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Chapter 12. Light scattering (determination of MW without calibration). Electromagnetic radiation 과 물질과의 상호작용의 결과. 네 가지 현상 : transmission: transmitted radiation passes through the medium unaltered. - PowerPoint PPT Presentation
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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) re-emitting 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: , where Is and It are the intensity of the scattered and transmitted light, respectively.
tso III
Io
Elastic Scattering
Transmission
I =Is+It
Scattering
cos1
] [2
2
rI
I
o
1 + cos2
0.00.20.4
0.60.8
1.01.2
1.41.6
1.82.0
010
2030
40
50
60
70
80
90
100
110
120
130
140150
160170
180190
200210
220
230
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260
270
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290
300
310
320330
340350
(2) ] [ cos1
2
2
r
I
I
o
Constant, K
(3) 2
] [2
4
2
T
oAo c
RTcdc
dnnN
• Oscillating electric field of incident beam interacts with scattering center, induces a synchronous oscillating dipole, which re-emits electromagnetic energy in all directions.
• 1944, Debye
• Rearrange:
λo = 입사광파장 , dn/dc = refractive index increment
no: 용매의 refractive index, π= 삼투압 , c= 시료농도 [g/mL]
Rayleigh scattering 에 의한 산란광의 세기는 측정 위치에 따라 변한다 : (1+cos2θ) 에 비례하고 , scattering center 와 observer 사이의 거리 (r) 의 제곱에 반비례 .
T
oAoo c
RTcdc
dnnN
r
I
I 2
4
2
2
2 2
cos1 Then
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.).
Define “Rayleigh ratio” Rθ
measured얻고자 하는 정보 포함
T
oAoo c
RTcdc
dnnN
r
I
I 2
4
2
2
2 2
cos1
T
oAo c
RTcdc
dnnN
R2
4
22
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
Photomultiplier tube measures It = 1 - (It/Io) = (16/3)
R
• "Turbidity", τ = fraction of incident light which is scattered out = 1-(It/Io)
• τ is obtained by integrating Iθ over all angles: R
3
16
T
oAVo c
RTcdc
dnnN
R
2
4
2
3
32 : Substitute
BcM
RTc1
Substitute:
.......213
322
4
3
BcM
c
dc
dnn
No
avo
cAM
Hc
dc
dnn
NH o
avo 2
2
4
3
21
3
32 :Define
Solution is dilute, so higher order concentration terms can be ignored.
cAM
Hc
cAM
Hc2
2
21
21
Turbidity Data Processing
Concentration, c
Hc/
t
Intercept=
Slope=2
Procedure: Measure τ at various conc. Plot Hc/T vs. c (straight line) Determine
M from intercept, 2nd virial coeff., B from slope
(5) 1
2
cAM
RTc
(6) 1
2
cAM
RTc T
(4)
Tc
RTc
K
R
식 6 을 식 4 에 대입 : (7) 21
2cAMR
Kc
Light Scattering Data Processing
Concentration, c
Kc
/R
Intercept=1/M
Slope=2A2
* 반경이 파장의 약 5% (λ/20) 이하인 경우에 국한됨 – “ Rayleigh limit”
2. Light Scattering experiment (measure Iθ at certain θ and r)
Light Scattering Data ExamplesPS in cyclohexane
2.4
2.9
3.4
3.9
4.4
0 0.1 0.2 0.3 0.4 0.5 0.6
Concentration, c x 103 (g/cm3)
(Kc/
R)x
107
T(oC)=55
T(oC)=45
T(oC)=38
T(oC)=34
T(oC)=32.5
The slope of the plot cvsR
Kc .
can be either positive or negative.
θ-condition 에서 기울기 =0.
< 참고 > For polydisperse sample, Turbidity ( 혹은 light scattering) is contributed by molecules of different MW.
Define: τi= 분자량 Mi 를 갖는 분자들에 의한 turbidity →
i
i
ii
BcM
Hc
21
iiiiitotaliiiii McHMHcMHccAc 02 0 If 2
(Hc)/τ vs. c 그래프의 절편 =1/M 이므로
0ctotal
total
Hc
평균분자량
i
ii
i
ii
c
Mc
cH
McH
V
mc i
i Substitute
tconsV
i
ii
Vm
MVm
tan 평균분자량
MWaverageweight
MN
MN
MN
MMN
m
Mm
ii
ii
ii
iii
i
ii 2
평균분자량
따라서 turbidity 나 light scattering 실험에서 얻는 분자량은 weight-average MW 이다 .
Rayleigh-Gans-Debye (RGD scattering) : when the scattering centers are larger than Rayleigh limit
Plain Polarized Light
1 2
A
B
Different part of more extended domain (B) produce scattered light which interferes with that produced by other part (A) - constructive or destructive
0.0
0.5
1.0
1.5
2.00
10 2030
4050
60
70
80
90
100
110
120
130140
150160170
180190200
210220
230
240
250
260
270
280
290
300
310320
330340 350
Small ParticlesLarge Particles
Effect of particle size on intensity distribution
(8) )( PRR RayleighRGD
(9)
51
1 2
Qa
P
a = 반경Q = scattering vector = (4π/λ)sin(θ/2)rg (10)
구형입자의 경우 :
(11) 3
5 21
gra
Random coil 고분자의 경우 ,
Distribution is symmetrical for small particles (<λ/20). For larger particles, intensity is reduced at all angles except zero.
Contributions from two scattering centers can be summed to give the net scattering intensity. The result is a net reduction of the scattered intensity
Pθ = "shape factor" or "form factor"
)(8' 211
BcMPR
Kc
Always Pθ < 1, function of size and shape of scattering volume. Now we start
seeing the angle dependence of the scattered light !
Effect of Angular Asymmetry on MW Measurements
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90
Scattering Angle,
Sca
tte
rin
g f
act
or,
P
()
MW
200k
400k
600k
100k10k
• p(θ) decreases with θ.• p(θ) decreases more for higher MW.
Effect of MW and Chain Conformation on Pθ, and on measured MW at 90o.
Conformation MW (g/mol) RG (nm) P(90o) MW(90o)
Random coil
Polystyrene 51K 8 0.98 51K
Polystyrene( condition) 420K 19 0.95 400K
PMMA 680K 36 0.70 480K
Polyisoprene(~70% cis) 940K 48 0.56 530K
Spherical
Bovine serum albumin 66K 3 1.00 66K
Bushy stunt virus 10700K 12 0.98 10500K
Rod shaped
Poly- -benzyl-L-glutamate 130K 26 0.91 118K
Myosin 493K 47 0.74 365K
DNA 4000K 117 0.35 1400K
[Case 2] c→0:
cAMR
Kc22
1
두 가지 극한 상황 :
Plot Kc/Rθ vs. c: y- 절편 =1/M, 기울기 =2A2
2sin
3
161
1 222
2
grMMR
Kc
Plot Kc/Rθ vs. sin2(θ/2): y- 절편 =1/M, 기울기 = (16π2/3Mλ2) rg2
Three information!
[Case 1] θ→0:
(11) 3
5 21
gra
Random coil 고분자의 경우 ,
(12) 2sin3
1612
1 222
2
2
grcA
MR
Kc식 (11) 을 식 (9) 에 대입한 후 식 (7’) 에 대입 :
Final Rayleigh equation for random coil polymer
빛산란 실험 방법(1) 다양한 각도와 농도에서 Rθ 측정 .
(2) Kc/Rθ vs. c, Kc/Rθ vs. sin2(θ /2) plot 작성 .
(3) θ =0 와 c =0 로 extrapolate.
Kc/Rθ vs. sin2(θ /2)Kc/Rθ vs. c
Zimm plot:
채워진 점 : 실험 데이터 .빈 점 : extrapolated points
Cases
1. Small polymers: 각도의존성 없음 . (Horizontal line)
- 다섯 농도에서 측정한 데이터 .
- Mw 와 A2 결정 가능- 분자크기 측정 불가능 .
Zimm plot for PMMA in butanoneλo=546 nm, 25 , ℃ no ~1.348, dn/dc = 0.112 cm3/g
(Kc/Rθ) vs. c
-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.
2. Small polymers in θ-solvent: 각도 및 농도 의존성 없음 .
Zimm plot of poly(2-hydroxyethyl methacrylate) in isopropanolλo=436 nm, 25 , ℃ no ~1.391, dn/dc = 0.125 cm3/g
θ-solvent : A2=0 가 되는 용매 , 고분자 -
고분자 , 고분자 - 용매분자간 상호작용의 에너지가 동일 , 이상용액과 같이 행동 .
3. Larger polymers in good solvent: 각도 및 농도에 의존 .
Zimm plot of polystyrene in tolueneλo=546 nm, 25 , ℃ no ~1.498, dn/dc = 0.110 cm3/g
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 , aggregate 과 같은 큰 입자 존재 .
- Curve-fitting 에 주의를 요함 .
분자크기 측정의 정확도에 영향 .
- 분자가 커지면 good solvent 에서도 직선성을 벗어날 수 있다 .
- 분자량 약 2x105 이상의 경우 ,
Kc/Rθ 는 양의 기울기 (A2= 양수 ) 를 가진다 .
- Athermal Condition - No effect of temperature on polymer structure
<Stand-alone mode>
• Stand-alone mode: LS instrument is used it
self.
• Zimm plot 을 이용 M, A2, Rθ 를 결정
<On-line mode>
• LS instrument is used as a detector for a sep
arator.
• c=0 이라 가정 .
• 각 slice 에 대해 Kc/Rθ vs. sin2(θ/2)
그래프를 이용 , y- 절편으로부터 분자량 (M), 초기기울기로부터 rg 를 결정 . y-
절편 =1/M, 초기기울기 = (16π2/3Mλ2) rg2
• 각 slice 가 monodisperse 하다고 가정하고 평균분자량과 평균크기를 계산 . 따라서 높은 분리도가 요구됨 ( 분리방법선택 및 분리최적화가 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)<rg
2>i]/Σ(ci/Mi)
2.Wt-average: <rg2>w= Σ(ci<rg
2>i)/Σci
3.Z-average: <rg2>z=Σ(ciMi<rg
2>i)/Σ(ciMi)
Stand-alone vs. On-line MALS
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 2sin . 2
vsR
Kc
For each slice. Determine M from intercept (intercept = 1/M), rg from slope (slope = )2
2
2
3
16gr
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 2sin . 2
vsR
Kc 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 int
ercept and slope.
(2) On-line mode: determine Kc/Rθ for each slice ( calculate M). Considering each slic
e is narrow distribution, let Mw ( Mi, from which average MW's can be calculated (as le
arned 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., les
s than 200K of linear random polymer)
• RALLS combined with differential viscometer (commercially available from Visc
otek, "TRISEC")
<TRISEC 이용 방법 >
Assume Pθ = 1 and A2 = 0. Determine Mest.
Kc
RMest
BcMPR
Kc2
11 From
RG can be obtained using the Flory-Fox equation: 31
6
1
MRG
[η] is determined by differential viscometer, and M determined in step 2.
Calculate new MW by 90
P
MM est
est
Go to step 2. Repeat until Mest does not change.
sin4
x where,12 2
1
2 go
ox rn
xex
P
Calculate P(θ=90).
<Light scattering experiment 에 필요한 상수들 >
2sin
3
1612
1 222
2
grBc
MR
Kc
에서 K 와 B 를 제외한 모든 parameter 는 이미 알고 있다 . 그런데
2
4
32
dc
dnn
NK o
avo
이므로 다음 세 개의 상수가 필요 .
1. n: 용매의 refractive index
2. dn/dc : Specific refractive index increment
3. B: 2nd virial coefficient (Static mode 에서는 B 를 실험에 의해 결정할 수 있기 때문에 Static mode 는 제외 ).
1. 용매의 Refractive Index
거의 모든 용매에 대해 RI 값들이 알려져 있음 .
자주 쓰이는 용매들 (R 가 감소하는 순 )
Solvent RI R x 106 [cm-1]
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
• 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.
• Extrapolation 에 관한 reference: Johnson, B. L.; Smith, J. "Light Scattering from Polymer solutions" Huglin, M. B. ed., Academic press, New York, 1972, pp 27
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
• DRI 를 이용
• 몇 가지 다른 농도에서 (n2-n1) 을 측정 (recommended conc. = 2, 3, 4, 5 x 10-3 g/m
L) → (n2-n1)/c2 vs. vs. c2 를 plot → zero concentration 으로 extrapolate → dn/dc는 intercept 로 부 터 구한다 .
02
12
cc
nn
dc
dn
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 그래프의 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/RI 를 이용
이미 배운 바와 같이 iRi cdc
dnkR
Ri = detector signal at the slice I
kR = RI const
ci = conc. (g/mL) of the slice i)
먼저 dn/dc 를 아는 표준시료를 주입하여 kR· 을 계산 :
stdstd
stdR
cdc
dn
Areak
→ 시료를 주입 , dn/dc 계산 : 시료
시료
시료 ck
Area
dc
dn
R
• 문헌이나 실험에 의해 구할 수 없는 경우 estimate 을 할 수도 있다 .
• extrapolate to desired wavelength:
k
kdc
dn
혹은
2) polymer 와 용매의 refractive index 로 부터 estimate: 122 nndc
dn
여기에서 n2 는 polymer 의 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 의해 측정 (stand-alone Light scattering)
• 2nd Virial Coefficient 는 Solute-Solvent interaction 의 척도 .
+: Polymer-solvent interaction, good solvent (the higher, the better solvent).
0: Unperturbed system
-: Polymer-polymer interaction, poor solvent.
• A2 는 분자량의 함수 : A2 = b M-a log A2 vs. log M 은 직선 . 보통 기울기는 음수 , 즉 분자량에 반비례 .
• dn/dc 와 A2· 의 중요성에 관한 참고문헌 : S. Lee, O.-S. Kwon, "Determination of
Molecular Weight and Size of Ultrahigh Molecular Weight PMMA Using Thermal Field-
Flow 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, A2 가 필요 .
• As dn/dc increases, calculated MW decrease, calculated mass decrease, and no effect on calc
ulated 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 recomme
nded 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 usuall
y 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 i
mportant to use the same sensitivity setting of RI detector as that used when the detector was calibr
ated.
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 differ
ential 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 MALS 는 약 10 nm (about 100K MW)
• Inter-detector volume must be known accurately.
Comparison of online LS vs. viscometer
LS Viscometer
MWD 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.
[η] distribution indirect from universal calibration
direct, independent of separation mechanism
RG 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 [η ] vs. M plot (M-H coefficients can be obtained) RG vs. M plot.
Branching g obtained directly from MALS, indirectly from LALLS & universal calibration
g' obtained directly
heterogeneous samples
limited because of dn/dc uncertainty
directly applicable with univ. calib., but the change in dn/dc will affect DRI responses
Lower MW detectability
~2K. depends on dn/dc and polydispersity
as low as 300-400 has been reported
Response to particle
contamination
LALLS: highly sensitive, MALLS: less sensitive
Insensitive
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 center 로부터 산란 되는 빛의 세기의 합 (algebraic summation).
• 이러한 algebraic summation 의 관계는 각 입자들이 random 하게 array 되어있고 , 또한 p
hase relationship 이 scattering volume dimension 에 비해서 훨씬 작은 공간에 국한됨으로써 모든 interference effect 들이 average-out 되기 때문에 성립되는 것이다 .
• Scattering volume dimension 이 작을 때에는 , 산란광의 세기는 각 scattering center 로 부터 산란 되는 빛이 서로 어떻게 interfere (constructive or destructive) 하느냐에 따라 달라지며 따라서 입자들의 상대적인 위치에 따라 달라진다 .
• 각 입자들은 Brownian motion (diffusion) 에 의해 계속 움직이므로 입자들의 상대적인 위치 또한 계속 움직인다 . 따라서 측정되는 산란광의 세기는 시간에 따라 fluctuate 한다 .
• Fluctuate 하는 속도는 입자들의 diffusion rate 에 의존 (diffusion rate 이 빠를수록 빠르게 fl
uctuate).
• nanometer 에서 micron 범위의 크기를 가지는 입자들이 물의 viscosity 와 비슷한 viscosity
를 가지는 media 에 disperse 되어 있을 때 , 산란광의 세기의 변화 시간 (fluctuation) 은 mi
crosecond 내지 millisecond 이다 .
• 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 Stokes-Einstein equation, D=(kT)/(6πηd), where k = Boltzman's constant, d= equivalent spherical hydrodynamic diameter. 따라서 diffusion coefficient (D) 를 결정함으로써 입자 크기 ( 혹은 분자량 ) 을 결정할 수 있다 .
• DLS 실험을 할 때에는 정해진 시간 동안 계속해서 일정한 시간 간격 (τ = time interval) 에서 산란광의 세기를 측정한다 . 입자들의 위치가 변화하는 시간에 비해서 τ 가 작을 때 , I(0) 와 I(τ) 는 같다 . 만약 짧은 시간 interval 을 두고 계속해서 I(0) 와 I(τ) 를 측정할 때 intensity product, I(0)I(τ) 의 평균값은 <I2(0)>, 즉 average of the square of the instantaneous intensity 와 같아진다 - 이때 "I(0) 와 I(τ) 는 correlate 되어있다 " 라고 한다 . 입자들의 위치가 변화하는 시간에 비해서 τ 가 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 un-correlate 되어있다 " 라고 한다 . 이때에는 intensity product, I(0)I(τ) 의 평균값은 단순히 <I2>, 즉 square of the long-time averaged intensity 가 된다 . 입자들의 위치가 변화하는 시간에 비해서 τ 가 작지도 크지도 않을 때 , "I(0) 와 I(τ) 는 부분적으로 correlate 되어있다 ".
• 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 Correlation 의 정도를 결정할 수 있다 . correlation 의 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) rem
aining 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 deca
y constant, .
• Usefulness of G(t): directly relatable to the particle diffusivity
• For monodisperse samples,
2AeAG o , where Ao = background signal, A: instrument constant,
2 =constant decay DQ
d
kT= D
6t coefficiendiffusion =
2sin
4 = vector scattering
Q
실험 과정
실험에 의해 다양한 interval 에서 autocorrelation function, G(τ) 를 얻는다
G(τ) vs. τ 의 그래프를 얻는다Exponential function 을 이용하여 G(τ) 를 fit 한다 .
2AeAG o 을 이용 , Γ 를 결정
T
T dtItIT
G0
1lim 를 이용 , G(τ) 를 계산 .
2DQ 을 이용 , D 결정
Rh 를 이용 , 분자량 결정
정리하면 : Measure I(τ) at various G(τ) → dD
참고 : DLS 의 응용은 입자들의 diffusion 이 서로 방해를 받지 않는 묽은 dispersion ( ≤0.03) 인 경우에 국한됨 . = volume fraction of suspended spheres.
, where N = Avogadro's no., M = MW, Vh = hydrodynamic vol.). Infinite dilution D 값을 얻기 위해서는 보통 ≤ 0.005 가 만족 되어야 한다 .
D
kTa
6Stokes-Einstein 공식
을 이용하여 입자크기를 결정(a = 입자반경 or hydrodynamic radius, R
h)
hVM
Nc
구형 입자의 경우 ,
참고 : Narrow, mono-modal distribution 시료의 경우 , "method of cumulant" 를 이용 , 다음과 같이 표현할 수 있다 .
242
2ln Q
BAQG A, B - coefficients related to the moments of the size distri
bution, f(a).
• 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 samp
le polydispersity.
참고 : Polydispersed 시료의 경우 :
daaIaf
daeaIafG
QaD
,
,22
으로 표현된다 .
여기에서 f(a) = distribution function, I(a,θ) = scattering intensity function for RGD spheres.
PC 를 이용 , normal 혹은 log-normal distribution function 을 G(τ) 에 fit 한다 .
6
5
6 a
akTA
and 1
2
5
46
2
a
aa
A
BFor spherical Rayleigh scatterer, 으로 주어짐 .
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 conver
sion 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 value
s 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 mediu
m is need for unimodal analysis.
MRD h
• 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.
DLS summary
< 참고 >Particle Size Conversion Table
Mesh size Approximate μ size
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