Kolligative Eigenschaften der Makromolekülen

Kolligative Eigenschaften (colligere = sammeln) Gefrierpunkterniedrigung, Siedepunkterhöhung, Dampfdruckerniedrigung, Osmotischer Druck Kolligative Eigenschaften von PS in Cyclohexan (c = 0,01 g/g) Molmasse M n [g/mol] 10 000 100 000 1 000 000 Osmotischer Druck bei 34 C [Pa] (cm Lösungssäule) 2,5 x 10 3 2,5 x 10 2 25 (33) (3,3) (0,3) Gefrierpunktserniedrigung [grd] 2 x 10-2 2 x 10-3 2 x 10-4 Siedepunktserhöhung [grd] 2,7 x 10-3 2,7 x 10-4 2,7 x 10-5 Kryoskopie Ebullioskopie geeginet für MM < 10 3 g/mol

chemie.uni-hamburg.de Aus der Flory-Huggins Theorie

Van t Hoff, Flory-Huggins Zur Bestimmung von M n und A 2 LM < LSG M n,1 < M n,2 A M n,1 2 > 0 M n,2 A 2 < 0 M n,2 ; A 2 > 0

Statische Lichtstreuung

What is Light Scattering? Blue sky, red sunset Automobile headlights in fog Laser beam in a smoky room Reading from an illuminated page Dust particles in beamer light

What is Light Scattering? Absolute molar mass measurement by the use different angles.

Light and its Properties Light is an oscillating wave of electric and magnetic fields Polarization: direction of electric field oscillation Intensity: I E 2 1864-1873 by J. C. Maxwell

How does light scatter? When light interacts with matter, it causes charges to polarize. The oscillating charges radiate light. How much the charges move, and hence how much light radiates, depends upon the matter s polarizability.

Index of refraction The polarizability of a material is directly related to its index of refraction n. a n 2-1 The index of refraction is a measure of the velocity of light in a material. e.g., speed of light Snell s law For solutes, the polarizability is expressed as the specific refractive index increment, dn/dc.

Adding light Incoherent sum Coherent sum Interference:

How Light Scattering measures Mw coherent: incoherent: I (E + E) = 4 E total 2 2 2 2 I total E + E = 2 E 2 I scattered dn Mc dc 2

Isotropic Scattering For particles much smaller than the wavelength of the incident light ( <10 nm for l = 690 nm), the amount of radiation scattered into each angle is the same in the plane perpendicular to the polarization.

Isotropic and anisotropic Scattering small particles D < l/20 isotropic scattering anisotropic scattering large particles D >l/20

Rayleigh-Ratio R ( ) Lord John W. S. Rayleigh (1842-1919) Studied scattering of light by particles much smaller than a wavelength, discovered strong dependence of scattering on wavelength (1/l 4 ). R ( I = I ) ( I I 2, Sovent r, Solvent) = A 0 V I0 I The Rayleight-Ratio R is the fraction of light scattered, in excess of the light scattered by the solvent times the square of the distance between the scattering center and the detector divided by the incident intensity and the volume of the cell illuminated by the laser and seen by the detector.

Charakterisierungsmethoden Statische Lichtstreuung

Angular dependence of Light Scattering detector at 0 scattered light in phase detector at, scattered light out-of-phase Intermolecular interference leads to a reduction in scattering intensity as the scattering angle increases.

Scattering Function P(θ) P( ) form factor or scattering function describes how the scattered light varies with angle. This variation is affected by <r g2 >, the mean square radius. The greater <r g2 >, the larger the angular variation. Note that P(0 ) = 1

Molar mass and radius r g < 10 nm isotropic scatterer r g > 10 nm Non isotropic scatterer

How Light Scattering measures r g To calculate the angular distribution of scattered light, integrate over phase shifts from extended particle. No dn/dc and no Concentration in formalism. Integrating over extended particle involves integrating over mass distribution.

Interpretation of r g hollow sphere: solid sphere: Random coil polymer with average end to end length L:

Basic Light Scattering Principles The amount of light scattered is directly proportional to the product of the molar mass and the molecular concentration The amount of light scattered (divided by the incident light intensity) by a solution into a particular direction per unit solid angle in excess of the amount scattered by the pure solvent is directly proportional to the product of the weight-average molar mass and the concentration. R( ), in limit as 0, Mc The variation of scattered light with scattering angle is proportional to the average size of the scattering molecules. The variation of light scattered with respect to sin 2 /2, in the limit as 0, is directly proportional to the average molecular mean square radius. dr( )/dsin 2 /2 <r g2 >

What Do We Mean By ABSOLUTE? NO Reference to standards of mass ALL parameters measured directly from 1st principles Refractive indices geometries of cell and detector wavelength concentrations detector response temperature dn/dc NO assumptions of molecular model/conformation There are 4 Absolute Methods 1) Membrane Osmometry (Number Average MM) 2) Light Scattering (Weight Average MM) 3) Sedimentation Equilibrium (Ultracentrifugation) (z-average MM) 4) Mass spectrometry

Zimm Equation The Zimm formalism of the Rayleigh-Debye- Gans light scattering model for dilute polymer solutions: K* c 1 = + R( ) M P( ) w 2 2 A c J. Chem. Phys. 16, 1093-1099 (1948) This model embodies the two principles and addresses both intermolecular scattering and intramolecular scattering.

Zimm Equation K* c 1 = + R( ) M P( ) w K* = 4p 2 (dn/dc) 2 n 0 2 N A -1 l 0-4 2 2 A c n 0 is the refractive index of the solvent N A is Avogadro s number. l 0 is the vacuum wavelength of the incident light. dn/dc is the refractive index increment, which tells how much the refractive index of the solution varies with solute concentration. c is the concentration of the solute molecules (g/ml). R( ) is the fraction of light scattered, in excess of the light scattered by the solvent times the square of the distance between the scattering center and the detector divided by the incident intensity and the volume of the cell illuminated by the laser and seen by the detector.

Zimm Equation K* c 1 = + R( ) M P( ) w 2 2 A c M w is the weight-average molar mass. A 2 is the second virial coefficient (a measure of solvent-solute interaction) (A 2 >0 good solvent for the sample). P( ) is the form factor or scattering function, telling how the scattered light varies with angle. This variation is determined by <r g2 >, the mean square radius. The bigger <r g2 >, the greater the angular variation. Note that P(0 ) = 1.

Mathematical Solution We know: We don t know: R( ), c, K*, l 0 (l=l 0 /n0), Mw, <r g2 >, A 2 Three Limits of Interest: Low concentration limit (c 0) K*c R 1 = M W P Low angle limit ( 0) K*c R 0 = 1 + 2 A2c M W Low concentration and low angle (c 0, 0) K*c = 1 R 0 M W

Plot * K c R( ) The Zimm Plot vs. sin 2 ( /2)+kc where k is a stretch factor selected to put kc and sin 2( /2) into the same numerical range. Final results are independent of this factor. Initial slope of = 0 line gives A 2 Initial slope of c = 0 line gives <r g2 > = 0, c = 0 point gives Mw

SEC gekoppelt an MALLS (multi angle laser light scattering) Detektor Absolute MM-Bestimmung SEC LS I i ~ M i RI I i ~ C i Data Processing AS1, AS2 Wyatt MMD, LCB, R g MM Molmasse MMD Molmassenverteilung LCB Lankettenverzweigung Rg Gyrationsradius

R ( I = I I LM 0. V ). R 2 Statische Lichtstreuung Absolute M w!!! Mehrwinkel-Lichtstreudetektion: Multi Angle Laser Light Scattering (MALLS) detection isotrope Streuung d << l/20 anisotrope Streuung d l/20

SEC-MALLS

Dynamische Lichtstreuung

Light scattering Static Light Scattering (SLS): R g or RMS Radius mass averaged distance of each point in a molecule from the molecule s center of gravity. lower limit 10 nm Dynamic Light Scattering (DLS) R h or Hydrodynamic Radius radius of a sphere with the same diffusion coefficient as our sample. lower limit ~ 0.5 nm R h

An Example: Lysozyme Lysozyme M w = 14,300 Da Result with DynaPro: 1.9 nm

Conformation: r h vs. r g solid sphere 3-arm star polymer = r g = 0.77 = r g 1.4 r h r h

Hydrodynamic Radius Theoretical Examples R h + H 2 O H 2 O + H 2 O + + H 2 O + H 2 O R h R h

Dynamic light scattering Quasi elastische Lichtstreuung Diffusion constant, D T Size, R h Polydispersity, PD%

DLS Instrument Sample Laser Dynamic LS: Fluctuation s in scattered Static LS: light Averaged intensity scattered light intensity

What is a QELS Experiment? Scattered light intensity is measured through time.

How QELS Works: Interference of Light Diffusion! Particles Constructive diffuse interference due to Brownian motion, resulting Destructive in light interference intensities which fluctuate with time.

Intensity Fluctuations The rate at which particles diffuse is related to their size, all other things constant.

Autocorrelation Function

Form of the Autocorrelation Linear time axis Function Log time axis R h = 9nm (latex spheres)

Autocorrelation Function Inflection point = Diffusion coefficent R h = 9 nm latex spheres Width = Polydispersity Autocorrelation function: Laser Sample I 0 D t : Diffusion coefficient I S q Scattering vector:

What affects translational diffusion? Extrinsic factors D T 1/f h Intrinsic factors D T 1/ Attached solvent and/or interparticle interactions create drag D T 1/f s Viscous solvent slows it down. and if concentration too high, viscosity effects D T 1/R Asphericity slows it down D T T Small particles move faster High temperature speeds it up

Timescale of Motion From diffusion coefficient to R h? Stokes - Einstein Relation k B T D t R h R h = k T b p 6 D Boltzmann s constant temperature (Kelvin) viscosity of solvent diffusion coefficient hydrodynamic radius t

Distribution of Particle Sizes R h = 9 nm + 50 nm particles Fitting to a single exponential yields R h = 20 nm!

Monomodal Analyzing Particle Size Distributions Monodisperse Polydisperse Cumulants: Multimodal Monodisperse Polydisperse Assumes a Gaussian distribution of diffusion constants, and fits to obtain the mean and distribution of the diffusion constants (R h ) Regularization: Attempts to fit the distribution of exponentials to obtain an approximation of the real distribution of R h.

Cumulant Example R h = 3.5 nm BSA with 7 % Dimer. Time Temp Radius %PD Mw-R (s) ( C) (nm) (kda) Acq 1 69.8 22.7 3.48 9.1 62 Acq 2 72.8 22.8 3.69 0.0 72 Acq 3 75.9 22.8 3.56 1.8 66 Acq 4 79.0 22.8 3.64 0.0 69 Acq 5 82.0 22.8 3.58 6.4 67 Acq 6 85.1 22.8 3.53 14.4 64 Acq 7 88.1 22.8 3.52 10.3 64 Acq 8 91.2 22.8 3.65 0.0 70 Acq 9 94.3 22.8 3.54 7.4 65 Acq 10 97.3 22.8 3.56 5.7 66 - High reproducibility of the fit in each aquisition. - Cumulant fit is very robust.

Regularization Example R h = 9 nm + 50 nm PS latex particles Fit result: Peak 1: R h = 7 nm, width = 2 nm Peak 2: R h = 40 nm, width = 11 nm The peak width in the regularization limits the resolution of species are less than a factor of 5 different in R h

% Intensity % Mass Light scattering intensity sample sample 60 50 40 Radius(nm): 5.98 %Pd: 16.0 %Mass: 98.9 50 40 30 Radius(nm): 5.55 %Pd: 15.6 %Mass: 98.9 30 20 20 10 10 0 1. 00 10. 00 100. 00 Radius(nm) 0 1. 00 10. 00 100. 00 Radius(nm) % Intensity and % Mass take the Mw-R model in to account.