Compilation of Standard Methods of Molecular Weight Measurement

Standard Methods of Molecular Weight Measurement: Part II

3.1 Sedimentation Equilibrium and Sedimentation Velocity

Both sedimentation methods referenced in the heading can be performed in the same standard analytical ultracentrifuge, so the two will treated together, although the physics of M measurement are distinct.

As sketched below, an analytical ultracentrifuge rotates a measuring cell at extremely high angular velocities. The rotor makes ~60,000-70,000 revolutions per minute, and with the cell located off-axis by ~5 cm, the sample in the cell experiences a centrifugal force of the order 200,000-400,000 g, where g is the acceleration of gravity. The surrounding enclosure maintains constant temperature and acts as a barrier should anything fly loose! The limit to the g-force is set by the strength of the rotor material, which is usually titanium or an esoteric metal alloy.

1=motor and gear; 2=rotor; 3=measuring cell(s); 4=quartz window; 5=optics; 6=light source; 7=optical detector; 8=thermostated enclosure

[E. Schröder, G. Müller, and K.-F. Arndt, Polymer Characterization, Hanser, 1989, p 80]

The distribution of polymer along the axis of the measuring cell can be determined in a number of ways, but most commonly, UV absorbance or Schlieren optics are chosen. The optical path for either detection method follows the sketch, with a light beam traversing through windows that form the top and bottom of the measuring cell. UV absorbance is potentially the more sensitive of the two approaches, but many polymers don’t absorb UV or visible light significantly relative to solvent, in which case, this detection can’t be selected. Schlieren optics won’t be described here in any detail, but for this approach, the light detector measures a signal proportional to the gradient of refractive index (dn/dx, where x is measured outward from the center of rotation), and thus indirectly, of mass concentration c.

3.1.1 Theory.

The centrifugal force Fbexerted on a dissolved polymer molecule arises from the polymer’s buoyancy, i.e., the difference in density between polymer and solvent,

where meff is the effective mass of one molecule in the solvent,  is the angular velocity of the rotor, NA is Avogadro’s number, 1 is the solvent density, and is the partial specific volume of polymer, effectively the reciprocal of the polymer density when dissolved in the solvent. For simplicity, a polymer denser than solvent is assumed in the subsequent analysis, directing Fboutward, in the direction of increasing x. The opposite case is almost as common.

A balance of Fb with the polymer’s hydrodynamic resistance Fh to translational motion through stationary solvent sets the sedimentation velocity dx/dt,

where f is the chain’s friction coefficient. The sedimentation coefficient s relates dx/dt to the centrifugal acceleration 2x,

s is commonly expressed in Svedberg units, where 1 Svedberg equals 10-13 s. Equating Fh to Fb, an expression for M is obtained

By adopting the Einstein equation for solute friction, f = kT/D, where k is the Boltzmann constant, T is temperature, and D is the polymer’s tracer diffusion coefficient in the solvent (evaluated at dilute conditions), the so-called Svedberg equation is obtained,

which connects sedimentation to tracer diffusion while offering a route to M if s is measured, and D and are known (or measured separately).

Solving for x in the equation that defines s, one finds that a narrow band of homogeneous polymer migrates such that band position changes exponentially in time t.

Diffusion about the mean band position during centrifugal migration has been ignored so far. Adding diffusion to the force balance leads to a sedimentation-diffusion governing equation (Lamm’s equation) in the form of a differential equation,

When D is set to zero, this equation describes migration of a solute concentration pulse at a velocity s2x, reproducing the result of the simpler analysis given above.

When equilibrium across the cell is achieved at large t, i.e, when dc/dt=0 at all x, outward sedimentation balances inward diffusion down a steady c gradient. One can solve the same differential equation for c and then eliminate D and s through the Svedberg equation, finding for the steady c profile,

where at x=xb, the position of the outer end of the measuring cell, the no polymer flux condition allows the polymer to accumulates to c=cb, a parameter dependent on the original mass of polymer added to the cell. With known by a separate density measurement, M can be calculated at equilibrium by measuring c at any two positions x,

where “1” and “2” designate c and x at the two positions.

For a polydisperse system at equilibrium, each component of molecular weight Mi and concentration ciobeys a concentration profile analogous to that given above, and adding these profiles together to obtain c,

This equation can be manipulated to find various molecular weight averages; the formulae will not be given here.

The first equation given for M, derived in absence of diffusion, suggests that for large M (a condition which restricts diffusion of away from the mean band position because D is small) s should correlate with M similarly to f, i.e., with a power law,

Coefficients K and a, derived from a set of nearly monodisperse samples, are tabulated for many polymer/solvent combinations. When the two parameters are available in this manner, the velocity sedimentation method provides the full M distribution if the distribution of s is measured.

3.1.2 Practice

Measuring cells are from 0.2 to 3 cm in length and have a segment shape such that cell width increases toward the bottom (sector angle ~ 2-4º). The initial distribution of polymer within the cell is uniform, i.e., c is not a function of x.

To measure sedimentation velocity, the rotor speed is set so high that diffusion of polymer is negligible. Then, at some intermediate time t, well before equilibrium across the cell is achieved, one can envisage transient c profiles much as drawn below for the case of two discrete M fractions in the sample.

The top curve corresponds to complete neglect of diffusion. Each “step” corresponds to a sharp interface between two regions, one in which the concentration of a specified polymer fraction is uniform (for x greater than the step position) and one where this polymer fraction is absent (for x less than the step position). Adding diffusion, these interfaces are smeared, as shown in the middle sketch; the new profile parallels an the absorbance measurement of c. With Schlieren optics, which detects the gradient of c, measurement yields the bottom sketch.

Considering a single component polymer solution, as the c step approaches the bottom of the cell at larger t, its character changes, reflecting approach to the equilibrium c profile. This time evolution is sketched below:

Whereas initial concentration co was uniform, at intermediate t, a c step migrates to the right. Simultaneous to migration, at the bottom of the cell, c rises above co as polymer arrives in this region and is trapped by the closed cell end. Eventually, the migrating step merges with the perturbed bottom region and a continuously increasing c profile is obtained. Eventually, at equilibrium, the profile becomes time-invariant.

Only at intermediate t will a c step migrate as described in the theory derived without diffusion. Measurements in this t domain made with Schlieren optics for the theta system of polystyrene migrating in cyclohexane are shown below. As the simplified theory predicts, polymers move such that their x-positions move exponentially in t. The linearity of the semilog plot for specified integrated mass fractions I (M) are shown further down; slope differences among the I(M) linesreveal there is a distribution of M, the different fractions migrating at different s.

For this model system,

where M is in g/mol and s is in units of Svedbergs. The M distribution calculated using this s-M correlation is given in cumulative form at the top of the next page.

As described above: t is time in minutes

As described above: I is integrated mass fraction

Integral weight fraction distribution of M. The five points correspond to the 5 slopes of the preceding plot.

[all plots from: E. Schröder, G. Müller, and K.-F. Arndt, Polymer Characterization, Hanser, 1989, pp 88-89]

Issues:

An analytical ultracentrifuge is very expensive, ~$300-400,000.
The volume of sample is small (<1 ml of dilute solution).
One must have a pycnometer to measure .
The velocity method is fast (hours) but the equilibrium method is slow (days); for this reason, the equilibrium method is rarely performed.
There is really no upper M limit, especially for the velocity method.
The low M limit is established by the magnitude of g-force available.
The larger is and the larger is dn/dc (or absorbance), the better is accuracy. Accuracies of 5-10% are typical for model systems.
Unless s is known as a function M by correlation with M standards, the velocity method requires input of D. Ideally, one can get D for a narrow distribution sample by the width of the step of c. For a polydisperse sample, the M-dependence of D convolutes onto the M-dependence of s, making the velocity measurement hard to interpret.
Some investigators have connected a light scattering detector to the measuring cell such that unmixed sample fractions are streamed to the light scattering detector for absolute M measurements. The combination gives the absolute M distribution without further assumptions, but operation is tricky.
Equilibrium sedimentation in a solvent with a density gradient has often been used to study biopolymers. Use of this approach for synthetic polymers has been extremely rare. A density gradient can be created by adding a cosolute such as sucrose, which rapidly sediments to its own equilibrium concentration profile.
Data analysis is more straightforward for a system containing a few discrete molecular species than for a system containing a distribution of species. This makes the method more popular for biopolymers than for synthetic polymers.
Although seemingly suitable for studying composition distributions in copolymers, especially graft and block copolymers, over the past ten-fifteen years, I’ve only seen sedimentation used to study branched polymers. In the latter case, measuring M was not the objective.

3.2 Solution Viscometry

Dilute solution viscometry, the first relative M method to be described, is based on measuring the increase in viscosity  as a polymer is added at varying dilute concentrations c to a solvent. The normalized relative change in viscosity is referred to as the “specific” viscosity sp,

where o is the pure solvent viscosity, and rel (=/o) is termed the “relative” viscosity. Dividing sp by c, one obtains the “reduced” viscosity red, which varies linearly with c when c is low. The linear increase is captured in the empirical Huggins equation,

where [] is the “intrinsic” viscosity and kH is the Huggins coefficient. The latter has a value of ~0.3-0.4 in a good solvent and ~0.5 in a theta solvent.

The value of [] captures the degree to which a small increment in c raises . Adding larger polymers at the same c increment causes a greater increase in , as derived in the next subsection.

3.2.1 Theory of Dilute Solution Viscosity

To understand the viscosity method, we begin with a relationship derived by Einstein for the viscosity at volume fraction  of a dilute suspension of uniformly sized spheres. He found by difficult theory [difficult enough that Einstein made an math error that he had to correct in a 2nd publication] a rather simple result:

 = o[1 + 2.5 + …]

To connect to polymer solution measurements, we must relate the actual experimental variable c to , which doesn’t have a clear definition for flexible polymer molecules. Assuming that the molecules behave hydrodynamically as rigid spheres, we can assign to each chain in dilute solution a “hydrodynamic radius” Rh. This radius can be interpreted as the size of the sphere that would give a contribution to  equal to that from a single polymer. We do not know precisely how Rh relates to the size and structure of the dissolved polymer, but we can be confident that Rh is proportional to any choice of average coil size. Letting C be an undetermined coefficient, we can write, for example,

Rh = CRg

The effective “hydrodynamic volume” Vh of one chain is then,

Vh= (4/3)Rh3 = (4/3)C3 Rg3

Adopting this volume, we next transform c [mass/volume] into , effectively making this step a units conversion,

Eliminating  and Vh,

[] = 2.5 (4/3)C3NA[Rg 3/M]

The multiplicative factors outside of the […] on the right-hand-side define a universal constant of unknown value. For long enough linear polymers, Rg depends on M raised to a power,

Rg M

Combining the two previous formulae and gathering the incompletely specified constants into empirical parameters, K and a,

[] = KMa

where a=3-1. Many values of the constants K and a, called “Mark-Houwink coefficients”, are tabulated in reference books.

If K and a are known a priori for a particular polymer chemistry and solvent, and [] is measured for a sample of the same chemistry in the same solvent, we deduce the sample’s M by direct substitution into the preceding formula.

The value of M obtained in this manner is an average over the sample’s M distribution. To be specific, we subscript the value of M determined in this fashion with a “v”, terming Mv the “viscosity-average” molecular weight, calculated from the M distribution as

Experimental determination of v also reveals whether a solvent is a good, marginal, or theta solvent for the tested polymer.

The preceding analysis of polymer dilute solution viscosity assumed that a coiled polymer behaves hydrodynamically as a rigid sphere, an assumption known as the “nonfree-draining” hypothesis since it implies that a nearly time-invariant spherical solvent domain is dragged along with the polymer during flow. One might guess that this hypothesis is awful, as the segment density inside a coil can be quite small. An opposing hypothesis regards polymer chains as “free-draining”, with solvent readily able to penetrate and flow within the coil. To judge between the nonfree-draining and free-draining hypotheses, solvent flow must be analyzed within a polymer coil, a theoretically difficult problem. Those who do this sort of analysis report that the free-draining approach fails for dissolved high molecular weight polymers. Although many texts debate the two draining hypotheses at length, we need only to recognize that the nonfree-draining hypothesis is correct. Perhaps a physical analogy can best explain the result, that a tenuous porous solute effectively immobilizes all solvent in its interior: Although the branches and leaves of a bush might be sparse (comparable to the sparseness of segments within the envelop of a polymer coil), the branches and leaves effectively screen a strong wind so that air at the bush’s center is nearly stagnant.

For linear homopolymers, the Mark-Houwink relationship is only valid asymptotically, i.e., at large M, where the exponent a must have one of three values, 0.33, 0.5, or 0.6, corresponding to poor, theta, and good solvent conditions. Although other values of a are reported, any value other than these three implies that measurements were not made in the asymptotic high M regime. If so, the power law relationship is only approximate, extending over a limited range of M. The values of K and a are typically determined by examining a series of M standard polymers. Molecular weights of the standards could be determined by any method but typically these values are determined by light scattering.

3.2.2 Practice of Dilute Solution Viscometry

To assure that the tested polymer solution is dilute, viscosity measurements are performed over the range 1.0rel<2.0 (the coil overlap concentration c* is commonly defined by this upper limit). The solution thus has a viscosity within a factor of 2 of the solvent, and since viscosities of most simple solvents are in the range of centipoises, capillary viscometers are preferred for intrinsic viscosity measurements. Other types of commercially available viscometers don’t have the accuracy for such low viscosity fluids. For very high M polymers, however, as discussed below, one might be able to use a rotational viscometer with cylindrical Couette fixtures. Viable viscometers of this type were once commercial, but few remain in use today.

Two capillary viscometers are popular, Ubbelhode and Ostwald, shown left and right, respectively on the following sketch:

In both types, one measures the time t needed for a solution or solvent to drain through the test section of a glass capillary under the action of gravity. The Ubbelhode type is superior, since the drainage height doesn’t vary over the duration of the experiment. Texts often describe various hydrodynamic corrections for entrance/exist effects, but in modern capillary viscometers, assuming laminar flow,  is almost perfectly proportional to t, so these corrections aren’t needed.

However, appropriate values of t ARE necessary. If t for solvent is 60 s, the lowest solution t will be about 66 sec (i.e., rel=1.1). If the measurement uncertainty is 0.2 s (the human reaction time during use of a stop watch), an error in sp of the order 0.4/6= will be incurred, about 5-10%. Upon extrapolation to zero c, this level of error can generate larger uncertainty in [], perhaps of the level 50%. The consequential error in M is even larger. If a stopwatch is to be used, I recommend selection of a viscometer such that t is greater than 150 s. The timing problem is much reduced if t is measured automatically, as done in some commercial instrument packages. Drainage time is typically controlled through the length and diameter of the capillary, so different capillaries are often stocked in a lab that does these measurements routinely.

Many instruments allow dilution inside the viscometer; this feature requires a larger bottom bulb. One simple adds an aliquot of solvent after the preceding measurement is complete.

Issues:

Temperature control can be crucial, since viscosities of simple fluids vary exponentially with temperature. For example, the viscosity of water drops by more than 10% as temperature rises from 20ºC to 25ºC. Temperature control is typically achieved by immersing the viscometer in a transparent fluid bath with temperature controlled to better than 0.1ºC. Measurements in open air are only qualitative, and quantitative measurements in a bath at high temperature are extremely difficult.
Dust has a large effect, increasing artificially raising t. The problem is exacerbated by the small diameter of the capillary. Solutions/solvents should be filtered directly into the viscometer.
Since the solution moves up and down, polymer can readily be adsorbed or deposited on the capillary walls as the solution dries between runs. There is no easy way to clean the walls other than to add an aggressive solvent.
Co-solutes can be a killer. I tried to measure the molecular weight of a polymer that was dissolved in a surfactant solution, but I could never eliminate the bubbles generated by siphoning the solution up the capillary. The presence of even a minute bubble substantially raised t.
Polymers larger than about 0.5-1x106 g/mol generate shear-thinning solutions, and the appropriate viscosity for M determination is that at zero shear rate. Many capillaries generate shear rates of several hundred s-1. To avoid shear thinning, one commercial Ubbelhode design has four test sections in sequence, each with a different capillary diameter and thus producing a different shear rate. The four measured drainage times can be extrapolated to the drainage time corresponding to zero shear rate. Often, however, a large extrapolation is necessary, making the final measured quite uncertain. I would never attempt the solution viscosity method with standard capillaries for M greater than 107 g/mol even if supporting literature exists. Values above M~106 g/mol are open to question. Several tactics have been suggested to extend solution viscosity measurements to higher M, including accordion-like capillaries that drain on the tens-of-minute timescale.
To avoid the problems of the last two bullets, I have used a rotational viscosmeter made by Contraves, and I have also built a so-called Zimm-Crothers viscometer. Both viscometers have extremely high torque sensitivity and can monitor viscosities of the appropriate magnitude at shear rates of 1 s-1. (Indeed, I could measure the viscosity of simple solvents to an accuracy of better than 1% in the latter instrument. Unfortunately, it was extremely difficult to operate and clean.) No appropriate rotational viscometer for intrinsic viscosity measurement is currently commercial available.
In absence of calibration standards, what can one do? Not much. Unlike some other relative methods, one cannot make use of calibration standards of a different chemical type – the value of K varies too widely. Many industries characterize the size of their polymer products through specific viscosity at some standard polymer concentration.
One should never extrapolate to find M outside the range in which the Mark-Houwink coefficients were obtained.
Although solution viscosity dominated M measurements from the period 1940-1990, usage has dropped rapidly with the advent of simpler/cheaper light scattering instruments and GPCs. Most uses today relate to the probing of polymer architecture, not the measurement of M.

3.3 Gel Permeation Chromatography