CHAPTER
3.1 M n Methods
3.1.5 Membrane Osmometry
The practical range of molecular weights that can be measured with this method is approximately 30,000 to 1 million. The upper limit is set by the smallest osmotic pressure that can be measured at the concentrations that can be used with polymer solutions. The lower limit depends on the permeability of the membrane toward low-molecular-weight polymers. The rule that “like dissolves like” is gen- erally true for macromolecular solutes, and so the structure of solvents can be similar to that of oligomeric species of the polymer solute. Thus low-molecular- weight polystyrenes will permeate through a membrane that passes a solvent like toluene. The net result of this less than ideal semipermeability of real membranes is a tendency for the observed osmotic pressure to be lower than that which would be read if all the solute were held back. From Eq. (3-17), the molecular weight calculated from the zero concentration intercept will then be too high. Membrane osmometry is normally not used with lower molecular weight polymers for which vapor phase osmometry (Section 3.1.2) is more suitable for Mn measurements.
The membrane leakage error is usually not serious with synthetic polymers with Mn.~30000:
Osmometers consist basically of a solvent compartment separated from a solu- tion compartment by a semipermeable membrane and a method for measuring the equilibrium hydrostatic pressures on the two compartments. In static osmometers this involves measurements of the heights of liquid in capillary tubes attached to the solvent and solution cells (Fig. 3.1).
Osmotic equilibrium is not reached quickly after the solvent and solution first contact the membrane. Periods of a few hours or more may be required for the pressure difference to stabilize, and this equilibration process must be repeated for each concentration of the polymer in the solvent. Various ingenious proce- dures have been suggested to shorten the experimental time. Much of the interest in this problem has waned, however, with the advent of high-speed automatic osmometers.
Modern osmometers reach equilibrium pressure in 1030 min and indicate the osmotic pressure automatically. Several types are available. Some commonly used models employ sensors to measure solvent flow through the membrane and adjust a counteracting pressure to maintain zero net flow. Other devices use strain gauges on flexible diaphragms to measure the osmotic pressure directly.
The membrane must not be attacked by the solvent and must permit the sol- vent to permeate fast enough to achieve osmotic equilibrium in a reasonable time.
If the membrane is too permeable, however, large leakage errors will result.
Cellulose and cellulose acetate membranes are the most widely used types with synthetic polymer solutions. Measurements at the relatively elevated temperatures needed to dissolve semicrystalline polymers are hampered by a general lack of membranes that are durable under these conditions.
Membrane osmometry provides absolute values of number average molecular weights without the need for calibration. The results are independent of chemical
heterogeneity of the polymer, unlike light scattering data (Section 3.2).
Membrane osmometry measures the number average molecular weight of the whole sample, including contaminants, although very low-molecular-weight mate- rials will equilibrate on both sides of the membrane and may not interfere with the analysis. Water-soluble polyelectrolyte polymers are best analyzed in aqueous salt solutions, to minimize extraneous ionic effects.
Careful experimentation will usually yield a precision of about 65% on replicate measurements of Mn of the same sample in the same laboratory.
Interlaboratory reproducibility is not as good as the precision within a single loca- tion and the variation in second virial coefficient results is greater than in Mn
determinations.
The raw data in osmotic pressure experiments are pressures in terms of heights of solvent columns at various polymer concentrations. The pressure values are usually in centimeters of solvent (h) and the concentrations, c, may be in grams per cubic centimeter, per deciliter (100 cm3), or per liter, and so on. The most direct application of these numbers involves plotting (h/c) againstcand extrapo- lating to (h/c)0 at zero concentration. The column height h is then converted to osmotic pressureπby
π5ρhg (3-28)
where ρis the density of the solvent and g is the gravitational acceleration con- stant. The value ofMnfollows from
ðπ=cÞ05RT=Mn (3-29)
(cf. Eq. 3-26). It is necessary to remember that the units ofRmust correspond to those of (π/c)0. Thus, with h (cm), ρ (g cm23), and g (cm s22), R should be in ergs mol21K21. ForRin J mol21K21,h,ρ, andgshould be in SI units.
EXAMPLE 3-1
The following data is collected from an osmotic pressure experiment conducted at 298.2 K:
C23103(g/cm3) 1.5 2.1 2.5 4.9 6.8 7.9
π(cm toluene) 0.30 0.45 0.55 1.20 2.00 2.40
whereC2is the concentration of a polystyrene sample in toluene, andπis the osmotic pres- sure. FindMn.
Solution
Given lim
C2-0
π C2 5RT
Mn.Mn5 RT
clim2-0 Cπ2
98 CHAPTER 3 Practical Aspects of Molecular Weight Measurements
Using the given data, plotπ/C2againstC2and find lim
C2-0
π C2
by extrapolation:
4
3
2
1
0
0 1 2 3 4
C2×103 (g/cm3) 1.77
5 6 7 8
g/cm3 Cπ2× 10–2 cm toluene
From the plot,
clim2-0
π
C2 51:773102 cm toluene=ðg=cm3Þ To convert the above value to SI units,
ρtoluene50:8610 g=cm3andρHg513:53 g=cm3 1 cm of toluene5ð10Þð0:8610Þ=13:5350:6364 mm Hg In addition;1 mm Hg5133:3 Pa and 1 g=cm3513103 kg=m3
clim2-0
π C2
0
@ 1
A5ð1:773102Þð0:6364Þð133:3Þ 13103
51:53101 Pa m3 kg21 Therefore,
MN5ð8:314Þð298:2Þ 1:53101 51:653102 kg=mol
The second virial coefficient follows from the slope of the straight-line portion of the (π/c)c plot essentially by dropping the c2 terms in Eqs. (3-24)(3-26).
It is to be expected that measurements of the osmotic pressures of the same poly- mer in different solvents should yield a common intercept. The slopes will differ
(Fig. 3.2a), however, since the second virial coefficient reflects polymersolvent interactions and can be related, for example, to the FloryHuggins interaction parameterχ(Chapter 5) by
A25 1 2 2χ
LV10υ22 (3-30)
Here υ2is the specific volume of the polymer, V10 is the molar volume of the solute, and χ is an interaction energy per mole of solvent divided by RT. When χ50.5, A250 and the solvent is a theta solvent for the particular polymer.
Better solvents have lowerχvalues and higher second virial coefficients.
It may be expected also that different molecular weight samples of the same poly- mer should yield the same slopes and different intercepts when the osmotic pressures of their solutions are measured in a common solvent. This situation, which is shown inFig. 3.2b, is not realized exactly in practice because the second virial coefficient is a weakly decreasing function of increasing polymer molecular weight.