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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.
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Electromagnetic radiation 과 물질과의 상호작용의 결과
Oscillating electric field of incident beam interacts with scattering center, induces a synchronous oscillating dipole, which re-emits electromagnetic energy in all directions.
Rayleigh scattering에 의한 산란광의 세기는 측정 위치에 따라 변한다: (1+cos2θ)에 비례하고, scattering center와 observer사이의 거리(r)의 제곱에 반비례.
λo = 입사광파장, dn/dc = refractive index increment
no: 용매의 refractive index, π=삼투압, c=시료농도[g/mL]
얻고자 하는 정보 포함
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.).
"Turbidity", τ = fraction of incident light which is scattered out = 1-(It/Io)
1. Turbidimeter experiment (Transmitted light intensity, It is measured)
Solution is dilute, so higher order concentration terms can be ignored.
Procedure: Measure τ at various conc. Plot Hc/T vs. c (straight line) Determine M from intercept, 2nd virial coeff., B from slope
식6을 식4에 대입:
*반경이 파장의 약 5% (λ/20) 이하인 경우에 국한됨 – “Rayleigh limit”
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 →
따라서 turbidity나 light scattering실험에서 얻는 분자량은 weight-average MW이다.
Rayleigh-Gans-Debye (RGD scattering) : when the scattering centers are larger than Rayleigh limit
Different part of more extended domain (B) produce scattered light which interferes with that produced by other part (A) - constructive or destructive
Q = scattering vector = (4π/λ)sin(θ/2)rg (10)
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"
Always Pθ < 1, function of size and shape of scattering volume. Now we start seeing the angle dependence of the scattered light !
Random coil 고분자의 경우,
Final Rayleigh equation for random coil polymer
두 가지 극한 상황:
Plot Kc/Rθ vs. c: y-절편=1/M, 기울기=2A2
Plot Kc/Rθ vs. sin2(θ/2): y-절편=1/M, 기울기= (16π2/3Mλ2) rg2
(1) 다양한 각도와 농도에서 Rθ측정.
(2) Kc/Rθ vs. c, Kc/Rθ vs. sin2(θ /2) plot 작성.
(3) θ =0와 c =0로 extrapolate.
Kc/Rθ vs. sin2(θ /2)
Kc/Rθ vs. c
채워진 점: 실험 데이터.
빈 점: extrapolated points
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
- 다섯 농도에서 측정한 데이터.
- Mw 와 A2결정 가능
- 분자크기 측정 불가능.
2. Small polymers in θ-solvent: 각도 및 농도 의존성 없음.
θ-solvent: A2=0가 되는 용매, 고분자-고분자, 고분자-용매분자간 상호작용의 에너지가 동일, 이상용액과 같이 행동.
Zimm plot of poly(2-hydroxyethyl methacrylate) in isopropanol
λo=436 nm, 25℃, no ~1.391, dn/dc = 0.125 cm3/g
Zimm plot of polystyrene in toluene
λo=546 nm, 25℃, no ~1.498, dn/dc = 0.110 cm3/g
- 분자량 약 2x105이상의 경우,
Kc/Rθ는 양의 기울기 (A2=양수)를 가진다.
- Athermal Condition - No effect of temperature on polymer structure
4. Polymers in poor solvent: A2가 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
For each slice. Determine M from intercept (intercept = 1/M), rgfrom slope (slope = )
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
Assuming each slice is narrow distribution, Mw Mi
Average M can be calculated. It is therefore very important to have a good resolution.
deviates from linearity
Angular Dependence of Kc / R(시료= high molecular weight DNA)
Note the delicacy of extrapolation to zero angle from larger distances.
Assume Pθ = 1 and A2 = 0. Determine Mest.
[η] is determined by differential viscometer, and M determined in step 2.
Calculate new MW by
Go to step 2. Repeat until Mest does not change.
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. c2그래프의 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
kR = RI const
ci = conc. (g/mL) of the slice i)
→ 시료를주입, dn/dc 계산:
2) polymer와 용매의 refractive index로 부터 estimate:
여기에서n2는polymer의partial specific volume [mL/g]이다. 보통n2 1.
Light scattering 실험을 할 때 고려 해야 할 점들 (concerns)
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.
Chemical heterogeneity within each slice leads to non-defined dn/dc → Quantitation of chemical heterogeneous samples is very difficult.
Primary information: high precision and accuracy, insensitive to SEC variables, requires no SEC column calibration.
Both Viscometer and LS are insensitive to experimental conditions and separation mechanism
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)“
를이용, G(τ) 를계산.
(a = 입자반경or hydrodynamic radius, Rh)
실험에의해다양한interval에서autocorrelation function, G(τ)를얻는다
G(τ) vs. τ의그래프를얻는다
Exponential function을 이용하여 G(τ)를 fit한다.
정리하면: Measure I(τ) at various G(τ) →
참고: DLS 의응용은입자들의diffusion이서로방해를받지않는묽은dispersion ( ≤0.03)인경우에국한됨. = volume fraction of suspended spheres.
, where N = Avogadro's no., M = MW, Vh = hydrodynamic vol.). Infinite dilution D값을얻기위해서는보통≤0.005가만족되어야한다.
A, B - coefficients related to the moments of the size distribution, f(a).
For spherical Rayleigh scatterer,
여기에서f(a) = distribution function, I(a,θ) = scattering intensity function for RGD spheres.
PC를이용, normal혹은log-normal distribution function을G(τ)에fit한다.
참고: Narrow, mono-modal distribution 시료의경우, "method of cumulant"를이용, 다음과같이표현할수있다.
Particle Size Conversion Table