Common Techniques for the Determination of Molecular Weight

Γ ² 6 1. 52.

Here we see that translational diffusion leads to a linear increase of the decay rates with q², whereas the contribution of rotational motion is irrespective to the scattering angle. At small scattering angles large distances within the sample are probed. Therefore time correlations are present for longer lag times τ, since the particles need more time to rearrange within the probed dimensions. In contrast internal motions are irrespective to the probed distances.

For polydisperse systems the analysis can be modified, since the decay in the autocorrelation function is a superposition of single exponential decays from different species:

∫

⋅ Γ

− Γ

⋅ Γ

=

0 )

1

( ( , ) G( ) e d

g τ θ ^τ 1. 53.

G(Γ) is a probability density function. There are two established procedures for its analysis:

Cumulant and CONTIN. CONTIN is a modified Laplace transformation of g⁽¹⁾(τ) and is therefore a suitable method to determine multimodal size distributions. The primary distributions are intensity weighted distributions. The cumulant method is a polynomial fitting procedure of ln(g⁽¹⁾(τ)) = A + B ^. τ+ C ^.τ ² + D^. τ³, which gives the average hydrodynamic radius along with its polydispersity (A, B, C and D are fitting parameters).

1.4.4. Common Techniques for the Determination of Molecular Weight Characterization in terms of molecular weight is a prerequisite before starting investigations of other physical properties. A structure-behavior relationship can only be obtained after careful characterization of the prepared macromolecules. Therefore a short, schematic description is given for the most common methods used in this thesis.⁵⁰

1.4.4.1. Gel Permeation Chromatography

Size Exclusion Chromatography (SEC or GPC, which stands for Gel Permeation Chromatography) is a suitable relative method to determine the molecular weight distribution and therefore molecular weight averages.

The SEC column separates different macromolecules with regard to their size and therefore regarding their hydrodynamic volume, VH. VH is dependent on molecular weight M, so the elution-times correlate with the molecular weight (see equation 1. 55.).

Flow

Figure 1. 13: Principle of seperation inside a GPC column

Smaller molecules spend more time in the pores of the separating column-gel than larger sized particles. Therefore the elution-volumes Ve are smaller for expanded molecules. If the molecules are too large, none of them will fit into the pores. They can only use the space between the resin-beads Vv. Small molecules can however occupy both Vv and Vi (space in the pores). So every fraction has its separation constant KSEC:

Ve = Vv + KSEC Vi 1. 54.

The exact elution volume of a polymer is dependent on the chemical nature of the polymer, the solvent and the polymer’s topology. A prerequisite for meaningful results is the good solubility of the polymer in the solvent and the absence of adsorption of the polymer on the column. Other parameters are the length, width of the column, flow rate and temperature. The flow rate needs to be optimized, aiming for best separation and low amount of axial dispersion (diffusion along the column axis). Due to the many parameters it is crucial to calibrate the setup with polymers of known molecular weight, keeping all other parameters constant. The elution times of almost monodisperse samples are recorded to arrange a semi-logarithmic calibration plot. This is suitable as within certain limits a semi-logarithmic equation has been found empirically.

log M = K´ - k´ Ve 1. 55.

We mentioned that VH is the separation parameter and according to Einstein equation VH is proportional to [η] M.^{15, 16} This gives the opportunity for universal calibration:

log VH ~ log ([η] M) = log (K ^. M^α+1)= K* - k* Ve 1. 56.

During calibration with polymers, whose Kuhn-Mark-Houwink parameters K and α are known,¹⁴ each elution volume Ve can be assigned a certain hydrodynamic volume Vh. Once a universal calibration is established, one need to know only the Kuhn-Mark-Houwink parameters K and α of another polymer, to obtain the real molecular weight distribution of that polymer without performing a new calibration. The only drawback: Kuhn-Mark-Houwink equation¹⁴ holds only true for polymers larger than 20000 g/mol. The most elegant way making use of the universal calibration is the use of a viscosity detector, which detects the specific viscosity ηsp of any fraction. Knowing the mass concentration c in each fraction by use of concentration sensitive detectors (UV or RI), the intrinsic viscosity [η] is approximated with the reduced viscosity ηred = ηsp/c.

Other useful GPC setups include GPC-light-scattering or GPC-MALDI-ToF coupling. Both give directly molecular weight distributions without any calibration, as long as the column guarantees effective separation of different species.

1.4.4.2. Osmometry

Number averaged molecular weights Mn are obtained by osmometry. The result is irrespective to the shape of the polymer, since osmometry simply counts the number of molecules and is therefore an absolute method. Only the choice of solvent and membrane requires careful considerations. The solvent needs to be chosen in a way that the polymer is not charged up by reactions involving the solvent (e.g. avoid protic solvents for PDMAEMA).

Also aggregation must not occur in order to obtain the molecular weight of the single chains.

The molecular weight cut-off, MWCO, of the membrane should be considerably smaller than the molecular weight at the onset of the molecular weight distribution. For details on the theory please refer to chapter 1.4.2.

1.4.4.3. Static Light Scattering

Static light scattering (SLS) gives the weight-average molecular weight, Mw, besides the second virial coefficient, A2, of the osmotic pressure (see chapter 1. 4. 2.) and the radius of gyration, Rg. The oscillating electric field of light polarizes the illuminated medium. The electrons of the molecules are shifted compared to the nuclei. This results in a dipole,

oscillating with the same frequency than the incident beam. On the other hand the dipoles themselves act as transmitter of light with the same frequency. For polarized light the transmitter transmits light perpendicular to the dipole axis, irrespective to the orientation of the incident beam. This behaviour is called scattering. The scattered intensity is irrespective to the scattering angle θ, when the detection plane is perpendicular to the field vector of polarized light. This holds strictly true for scatterers, which are much smaller than the wavelength of light. For larger scatterers, the angular dependence of the scattered light is altered. Destructive interference leads to lower intensities at higher scattering angles. This can give information on the dimensions of the scatterer.

Θθ

Figure 1. 14: Principle of light scattering on nano-scaled objects

The more single scatterers are combined in one object the higher is its contribution to the total scattered intensity especially at low scattering angles. Therefore the molecules are weighed by light scattering, which leads to the weight-average molecular weight Mw.

The theoretical background of the scattering is given by the terminological description of the radiation characteristics of an oscillating dipole, which leads to the Rayleigh ratio R(θ) for non polarized light:⁸¹

4 0 2 4 2

0 2

/ ) 8

cos 1 ) (

( π α λ

θ θ =

= + I

r

R I^S 1. 57.

I0 is the intensity of the incident beam. IS assigns the intensity of the scattered light, detected at a distance r from the sample under a scattering angle θ. α assigns the polarizability of the scatterer for the wavelength λ .

In the frequency-averaged light scattering theory of Einstein,⁸² the sample solution is divided into many small volumes, to account for fluctuation in polarizability of those volumes.

Those fluctuations are vital to obtain a net scattering, since a microscopically homogenous solution cannot yield any net scattering due to destructive interference of the scattered light (a scattered beam always encounters another beam with the necessary phase-shift to obtain destructive interference). The fluctuations of the polarizability of those small volumes are caused by concentration fluctuations. Pressure or temperature fluctuations are canceled out, when the excess intensity ISexcess (= ISsolution - ISsolvent) instead of IS enters the scattering equations. Those fluctuations obey Boltzmann statistics and the extent of those fluctuations are coupled to the extent of the change of the solvent’s chemical potential with polymer’s concentration (δμ/δcm.,P) compared to thermal energy kT. Non-ideal behavior of the chemical potential is accounted with the introduction of the second virial coefficient A2. This yields following equation:

K is a constant, which includes the refractive index of the solvent and the refractive index increment, ∂n/∂cm.,P. To account for I0 (which can not be measured easily), the scattered intensity of a standard (toluene) is also recorded during SLS measurement.

Until now only interparticle interactions and the arrangement of particles have been addressed by introducing A2. As the intensity of the scattered radiation can be expressed by the product of form factor P(θ) (addresses the shape of a single molecule) and structure factor S(θ) (addresses the arrangement of molecules), we need to define P(θ).

As already explained, destructive interference leads to decreased intensity for large scattering angles, when particles are examined, which are larger than λ0/20. Debye^83-85 and Guinier^{86, 87} derived an equation for the form factor in dependence of the radius of gyration Rg

by use of geometrical considerations.

3 incident light; n: refractive index of medium used; θ: scattering angle) is introduced to give the final Zimm equation:⁸⁸

P m w

P

m A c

M q P q R Kc

, 2

, 1 2

) (