Polymer Molecular Weight Measurement, Special Case I ...



Polymer Molecular Weight Measurements: Special Cases

5.1 Polyelectrolytes

Because polyelectrolytes dissociate counterions, their solutions have at least three components: polymer, counterion, and solvent. The polyelectrolyte itself is the least abundant species (by number of molecules) in a dilute solution. If additional low molecular weight electrolyte is present, the number of components is greater. Many procedures associated with standard methods of M measurement assume just two solution components, polymer and solvent. The extra components in polyelectrolyte systems not only create complexity, they may render standard methods useless.

Most polyelectrolytes are studied in aqueous media, environments with properties unfavorable for M measurement. For example, water has a high surface tension and dielectric constant. The former makes water “attractive” to airborne particulates, which can be enormously troublesome (e.g., to light scattering and viscometry).

Because of their multiple charges, polyelectrolytes experience potentially strong electrostatic interactions, especially if low molecular weight electrolyte is not added to the solution to screen these interactions, and indeed, the distinct behaviors of polyelectrolytes are often said to manifest the long-range nature of their interactions. Although long-range interactions certainly influence polyelectrolyte properties, they don’t have a major impact on M measurement methods, excepting those that exploit a correlation between M and molecular size (e.g., intrinsic viscosity and diffusion). More often, the dominant polyelectrolyte difficulty for M measurements is the presence of counterions and low molecular weight electrolyte.

Although reasons differ, many standard M methods DO APPLY to polyelectrolytes dissolved dilutely in water alongside a large excess of low molecular weight electrolyte (i.e., at high salt). Although dependent on method, the threshold for added electrolyte is often about 0.1 M. The standard M methods most appropriate to polyelectrolytes are light scattering, GPC, membrane osmometry, sedimentation, and intrinsic viscosity. Electrophoresis, of course, is a polyelectrolyte-specific method; unlike the other listed methods, it works best when there is not a large excess of added low molecular weight electrolyte (conductive heating is thereby minimized). A method that fails completely in the context of polyelectrolytes is vapor phase osmometry; this method equally counts the nonvolatile dissociated counterions along with the polymer chains, and since there are many more counterions than chains, the calculated M is much too small.

The following sections highlight differences between neutral polymer and polyelectrolytes in the application of specific methods. A good reference is H. Dautzenberg et al., Polyelectrolytes: Formation, Characterization, and Application, Hanser, NY, 1994.

5.5.1 Membrane Osmometry and Light Scattering

For reasons that will become clear shortly, these methods will be discussed together.

Salt-free solutions. If we dissolve a monodipserse polyelectrolyte sample of molecular weight M in salt-free water, the osmotic pressure (( at concentration c will reflect the total number of dissolved solutes. Specifically, if the polyelectrolyte has Z ionizable units, all of which dissociate,

[pic]

Since Z is large and nearly proportional to M, the right-hand-side is essentially independent of M. One therefore cannot determine M from osmometry measurements on a salt-free polyelectrolyte solutions.

The above expression is usually corrected for counterion condensation, which reduces the number of osmotically active (“free”) counterions as assessed by (o, the osmotic coefficient. The non-osmotically active counterions are considered electrostatically “bound” to the polyelectrolyte such that their binding energy much exceeds kT. According to the Manning theory of counterion condensation, (o is much lower than unity and M-independent. We can thus write

[pic]

where (o captures the nonideality of counterions due to their attraction to the polyelectrolytes.. Even if counterion condensation is not accepted (the concept is controversial), the electrostatic interactions of a polyelectrolyte solution make the counterions highly nonideal, with (o much less than unity even without the condensation.

From the fluctuation theory of light scattering, the normalized scattered intensity from a solution at zero angle is inversely proportional to the osmotic compressibility, the factor in parenthesis on the right-hand-side of the equation below,

[pic]

where the other variables are as defined in the second handout. From the salt-free polyelectrolyte expression for ((,

[pic]

Once again, all M dependence is lost if Z is large and proportional to M. Further, the scattered intensity becomes extremely weak, more comparable to that of a small molecule than a polymer. Plugging in values for poly(styrene sulfonate) in water, if (o=0.2, dn/dc=0.2 ml/g, and c=1x10-3 g/ml, the calculated value of R( is much less than the scattered intensity of pure water, which scatters less than most organic solvents since hydrogen bonding suppresses solvent density fluctuations.

The M independence and low scattering of salt-free polyelectrolytes solutions can also be understood by recognizing that weight average molecular weight of all dissolved species – counting both polyelectrolytes and counterions – is much closer to that of counterion than to that of polyelectrolyte.

These considerations seem to disallow the two most important methods, osmometry and light scattering, for M determination of polyelectrolytes.

Also, an unexplained peak is found when intensity for a polyelectrolyte is plotted vs. q, suggesting that even in a very dilute solution, some type of electrostatic-induced ordering of polymer chains occurs, further affecting (( vs. c from that anticipated in the standard formula.

Salted solutions. The dim outcome of the salt-free case motivates a consideration of osmotic pressure and light scattering measurements taken in the presence of a large excess of added salt.

The added salt can equilibrate between compartments during the membrane osmometry experiment. Equilibration of a salt-containing solution such that a large charged solute is unable to pass between the two osmometer compartments defines “Donnan equilibrium”. With this equilibrium, an electrostatic potential difference spontaneously appears across the membrane. Derivations of the equations of Donnan equilibrium are not difficult but tedious, care taken that the chemical potentials of solvent and each salt ion are matched across the membrane and that charge neutrality is maintained in the two compartments.

In a 1:1 salt with univalent counterions for the polyelectrolyte, under Donnan equilibrium the expression for (( has the form,

[pic]

where

[pic]

where the subscript “d” reminds us that (( is to be evaluated at Donnan equilibrium, and cs is the molarity of the added 1:1 salt.

The factor [pic] is the Donnan second virial coefficient. This quantity is only an apparent virial coefficient, since it doesn’t reflect the interactions between polymer molecules but rather the c-dependence of equilibration across the membrane. The real virial coefficient A2 would be associated with a second c-dependent term (not shown) within the parenthesis.

Note that if cs is large (high salt), the Donnan term disappears and the usual equation for osmotic equilibrium allows determination of Mn from ((d.

The osmotic method has not been much used for polyelectrolytes. However, if sufficient salt is present, it works in exactly the same manner as for neutral polymers, and indeed, there are many good choices of membrane material.

The light scattering method is more frequently used for polyelectrolytes, and from the discussion just given, it shouldn’t be too surprising that, if salt is present in sufficient concentration, this method too works pretty much as for a neutral polymer.

[There is one subtle difference: in the polyelectrolyte measurement of M, each polyelectrolyte chain can be considered to be in Donnan equilibrium with its local environment. Thus, the value of dn/dc should be determined such that each polyelectrolyte solution has been equilibrated across a polymer-impermeable membrane with a salt solution of the desired high salt concentration. This condition keeps the salt’s chemical potential constant in the polyelectrolyte solutions as the polymer concentration changes. In practice, one must dialyze each polyelectrolyte solution of different c against a large excess of the salt solution and then determine dn/dc by referencing the polymer-salt solution refractive index against the equilibrated salt solution refractive index. Unfortunately, dialysis tends to dilute the polymer, and so c must be redetermined after dialysis, mandating a rather tedious set of steps. In practice, diluting the polyelectrolyte with a salt solution of fixed c causes only a small error ( ................
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