Chemical Engineering 160/260 Polymer Science and Engineering

Lecture 4 - Direct Measures of Molecular Weight: Osmometry and Light Scattering January 24, 2001 Outline

! Chemical potential of dilute solutions ! Raoult’s Law ! Osmotic pressure ! Van’t Hoff Equation ! Virial expansion ! Light scattering Experimental Approaches

! Direct measures of molecular weight may be obtained from osmometry,, lightlight scattering,scattering, and ultracentrifugation. ! Indirect measures of molecular weight, such as viscometry and gel permeation chromatography, yield relative estimates thatthat mustmust bebe calibrated.calibrated. Chemical Potential of Dilute Solutions The chemical potential of a solvent in a solution is o µµss=+RTln a s o µs = chemical potential for pure solvent at T = activity of solvent as If the solvent vapor obeys the ideal gas law, we have P PP< o o RT ln s ss µµss=+ o o sP µµss<

Ps = vapor pressure above the solution at T o Ps = vapor pressure above the pure solvent at T Raoult’s Law In general, the activity is related to vapor pressure by

Ps as = o Ps If the solution is sufficiently dilute, Raoult’s Law will be obeyed Ps axss==o Ps

xs = mole fraction of solvent in solution o µµss=+RTln x s Osmotic Pressure P Po + π o

Solution Pure solvent

Semipermeable membrane

The osmotic pressure π is the additional pressure that must be imposed to keep solvent and solution sections at the same level. This static method requires a long time to reach equilibrium. Semipermeable Membrane Construction

! Membranes for different solvents " Organic - cellulose, gel cellophane " Aqueous - cellulose acetate, nitrocellulose " Corrosive - glass ! Membrane porosity " Must consider pore size and its distribution ! Membrane conditioning " Membranes shipped in isopropanol/water " Programmed sequence of solvent mixtures " Dried-out membranes should be discarded Derivation of Van’t Hoff Equation At constant temperature, the chemical potential depends upon both pressure and composition.

µsp= fPx(, ) xxps= 1 − The total derivative of the chemical potential is then   d ∂ s  dP s dx µs =  µ  +   p  P  Tx,  xp  ∂ p ∂µ TP,

If no solvent flow occurs, dµs = 0 ∂ ∂    µs  dP s dx   = −  p  P  Tx x ∂ , p  ∂µp  TP, ∂ Derivation of Van’t Hoff Equation The chemical potential is defined by N  G  s = moles of solvent N = moles of polymer µs =  ∂  p  Ns  = Gibbs free energy ∂ TPx,,p G Partial differentiation with respect to P yields 2  ∂µs  G   =  P  ∂PN ∂ Tx, p s ∂∂ Derivation of Van’t Hoff Equation The volume of the system is given by  ∂G   = V  P  ∂ Tx, p

Partial differentiation with respect to Ns yields 2 ∂ G  V  = partial molar =  ∂  = Vs Vs ∂∂NPss N Tx,, P volume of solvent ∂ p Since the order of differentiation is immaterial, we have

 ∂µs    = V  P  s ∂ Tx, p Derivation of Van’t Hoff Equation

Recall the chemical potential

o  ∂µs  RT µµss=+RTln xs =  x  x  ∂ s  TP, s

dxs Note that xxsp= 1 − and = −1 dxp

Partial differentiation with respect to xp yields

 ∂µs   s   dxs  RT =   = −  xx ∂µ  dx  1 − x  ∂ p  TP  s  TP,  pp , ∂ Derivation of Van’t Hoff Equation Recall that for conditions of no solvent flow, dµ = 0 and ∂    µs  dP s dx   = −  p  P  Tx x ∂ , p  ∂µp  TP, ∂ V RT s − 1 − x Substitution leads to p x Po +π p dx VdP= RT p ∫∫s 1 x Po 0 − p Derivation of Van’t Hoff Equation If the partial molar volume is independent of pressure, we have RT π = −−ln() 1 xp Vs For the very dilute solutions that obey Raoult’s Law, 23 xxpp ln() 1− xxpp= −− − −≅−L x p 23 For the case where Ns >> Np, N N x = p ≅ p p Nss V= V s≅ V() solution NNsp+ Ns Derivation of Van’t Hoff Equation N π = RT p Substitution yields V() solution

Compare this expression to the ideal gas law.

Convert to concentration units of mass/volume and take the limit as concentration goes to zero  π  RT = number average lim Mn = molecular weight  c  Mn At last, we have the Van’t Hoff Equation. Note that it is applicable only at infinite dilution. Virial Expression for Osmotic Pressure In order to account for concentration effects in polymer solutions, a virial expression is often used.    π  RT 1 Ac Ac 2 =+++ 23L  c  M n 

A2 = second virial coefficient A3 = third virial coefficient Alternative expressions: The forms are ππ  2 = []1 ++ΓΓ23cc +L equivalent if: cc  c=0  RT  B== RTA Γ π  RT  2 22 M  =   ++Bc Cc  n  c  M  n Effect of Solvent Quality on Osmotic Pressure π solvent quality c RT Θ Mn →

c A2 > 0 for a good solvent A2 = 0 for a theta solvent A2 < 0 for a poor solvent Light Scattering

We will only summarize the results -- for background, see, e.g., Allcock and Lampe, 2nd ed., pp 348-363.

Iθ V =scattering V s s w volume Io

Light scattering arises from fluctuations in refractive index, which can be related to the osmotic pressure. Rayleigh’s Ratio, R(θ)

Hc 1   = ∂π  RRTc()θ   T ∂ 2 2 dn Iw 22  θ 2 π n R()θ = odc  IVos H = N 4 A λ w = scattering distance λ = incident wavelength

Vs = scattering volume no = refractive index at λ NA = Avogadro’s number Relationship to Weight Average Molecular Weight and Second Virial Coefficient

Hc 1 =+2A 2 R()θθ− R ( solvent ) Mw P ()

P(θ) is the single chain form factor, which accounts for the finite size of the macromolecule relative to the wavelength of incident light. Single Chain Form Factor for a Random Coil

For particles larger than λ/20, P(θ) is not unity. For small angles (in the Guinier region) and a random coil structure,

2 22 22 P(θ )= 44{}Rkgg−−[]1 exp( − Rk ) Rkg

Rg is the radius of gyration, which is a measure of the three-dimensional structure of the random coil.

4π k = sinθ  This is also referred to as q. λ  2 Zimm Plot

Extrapolations to zero angle and zero concentration

allow for determination of Mw, A2, and Rg. Eliminate shape effects:  Hc  1 Ac   =+2 2  RM()θ θ =0 w Eliminate intermolecular interactions:

3Hc c 16πθR() c = 0

θ 100c + sin2    2