1. Chapter 12. Light scattering (determination of MW without calibration) Electromagnetic radiation 과 물질과의 상호작용의 결과 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)

2. 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 (λ o ). 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 < 0.05λ o, less than 5% of λ o ). 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.

3. Constant, K 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: λ o = 입사광파장, dn/dc = refractive index increment n o : 용매의 refractive index, π= 삼투압, c= 시료농도 [g/mL] Rayleigh scattering 에 의한 산란광의 세기는 측정 위치에 따라 변한다 : (1+cos 2 θ) 에 비례하고, scattering center 와 observer 사이의 거리 (r) 의 제곱에 반비례.

4. I θ is inversely proportional to λ o. 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” R θ measured 얻고자 하는 정보 포함 Two ways to access the light scattering information experimentally: 1.Turbidimeter (or spectrophotometer) 2.Light scattering

5. 1. Turbidimeter experiment (Transmitted light intensity, I t is measured) "Turbidity", τ = fraction of incident light which is scattered out = 1-(It/Io) τ is obtained by integrating I θ over all angles:  Substitute: Solution is dilute, so higher order concentration terms can be ignored.

6. Procedure: Measure τ at various conc.  Plot Hc/T vs. c (straight line)  Determine M from intercept, 2nd virial coeff., B from slope

7. 식 6 을 식 4 에 대입 : * 반경이 파장의 약 5% (λ/20) 이하인 경우에 국한됨 – “Rayleigh limit” 2. Light Scattering experiment (measure I θ at certain θ and r)

8. The slope of the plot can be either positive or negative. θ-condition 에서 기울기 =0.

9. For polydisperse sample, Turbidity ( 혹은 light scattering) is contributed by molecules of different MW. Define: τ i = 분자량 M i 를 갖는 분자들에 의한 turbidity → (Hc)/τ vs. c 그래프의 절편 =1/M 이므로 따라서 turbidity 나 light scattering 실험에서 얻는 분자량은 weight-average MW 이다.

10. 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

11. a = 반경 Q = scattering vector = (4π/λ)sin(θ/2)r g (10) 구형입자의 경우 : 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 P θ = "shape factor" or "form factor" Always P θ < 1, function of size and shape of scattering volume. Now we start seeing the angle dependence of the scattered light !

12. p(θ) decreases with θ. p(θ) decreases more for higher MW.

13. Effect of MW and Chain Conformation on P θ, and on measured MW at 90 o. ConformationMW (g/mol)R G (nm)P(90 o )MW(90 o ) Random coil Polystyrene51K80.9851K Polystyrene(  condition)420K190.95400K PMMA680K360.70480K Polyisoprene(~70% cis)940K480.56530K Spherical Bovine serum albumin66K31.0066K Bushy stunt virus10700K120.9810500K Rod shaped Poly- -benzyl-L-glutamate130K260.91118K Myosin493K470.74365K DNA4000K1170.351400K

14. [Case 2] c→0: 두 가지 극한 상황 : Plot Kc/R θ vs. c: y- 절편 =1/M, 기울기 =2A 2 Plot Kc/R θ vs. sin 2 (θ/2): y- 절편 =1/M, 기울기 = (16π 2 /3Mλ 2 ) r g 2 Three information! [Case 1] θ→0: Random coil 고분자의 경우, 식 (11) 을 식 (9) 에 대입한 후 식 (7’) 에 대입 : Final Rayleigh equation for random coil polymer

15. 빛산란 실험 방법 (1) 다양한 각도와 농도에서 R θ 측정. (2) Kc/R θ vs. c, Kc/R θ vs. sin 2 (θ /2) plot 작성. (3) θ =0 와 c =0 로 extrapolate. Kc/R θ vs. sin 2 (θ /2) Kc/R θ vs. c Zimm plot: 채워진 점 : 실험 데이터. 빈 점 : extrapolated points

16. Cases 1. Small polymers: 각도의존성 없음. (Horizontal line) - 다섯 농도에서 측정한 데이터. - Mw 와 A 2 결정 가능 - 분자크기 측정 불가능. Zimm plot for PMMA in butanone λ o =546 nm, 25 ℃, n o ~1.348, dn/dc = 0.112 cm 3 /g (Kc/R θ ) vs. c -Calculated values : M w = 66,000 g/mol A 2 = 0 mol cm 3 /g 2 - Kc/R θ 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 isopropanol λ o =436 nm, 25 ℃, n o ~1.391, dn/dc = 0.125 cm 3 /g θ-solvent : A 2 =0 가 되는 용매, 고분자 - 고분 자, 고분자 - 용매분자간 상호작용의 에 너지가 동일, 이상용액과 같이 행동.

17. 3. Larger polymers in good solvent: 각도 및 농도에 의존. Zimm plot of polystyrene in toluene λ o =546 nm, 25 ℃, n o ~1.498, dn/dc = 0.110 cm 3 /g 4. Polymers in poor solvent: A 2 가 음수가 됨 ( 큰 음수는 될 수 없음. 더 이상 녹지 않기 때문 ) Zimm plot of polybutadiene in dioxane λ o =546 nm, 25 ℃, n o ~1.422, dn/dc = 0.110 cm 3 /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 2x10 5 이상의 경우, Kc/R θ 는 양의 기울기 (A 2 = 양수 ) 를 가진다. - Athermal Condition - No effect of temperature on polymer structure

18. Stand-alone mode: LS instrument is used itself. Zimm plot 을 이용 M, A 2, R θ 를 결정 LS instrument is used as a detector for a separator. c=0 이라 가정. 각 slice 에 대해 Kc/R θ vs. sin 2 (θ/2) 그래프를 이용, y- 절편으로부터 분자량 (M), 초기기울 기로부터 r g 를 결정. y- 절편 =1/M, 초기기 울기 = (16π 2 /3Mλ 2 ) r g 2 각 slice 가 monodisperse 하다고 가정하고 평 균분자량과 평균크기를 계산. 따라서 높 은 분리도가 요구됨 ( 분리방법선택 및 분리최 적화가 요구됨 ). Average Molecular Weights 1.No-average: M n =(Σc i )/(Σ(c i /M i )) 2.Wt-average: M w =Σ(c i M i )/ Σ(c i ) 3.Z-average: M z = Σ(c i M i 2 )/Σ(c i /M i ) Average Sizes (mean square radii) 1.No-average: n = Σ[(c i /M i ) i ]/Σ(c i /M i ) 2.Wt-average: w = Σ(c i i )/Σc i 3.Z-average: z =Σ(c i M i i )/Σ(c i M i ) Stand-alone vs. On-line MALS

19. 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, R g (2) On-line mode: Assume c=0, Plot For each slice. Determine M from intercept (intercept = 1/M), r g from slope (slope = ) Assuming each slice is narrow distribution, M w  M i Average M can be calculated. It is therefore very important to have a good resolution.

20. TALLS (Triple Angle): I  is measured at 45 o, 90 o, and 135 o Not useful when the plot of deviates from linearity

21. Angular Dependence of Kc / R  ( 시료 = high molecular weight DNA)

22. Effect of Particles/Gels on Light Scattering Measurement Note the delicacy of extrapolation to zero angle from larger distances.

23. DALLS (Dual Angle): I θ  is measured at 15 o and 90 o LALLS (Low Angle): Iθ is measured at one low angle (assume:  = 0) (1)Static mode: measure LS at a few c  Plot Kc/R θ vs. c  Determine M and B from intercept and slope. (2)On-line mode: determine Kc/R θ 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) I θ 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")

24. Assume P θ = 1 and A 2 = 0. Determine M est. R G can be obtained using the Flory-Fox equation: [η] is determined by differential viscometer, and M determined in step 2. Calculate new MW by Go to step 2. Repeat until M est does not change. Calculate P(θ=90).

25. 에서 K 와 B 를 제외한 모든 parameter 는 이미 알고 있다. 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 mode 에서는 B 를 실험에 의해 결정할 수 있기 때문에 Static mode 는 제외 ).

26. 1. 용매의 Refractive Index 거의 모든 용매에 대해 RI 값들이 알려져 있음. 자주 쓰이는 용매들 (R  가 감소하는 순 ) SolventRIR  x 10 6 [cm -1 ] Carbon disulfide1.620757.5 a-chloronaphthalene (140 o C)1.532352.8 1,2,4-Trichlorobenzene (135 o C)1.50235.7 Chlorobenzene1.518718.6 o-Xylene (35 o C)1.5015.5 Toluene1.4914.1 Benzene1.5012.6 Chloroform1.4446.9 Methylene chloride1.42236.3 Carbon tetrachloride1.466.2 Dimethyl formamide1.43 (589 nm)5.6 Cyclohexane1.4255.1 Cyclohexanone1.44664.7 Methyl ethyl ketone1.384.5 Ethyle acetate1.374.4 THF1.414.4 Acetone1.364.3 Dimethyl sulfoxide1.478 (589 nm)4.1 Methanol1.332.9 Water1.331.2 Except where otherwise noted, all measurements made at λ= 632.8 nm and T=23 o C. 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

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 (n 2 -n 1 ) 을 측정 (recommended conc. = 2, 3, 4, 5 x 10 -3 g/mL) → (n2-n1)/c2 vs. vs. c2 를 plot → zero concentration 으로 extrapolate → dn/dc 는 intercept 로 부 터 구한다.

28. For concentration ranges generally used, the refractive index difference, n 2 -n 1, is a linear function of concentration. In other words, (n 2 -n 1 )/c 2 is constant. 즉 (n 2 -n 1 )/c 2 vs. c 2 그래 프의 기울기 =0. This means that (n 2 -n 1 ) needs to be measured for only one or two different concentrations. If (n 2 -n 1 )/c 2 shows no significant dependence on c, then dn/dc can be obtained by averaging (n 2 -n 1 )/c 2 values

29. SEC/RI 를 이용 이미 배운 바와 같이 R i = detector signal at the slice I k R = RI const c i = conc. (g/mL) of the slice i) 먼저 dn/dc 를 아는 표준시료를 주입하여 k R · 을 계산 : → 시료를 주입, dn/dc 계산 : 문헌이나 실험에 의해 구할 수 없는 경우 estimate 을 할 수도 있다. 1)extrapolate to desired wavelength: 혹은 2) polymer 와 용매의 refractive index 로 부터 estimate: 여기에서 n 2 는 polymer 의 partial specific volume [mL/g] 이다. 보통 n 2  1. 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.

30. 3. Virial Coefficient, B or A 2 문헌에서 구할 수 있음 ( 예 : Polymer Handbook). 문헌에서 구할 수 없는 경우 실험에 의 해 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 A 2 vs. log M 은 직선. 보통 기울기는 음수, 즉 분자량에 반비례. dn/dc 와 A 2 · 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.

31. Light scattering 실험을 할 때 고려 해야 할 점들 (concerns) 정확한 dn/dc, RI constant, A 2 가 필요. As dn/dc increases, calculated MW decrease, calculated mass decrease, and no effect on calculated R G. As RI constant increases, calculated MW decreases, calculated mass increases, and no effect on R G. As A 2 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.

32. 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

33. 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. M n (exp)>M n (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 R G with MALS 는 약 10 nm (about 100K MW) Inter-detector volume must be known accurately.

34. Comparison of online LS vs. viscometer LSViscometer MWDAbsoluteRelative need precise n and dn/dcUniversal 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] distributionindirect from universal calibration direct, independent of separation mechanism RGRG direct from MALS (limited to >10 nm) indirect from universal cal. and Flory-Fox eqn. applicable to linear molecules only Chain conformationMALLS: R G vs. M plot[η ] vs. M plot (M-H coefficients can be obtained) R G vs. M plot. Branchingg 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

35. Information Content PrimarySecondary LALLSM MALLSMRGRG PCSDR h, M Viscometer[η ]M, R G Primary information: high precision and accuracy, insensitive to SEC variables, requires no SEC column calibration.

36. Features: MWD measured by LS IVD measured by Viscometer

37. 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

38. 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 phase 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 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 fluctuate 한다. Fluctuate 하는 속도는 입자들의 diffusion rate 에 의존 (diffusion rate 이 빠를수록 빠르게 fluctuate). 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 microsecond 내지 millisecond 이다.

39. 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 분자 량 ) 을 결정할 수 있다. DLS 실험을 할 때에는 정해진 시간 동안 계속해서 일정한 시간 간격 (τ = time interval) 에서 산란광 의 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 평균 값은, 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 correlate 되어있지 않다 " 혹은 "I(0) 와 I(τ) 는 un- correlate 되어있다 " 라고 한다. 이때에는 intensity product, I(0)I(τ) 의 평균값은 단순히, 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 ".

40. 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” = : 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,, where A o = background signal, A: instrument constant,

41. 실험 과정 실험에 의해 다양한 interval 에서 autocorrelation function, G(τ) 를 얻는다 G(τ) vs. τ 의 그래프를 얻는다 Exponential function 을 이용하여 G(τ) 를 fit 한다. 을 이용, Γ 를 결정 를 이용, G(τ) 를 계산. 을 이용, D 결정 R h 를 이용, 분자량 결정 정리하면 : Measure I(τ) at various   G(τ) → 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, V h = hydrodynamic vol.). Infinite dilution D 값을 얻기 위해서는 보통  ≤0.005 가 만족 되어야 한다. Stokes-Einstein 공식 을 이용하여 입자크기를 결정 (a = 입자반경 or hydrodynamic radius, R h ) 구형 입자의 경우,

42. 참고 : Narrow, mono-modal distribution 시료의 경우, "method of cumulant" 를 이용, 다음과 같이 표현할 수 있다. A, B - coefficients related to the moments of the size distribution, f(a). 여기에서 a n = n th 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 : Polydispersed 시료의 경우 : 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. and For spherical Rayleigh scatterer, 으로 주어짐.

43. 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. 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

44. Particle Size Conversion Table Mesh sizeApproximate  μ size 44760 63360 82380 121680 161190 20840 30590 40420 50297 60250 70210 80177 100149 140105 20074 23062 27053 32544 40037 62520 125010 25005