Chapter 3Chapter 3
Molecular WeightMolecular Weight
1. Thermodynamics of Polymer Solution2. Mol Wt Determination
Ch 3-1 Slide 2
1. Weight, shape, and size of polymers1. Weight, shape, and size of polymersmonomer – oligomer – polymer
dimer, trimer, ---
telomer ~ oligomer from telomerization (popcorn polymerization)
telechelic polymer ~ with functional group
macro(mono)mer ~ with polymerizable group [wrong definition p72]
pleistomer ~ mol wt > 1E7
usual range of mol wt of polymers25000 ~ 1E6
mol wt of chain polymers are higher
molecular weight molecular sizeLS (solution) and SANS (bulk) determine size.
DSV and GPC utilize this relation in solution.
conformation shape
Ch 3-1 Slide 3
2. Solution2. SolutionΔGm = ΔHm – T ΔSm
ΔSm > 0 always
ΔHm > 0 almost always
“like dissolves like”
ΔHm = 0 at best (when solute is the same to solvent)
if not, ΔHm > 0
ΔHm < 0 only when specific interaction like H-bonding exists
For solution, ΔHm < T ΔSm
m for mixingf for melting (fusion)
Ch 3-1 Slide 4
Solubility parameterSolubility parameter
ΔHm = Vm [(ΔE1/V1)½ – (ΔE2/V2)½]2 v1v2
= Vm [δ1 – δ2]2 v1v2
ΔE ~ cohesive energy ~ energy change for vaporization
ΔE = ΔHvap – PΔV ≈ ΔHvap – RT [J]
ΔE/V ~ cohesive energy density [J/cm3 = MPa]
δ ~ solubility parameter [MPa½]
[MPa½] = [(106 N/m2)½] = [(J/cm3)½] ≈ [(1/2)(cal/cm3)½]
1: solvent2: solute
Table 3.1 & 3.2
Ch 3-1 Slide 5
Determination of δfrom ΔHvap data ~ for low mol wt, not for polymers
with solvent of known δswelling ~ Fig 3.1
viscosity ~ Fig 3.2
group contribution calculationδ = ρ ΣG / M ~ Table 3.3
G ~ group attraction constant
example p79
ΔE = ΔEdispersion + ΔEpolar (+ ΔEHB)δ2 = δdispersion
2 + δpolar2 (+ δHB
2 )
Ch 3-1 Slide 6
For solution,ΔHm < T ΔSm
without specific interaction
δ1 = δ2 at best ΔHm = 0 ΔGm < 0
Δδ < 20 MPa½ (?) ~ for solvent/solvent solution
Δδ < 2 MPa½ ~ a rough guide for solvent/polymer solutionΔSm smaller
Δδ < 0.1 MPa½ ~ for polymer/polymer solution
semicrystalline polymers not soluble at RTpositive ΔHf ΔHf + ΔHm > T ΔSm
Table 3.2 ~ δ for amorphous state at 25 °C
Ch 3-1 Slide 7
3. Thermodynamics of polymer solution3. Thermodynamics of polymer solutionTypes of solutions
ideal soln ΔHm = 0, ΔSm = – k (N1 ln n1 + N2 ln n2)
regular soln ΔHm ≠ 0, ΔSm = – k (N1 ln n1 + N2 ln n2)
athermal soln ΔHm = 0, ΔSm ≠ – k (N1 ln n1 + N2 ln n2)
real soln
ideal solutionΔG1 = μ1 – μ1
o = RT ln n1
ΔG2 = μ2 – μ2o = RT ln n2
ΔGm = (N1/NA)ΔG1 + (N2/NA)ΔG2
= kT (N1 ln n1 + N2 ln n2)
ΔHm = 0 ΔSm = – k (N1 ln n1 + N2 ln n2)
Eqn (3.9) corrected
n: mol fractionN: number of moleculesn1 = N1/(N1+N2)
Eqn (3.12)
Ch 3-1 Slide 8
ΔΔSSmm from statistical thermodynamicsfrom statistical thermodynamicsLattice model
Filling N1 & N2 molecules in N1+N2 = N0 cellsvolume of 1 ≈ volume of 2
Boltzmann relation, Sconfigurational = k ln ΩΩ ~ number of (distinguishable) ways
Ω12 = (N1+N2)!/N1!N2!
ΔSm = S12 – S1 – S2
= k ln Ω12 = k ln [(N1+N2)!/N1!N2!]Sterling’s approximation, ln x! = x ln x – x
ΔSm = k [(N1+N2) ln (N1+N2) – (N1+N2) – N1 ln N1 + ---= – k (N1 ln n1 + N2 ln n2)
Fig 3.3(a)
S1 = k ln Ω1 = k ln (N1!/N1!) = 0 = S2
n1 = N1/(N1+N2)
Ch 3-1 Slide 9
ΔΔSSmm of polymer of polymer solnsoln from stat thermofrom stat thermodeveloped by Flory & Huggins
polymer soln = mixture of solvent/polymervolume of 1 << volume of 2 (by x)
A polymer molecule with x mers (repeat units) takes x cells.volume of 1 mer ≈ volume of 1 solvent molecule
Filling N1 solvents & N2 polymers in N1+ xN2 = N0 cellsΔSm = S12 – S1 – S2 = k ln [Ω12/Ω1Ω2]
Number of ways to fill the (i+1)th chain in N0 cells
νi+1 = (N0-xi) z(1-fi) (z-1)(1-fi) ----- (z-1)(1-fi)
1st 2nd 3rd xth segment (mer)
z ~ coordination number (# of nearest neighbor)fi ~ probability of a site not available ≈ xi/N0
Fig 3.3(c)
Ch 3-1 Slide 10
(cont’d)
νi+1 = (N0-xi) z(1-fi) (z-1)(1-fi) ----- (z-1)(1-fi)
= (N0–xi) z (z–1)x-2 [1–(xi/N0)]x-1
= (N0–xi) (z–1)x-1 [(N0–xi)/N0]x-1
= (N0–xi)x [(z–1)/N0]x-1
= {(N0–xi)!/[N0–x(i+1)]!} [(z–1)/N0]x-1
(N0-xi)! / [N0-x(i+1)]! =
(N0-xi)(N0-xi-1)(N0-xi-2)----(3)(2)(1)(N0-xi-x)(N0-xi-x-1)-----(3)(2)(1)
= (N0-xi)(N0-xi-1)----(N0-xi-x+1)
≈ (N0-xi)x
Ch 3-1 Slide 11
ΔSm = S12 – S1 – S2 = k ln [Ω12/Ω1Ω2]
Ω12 ~ # of ways to fill N1+N2 molecules in N0 cells
= (1/N2!) Π νi+1 (from i = 0 to N2-1) (x 1)
= (1/N2!) {[N0!/(N0–x)!][(N0–x)!/(N0–2x)!] -----
[(N0–(N2–1)x)!/(N0–N2x)!]} [(z–1)/N0]N2(x-1)
= (1/N2!) [N0!/(N0–N2x)!] [(z–1)/N0]N2(x-1)
= [N0!/ N1!N2!] [(z–1)/N0]N2(x-1) << [N0!/ N1!N2!]
Ω1 ~ # of ways to fill N1 solvent molecules in N1 cells = 1
Ω2 ~ # of ways to fill N2 polymer molecules in xN2 cells
~ xN2 mers in xN2 cells ~ Ω2 = 1? No
= (1/N2!) [(xN2)!/(xN2–N2x)!] [(z–1)/xN2]N2(x-1)
= [(xN2)!/N2!] [(z–1)/xN2]N2(x-1)
Ch 3-1 Slide 12
Allcock p412
Sc = ΔSdis + ΔSm = (a) + (b)
ΔSdis for disorientation ~ equiv to S2 (Ω2) ~ Ω with N1 = 0
ΔSm = Sc – ΔSdis
ΔSmΔSdis
S = 0 S = Sc = S12
Ch 3-1 Slide 13
(con’t)
ΔSm = k ln [Ω12/Ω2]
= k ln {[N0!/N1!xN2!] [xN2/N0]N2(x-1)}Sterling’s approximation, ln x! = x ln x – x
= k {– N1 ln [N1/N0] – N2 ln [xN2/N0]}
= – k [N1 ln v1 + N2 ln v2]
x (mol wt ) N2 ΔSm
for polymer/polymer soln, ΔSm even smaller (N1 & N2 )
Flory-Huggins theory
volume fraction instead of mole fraction
Eqn (3.16)
v ~ volume fraction
Eqn (3.19)
Ch 3-1 Slide 14
ΔSm = – k [N1 ln v1 + N2 ln v2]
x (mol wt ) ΔSm
for polymer/polymer soln, ΔSm even smaller
Examples (for the same v1 = v2 = .5)
case 1: N1=10000, N2=10000, x1 = x2 = 1
ΔSm = – k [10000 ln .5 + 10000 ln .5] = – 20000 k ln .5
case 2: N1=10000, N2=100, x2 = 100; ΔSm = – 10100 k ln .5
case 3: N1=10000, N2=10, x2 = 1000; ΔSm = – 10010 k ln .5
case 4: N1=10, N2=10, x1 = x2 = 1000; ΔSm = – 20 k ln .5
more examples p85
Ch 3-1 Slide 15
ΔΔHHmm of polymer of polymer solnsolnregular solution
ΔHm ≠ 0, ΔSm = – k (N1 ln n1 + N2 ln n2)
ΔHm = N1 z n2 ΔwΔw ~ energy change per contact = w12 – [(w11+w22)/2]
for polymer solutionΔHm = k T N1 v2 χ
χ ~ Flory-Huggins interaction parameter [dimensionless]
kTχ ~ interaction energy (solvent in soln – in pure solvent)χ ΔHm solvent power
ΔHm = Vm [δ1 – δ2]2 v1v2 ~ k T N1 v2 χχ = β1 + (V1/RT) [δ1 – δ2]2
β1 ~ entropic ≈ 0
χ = χ1 = χ12Eqn (3.21)
Table 3.4
Eqn (3.28)
1---11---2
2---2
See Young pp143-144
Ch 3-1 Slide 16
ΔΔGGmm ~ Flory~ Flory--Huggins Huggins EqnEqnΔGm = ΔHm – T ΔSm
= kT [N1 ln v1 + N2 ln v2 + χN1v2]
useful for predicting miscibility (solubility)
drawbacksno volume change
self-intersection
for concentrated solutions only (high v2)χ is not purely enthalpic
example calculation p85
Eqn (3.22)
See Young p145
Ch 3-1 Slide 17
Partial molar free energy of mixing for solventPartial molar free energy of mixing for solvent
ΔG1 = ∂ΔGm/∂m1
from Flory-Huggins eqn
ΔGm = kT [N1 ln v1 + N2 ln v2 + χN1v2]
N1 = NAm1, v1 = m1/(m1+xm2), v2 = xm2/(m1+xm2), kNA = R
ΔG1 = RT [ln (1 – v2) + (1 – 1/x)v2 + χv22]
other form of Flory-Huggins eqn
ΔG1 = μ1 – μ1o = RT ln a1 = RT ln n1γ1
ΔG1 = μ1 – μ1o = (μ1–μ1
o)ideal + (μ1–μ1o)xs
ideal: (μ1–μ1o)ideal = RT ln n1
excess: (μ1–μ1o)xs = RT ln γ1
m: # of moles
Eqn (3.23)
a: activityγ: activity coeff.n: mol fraction
Eqn (A)
Sup 2 Young p145-149
Ch 3-1 Slide 18
Thermo of Thermo of dilutedilute polymer polymer solnsolndilute polymer soln
polymer chains separated by solvent
FH theory does not holdIn FH theory, chains are placed randomly
Modification ~ Flory-Krigbaum theory
for dil polym solnn2 = v2/x
v2 = xN2/(N1+xN2) ≈ xN2/N1 (N1 >> xN2)
n2 = N2/(N1+N2) ≈ N2/N1 (N1 >> N2)
ln v1 = ln (1 – v2) = – v2 – v22/2 – v2
3/3 – ---ln n1 = ln (1 – n2) = – n2 – n2
2/2 – n23/3 – ---
= – v2/x – (v2/x)2/2 – ---
Ch 3-1 Slide 19
from Eqn (3.23)ΔG1 = μ1–μ1
o = RT [– v2 – v22/2 + v2 + v2/x + χv2
2]= –RT(v2/x) + RT(χ – ½)v2
2 Eqn (3.23-1)
from Eqn (A)ΔG1 = μ1–μ1
o = RT ln n1 + (μ1–μ1o)xs
= –RT(v2/x) + (μ1–μ1o)xs
By Flory-KrigbaumΔG1
xs = (μ1–μ1o)xs = ΔHxs – T ΔSxs
= RTκ v22 – T Rψ v2
2 = RT(κ – ψ) v22
ΔG1xs = RTψ [(θ/T) – 1] v2
2 = RT (χ – ½) v22
When T = θ, χ = ½ ΔG1xs = 0 ΔG1= ΔG1
ideal
θ-condition (Flory condition) ~ becomes ideal solutionWhen T > θ, χ < ½ ΔG1
xs < 0 soluble
κ = ψθ/T
Table 3.4
–
χ = ½ for ideal
Chapter 3Chapter 3
Molecular WeightMolecular Weight
1. Thermodynamics of Polymer Solution
2. Mol Wt Determination
Ch 3-1 Slide 21
4. Mol wt and mol wt distribution4. Mol wt and mol wt distributionmol wt distribution
xi ~ number (mol) of i = Nii ~ molecule having Mi
wi ~ weight (amount) of i = NiMi
Usually xi and wi are fractionsxi = Ni/ΣNi , wi = NiMi/ΣNiMiNot in this textbook
Ni
xi
Mi
Mi
wi
NiMi
Ch 3-1 Slide 22
mol wt averagesmol wt averagesnumber-average mol wt (수평균 분자량)
weight-average mol wt (중량평균 분자량)
z-average mol wt
viscosity-average mol wt (점도평균 분자량)
= total weight/total number~ weight of 1 molecule
a dep on solvent & tempMv is not an absolute mol wt.
Mn, Mw, Mz are absolute mol wts.
z+1-average mol wt, etc
Eqn (3.31-34)
Ch 3-1 Slide 23
mol wt distribution (MWD)mol wt distribution (MWD)Mol wt of polymers almost always has a distribution.
polydisperse (다분산성) ↔ monodisperse (단분산성)
polydispersity index (PDI) = Mw/Mn
other indexes; Mz/Mw, Mz+1/Mz …
Most probable distribution (Flory(-Schultz) distribution)
Mn/Mw/Mz = 1/2/3
ideal, not probable
practically Mw/Mn > 2
p86 wrong!
Fig 3.4
Ch 3-1 Slide 24
mol wt & propertiesmol wt & propertiesmol wt independent properties
density, refractive index, solubility, stability, etc
dep on repeat unit (chemical) structure
Mn dependent propertiesthermal and mechanical properties
Tg, Tm, strength, modulus, etc
dep on segmental motion, chain-end concentration
Tg = Tg∞ – A/Mn
Tg
Mn
Ch 3-1 Slide 25
mol wt & properties (2)mol wt & properties (2)Mw dependent properties
(melt) viscosity
dep on whole chain motion
MWD dependent propertiesshear-rate sensitivity of viscosity
dep more on larger molecules
log η
log Mw
Ch 3-1 Slide 26
5. Determination of 5. Determination of MMnn
end-group analysesstep polymers
HOOC------COOH H2N-----NH2 HO-----OH
titration or spectroscopic methods
chain polymers
RMMMM----- (R=initiator fragment)
spectroscopic methods
accurate but limited
Ch 3-1 Slide 27
Colligative property measurementscolligative (collective) property ~ property that depends only on the number of molecules
osmotic pressure, boiling point, freezing point, etc
counting number & measuring weight Mn
ΔG1 = μ1 – μ1o = RT ln a1 = RT ln γ1n1
For dilute polymer solution (c2 0)
solvent behaves ideally, a1 ≈ n1
μ1 – μ1o = RT ln n1 = RT ln (1–n2)
= –RT[n2 + n22/2 + n2
3/3 + -----]
a: activityγ: activity coeff.n: mol fractionc: wt conc’n
Ch 3-1 Slide 28
n2 = N2/(N1+N2) ≈ N2/N1 = (N2/NA)/(N1/NA) = m2/m1 = (m2/L)/(m1/L) = (c2/M2)/(1/V1
0) [(g/L)/(g/mol)]/[(1/(L/mol)]= (c2V1
0)/M2
μ1 – μ1o = –RT[n2 + n2
2/2 + n23/3 + ----]
= –RTV10[(1/M2)c2 + (V1
0/2M22)c2
2 + (V102/3M2
3)c23 --]
–(μ1 – μ1o)/V1
0 = RT [(1/M2) c2 + A2 c22 + A3 c2
3 + ----]
virial equationA2 ~ 2nd virial coeff, A3 ~ 3rd virial coeff
for dilute polymer soln, c2 0[CP/c]c 0 = RT/Mn
colligative property (CP)CP/c
c
A2RT/Mn
c: wt conc’nm: # of molesV1
0: molar volM: mol wt
Ch 3-1 Slide 29
ebulliometry (bp elevation)ΔTb/c = Ke [(1/Mn) + A2 c + A3 c2 + ----]
Ke calibrated with known mol wt
limited by precision of temperature measurementuseful only for Mn < 30000
not used these days
cryoscopy (fp depression)ΔTf/c = Kc [(1/Mn) + A2 c + A3 c2 + ----]
Kc calibrated with known mol wt
limited by precision of temperature measurementuseful only for Mn < 30000
not used these days
Eqn (3.35)
Eqn (3.36)
Ch 3-1 Slide 30
membrane osmometry
static or dynamic method
useful for 30000 < Mn < 10E6diffusion of solutesmall signal (π)
h ρgh = π ~ osmotic pressure
μ1(1,P) = μ1(n1,P+π) μ1
0(P) = μ10(P) + ∫PP+π V1
0dP + RT ln a1
πV10 = RTV1
0 [(1/Mn)c + A2 c2 + A3 c3 ----]π/c = RT [(1/Mn) + A2 c + A3 c2 + ----]
Eqn (3.41)
Ch 3-1 Slide 31
Determination without extrapolation?πV1
0 = RTV10 [(1/Mn)c+A2 c2+ --] = –RT ln a1 = –(μ1–μ1
o)
μ1–μ1o = –RT(v2/x) + RT(χ – ½)v2
2
π = RT(v2/xV10) + RT(χ – ½)v2
2/V10
v2 ≈ xN2/N1, V = (N1/NA)V10, Mn = ΣNiMi/ΣNi = M2/(N2/NA)
c2 = M2/V = MnN2/NAV, ρ2 = V2/Mn, x = V2/V1
π/c = RT(1/Mn) + RT (χ – ½)(1/V1ρ22) c
= RT [1/Mn + A2 c]At θ-condition, χ = ½ , A2 = 0
no conc’n dependence
determination at 1 conc’n ~ need no extrapolation
hard to do ~ not a good solvent (ppt)
Eqn (3.23-1) dil soln
Eqn (3.26)
Fig 3.5
Ch 3-1 Slide 32
vapor phase (pressure) osmometry (VPO)P1
0 –P1 = ΔP~ vapor pressure drop
due to solute
ΔP ΔT Δr
Δr/c = KVPO[(1/Mn) + A2 c ----]KVPO calibrated with known mol wt at the same temp,drop size, time
Useful for Mn < 30000small signal (Δr)
comparison of the methods Table 3.5
Ch 3-1 Slide 33
6. Determination of M6. Determination of Mww
light scattering (LS)Light scattered by fluctuation in
refractive index (n) concentration mol wt
Hc/Rθ = 1/Mw + 2 A2c + 3 A3c2 + ---H = 2π2n0
2(dn/dc)2/NAλ4
Why Mw? intensity ∝ (amplitude)2 ∝ (mass)2
[Hc/Rθ]c 0 = 1/Mw for small molecules, not for polymers
θr
I0
λ
iθiθ/I0 = f (dn/dc, M, λ, n0)
Rayleigh ratio, Rθ = (iθ/I0)r2
Eqn (3.43)
Ch 3-1 Slide 34
for large molecules (D > λ/20)Hc/Rθ = 1/(MwP(θ)) + 2 A2c + ---
P(θ) = scattering (form) factor = Rθ/R0
1/P(θ) = 1 + (8π2/9λ2)<r2>sin2(θ/2)= 1 + (16π2/3λ2)<Rg
2>sin2(θ/2)r = end-to-end distance
Rg = radius of gyration
<r2>0 = 6 <Rg2>0
Hc/Rθ = 1/Mw + (16π2/3λ2Mw)<Rg2>sin2(θ/2) + 2 A2c + ---
[Hc/Rθ]θ=0 = 1/Mw + 2 A2c + ---
[Hc/Rθ]c=0 = 1/Mw + (16π2/3λ2Mw)<Rg2>sin2(θ/2)
[Hc/Rθ]c=0, θ=0 = 1/Mw ‘Zimm Plot’
r
Rg
Eqn (3.50), Fig 3.10(c)(d)
i30 ≠ i45
Eqn (3.51), Fig 3.10(a)(b)Fig 3.11
Eqn (3.61)
Ch 3-1 Slide 35
7. MW of common polymers7. MW of common polymersMW of commercial polymers
step polymers: 20000 – 40000
chain polymers: 20000 – 1000000
MWDFlory-Schultz distribution: PDI = 2
when ideal
Poisson distribution: PDI = 1anionic living polymerization
In most polymerizations: PDI > 2Table 3.9
PDI(chain polymers) > PDI(step polymers)
Ch 3-1 Slide 36
8. Determination of 8. Determination of MMvv
dilute solution viscometry (DSV)viscosity size mol wt
measures molecular size, not weight
not an absolute method, but a relative method
viscosity, ηη = η0 (1 + ωv2) Einstein eqn
ω = 2.5 for sphere
v2 ∝ size of solute
η/η0 – 1 = 2.5 N2Ve/V
ηrel – 1 = ηsp = 2.5 cNAVe/M (g/L)(1/mol)(L)/(g/mol)
[ηsp/c]c 0 = 2.5 NAVe/M = [η] ‘intrinsic viscosity’ (dL/g)
η0: solvent onlyv2: vol fraction of soluteV: vol of solnVe: vol of equiv. spherec: wt conc’nM: mol wtfor η’s, see handout p92Fig 3.13
Fig 3.12shape
Ch 3-1 Slide 37
[η] mol wtVe = (4/3)πRe
3 = (4/3)πH3Rg3 ∝ (4/3)πH3(M/M0)3/2
Re = HRg, Rg ∝ (M/M0)½ at θ-condition
[η]θ = 2.5 NAVe/M ∝ (10πNAH3/3M03/2)M½
[η]θ = Kθ M0.5 at θ-condition
in good solvent, Rg = α Rgθ
[η] = α3 [η]θ = α3 Kθ M0.5 = λ3 Kθ M(0.5+3Δ) = K Ma
α = λ MΔ
[η] = K Mva ‘Mark-Houwink-Sakurada (MHS) eqn’
Mv~ viscosity-average mol wta ≥ 0.5
Ch 3-1 Slide 38
Mv ~ viscosity-average mol wtηsp = Σ(ηsp)i ~ Ni moles of Mi mol wt
= Σ ci [η]i = Σ (NiMi) (KMia) = K ΣNiMi
1+a
[η] = [ηsp/c]c 0 = K ΣNiMi1+a/ΣNiMi = K Mv
a
Mv = [ΣNiMi1+a/ΣNiMi]1/a
0.8 ≥ a ≥ 0.5
0.5 at θ-condition
when a = 1, Mv = Mw
when a = –1, Mv = Mn
Mv close to MwFig 3.4
Ch 3-1 Slide 39
DSV experiment capillary viscometer Poiseulli eqn, Q = V/t = πr4P/8ηLη ∝ t η/η0 = t/t0Procedure
measure t0, t1, t2 --- at c0, c1, c2 --- (0 ~ solvent)[η] = [ηred]c 0 = [ηsp/c]c 0 = [(ηrel – 1)/c]c 0
= [(η/η0 – 1)/c]c 0 = [(t/t0 – 1)/c]c 0
or [η] = [ηinh]c 0 = [ln ηrel/c]c 0= [ln (η/η0)/c]c 0 = [ln (t/t0)/c]c 0
[η] = K Mva
K, a from handbook at the same temp and solvent
Cautions: temp control < 0.2 Kt0 > 100 s (laminar) c < 1 g/dL (Newtonian)
Table 3.10
Fig 3.14
Ch 3-1 Slide 40
9. Gel Permeation Chromatography (GPC)9. Gel Permeation Chromatography (GPC)size exclusion chromatography
separation by size using porous gel substrate
Larger molecules elute earlier.
Instrumentationinjector – column(s) – detector
A chromatogram
Fig 3.16-18
VR: retention volumeVR = tR x flow rate
VR ~ size mol wt (Mi)Hi ~ amount NiMiNeeds calibration
Ch 3-1 Slide 41
Universal calibrationWith the same instrument, column, and solvent, the same VR represents the same hydrodynamic volume.
[η] M = [2.5 NA V] = K Mva+1
Many polymers fall on the same curve on the [η]M – VR
plot ~ universal calibration curve
Procedure of an experiment1. From the chromatogram,
read VRi and Hi
(column 1 and 2).
2. Run the same experimentwith polystyrene standards.
Fig 3.23
Ch 3-1 Slide 42
2 (cont’d) PS standard anionically polymerized with known mol wt
3. Draw a calibration curve (Mi vs VRi) for PS.
4. Read Mi (PS) for each VRi (column 1 , sample).
Ch 3-1 Slide 43
5. M (PS) M (sample)
[η]PSMPS = [η]sampleMsample
KPSMPSa(PS)+1 = KsampleMsample
a(sample)+1
6. Ni = Hi / Mi (column 2 / column 3)
7. Calculate Mn, Mw, MWD.
Are Mn and Mw obtained absolute? No.
from handbookfrom calibration curve
column 3
Ch 3-1 Slide 44
10. Mass spectrometry10. Mass spectrometryMS determines mol wt
by detecting molecular ion, M+
in vapor phase
ionization of polymers in gas phase?
MALDI-TOF techniquesoft ionization
choice of the matrix critical
for not-too-high mol wt
useful for (highly) branched polymers