160
Then
 
  2 2  dn
I 
r2
 no


dc
Io  1  cos2    o4 N A 


 RTc 
 

c T 

 
2

Iθ is inversely proportional to λo. Shorter wavelength scatters more than longer wavelength
Assume: system is dilute, the net signal at the point of observation is sum of all scattering
intensities from individual scatterer - no multiple scattering (scattered light from one center
strike another center causing re-scattering, etc.).
 
  2 2  dn
I 
r2
 no


dc
Io  1  cos2    o4N A 
Define “Rayleigh ratio” Rθ
measured
 
 2 2  dn
 no
R   4

dc

N
 o A

얻고자 하는 정보 포함


RTc


 

c T 

 
2


 RTc 
 

c T 

 
2

Two ways to access the light scattering information experimentally:
1. Turbidimeter (or spectrophotometer)
2. Light scattering
1. Turbidimeter experiment (Transmitted light intensity, It is measured)
Sample Cell
Monochromatic
light source
= 1 - (It/Io) = (16/3)
R
Photomultiplier tube
measures It
• "Turbidity", τ = fraction of incident light which is scattered out = 1-(It/Io)
• τ is obtained by integrating Iθ over all angles:   16 R
 3 
 32 2
Substitute R :   
4
 3o N AV
 1 

  RTc   Bc    
 M 




2
RTc
dn


 no

dc  


c T 

 
 
 Substitute:


 32 3    dn  2 
c




 
4
 no  dc   1  2Bc  ....... 
3

N
 o av 
 M

 32 3   dn 
Hc
no     
Define : H  
4

1  2A c    
 3o Nav   dc 
2
M
2
Solution is dilute, so higher order concentration terms can be ignored.

Hc
1  2A2c    
M

Hc 1
  2A2c
 M
Procedure: Measure τ at various conc.  Plot Hc/T vs. c (straight line)  Determine
M from intercept, 2nd virial coeff., B from slope
Turbidity Data Processing
Hc/t
Slope=2
Intercept=
Concentration, c
2. Light Scattering experiment (measure Iθ at certain θ and r)
R
RTc


K
c T


(4)
Light Scattering Data Processing
 1 

  

  RT    A2c    
 c T
 M 

Kc /R 
 1 

  RTc   A2c     (5)
 M 

(6)
Slope=2A2
Intercept=1/M
식6을 식4에 대입:
Kc 1
  2A2c
R M
(7)
Concentration, c
* 반경이 파장의 약 5% (λ/20) 이하인 경우에 국한됨 – “Rayleigh limit”
The slope of the plot
Kc
vs. c
R
θ-condition에서 기울기=0.
can be either positive or negative.
<참고> For polydisperse sample, Turbidity (혹은 light scattering) is contributed by molecules of
different MW.
Define: τi= 분자량 Mi를 갖는 분자들에 의한 turbidity →  i
Hci
1
 2Bci    
Mi
If ci  0  2A2ci  0  i Hci Mi   total  i   Hci Mi   H  ci Mi 
(Hc)/τ vs. c 그래프의 절편 =1/M 이므로
 
평균분자량   total
 Hctotal
m
Substitute ci 
mi
V
 평균분자량 

 
c 0
H  ci Mi 

H  ci

  V i Mi 
m 
  V i 
V  cons tan t

 
m M 
N M M 
N M 
평균분자량   i i   i i i   i i  weight  average MW
 mi 
 Ni Mi   Ni Mi 
2
따라서 turbidity나 light scattering실험에서 얻는 분자량은 weight-average MW이다.
 ci Mi 
 ci
Rayleigh-Gans-Debye (RGD scattering) : when the scattering centers are larger than
Rayleigh limit
Plain Polarized Light
A
B
2
1
Different part of more extended domain (B) produce scattered light which interferes with that
produced by other part (A) - constructive or destructive
Effect of particle size on intensity distribution
0
330
320
310
300
340 3502.0
10 20
30
Distribution is symmetrical for small
particles (<λ/20).
For larger particles, intensity is
reduced at all angles except zero.
40
1.5
50
60
1.0
290
70
0.5
280
270
80
0.0
R RGD  R RayleighP ( )
90
260
(8)
100
250
Contributions from two scattering
centers can be summed to give
the net scattering intensity. The
result is a net reduction of the
scattered intensity
110
240
230
220
210
200 190
170 160
120
130
140
150
Small Particles
Large Particles
180
Pθ = "shape factor" or "form factor"
Always Pθ < 1, function of size and shape of scattering volume.
seeing the angle dependence of the scattered light !
Now we start
Kc
11


  2Bc  (8' )
R P  M

Qa        (9)
1
 1
P
5
2
구형입자의 경우:
a =반경
Q = scattering vector = (4π/λ)sin(θ/2)rg (10)
1
Random coil 고분자의 경우,
5 2
a    rg
3
(11)
Scattering factor, P ( )
Effect of Angular Asymmetry on MW Measurements
MW
1
10k
100k
200k
400k
0.9
600k
0.8
0.7
0.6
0
10
20
30
40
50
60
Scattering Angle, 
• p(θ) decreases with θ.
• p(θ) decreases more for higher MW.
70
80
90
Effect of MW and Chain Conformation on Pθ, and on measured MW at 90o.
MW (g/mol)
RG (nm)
P(90o)
MW(90o)
51K
8
0.98
51K
420K
19
0.95
400K
PMMA
680K
36
0.70
480K
Polyisoprene(~70% cis)
940K
48
0.56
530K
66K
3
1.00
66K
10700K
12
0.98
10500K
Poly- -benzyl-L-glutamate
130K
26
0.91
118K
Myosin
493K
47
0.74
365K
DNA
4000K
117
0.35
1400K
Conformation
Random coil
Polystyrene
Polystyrene(
condition)
Spherical
Bovine serum albumin
Bushy stunt virus
Rod shaped
1
Random coil 고분자의 경우,
5 2
a    rg
3
(11)
 
2
Kc  1
  16  2 2  
   2A2c 1  
rg sin
2
2 
R  M
  3 

식(11)을 식(9)에 대입한 후 식(7’)에 대입:
(12)
Final Rayleigh equation for random coil polymer
두 가지 극한 상황:
[Case 1] θ→0:
Kc  1

   2 A2c 
R  M

Plot Kc/Rθ vs. c: y-절편=1/M, 기울기=2A2
[Case 2] c→0:
 
Kc 1   16 2  2 2  
rg sin
 1  
2 
R M   3M2 

Plot Kc/Rθ vs. sin2(θ/2): y-절편=1/M, 기울기= (16π2/3Mλ2) rg2
Three information!
빛산란 실험 방법
(1) 다양한 각도와 농도에서 Rθ측정.
(2) Kc/Rθ vs. c, Kc/Rθ vs. sin2(θ /2) plot 작성.
(3) θ =0 와 c =0 로 extrapolate.
Kc/Rθ vs. c
Kc/Rθ vs. sin2(θ /2)
Zimm plot:
채워진 점: 실험 데이터.
빈 점: extrapolated points
Cases
1. Small polymers: 각도의존성 없음. (Horizontal line)
Zimm plot for PMMA in butanone
λo=546 nm, 25℃, no ~1.348, dn/dc = 0.112 cm3/g
(Kc/Rθ) vs. c
- 다섯 농도에서 측정한 데이터.
- Mw 와 A2 결정 가능
- 분자크기 측정 불가능.
2. Small polymers in θ-solvent: 각도 및 농도 의존성 없음.
Zimm plot of poly(2-hydroxyethyl methacrylate) in isopropanol
λo=436 nm, 25℃, no ~1.391, dn/dc = 0.125 cm3/g
θ-solvent : A2=0가 되는 용매, 고분자-고분
자, 고분자-용매분자간 상호작용의 에
너지가 동일, 이상용액과 같이 행동.
-Calculated values : Mw = 66,000 g/mol
A2 = 0 mol cm3/g2
- Kc/Rθ at small angles fall mostly below
the horizontal line plotted through the
points from medium and large angles.
3. Larger polymers in good solvent: 각도 및 농도에 의존.
Zimm plot of polystyrene in toluene
λo=546 nm, 25℃, no ~1.498, dn/dc = 0.110 cm3/g
- 분자량 약 2x105 이상의 경우,
Kc/Rθ 는 양의 기울기 (A2=양수)를 가진다.
- Athermal Condition - No effect of temperature
on polymer structure
4. Polymers in poor solvent: A2 가 음수가 됨 (큰 음수는 될 수 없음. 더 이상 녹지 않기 때문)
Zimm plot of polybutadiene in dioxane
λo=546 nm, 25℃, no ~1.422, dn/dc = 0.110 cm3/g
- 각도의존성이 직선이 아님 (nonlinear).
- 이유: microgel, 먼지, aggregate과 같은 큰 입자 존재.
- Curve-fitting에 주의를 요함.
분자크기 측정의 정확도에 영향.
- 분자가 커지면 good solvent에서도
직선성을 벗어날 수 있다.
Stand-alone vs. On-line MALS
<Stand-alone mode>
<On-line mode>
• Stand-alone mode: LS instrument is used
itself.
• LS instrument is used as a detector for a
separator.
• Zimm plot 을 이용 M, A2, Rθ를 결정
• c=0 이라 가정.
• 각 slice에 대해 Kc/Rθ vs. sin2(θ/2) 그래프를
이용, y-절편으로부터 분자량 (M), 초기기울
기로부터 rg를 결정.
y-절편=1/M, 초기기
울기= (16π2/3Mλ2) rg2
• 각 slice가 monodisperse하다고 가정하고 평
균분자량과 평균크기를 계산.
따라서 높
은 분리도가 요구됨 (분리방법선택 및 분리최
적화가 요구됨).
Average Molecular Weights
1.No-average: Mn=(Σci)/(Σ(ci/Mi))
2.Wt-average: Mw=Σ(ci Mi)/ Σ(ci)
3.Z-average: Mz= Σ(ci Mi2)/Σ(ci/Mi)
Average Sizes (mean square radii)
1.No-average: <rg2>n= Σ[(ci/Mi)<rg2>i]/Σ(ci/Mi)
2.Wt-average: <rg2>w= Σ(ci<rg2>i)/Σci
3.Z-average: <rg2>z=Σ(ciMi<rg2>i)/Σ(ciMi)
Light scattering instruments
MALLS (Multi Angle Laser Light Scattering) : I is measured at 15 angles
(1) Stand-alone mode: Measure scattered light at different angles for different concentrations 
Make a Zimm plot  Determine M, B, Rg
(2) On-line mode: Assume c=0, Plot
For each slice.
 
Kc
vs. sin2 
2
R
 16 2  2
r
Determine M from intercept (intercept = 1/M), rg from slope (slope = 
)
2  g
3
M



Assuming each slice is narrow distribution, Mw  Mi
Average M can be calculated. It is therefore very important to have a good resolution.
TALLS (Triple Angle): I is measured at 45o, 90o, and 135o
• Not useful when the plot of
 
Kc
vs. sin2 
2
R
deviates from linearity
Angular Dependence of Kc / R (시료 = high molecular weight DNA)
Effect of Particles/Gels on Light Scattering Measurement
Note the delicacy of extrapolation to zero angle from larger distances.
DALLS (Dual Angle): Iθ is measured at 15o and 90o
LALLS (Low Angle): Iθ is measured at one low angle (assume:
= 0)
(1) Static mode: measure LS at a few c  Plot Kc/Rθ vs. c  Determine M and B from
intercept and slope.
(2) On-line mode: determine Kc/Rθ for each slice ( calculate M). Considering each slice
is narrow distribution, let Mw ( Mi, from which average MW's can be calculated (as
learned in chapter 1). It is therefore again very important to have a good resolution.
RALLS (Right Angle)
• Iθ is measured at 90o.
• Simple design
• Higher S/N ratio, Application is limited to cases where Pθ is close to 1 (e.g., less
than 200K of linear random polymer)
• RALLS combined with differential viscometer (commercially available from
Viscotek, "TRISEC")
<TRISEC 이용 방법>
Assume Pθ = 1 and A2 = 0.
Mest 
R
Kc
From
Determine Mest.
Kc
11


  2Bc 
R P  M

1   M 
RG can be obtained using the Flory-Fox equation: RG  


 6   
1
3
[η] is determined by differential viscometer, and M determined in step 2.


1
 4no 
 2  x
rg sin
Calculate P(θ=90). P   2  e  x  1 , where x 2  
x 
 o 
Calculate new MW by Mest 
Mest
P   90
Go to step 2. Repeat until Mest does not change.
<Light scattering experiment에 필요한 상수들>
 
2
Kc  1
  16  2 2  
   2Bc 1  
rg sin
2
2 
R  M
  3 

에서 K와 B를 제외한 모든 parameter는 이미 알고 있다.
 2 3    dn  2
 no  
K   4


N
 o av    dc 
그런데
이므로 다음 세 개의 상수가 필요.
1. n: 용매의 refractive index
2. dn/dc : Specific refractive index increment
3. B: 2nd virial coefficient (Static mode에서는 B를 실험에 의해 결정할 수 있기 때문에
Static mode는 제외).
1. 용매의 Refractive Index
거의 모든 용매에 대해 RI 값들이
알려져 있음.
• Except where otherwise noted, all
measurements made at λ= 632.8
nm and T=23 oC. RI at 632.8 nm
calculated by extrapolation from
values measured at other
wavelengths.
• Extrapolation에 관한 reference:
Johnson, B. L.; Smith, J. "Light
Scattering from Polymer solutions"
Huglin, M. B. ed., Academic press,
New York, 1972, pp 27
자주 쓰이는 용매들 (R가 감소하는 순)
x 106 [cm-1]
Solvent
RI
R
Carbon disulfide
1.6207
57.5
a-chloronaphthalene (140 oC)
1.5323
52.8
1,2,4-Trichlorobenzene (135 oC)
1.502
35.7
Chlorobenzene
1.5187
18.6
o-Xylene (35 oC)
1.50
15.5
Toluene
1.49
14.1
Benzene
1.50
12.6
Chloroform
1.444
6.9
Methylene chloride
1.4223
6.3
Carbon tetrachloride
1.46
6.2
Dimethyl formamide
1.43 (589 nm)
5.6
Cyclohexane
1.425
5.1
Cyclohexanone
1.4466
4.7
Methyl ethyl ketone
1.38
4.5
Ethyle acetate
1.37
4.4
THF
1.41
4.4
Acetone
1.36
4.3
Dimethyl sulfoxide
1.478 (589 nm)
4.1
Methanol
1.33
2.9
Water
1.33
1.2
2. Specific refractive Index, dn/dc
• 문헌에서 구할 수 있다 (Polymer Handbook, Huglin, ed., Light Scattering from Polymer
Solutions, Academic Press, 1972)
• 문헌에서 구할 수 없는 경우 실험에 의해 측정
• Conventional method
• DRI를 이용
• 몇 가지 다른 농도에서 (n2-n1)을 측정 (recommended conc. = 2, 3, 4, 5 x 10-3 g/mL)
→ (n2-n1)/c2 vs. vs. c2를 plot → zero concentration으로 extrapolate →
dn/dc는 intercept로 부 터 구한다.
dn  n 2  n1 


dc  c 2  c 0
For concentration ranges generally used, the refractive index difference, n2-n1, is a linear
function of concentration. In other words, (n2-n1)/c2 is constant. 즉 (n2-n1)/c2 vs. c2 그래
프의 기울기=0.
This means that (n2-n1) needs to be measured for only one or two different
concentrations. If (n2-n1)/c2 shows no significant dependence on c, then dn/dc
can be obtained by averaging (n2-n1)/c2 values
• SEC/RI를 이용
Ri = detector signal at the slice I
kR = RI const
ci = conc. (g/mL) of the slice i)
이미 배운 바와 같이 Ri  k R 
dn 
c i
 dc 
먼저 dn/dc를 아는 표준시료를 주입하여 kR·을 계산: k R 
→ 시료를 주입, dn/dc 계산:
•
Area 시료
 dn 

 
k R c시료
 dc  시료
Area std
 dn 
  c std
 dc  std
문헌이나 실험에 의해 구할 수 없는 경우 estimate을 할 수도 있다.
1)
extrapolate to desired wavelength:dn  k  k
dc

혹은
2) polymer와 용매의 refractive index로 부터 estimate:
dn
  2 n 2  n1 
dc
여기에서 n2는 polymer의 partial specific volume [mL/g]이다. 보통 n2  1.
<유의사항>
• dn/dc 는 파장의 함수이므로 light scattering 실험을 하는 기기의 광원의 파장과 같은 파장에서
측정해야 한다.
• Dn/dc 는 파장이 짧아질수록 증가하는 경향이 있다. Dn/dc 는 분자량의 함수.
• 정확한 dn/dc 값이 필요. 분자량이 커질수록 더욱 중요해 진다.
3. Virial Coefficient, B or A2
• 문헌에서 구할 수 있음 (예: Polymer Handbook). 문헌에서 구할 수 없는 경우 실험에 의
해 측정 (stand-alone Light scattering)
• 2nd Virial Coefficient는 Solute-Solvent interaction의 척도.
+: Polymer-solvent interaction, good solvent (the higher, the better solvent).
0: Unperturbed system
-: Polymer-polymer interaction, poor solvent.
• A2는 분자량의 함수: A2 = b M-a
분자량에 반비례.
 log A2 vs. log M은 직선. 보통 기울기는 음수, 즉
• dn/dc와 A2·의 중요성에 관한 참고문헌: S. Lee, O.-S. Kwon, "Determination of
Molecular Weight and Size of Ultrahigh Molecular Weight PMMA Using Thermal FieldFlow Fractionation/Light Scattering" In Chromatographic Characterization of Polymers.
Hyphenated and Multidimensional Techniques, Provder, T., Barth, H. G., and Urban, M.
W. Ed.; Advances in Chemistry Ser. No. 247; ACS: Washington, D. C., 1995; pp93.
Light scattering 실험을 할 때 고려 해야 할 점들 (concerns)
• 정확한 dn/dc, RI constant, A2가 필요.
• As dn/dc increases, calculated MW decrease, calculated mass decrease, and no effect on
calculated RG.
• As RI constant increases, calculated MW decreases, calculated mass increases, and no effect
on RG .
• As A2 increases, calculated MW increases, no effect on calculated mass, RG slightly increases.
Refractive Index Detector Calibration 시 알아두어야 할 점들
• RI Calibration constant: inversely proportional to the detector sensitivity.
• Sensitivity of most RI detector is solvent-dependent.
• A calibration constant measured in a solvent may not be accurate for other solvents. It is
recommended to use a solvent that will be used most often (e.g., THF or toluene).
• For RI calibration, only the RI signal is used. Light scattering instrument calibration is not needed.
• Concentration of standards should be such that the output of RI detector varies between about 0.1 1.0 V and should correspond to normal peak heights of samples (For a Waters 410 RI at sensitivity
setting of 64, this corresponds roughly to concentrations of 0.1 - 1.0 mg/mL. RI output can be
usually monitored by light scattering instrument (e.g., channel 26 of DAWN).
• Use NaCl in water as a standard for aqueous system.
• The RI calibration constant will change if you change the sensitivity setting of the detector: So it is
important to use the same sensitivity setting of RI detector as that used when the detector was
calibrated.
RI calibration preparation: One Manual injector with at least 2 mL loop, Five or more known
concentrations (0.1 - 1 mg/mL) of about 200 K polystyrene in THF.
RI calibration Procedure
1. Remove columns. Place manual injector with loop.
2. Pump THF through a RI detector at normal flow rate (about 1 mL/min). Purge both reference
and sample cells of detector until baseline becomes flat & stable.
3. Stop purging and wait till baseline becomes stable.
4. Set up the light scattering data collection software (enter filename, dn/dc, etc.) Enter 1 x 10-4
for RI constant (light scattering instrument usually requires the RI constants to be entered).
Set about 60 mL for Duration of Collect .
5. Begin collecting data with ASTRA.
6. Inject pure solvent first followed by stds from low to high conc, and finish with pure solvent.
7. Repeat the measurements if you want.
8. Data Analysis: (1)set baseline using signals from pure solvent at the beginning and the end
(2)calculate each concentration as a separate peak by marking exactly 1 mL as peak width (or
30 slices at 1 mL/min, 2 seconds of collection interval).(3)calculate the mass of the peak
(4)plot the injected mass (y-axis) vs. calculated mass (x-axis) (5)do linear regression on data
by forcing the intercept be zero (6)calculate RI constant using RI constant = slope x 1x10-4
• Chemical heterogeneity within each slice leads to non-defined dn/dc → Quantitation of
chemical heterogeneous samples is very difficult.
• Limited sensitivity to low MW components. Mn(exp)>Mn(true). The same concern with
differential viscometer experiments.
Limited Sensitivity of Light Scattering and RI Detector
• g' values may be in error if each peak slice contains both linear and branched polymer or
different types of long-chain branching: g' will be overestimated.
• Quality of data is highly affected by the presence of particles.
• Lower limit of RG with MALS는 약 10 nm (about 100K MW)
• Inter-detector volume must be known accurately.
Comparison of online LS vs. viscometer
LS
MWD
[η] distribution
RG
Absolute
Relative
need precise n and dn/dc
Universal calibration must be valid or need M-H
coefficient
independent of separation
mechanism
Independent of separation mechanism if M-H
coefficients are used. Dependent on separation
mechanism if universal calibration is used.
indirect from universal
calibration
direct, independent of separation mechanism
direct from MALS (limited
to >10 nm)
indirect from universal cal. and Flory-Fox eqn.
applicable to linear molecules only
Chain conformation MALLS: RG vs. M plot
Branching
heterogeneous
samples
Lower MW
detectability
Response to
particle
contamination
Viscometer
[η ] vs. M plot (M-H coefficients can be obtained)
RG vs. M plot.
g obtained directly from MALS, g' obtained directly
indirectly from LALLS &
universal calibration
limited because of dn/dc
uncertainty
directly applicable with univ. calib., but the
change in dn/dc will affect DRI responses
~2K. depends on dn/dc and
polydispersity
as low as 300-400 has been reported
LALLS: highly sensitive, MALLS: Insensitive
less sensitive
Information Content
Primary
Secondary
LALLS
M
MALLS
M
RG
PCS
D
Rh, M
Viscometer
[η ]
M, RG
Primary information: high precision and accuracy, insensitive to SEC variables,
requires no SEC column calibration.
<SEC-VISC-LS instrument>
Features:
• MWD measured by LS
• IVD measured by Viscometer
• Both Viscometer and LS are insensitive to experimental conditions and separation mechanism
• No band broadening corrections are needed for Mw, [η ], a, k, and g‘
• Precise and accurate calculation of hydrodynamic radius distribution, M-H constants, and
Branching distribution
Dynamic light scattering (DLS, QELS, PCS)
• Classical light scattering: "time-averaged scattering intensity"를 측정 – 산란광의 세기는 각
scattering center로부터 산란 되는 빛의 세기의 합 (algebraic summation).
• 이러한 algebraic summation의 관계는 각 입자들이 random하게 array되어있고, 또한 phase
relationship이 scattering volume dimension에 비해서 훨씬 작은 공간에 국한됨으로써 모든
interference effect 들이 average-out되기 때문에 성립되는 것이다.
• Scattering volume dimension이 작을 때에는, 산란광의 세기는 각 scattering center로 부터 산
란 되는 빛이 서로 어떻게 interfere (constructive or destructive) 하느냐에 따라 달라지며 따라
서 입자들의 상대적인 위치에 따라 달라진다.
• 각 입자들은 Brownian motion (diffusion) 에 의해 계속 움직이므로 입자들의 상대적인 위치 또
한 계속 움직인다. 따라서 측정되는 산란광의 세기는 시간에 따라 fluctuate한다.
• Fluctuate하는 속도는 입자들의 diffusion rate에 의존 (diffusion rate이 빠를수록 빠르게
fluctuate).
• nanometer 에서 micron범위의 크기를 가지는 입자들이 물의 viscosity와 비슷한 viscosity를
가지는 media에 disperse되어 있을 때, 산란광의 세기의 변화 시간 (fluctuation)은
microsecond 내지 millisecond이다.
• A vertically polarized laser beam is scattered from a colloidal dispersion. The photomultiplier
detects single photons scattered in the horizontal plane at an angle  from the incident beam,
and the technique is referred to as "photon correlation spectroscopy (PCS)“
• Because the particles are undergoing Brownian motion, there is a time fluctuation of the
scattered light intensity, as seen by the detector. The particles are continually diffusing about
their equilibrium positions. Analyzing the intensity fluctuations with a correlator yields the effect
diffusivity of the particles.
• Measured intensity, I = vector sum of scattering from each particle
• Brownian motion: motion caused by thermal agitation, that is, the random collision of particles
in solution with solvent molecules. These collisions result in random movement that causes
suspended particles to diffuse through the solution. For a solution of given viscosity, η, at a
constant temperature, T, the rate of diffusion (diffusion coefficient) D is given by the StokesEinstein equation, D=(kT)/(6πηd), where k = Boltzman's constant, d= equivalent spherical
hydrodynamic diameter. 따라서 diffusion coefficient (D)를 결정함으로써 입자 크기 (혹은 분자
량)을 결정할 수 있다.
• DLS실험을 할 때에는 정해진 시간 동안 계속해서 일정한 시간 간격(τ = time interval)에서 산란광
의 세기를 측정한다. 입자들의 위치가 변화하는 시간에 비해서 τ가 작을 때, I(0)와 I(τ)는 같다.
만약 짧은 시간 interval을 두고 계속해서 I(0)와 I(τ)를 측정할 때 intensity product, I(0)I(τ)의 평균
값은 <I2(0)>, 즉 average of the square of the instantaneous intensity 와 같아진다 - 이때 "I(0)와
I(τ)는 correlate되어있다"라고 한다. 입자들의 위치가 변화하는 시간에 비해서 τ가 클 때, I(0)와
I(τ)는 아무런 관계도 같지 않는다 - "I(0)와 I(τ)는 correlate되어있지 않다" 혹은 "I(0)와 I(τ)는 uncorrelate 되어있다" 라고 한다. 이때에는 intensity product, I(0)I(τ)의 평균값은 단순히 <I2>, 즉
square of the long-time averaged intensity가 된다. 입자들의 위치가 변화하는 시간에 비해서 τ
가 작지도 크지도 않을 때, "I(0)와 I(τ)는 부분적으로 correlate되어있다".
• Measured intensity, I = vector sum of scattering from each particle
• Measure I at various time interval, ,
• I(0) = I(τ) for short τ  “correlated”, correlation decreases as  increases.
• I(0)와 I(τ)를 비교함으로써 Correlation의 정도를 결정할 수 있다. correlation의 정도를
결정하기 위해 average of the intensity product, G(τ)를 결정한다.
• 정의: G(τ)=“Anto correlation function” = <I(t)I(t+τ)> : average of the intensity product.
• 이미 배웠듯이 τ 가 증가함에 따라 G(τ)는 감소.
• G(τ) is high for high correlation, and is low for low correlation.
• High correlation means that particles have not diffused very far during τ.
Thus G(τ)
remaining high for a long time interval indicates large, slowly moving particles.
• The time scale of fluctuation is called "decay time“
• Decay time is directly related with the particle size. The inverse of decay time is the decay
constant, .
• Usefulness of G(t): directly relatable to the particle diffusivity
• For monodisperse samples,
G   Ao  Ae 2
, where Ao = background signal, A: instrument constant,
  decay constant = DQ 2
D = diffusion coefficient =
kT
6 d
 4    
Q  scattering vector =   sin 
   2
실험 과정
실험에 의해 다양한 interval에서 autocorrelation function, G(τ)를 얻는다
G(τ) vs. τ의 그래프를 얻는다
Exponential function을 이용하여 G(τ)를 fit한다.
G   Ao  Ae 2 을 이용, Γ를 결정
1 T
G   limT    I t I t   d
를 이용, G(τ) 를 계산.
T  0
  DQ 2 을 이용, D결정
Stokes-Einstein공식 a 
kT
6 D
을 이용하여 입자크기를 결정
(a = 입자반경 or hydrodynamic radius, Rh)
Rh를 이용, 분자량 결정
정리하면: Measure I(τ) at various   G(τ) →
Dd
참고: DLS 의 응용은 입자들의 diffusion이 서로 방해를 받지 않는 묽은 dispersion (  ≤0.03)인
경우에 국한됨.  = volume fraction of suspended spheres.
구형 입자의 경우,
 Nc 
   Vh
M 
, where N = Avogadro's no., M = MW, Vh = hydrodynamic vol.).
위해서는 보통  ≤0.005가 만족 되어야 한다.
Infinite dilution D값을 얻기
 2 D  a Q 2
참고: Polydispersed 시료의 경우: G   
 f a I a,  e
 f a I a,  da
da
으로 표현된다.
여기에서 f(a) = distribution function, I(a,θ) = scattering intensity function for RGD spheres.
PC를 이용, normal혹은 log-normal distribution function을 G(τ)에 fit한다.
참고: Narrow, mono-modal distribution 시료의 경우, "method of cumulant"를 이용, 다음과 같이
표현할 수 있다.
B
ln G    AQ 2   Q 4 2    
2
For spherical Rayleigh scatterer,
A, B - coefficients related to the moments of the size
distribution, f(a).
5
 kT  a 
A  
 6
6


 a 
and
6 4
B  a a

2
 a5
A

2

 1


으로 주어짐.
• 여기에서 an = nth moment of f(a).
• We see that DLS yields a somewhat unusual average radius (the inverse "z-average", and
one which is quite highly sensitive to the presence of outsized particles.
• DLS uses a single exponential decay function, and thus it does not give information on
sample polydispersity.
참고
• RI values of medium and sample are needed for DLS experiments.
• RI = 1.333 for water, and 1.5 - 1.55 for typical polymers and proteins.
• RI of sample is needed only when the intensity weight needs to be converted to the volume
weight (e.g., for samples having broad distributions).
• Theory to convert the intensity % to the volume % is only for solid particles. So the
conversion will not be accurate for samples such as liposome’s which are hollow inside.
• For samples such as liposome, a value between 1.5 - 1.55 can be used as it is typical
values for polymers and proteins.
• For samples having narrow distributions, only the unimodal analysis is performed, and thus
there is no need to convert the intensity % to the volume %.
• RI value will not make any difference in the average size data because only the RI of
medium is need for unimodal analysis.
DLS summary
D  Rh  M
• D depends on MW and conformation
• Diffusion coefficient distribution can be obtained
• D is independent on chemical composition. D can be obtained without knowing
chemical composition.
• Concentration is not needed to determine D
• Input parameters (T, n,  ) are easily measured.
• Concerns: sensitivity, interference from particulates, inconsistency, not very useful for
polydispersed or multi-modal distributions.
<참고>
Particle Size Conversion Table
Mesh size
Approximate
4
4760
6
3360
8
2380
12
1680
16
1190
20
840
30
590
40
420
50
297
60
250
70
210
80
177
100
149
140
105
200
74
230
62
270
53
325
44
400
37
625
20
1250
10
2500
5
μ size