Membrane Osmometry ( , A2, χ)• Osmotic pressure, π, is a colligative property which depends only on

the number of solute molecules in the solution.• In a capillary membrane osmometer, solvent flow occurs until π

increases to make the chemical potential μ(π) = μ1o

μ1° = μ1 +

∂μ1

∂PP0

P0 +π

∫ dP

∂μ1

∂P=

∂∂P

∂G∂n1

⎛

⎝ ⎜ ⎞

⎠ ⎟ =

∂∂n1

∂G∂P

⎛ ⎝

⎞ ⎠

dG = Vdp− sdT + μdn

∂G∂P

= V

∂∂n1

(V )≡ V 1

so

(partial molar volume of pure solvent)

Mn

since T, n fixed

π

Solution Pure solvent

Semi-permeable membrane

Polymer chains

Figure by MIT OCW.

Membrane Osmometry cont’d

μ1° = μ1 +

P0

P0 +π

∫ V 1dP = μ1 +πV 1

μ1 −μ1

o =− πV 1

Recall F-H expression for a dilute solution (φ2 small):

μ1 −μ1° = RT

−φ2

x2

+(χ −1/ 2)φ22⎡

⎣ ⎢ ⎤

⎦ ⎥

Rearranging:

π = RTφ2

V 1x2

⎛

⎝ ⎜ ⎞

⎠ ⎟ +

12

− χ⎛ ⎝

⎞ ⎠

φ22

V 1

⎛ ⎝ ⎜ ⎞

⎠ ⎡

⎣ ⎢ ⎤

⎦ ⎥

Since solution is dilute: V ≅ n1V1 so V 1 ≅ V1

π = RTn2

V⎛ ⎝

⎞ ⎠ +

12

− χ⎛ ⎝

⎞ ⎠

n2

V⎛ ⎝

⎞ ⎠

2

V1x22⎡

⎣ ⎢ ⎤

⎦ ⎥

The osmotic pressure is a power series in i.e., π depends on the number of molecules of solute per unit volume of the solution.

n2V

⎛ ⎝ ⎜

⎞ ⎠ ⎟

φ2 ≅ n2 x2

n1

recall

Osmotic Pressure cont’d• For a polydisperse polymer solution, where ni is the number of moles

of polymer of molecular weight Mi in the solution. The number average molecular weight of the polymer is simply

• The concentration in grams/cm3 of the polymer in the solution is just

• The “reduced osmotic pressure” when plotted as a function of the solute (i.e. polymer) concentration will give a straight line with:

n2 = Σi

ni

M n =∑niMi

∑ni

=mn2

c2 =mV

=n2 M n

Vor

n2

V⎛ ⎝

⎞ ⎠ =

c2

M n πc2

=RT 1M n

+12

− χ⎛ ⎝ ⎜

⎞ ⎠ ⎟ V1x2

2

M n2 c2

⎡

⎣ ⎢

⎤

⎦⎥

πc 2

c2

χ < 1/2 (good solvent)

χ = 1/2 (θ condition)

> 1/2 (poor solvent)

xx

x xx

x x x x x

xxxxcondition)

χ > 1/2 (poor solvent)

xx

x xx

x x x x x

xxxx2

221

21

n

n

MxV

MRT

⎟⎠⎞

⎜⎝⎛ − χ

Intercept:

Hence the osmotic pressure can be written

Three types of behavior:Key Information:

Slope:

Osmometry cont’d

• At the θ condition, 2nd term disappears (recall χ = 1/2) and solution is ideal

• The slope of the reduced osmotic pressure yields a measure of the polymer segment-solvent F-H interaction parameter, ie we have a way to experimentally determine χ.

• For gases, the pressure is often developed as a function of concentration

• We can therefore identify the virial coefficients A1 and A2 as:

πV ≅ n2RT

A1=1

M nA2 =

12

− χ⎛ ⎝ ⎜

⎞ ⎠ ⎟

ρ22V1

...)( 33

221 +++= cAcAcARTP

πc2

=RT 1M n

+12

− χ⎛ ⎝ ⎜

⎞ ⎠ ⎟ V1x2

2

M n2 c2

⎡

⎣ ⎢

⎤

⎦⎥

Comments about Osmometry1. We assumed mean field conditions - local environment is similar

everywhere.2. We employed a dilute solution approximation with φ2<<1 .

The requirements force the experimental conditions to be chosen such that:

(1) System is “thermodynamically concentrated” c2 > c2*

(2) System is “mathematically dilute” φ2 << 1

c2* is the so called overlap concentration; the polymer concentration at which the coils just begin to touch and hence the solution is reasonably uniform in composition (i.e. mean field situation, no large regions of pure solvent)

c2* ≡ concentration at overlap ≅ 32/12

/

⎟⎠⎞⎜

⎝⎛

=r

NMcoilofvolumemoleculepolymerofmass A

c2 ≈c2*

*22 cc >

Schematic Polymer-Solvent Structure at Various c2

C OverlapConcentration

Semi-ConcentratedSolution

Figure by MIT OCW.

SEC/GPC (Size Exclusion Chromatography)/(Gel Permeation Chromatography)

• SEC allows measurement of the entire molar mass distribution enabling all molecular weight averages to be computed for comparison with other techniques. Preparative GPC uses 25-50 mg of a polymer per run and allows one to fractionate polydisperse samples.

• Apparatus: Set of columns (2-6) containing solvent swollen crosslinked PS beads. The beads are selected to have pore sizes in the range of 102 –105Å. A run consists of injecting 0.05 ml of a dilute solution (requires only ~0.1 mg of polymer !), then allowing the solution to transverse the columns and monitoring the eluting polymer molecules using various detectors. In order to “see” the molecules, one needs sufficient “contrast” between the solvent and solute (polymer). Three types of detectors are commonly used:

– Ultra violet absorption : UV Detector– Infrared absorption : IR Detector– Refractive index changes : RI Detector

• The solvent is chosen to be a good solvent for the polymer, to have a different refractive index from the polymer and to not have light absorption in the range for which the polymer is absorptive.

Schematic of SEC Separation Process

Porous Gel

Magnified

Cross linked PSbeads in well packed column

Pore

Diffusing chain

PS chain network

102 A - 105 A∼

Greatly magnified

GPC Column

Polymer molecules outside beads

Porous Interior of PS BeadSwollen PS Xlinked Bead

Figure by MIT OCW.

Principles behind SEC involves…• Competition between 2 types of entropy

1. ΔSmix favorable entropy of mixing of polymer and solvent inside pores.2. ΔSconf unfavorable loss of conformation entropy of large size polymers (high MW)

when entering smaller pores in the gel (column packed with swollen Xlinked PS beads).

• The interplay of both types of entropies results in substantially different residencetimes in the column for the different polymer molar masses.

1. Mixing entropy gain drives smaller chains to enter and diffuse throughout entire gel2. Intermediate size chains access a portion (the larger pores) of the pore volume in the

gel3. Conformational entropy loss prevents larger chains from entering the gel

• GPC is not an absolute method so it is necessary to calibrate the MW vs. elution volume curve using known narrow fraction samples of the same polymer in the same solvent at the same temperature. Normally researchers employ PS in THF at 23°C for PS-based calibration. Thus samples are referred to as eg “60,000 g/mol on a PS-basis,” meaning that the particular sample exited the column at an elution volume corresponding to a 60,000 g/mol PS sample going through the same column using THF at 23°C (a good solvent) as the carrier medium.

A Simple GPC Chromatographsolventcarrier

reservoirpolymer solution

of mixed Mi

inject(typical flowrate ~ 1 cm3/min)

hν ConcentrationDetector

GPC column(s): 4-6 columns in series

packed column of 10μm diam. crosslinked PS gel beadspore size ~ 102Å – 105Å

UV, IR, or RI signal

Preparative GPC(collect fractions of eluted solution)

solventcarrier

reservoirpolymer solution

of mixed Mi

inject(typical flowrate ~ 1 cm3/min)

hν ConcentrationDetector

GPC column(s): 4-6 columns in series

packed column of 10μm diam. crosslinked PS gel beadspore size ~ 102Å – 105Å

UV, IR, or RI signal

Preparative GPC(collect fractions of eluted solution)

UV, IRor R.I.signal

Elution volume →time →

high MW low MW

UV, IRor R.I.signal

Elution volume →time →

high MW low MW

For a Polymer with Narrow Weight Distribution

SEC cont’d• The volume eluted Ve consists of 2 parts:

External to the Beads: “Void” Volume Vo (outside volume)Internal to the Beads: Pore Volume Vi (inside volume)

• Depending on the size of the molecule passing through the column all or only a portion of the total volume is visited/explored (called pore permeation) by the diffusing molecules.

pure solvent eluted volume Ves = Vo + Vi polymer solution eluted volume Vep = Vo + ViKse

• where Kse is the size exclusion equilibrium constant.

Kse = 1 for x1 = 1 solvent, i.e. no exclusionKse = f(x2) < 1 for x2 > x1 Kse → 0, ie total exclusion from pores for x2→∞

ΔGpp = Go – Gi change in Gibbs free energy for pore permeation by polymer

• Solvent and column material are chosen such ΔHpp of the polymer is = 0Δ Gpp = -TΔSpp = -RT ln Kse

ΔSpp is the change in entropy for pore permeation. This depends mainly on the relative sizes of the polymer chain and the pore. solving

Kse (x2) ≡polymer concentration inside beadpolymer concentration outside bead

=c2i(x2)c2o(x2)

Kse = eΔS/R

SEC cont’d

ΔSpp = − R pore surface areavolume of pore

⎛

⎝ ⎜

⎞

⎠ ⎟ diameter of chain( )

⎡

⎣ ⎢

⎤

⎦ ⎥ ⎟

⎠⎞

⎜⎝⎛−

+=2/12rA

ioeps

eVVV

Elution Volume vs. MW

first out

last out

useful rangeof separation

V0 + Vi

V0

Vep

Chain size(Molecular Weight) Elution volume →

log

MW

→

PS CalibrationPSPIPBPVCH

Elution volume →

log

MW

PS CalibrationPSPIPBPVCH

PSPIPBPVCH

GPC Calibration

Furthermore, we can approximate:

As depends on the pore/column and <r2>1/2 depends on solvent, T and the MW of that polymerchain entering the pore

THF solvent

here Vep

Vep = Vo + ViKse

is the elution volume of the polymer and As is the pore S/V ratio

Detector Sensitivity/Selectivity: An Example

6 Column GPC (high resolution)

PDI >> 1.12!!

2 Column GPCPDI = 1.12, apparently

a nice sample…

[The Truth hurts!!]

Note very different elution volume values

Polydispersity Index (PDI)

PDI = Mw

Mn

10.0 11.2 12.4 13.6 14.8 16.0 17.2

Retention Volume (ml)18.4 19.6 20.8 22.0

0.0

4.4

8.8

13.2

17.6

22.0

Res

pons

e

38 39 40 41 42 43 44 45 46 47

GPC result

Elution Volume (ml)

Figure by MIT OCW.

Scattering of Radiation: Dilute Polymer Solution

• , A2(χ) and <Rg2> (Zimm plot)

• SALS (small angle light scattering)SAXS (small angle X-ray scattering) SANS (small angle neutron scattering)

M w

q = 4πλ

no

sinθ 2I0

I0

r

Figure by MIT OCW.

Scattering Intensity from a Solution of Polymer Chains in Solvent

• Geometry of apparatus: sample detector distance r, scattering angle θ

• Optical constants and material properties: λ, n(or α), dn/dc2

• Thermodynamics of polymer-solvent system: c2, and A2(chi)wM

Basic Scattering Equation

Scattering arises from…Light (Δα)2 polarizability fluctuationsX-ray (Δρ)2 electron density variationsNeutron (Δb)2 neutron scattering length variation

solventsolution

w

RRRV

A

PcK

cAMPR

Kc

−==

=⎟⎠⎞

⎜⎝⎛ −

=

===

⎥⎦⎤

⎢⎣⎡ ++=

ratioRaleigh excess)(

tcoefficien virial221

geometries particle sfor variouknown factor, scattering particle)(ionconcentratpolymer

constants optical

....21)(

1)(

nd

12

22

2

222

θρ

χ

θ

θθ

Δ

Δ We will derive this. Note thenice set of variables…that wewould like to be able to determine

~ excess scattering intensity

Scattering I: from Density FluctuationsA dilute gas in vacuum• Consider small particles: dp << λradiation

– (situation:~ point scatterers)

• Scattered Intensity at scattering angle θ to a detector r away from sample:

• For N particles in total volume V (assume dilute, so no coherent scattering)

dp

λ

Incident radiationParticles

dp

λ

Incident radiationParticles

Iθ =I0 8π 4 1+ cos2 θ( )

λ4r 2 α 2 α = polarizability of molecule

I0 = incident beam intensity

Iθ' =

NV

Iθ α⎟⎠⎞

⎜⎝⎛π+=ε

VN41 ε = dielectric constant

ε = n2, ε(ω) = frequency dependent

Fundamental relationship: index of refraction polarizability

απ+=VNn 41

cdcdn1ngas +≅

Can approximate

dcdn = refractive index increment

c = conc. of particles per unit volume

...cdcdn21c

dcdn1n

22gas +⎟

⎠⎞

⎜⎝⎛+≅⎟

⎠⎞

⎜⎝⎛ +=

So by analogy for a polymer-solvent solution:

n ≅ n0 +dn

dc2

c2n2 ≅ n0

2 + 2n0dn

dc2

c2

α = 12π

dn / dc( ) cN /V( )

Solving givesthe polarizability

Rayleigh and Molecular Weight of GasesIθ

' = NV

⎛ ⎝ ⎜

⎞ ⎠ ⎟

8π 2 1+ cos 2 θ( )λ4 r 2

(dn / dc) c2π (N /V )

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

and sinceIθ' =

I0 2π 2 1+ cos2 θ( )λ4 r 2 (N /V )

dndc

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

c2

This expression contains several parameters dependent on scattering geometry, so we define

Rayleigh Ratio, R as R = Iθ'

I0 1+ cos2 θ( )/r 2

AV4

22

N

cMdcdn2

Rλ

⎟⎠⎞

⎜⎝⎛π

=which equals R = K•M•c

Where K is a lumped optical constant

AVNdcdn

K 4

222

λ

π ⎟⎠⎞

⎜⎝⎛

≡

Note, for polymer-solvent solution:

K ≡2π 2n0

2 dndc2

⎛

⎝ ⎜

⎞

⎠ ⎟

2

λ4 NAV

Rayleigh measured the molecular weight of gas molecules using light scattering!

AVNMc

VN

/=

Or just

simplifying