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

240

250

260

270

280

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