Polymer Characterization by Analytical Ultracentrifugation

In analytical ultracentrifugation experiments, dissolved or dispersed samples inside AUC measuring cells are exposed to high gravitational fields induced by the spinning of the centrifuge rotor. It must be emphasized that, inside a rotating AUC cell, fractionation of the dissolved particles takes place according to their size and density. Thus, not only the average of physicochemical parameters can be measured, but also their distributions, such as the molar mass distribution MMD, and the PSD.

The basic information obtained in an AUC experiment is the change of sample concentration as a function of radius and time, c(r, t). This change of sample concentration basically holds for all different types of AUC experiments.

Analytical ultracentrifugation (AUC) is the leading technique for determining the molar mass, state of association in solution, and association constants of macromolecules. Even though it is little used for supramolecular polymers, this technique has already been shown to represent one of the best options in the characterization of these systems.

The most encountered applications of AUC are the determination of the average molar mass and the state of association, for the latter one AUC being the leading technique. Determination of the sedimentation coefficient is the primary method when sedimentation velocity measurements are analyzed, which also allow the obtaining of the diffusion coefficient, friction factor, and hydrodynamic radius (for spherically-shaped species).

Several techniques are frequently employed for the characterization of polymers to obtain the molar mass distribution as well as size and structural information. Obtaining reliable results requires confirmation by other techniques, so that AUC is at least a candidate in all these cases.

More problematic can be the determination of the average molar mass and especially the state of association in solution, which made AUC the leading technique for such determinations.

The molar mass distribution can be determined from gel permeation chromatography (GPC) benefiting from a fast analysis. GPC is the most frequently applied technique for determining the molar mass distribution. Once the distribution is known, both the number-average molar mass Mn and the weight-average molar mass Mw can be calculated. However, a calibration of the measurements is required which is problematic for new types of polymeric species due to the lack of suitable calibration standards.

The main advantages of AUC can be summarized as follows: it provides the absolute molar mass of species, a large range of molar masses can be analyzed (from hundreds up to several millions), weakly linked assemblies can be safely investigated, and interactions of the species with matrices or surfaces are not a limiting factor.

AUC offers several approaches for studying sedimenting species. The sedimentation velocity measurements and sedimentation equilibrium measurements are the most used ones.

There are five basic types of experiments that can be performed with an analytical ultracentrifuge. Each of these can deliver its own range of physicochemical information on samples.

In the sedimentation velocity measurements, a set of many intermediate profiles is recorded during the sedimentation process, while in the sedimentation equilibrium measurements the sedimentation–diffusion equilibrium profile is analyzed only. Analytical ultracentrifuges are equipped with optical systems which measure these sedimentation profiles, absorption optics and interference optics being common. Other experimental approaches (e.g. density gradients) as well as different optical systems have been much used.

Sedimentation equilibrium measurements allow the direct determination of average molar mass(es), state of association, association constants as well as molar mass distribution.

Sedimentation Velocity Experiment

The sedimentation velocity experiment carried out at high centrifugal fields is the most important AUC technique for macromolecule and nanoparticle characterization,especially for the measurement of particle size distributions. Here, the molecules/particles sediment according to their mass/size, density, and shape, without significant back-diffusion according to the simultaneously generated concentration gradient. Under such conditions, a fractionation of mixture components, mainly according to size, takes place, and one can detect this fractionation as a step-like or broadly distributed radial concentration profile c(r, t) in the ultracentrifuge cell.

These profiles usually exhibit an upper and a lower plateau. In most cases, these measured sedimentation profiles can be transferred into average values of s, M or dp, or into their corresponding distributions, such as MMD and PSD.

Sedimentation velocity measurements performed under ideal conditions are typically used for the determination of the sedimentation coefficient distribution, molar mass distribution (obtained by conversion of the sedimentation coefficient distribution, which conversion however leads to correct results in some particular cases only), even size distribution in the case of spherical species, as well as the average diffusion or friction coefficients.

In some situations (colloids, polyelectrolytes or some polymers), measurements under ideal conditions are not possible because the systems are not stable in the presence of salt so that non-ideal sedimentation theories must be applied.

The determinations mentioned above are currently done via numerical methods. The fundamental equation which describes the sedimentation profiles of monodisperse species, with sedimentation coefficient S and diffusion coefficient D, is the Lamm equation. The sedimentation coefficient is defined by S = v/(v2r), where v is the sedimentation velocity and v2r the centrifugal acceleration. At infinite dilution, it is

where NA is Avogadro’s number, r is the solvent density, v¯ is the partial specific volume, M is the molar mass and f is the friction coefficient.

S, D, and the molar mass M are linked by the Svedberg equation valid for non-interacting, ideally sedimenting species:

Where T is the absolute temperature and R the gas constant. For polymers an ultracentrifuge can be used to determine

their weight-average molecular masses using the modified Svedberg equation

The application of the sedimentation velocity method to calculate molecular mass distribution is extremely complex but has been successfully done in a number of instances.

Sedimentation Equilibrium In contrast to the sedimentation velocity run, a

sedimentation equilibrium experiment is carried out at moderate centrifugal fields. Here, the sedimentation of the dissolved sample is balanced by back-diffusion, according to Fick’s first law, caused by the established concentration gradient. This means that in Lamm’s Eq. ∂c/∂t = 0 is valid. After the equilibrium between these two transport processes is achieved, a radial exponential concentration gradient c(r) has formed in the ultracentrifuge cell. Therefore, the sedimentation equilibrium analysis is based on thermodynamics.

The radial concentration gradient c(r) contains information about the absolute molar mass M and the MMD of the dissolved sample, and of the interaction constants in case of interacting systems, independently of the shape of the dissolved macromolecules. An advantage is that the detection of the concentration gradient is possible without disturbing the chemical equilibrium even of weak interactions.

Schematic of a ultracentrifuge

Monodisperse (a) and polydisperse (b) polymers in a cell during ultracentrifugation.

Determination of an MMD For the determination of an MMD via s runs and a scaling

law, we now will use the well-known system polystyrene/cyclohexane at 25◦C, where also a scaling law exists:

s0 = 0.01343[S] · M0.50

Polystyrene exhibiting a molar mass of Mw = 106 000g/mol is used with cyclohexane as a solvent, because the measurements are performed with the digital XLA/I UV optics. Cyclohexane is UV-transparent at the chosen polystyrene absorption wavelength of λ = 257nm.We use the four-hole Ti rotor at 40 000rpm, and three very low concentrations, c0 = 0.2, 0.4, and 0.6g/l, as the UV optics is very sensitive.

The figure shows the XL-A/I absorption scans A(r) at 257nm as a function of the radial distance r at one concentration, c0 = 0.2g/l, and at four different, selected scanning times t = 133, 153, 173, and 193min (for the sake of clarity, not all scans at all times t are shown). The small points are the experimental values, and the dashed lines represent the smoothest curves obtained by a special spline-fitting procedure.

Now, to obtain Fig. b, all r values in Fig. a are transformed into s values with (3.2), and all A values are initially transformed into c values via Lambert-Beer’s law. These c values are then transformed by means of the equation below into radial dilution-corrected apparent integral sedimentation distribution G values:

The two prime symbols indicate that these G values are not corrected for diffusion and for concentration. The solid lines of Fig. 3.10b show some plots of G as a function of s for different scanning times t. To accomplish the diffusion correction t→∞ we use a modified correction method of Holde-Weischet both for concentration and diffusion (details elsewhere).

Finally, this g(s) curve was transferred by the scaling law into the differential MMD function w(M),which is presented in Fig. d The differential molar mass distribution w(M) is connected with g(s) via w(M) dM = g(s)ds.

Integration leads to W(M) = G(s)

with w(M) = dW(M)/dM, g(s) = dG(s)/ds and the different averages of the molar

masses Mβ (i.e., Mn, Mw, Mz, ...)

with the indices β = 0, 1, 2, ... (n, w, z, ...).Mn,Mw andMz are the three well-known average molar masses – the number-, the weight-, and the z-average molar masses (“z” is the abbreviation for the German word Zentrifuge = centrifuge)