Membrane Osmometry ( M n , 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
Polymer chains
∂μ1
dP
∂P
∂μ1
∂ ⎜⎛ ∂G ⎞⎟
∂ ⎛ ∂G ⎞
=
=
∂P ∂P ⎝ ∂n1 ⎠ ∂n1 ⎝ ∂P ⎠
P0 + π
μ = μ1 + ∫P
°
1
p
0
dG = Vdp − sdT + μdn
Solution
Pure solvent
Semi-permeable membrane
Figure by MIT OCW.
so
∂G
=V
∂P
∂
(V )≡ V1
∂n1
since T, n fixed
(partial molar volume
of pure solvent)
Membrane Osmometry cont’d
P0 +π
μ = μ1 + ∫P
°
1
0
V1 dP = μ1 +πV1
μ1 − μ1o = − πV1
Recall F-H expression for a dilute solution (φ2 small):
⎡ −φ 2
2⎤
μ1 − μ = RT⎢
+( χ −1/ 2) φ2
⎥⎦
⎣ x2
°
1
Rearranging:
⎡ ⎛ φ2 ⎞ ⎛ 1
⎛⎜ φ 22 ⎞ ⎤
⎞
⎟ + −χ
π = RT ⎢ ⎜
⎠ ⎝ V 1 ⎠ ⎥⎦
⎣ ⎝ V1 x 2 ⎠ ⎝ 2
Since solution is dilute:
recall
φ2 ≅
V ≅ n1V1 so V1 ≅ V1
n2 x 2
n1
2
⎡⎛ n2 ⎞ ⎛ 1
n
⎛
2⎤
2⎞
⎞
π = RT ⎢⎝ ⎠ +⎝ − χ ⎠ ⎝ ⎠ V1x 2 ⎥
V
2
⎣ V
⎦
The osmotic pressure is
a power series
⎛n ⎞
in ⎜⎝ 2 ⎟⎠ i.e., π depends
V
on the number of
molecules of solute
per unit volume of the
solution.
Osmotic Pressure cont’d
•
•
For a polydisperse polymer solution, n2 = Σ ni where ni is the number of moles
i
of polymer of molecular weight Mi in the solution. The number average
molecular weight of the polymer is simply
∑ ni Mi m
Mn =
=
∑ ni
n2
The concentration in grams/cm3 of the polymer in the solution is just
Hence the osmotic pressure can be
written
⎡
c
m n2 Mn
⎛n
c2 =
=
or ⎝ 2 ⎞⎠ = 2
V
V
V
Mn
•
⎞ V1 x 22 ⎤
1 ⎛1
= RT⎢
+⎜ − χ⎟
2 c2 ⎥
⎝
⎠
c2
Mn
⎦
⎣ Mn 2
π
The “reduced osmotic pressure” when plotted as a function of the solute (i.e.
polymer) concentration will give a straight line with:
Key Information:
π
c2
RT
Intercept:
Mn
Slope:
Three types of behavior:
x
x
x
χ < 1/2 (good solvent)
x
2
1 2
2
n
⎛1
⎞V x
χ
−
⎜
⎟
⎝2
⎠M
x
x
x
x
x x
x
x
x
x
χ = 1/2 (θ condition)
χ > 1/2 (poor solvent)
c2
Osmometry cont’d
⎡ 1 ⎛1
⎞ V1 x 22 ⎤
= RT⎢
+⎜ − χ⎟
2 c2 ⎥
⎝
⎠
c2
Mn
⎦
⎣ Mn 2
• At the θ condition, 2nd term disappears (recall χ = 1/2) and solution is
ideal
2
π
πV ≅ n RT
• 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
2
3
concentration
P = RT ( A1c+ A2c + A3c +...)
• We can therefore identify the virial coefficients A1 and A2 as:
A1 =
1
Mn
⎛1
⎞
⎜ − χ⎟
⎝2
⎠
A2 = 2
ρ 2V1
Comments about Osmometry
1.
2.
We assumed mean field conditions - local environment is similar
everywhere.
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 ≅
c2 ≈ c
*
2
M /NA
mass of polymer molecule
=
3
volume of coil
⎛ r 2 1/ 2 ⎞
⎟
⎜
⎠
⎝
c2 > c2*
Schematic Polymer-Solvent Structure at Various c2
C Overlap
Concentration
Semi-Concentrated
Solution
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
– Infrared absorption
– Refractive index changes
•
:
:
:
UV Detector
IR Detector
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
Swollen PS
Xlinked Bead
GPC Column
Porous Interior of PS Bead
Porous Gel
Greatly magnified
Magnified
Cross linked PS
beads in well
packed column
Diffusing chain
PS chain network
Pore
Polymer
molecules
outside beads
: 102 A - 105 A
Figure by MIT OCW.
Principles behind SEC involves…
•
Competition between 2 types of entropy
1.
2.
•
The interplay of both types of entropies results in substantially different residence
times in the column for the different polymer molar masses.
1.
2.
3.
•
ΔSmix favorable entropy of mixing of polymer and solvent inside pores.
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).
Mixing entropy gain drives smaller chains to enter and diffuse throughout entire gel
Intermediate size chains access a portion (the larger pores) of the pore volume in the
gel
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 Chromatograph
solvent
carrier
reservoir
polymer solution
of mixed Mi
inject
(typical flowrate ~ 1 cm3/min)
GPC column(s): 4-6 columns in series
packed column of 10μm diam. crosslinked PS gel beads
pore size ~ 102Å – 105Å
hν
Concentration
UV, IR, or RI signal
Detector
Preparative GPC
(collect fractions of eluted solution)
For a Polymer with Narrow
Weight Distribution
UV, IR
or R.I.
signal
high MW
low MW
Elution volume →
time →
SEC cont’d
•
The volume eluted Ve consists of 2 parts:
External to the Beads: “Void” Volume Vo (outside volume)
Pore Volume Vi (inside volume)
Internal to the Beads:
•
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.
K se (x 2 ) ≡
c (x )
polymer concentration inside bead
= 2i 2
polymer concentration outside bead
c 2o (x 2 )
Kse = 1 for x1 = 1 solvent, i.e. no exclusion
Kse = 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 = 0
Δ 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 K = eΔS/R
se
SEC cont’d
Furthermore, we can approximate:
⎡⎛ pore surface area ⎞
⎤
ΔS pp = − R⎢⎜
⎟ (diameter of chain)⎥
⎣⎝ volume of pore ⎠
⎦
here Vep
Vep = Vo + ViKse
Vep = Vo + Vi e
−⎛⎜ As r 2
⎝
1/ 2
⎞
⎟
⎠
is the elution volume of the polymer and As is the pore S/V ratio
As depends on the pore/column and <r2>1/2 depends on solvent, T and the MW of that polymer
chain entering the pore
GPC Calibration
Elution Volume vs. MW
PS Calibration
PS
PI
PB
PVCH
useful range
of separation
V0 + Vi
log MW →
last out
Vep
V0
first out
Chain size
(Molecular Weight)
THF solvent
Elution volume →
Detector Sensitivity/Selectivity: An Example
Polydispersity Index (PDI)
GPC result
22.0
Response
17.6
PDI =
13.2
Mw
Mn
2 Column GPC
8.8
PDI = 1.12, apparently
a nice sample…
4.4
0.0
10.0
11.2
12.4
13.6
14.8
16.0
17.2
18.4
19.6
20.8
22.0
Retention Volume (ml)
6 Column GPC (high resolution)
PDI >> 1.12!!
[The Truth hurts!!]
38
39
40
41
42
43
44
45
46
47
Elution Volume (ml)
Figure by MIT OCW.
Note very different elution volume values
Scattering of Radiation: Dilute Polymer Solution
• M w , A2(χ) and <Rg2> (Zimm plot)
• SALS (small angle light scattering)
SAXS (small angle X-ray scattering)
SANS (small angle neutron scattering)
I0
r
I0
Figure by MIT OCW.
q=
4π
λ
no
sin θ 2
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, M w and A2(chi)
Basic Scattering Equation
1 ⎡ 1
Kc2
⎤
2
....
=
+
+
A
c
2 2
⎥⎦
ΔR(θ ) P(θ ) ⎢⎣ M w
We will derive this. Note the
nice set of variables…that we
would like to be able to determine
K = optical constants
c2 = polymer concentration
P(θ ) = particle scattering factor, known for various particle geometries
⎞
⎛1
⎜ −χ⎟
2
A2 = ⎝ 2 ⎠ = 2 nd virial coefficient
ρ 2 V1
ΔR(θ ) = excess Raleigh ratio = Rsolution − Rsolvent
Light
X-ray
Neutron
~ excess scattering intensity
Scattering arises from…
(Δα)2
polarizability fluctuations
(Δρ)2
electron density variations
(Δb)2
neutron scattering length variation
Scattering I: from Density Fluctuations
A dilute gas in vacuum
• Consider small particles:
– (situation:~ point scatterers)
dp << λradiation
Particles
dp
Incident radiation
λ
• Scattered Intensity at scattering angle θ to a detector r away
from sample:
I 8π (1+ cos θ )
α = polarizability of molecule
Iθ =
α
I0 = incident beam intensity
λr
• For N particles in total volume V (assume dilute, so no coherent
4
2
0
2
4
2
scattering)
N
Iθ = Iθ
V
'
⎛N⎞
ε = 1 + 4 π⎜ ⎟ α
⎝V⎠
ε = dielectric constant
ε = n2, ε(ω) = frequency dependent
Fundamental relationship:
index of refraction
polarizability
N
n = 1+ 4π α
V
Can approximate
n gas
dn
≅ 1+
c
dc
dn
= refractive index increment
dc
c = conc. of particles per unit volume
2
⎛ dn ⎞
⎛ dn ⎞
n = ⎜1 +
c ⎟ ≅ 1 + 2⎜ ⎟ c + . . .
⎝ dc ⎠
⎝ dc ⎠
1 (dn / dc) c
Solving gives
α=
2 π (N /V )
the polarizability
2
gas
So by analogy for a polymer-solvent solution:
dn
n ≅ n0 +
c2
dc 2
n 2 ≅ n 02 + 2 n 0
dn
c2
dc 2
Rayleigh and Molecular Weight of Gases
2
2
π
1+
cos
θ ) ⎛ (dn / dc) c ⎞
8
⎛
⎞
N
(
'
Iθ = ⎜ ⎟
⎜
⎟
⎝V ⎠
λ4 r 2
⎝ 2π (N /V ) ⎠
simplifying
2
2
2
2
π
1+
cos
θ
I
⎛
⎞
dn
(
)
2
Iθ' = 0 4 2
⎜ ⎟c
λ r (N /V ) ⎝ dc ⎠
2
and since
N
c
=
V M / N AV
This expression contains several parameters dependent on scattering geometry, so we define
Iθ'
R=
I 0 (1+ cos 2 θ )/r 2
Rayleigh Ratio, R as
2
which equals
⎛ dn ⎞
2π 2 ⎜ ⎟ M c
⎝ dc ⎠
R=
λ4 N AV
Or just
Where K is a lumped optical constant
⎛ dn ⎞
2π 2 ⎜ ⎟
⎝ dc ⎠
K≡
λ4 N AV
2
R = K•M•c
Note, for polymer-solvent solution:
⎛
⎞
2 dn
2
2π n 0 ⎜
⎟
dc
⎝ 2⎠
K≡
λ4 N AV
Rayleigh measured the molecular weight of gas molecules
using light scattering!
2