Chapter 3. Molecular Weight. 1. Thermodynamics of Polymer Solution 2. Mol Wt Determination

Transcription

1 Chapter 3 Molecular Weight 1. Thermodynamics of Polymer Solution 2. Mol Wt Determination

2 1. Weight, shape, and size of polymers monomer 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 polymers ~ 1E6 mol wt of chain polymers are higher molecular weight conformation shape LS (solution) and SANS (bulk) determine size. DSV and GPC utilize this relation in solution. molecular size Ch 3-1 Slide 2

3 2. Solution ΔG m = ΔH m T ΔS m ΔS m > 0 always ΔH m > 0 almost always like dissolves like ΔH m = 0 at best (when solute is the same to solvent) if not, ΔH m > 0 ΔH m < 0 only when specific interaction like H-bonding exists For solution, ΔH m < T ΔS m m for mixing f for melting (fusion) Ch 3-1 Slide 3

4 Solubility parameter ΔH m = V m [(ΔE 1 /V 1 ) ½ (ΔE 2 /V 2 ) ½ ] 2 v 1 v 2 = V m [δ 1 δ 2 ] 2 v 1 v 2 ΔE ~ cohesive energy ~ energy change for vaporization ΔE = ΔH vap PΔV ΔH vap RT [J] ΔE/V ~ cohesive energy density [J/cm 3 = MPa] δ ~ solubility parameter [MPa ½ ] [MPa ½ ] = [(10 6 N/m 2 ) ½ ] = [(J/cm 3 ) ½ ] [(1/2)(cal/cm 3 ) ½ ] Table 3.1 & 3.2 1: solvent 2: solute Ch 3-1 Slide 4

5 Determination of δ from ΔH vap data ~ for low mol wt, not for polymers with solvent of known δ swelling ~ Fig 3.1 viscosity ~ Fig 3.2 group contribution calculation δ = ρ ΣG / M ~ Table 3.3 G ~ group attraction constant example p79 ΔE = ΔE dispersion + ΔE polar (+ ΔE HB ) δ 2 = δ dispersion2 + δ polar2 (+ δ HB2 ) Ch 3-1 Slide 5

6 For solution, ΔH m < T ΔS m without specific interaction δ 1 = δ 2 at best ΔH m = 0 ΔG m < 0 Δδ < 20 MPa ½ (?) ~ for solvent/solvent solution Δδ < 2 MPa ½ ~ a rough guide for solvent/polymer solution ΔS m smaller Δδ < 0.1 MPa ½ ~ for polymer/polymer solution semicrystalline polymers not soluble at RT positive ΔH f ΔH f + ΔH m > T ΔS m Table 3.2 ~ δ for amorphous state at 25 C Ch 3-1 Slide 6

7 3. Thermodynamics of polymer solution Types of solutions ideal soln ΔH m = 0, ΔS m = k (N 1 ln n 1 + N 2 ln n 2 ) regular soln ΔH m 0, ΔS m = k (N 1 ln n 1 + N 2 ln n 2 ) athermal soln ΔH m = 0, ΔS m k (N 1 ln n 1 + N 2 ln n 2 ) real soln ideal solution ΔG 1 = μ 1 μ 1o = RT ln n 1 ΔG 2 = μ 2 μ 2o = RT ln n 2 ΔG m = (N 1 /N A )ΔG 1 + (N 2 /N A )ΔG 2 = kt (N 1 ln n 1 + N 2 ln n 2 ) ΔH m = 0 ΔS m = k (N 1 ln n 1 + N 2 ln n 2 ) n: mol fraction N: number of molecules n 1 = N 1 /(N 1 +N 2 ) Eqn (3.9) corrected Eqn (3.12) Ch 3-1 Slide 7

8 ΔS m from statistical thermodynamics Lattice model Filling N 1 & N 2 molecules in N 1 +N 2 = N 0 cells volume of 1 volume of 2 Boltzmann relation, S configurational = k ln Ω Ω ~ number of (distinguishable) ways Ω 12 = (N 1 +N 2 )!/N 1!N 2! Fig 3.3(a) ΔS m = S 12 S 1 S 2 = k ln Ω 12 = k ln [(N 1 +N 2 )!/N 1!N 2!] S 1 = k ln Ω 1 = k ln (N 1!/N 1!) = 0 = S 2 Sterling s approximation, ln x! = x ln x x ΔS m = k [(N 1 +N 2 ) ln (N 1 +N 2 ) (N 1 +N 2 ) N 1 ln N = k (N 1 ln n 1 + N 2 ln n 2 ) n 1 = N 1 /(N 1 +N 2 ) Ch 3-1 Slide 8

9 ΔS m of polymer soln from stat thermo developed by Flory & Huggins polymer soln = mixture of solvent/polymer volume 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 N 1 solvents & N 2 polymers in N 1 + xn 2 = N 0 cells ΔS m = S 12 S 1 S 2 = k ln [Ω 12 /Ω 1 Ω 2 ] Number of ways to fill the (i+1) th chain in N 0 cells ν i+1 = (N 0 -xi) z(1-f i ) (z-1)(1-f i ) (z-1)(1-f i ) Fig 3.3(c) 1st 2nd 3rd x th segment (mer) z ~ coordination number (# of nearest neighbor) f i ~ probability of a site not available xi/n 0 Ch 3-1 Slide 9

10 (cont d) ν i+1 = (N 0 -xi) z(1-f i ) (z-1)(1-f i ) (z-1)(1-f i ) = (N 0 xi) z (z 1) x-2 [1 (xi/n 0 )] x-1 = (N 0 xi) (z 1) x-1 [(N 0 xi)/n 0 ] x-1 = (N 0 xi) x [(z 1)/N 0 ] x-1 = {(N 0 xi)!/[n 0 x(i+1)]!} [(z 1)/N 0 ] x-1 (N 0 -xi)! / [N 0 -x(i+1)]! = (N 0 -xi)(n 0 -xi-1)(n 0 -xi-2)----(3)(2)(1) (N 0 -xi-x)(n 0 -xi-x-1)-----(3)(2)(1) = (N 0 -xi)(n 0 -xi-1)----(n 0 -xi-x+1) (N 0 -xi) x Ch 3-1 Slide 10

11 ΔS m = S 12 S 1 S 2 = k ln [Ω 12 /Ω 1 Ω 2 ] Ω 12 ~ # of ways to fill N 1 +N 2 molecules in N 0 cells = (1/N 2!) Π ν i+1 (from i = 0 to N 2-1) (x 1) = (1/N 2!) {[N 0!/(N 0 x)!][(n 0 x)!/(n 0 2x)!] [(N 0 (N 2 1)x)!/(N 0 N 2 x)!]} [(z 1)/N 0 ] N 2 (x-1) = (1/N 2!) [N 0!/(N 0 N 2 x)!] [(z 1)/N 0 ] N 2 (x-1) = [N 0!/ N 1!N 2!] [(z 1)/N 0 ] N 2 (x-1) << [N 0!/ N 1!N 2!] Ω 1 ~ # of ways to fill N 1 solvent molecules in N 1 cells = 1 Ω 2 ~ # of ways to fill N 2 polymer molecules in xn 2 cells ~ xn 2 mers in xn 2 cells ~ Ω 2 = 1? No = (1/N 2!) [(xn 2 )!/(xn 2 N 2 x)!] [(z 1)/xN 2 ] N 2 (x-1) = [(xn 2 )!/N 2!] [(z 1)/xN 2 ] N 2 (x-1) Ch 3-1 Slide 11

12 Allcock p412 S c = ΔS dis + ΔS m = (a) + (b) ΔS dis for disorientation ~ equiv to S 2 (Ω 2 ) ~ Ω with N 1 = 0 ΔS m = S c ΔS dis S = 0 S = S c = S 12 ΔS dis ΔS m Ch 3-1 Slide 12

13 (con t) ΔS m = k ln [Ω 12 /Ω 2 ] = k ln {[N 0!/N 1!xN 2!] [xn 2 /N 0 ] N 2 (x-1) } Sterling s approximation, ln x! = x ln x x = k { N 1 ln [N 1 /N 0 ] N 2 ln [xn 2 /N 0 ]} = k [N 1 ln v 1 + N 2 ln v 2 ] v ~ volume fraction x (mol wt ) N 2 ΔS m for polymer/polymer soln, ΔS m even smaller (N 1 & N 2 ) Flory-Huggins theory volume fraction instead of mole fraction Eqn (3.16) Eqn (3.19) Ch 3-1 Slide 13

14 ΔS m = k [N 1 ln v 1 + N 2 ln v 2 ] x (mol wt ) ΔS m for polymer/polymer soln, ΔS m even smaller Examples (for the same v 1 = v 2 =.5) case 1: N 1 =10000, N 2 =10000, x 1 = x 2 = 1 ΔS m = k [10000 ln ln.5] = k ln.5 case 2: N 1 =10000, N 2 =100, x 2 = 100; ΔS m = k ln.5 case 3: N 1 =10000, N 2 =10, x 2 = 1000; ΔS m = k ln.5 case 4: N 1 =10, N 2 =10, x 1 = x 2 = 1000; ΔS m = 20 k ln.5 more examples p85 Ch 3-1 Slide 14

15 ΔH m of polymer soln regular solution ΔH m 0, ΔS m = k (N 1 ln n 1 + N 2 ln n 2 ) ΔH m = N 1 z n 2 Δw Δw ~ energy change per contact = w 12 [(w 11 +w 22 )/2] for polymer solution ΔH m = k T N 1 v 2 χ χ ~ Flory-Huggins interaction parameter [dimensionless] ktχ ~ interaction energy (solvent in soln in pure solvent) χ ΔH m solvent power ΔH m = V m [δ 1 δ 2 ] 2 v 1 v 2 ~ k T N 1 v 2 χ χ = β 1 + (V 1 /RT) [δ 1 δ 2 ] 2 β 1 ~ entropic 0 See Young pp Eqn (3.21) χ = χ 1 = χ 12 Eqn (3.28) Table 3.4 Ch 3-1 Slide 15

16 ΔG m ~ Flory-Huggins Eqn ΔG m = ΔH m T ΔS m = kt [N 1 ln v 1 + N 2 ln v 2 + χn 1 v 2 ] Eqn (3.22) useful for predicting miscibility (solubility) drawbacks no volume change See Young p145 self-intersection for concentrated solutions only (high v 2 ) χ is not purely enthalpic example calculation p85 Ch 3-1 Slide 16

17 Partial molar free energy of mixing for solvent ΔG 1 = ΔG m / m 1 from Flory-Huggins eqn m: # of moles Sup 2 Young p ΔG m = kt [N 1 ln v 1 + N 2 ln v 2 + χn 1 v 2 ] N 1 = N A m 1, v 1 = m 1 /(m 1 +xm 2 ), v 2 = xm 2 /(m 1 +xm 2 ), kn A = R ΔG 1 = RT [ln (1 v 2 ) + (1 1/x)v 2 + χv 22 ] other form of Flory-Huggins eqn Eqn (3.23) ΔG 1 = μ 1 μ 1o = RT ln a 1 = RT ln n 1 γ 1 ΔG 1 = μ 1 μ 1 o = (μ 1 μ 1o ) ideal + (μ 1 μ 1o ) xs Eqn (A) ideal: (μ 1 μ 1o ) ideal = RT ln n 1 excess: (μ 1 μ 1o ) xs = RT ln γ 1 a: activity γ: activity coeff. n: mol fraction Ch 3-1 Slide 17

18 Thermo of dilute polymer soln dilute polymer soln polymer chains separated by solvent FH theory does not hold In FH theory, chains are placed randomly Modification ~ Flory-Krigbaum theory for dil polym soln n 2 = v 2 /x v 2 = xn 2 /(N 1 +xn 2 ) xn 2 /N 1 (N 1 >> xn 2 ) n 2 = N 2 /(N 1 +N 2 ) N 2 /N 1 (N 1 >> N 2 ) ln v 1 = ln (1 v 2 ) = v 2 v 22 /2 v 23 /3 --- ln n 1 = ln (1 n 2 ) = n 2 n 22 /2 n 23 /3 --- = v 2 /x (v 2 /x) 2 /2 --- Ch 3-1 Slide 18

19 from Eqn (3.23) ΔG 1 = μ 1 μ 1 o = RT [ v 2 v 22 /2 + v 2 + v 2 /x + χv 22 ] = RT(v 2 /x) + RT(χ ½)v 2 2 Eqn (3.23-1) from Eqn (A) ΔG 1 = μ 1 μ 1 o = RT ln n 1 + (μ 1 μ 1o ) xs = RT(v 2 /x) + (μ 1 μ 1o ) xs By Flory-Krigbaum ΔG xs 1 = (μ 1 μ 1o ) xs = ΔH xs T ΔS xs κ = ψθ/t = RTκ v 22 T Rψ v 22 = RT(κ ψ) v 2 2 ΔG 1 xs = RTψ [(θ/t) 1] v 22 = RT (χ ½) v 2 2 When T = θ, χ = ½ ΔG 1 xs = 0 ΔG 1 = ΔG 1 ideal θ-condition (Flory condition) ~ becomes ideal solution When T > θ, χ < ½ ΔG 1 xs < 0 soluble χ = ½ for ideal Table 3.4 Ch 3-1 Slide 19

20 Chapter 3 Molecular Weight 1. Thermodynamics of Polymer Solution 2. Mol Wt Determination

21 4. Mol wt and mol wt distribution mol wt distribution x i N i x i ~ number (mol) of i = N i i ~ molecule having M i M i w i w i ~ weight (amount) of i = N i M i N i M i Usually x i and w i are fractions x i = N i /ΣN i, w i = N i M i /ΣN i M i Not in this textbook M i Ch 3-1 Slide 21

22 mol wt averages number-average mol wt ( 수평균분자량 ) = total weight/total number ~ weight of 1 molecule weight-average mol wt ( 중량평균분자량 ) Eqn ( ) z-average mol wt z+1-average mol wt, etc M n,m w,m z are absolute mol wts. viscosity-average mol wt ( 점도평균분자량 ) a dep on solvent & temp M v is not an absolute mol wt. Ch 3-1 Slide 22

23 mol wt distribution (MWD) Mol wt of polymers almost always has a distribution. polydisperse ( 다분산성 ) monodisperse ( 단분산성 ) polydispersity index (PDI) = M w /M n other indexes; M z /M w, M z+1 /M z Most probable distribution (Flory(-Schultz) distribution) M n /M w /M z = 1/2/3 ideal, not probable practically M w /M n > 2 p86 wrong! Fig 3.4 Ch 3-1 Slide 23

24 mol wt & properties mol wt independent properties density, refractive index, solubility, stability, etc dep on repeat unit (chemical) structure M n dependent properties thermal and mechanical properties T g, T m, strength, modulus, etc dep on segmental motion, chain-end concentration T g T g = T g A/M n M n Ch 3-1 Slide 24

25 mol wt & properties (2) M w dependent properties (melt) viscosity dep on whole chain motion log η log M w MWD dependent properties shear-rate sensitivity of viscosity dep more on larger molecules Ch 3-1 Slide 25

26 5. Determination of M n end-group analyses step polymers HOOC------COOH H 2 N-----NH 2 HO-----OH titration or spectroscopic methods chain polymers RMMMM----- (R=initiator fragment) spectroscopic methods accurate but limited Ch 3-1 Slide 26

27 Colligative property measurements colligative (collective) property ~ property that depends only on the number of molecules osmotic pressure, boiling point, freezing point, etc counting number & measuring weight M n ΔG 1 = μ 1 μ 1o = RT ln a 1 = RT ln γ 1 n 1 For dilute polymer solution (c 2 0) solvent behaves ideally, a 1 n 1 μ 1 μ 1o = RT ln n 1 = RT ln (1 n 2 ) = RT[n 2 + n 22 /2 + n 23 / ] a: activity γ: activity coeff. n: mol fraction c: wt conc n Ch 3-1 Slide 27

28 n 2 = N 2 /(N 1 +N 2 ) N 2 /N 1 = (N 2 /N A )/(N 1 /N A ) = m 2 /m 1 = (m 2 /L)/(m 1 /L) = (c 2 /M 2 )/(1/V 1 0 ) [(g/l)/(g/mol)]/[(1/(l/mol)] = (c 2 V 1 0 )/M 2 c: wt conc n m: # of moles V 10 : molar vol M: mol wt μ 1 μ 1o = RT[n 2 + n 22 /2 + n 23 / ] = RTV 10 [(1/M 2 )c 2 + (V 10 /2M 22 )c 22 + (V 1 02 /3M 23 )c 23 --] (μ 1 μ 1o )/V 10 = RT [(1/M 2 ) c 2 + A 2 c 22 + A 3 c ] colligative property (CP) virial equation A 2 ~ 2 nd virial coeff, A 3 ~ 3 rd virial coeff for dilute polymer soln, c 2 0 [CP/c] c 0 = RT/M n CP/c RT/M n A 2 c Ch 3-1 Slide 28

29 ebulliometry (bp elevation) ΔT b /c = K e [(1/M n ) + A 2 c+ A 3 c ] K e calibrated with known mol wt limited by precision of temperature measurement useful only for M n < not used these days cryoscopy (fp depression) ΔT f /c = K c [(1/M n ) + A 2 c+ A 3 c ] K c calibrated with known mol wt Eqn (3.35) Eqn (3.36) limited by precision of temperature measurement useful only for M n < not used these days Ch 3-1 Slide 29

30 membrane osmometry h ρgh = π ~ osmotic pressure μ 1 (1,P) = μ 1 (n 1,P+π) μ 10 (P) = μ 10 (P) + P P+π V 10 dp + RT ln a 1 πv 1 0 = RTV 10 [(1/M n )c + A 2 c 2 + A 3 c ] π/c = RT [(1/M n ) + A 2 c+ A 3 c ] static or dynamic method Eqn (3.41) useful for < M n < 10E6 diffusion of solute small signal (π) Ch 3-1 Slide 30

31 Determination without extrapolation? πv 1 0 = RTV 10 [(1/M n )c+a 2 c ] = RT ln a 1 = (μ 1 μ 1o ) μ 1 μ 1 o = RT(v 2 /x) + RT(χ ½)v 2 2 π = RT(v 2 /xv 10 ) + RT(χ ½)v 22 /V 1 0 v 2 xn 2 /N 1, V = (N 1 /N A )V 1 0, M n = ΣN i M i /ΣN i = M 2 /(N 2 /N A ) c 2 = M 2 /V = M n N 2 /N A V, ρ 2 = V 2 /M n, x = V 2 /V 1 π/c = RT(1/M n ) + RT (χ ½)(1/V 1 ρ 22 ) c = RT [1/M n + A 2 c] At θ-condition, χ = ½, A 2 = 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 31

32 vapor phase (pressure) osmometry (VPO) P 10 P 1 = ΔP ~ vapor pressure drop due to solute ΔP ΔT Δr Δr/c = K VPO [(1/M n ) + A 2 c ----] K VPO calibrated with known mol wt at the same temp, drop size, time Useful for M n < small signal (Δr) comparison of the methods Table 3.5 Ch 3-1 Slide 32

33 6. Determination of M w light scattering (LS) Light scattered by fluctuation in refractive index (n) concentration mol wt λ I 0 r θ i θ i θ /I 0 = f (dn/dc, M, λ, n 0 ) Rayleigh ratio, R θ = (i θ /I 0 )r 2 Hc/R θ = 1/M w + 2 A 2 c + 3 A 3 c H = 2π 2 n 02 (dn/dc) 2 /N A λ 4 Eqn (3.43) Why M w? intensity (amplitude) 2 (mass) 2 [Hc/R θ ] c 0 = 1/M w for small molecules, not for polymers Ch 3-1 Slide 33

34 for large molecules (D > λ/20) Hc/R θ = 1/(M w P(θ)) + 2 A 2 c i 30 i 45 P(θ) = scattering (form) factor = R θ /R 0 1/P(θ) = 1 + (8π 2 /9λ 2 )<r 2 >sin 2 (θ/2) = 1 + (16π 2 /3λ 2 )<R g 2 >sin 2 (θ/2) r = end-to-end distance R g = radius of gyration <r 2 > 0 = 6 <R g2 > 0 Eqn (3.61) r Hc/R θ = 1/M w + (16π 2 /3λ 2 M w )<R g2 >sin 2 (θ/2) + 2 A 2 c [Hc/R θ ] θ=0 = 1/M w + 2 A 2 c [Hc/R θ ] c=0 = 1/M w + (16π 2 /3λ 2 M w )<R g2 >sin 2 (θ/2) [Hc/R θ ] c=0, θ=0 = 1/M w Fig 3.11 Zimm Plot R g Eqn (3.50), Fig 3.10(c)(d) Eqn (3.51), Fig 3.10(a)(b) Ch 3-1 Slide 34

35 7. MW of common polymers MW of commercial polymers step polymers: chain polymers: MWD Flory-Schultz distribution: PDI = 2 when ideal Poisson distribution: PDI = 1 anionic living polymerization In most polymerizations: PDI > 2 Table 3.9 PDI(chain polymers) > PDI(step polymers) Ch 3-1 Slide 35

36 8. Determination of M v dilute solution viscometry (DSV) viscosity size shape mol wt measures molecular size, not weight not an absolute method, but a relative method viscosity, η η = η 0 (1 + ωv 2 ) Einstein eqn ω = 2.5 for sphere v 2 size of solute η/η 0 1 = 2.5 N 2 V e /V Fig 3.13 Fig 3.12 η 0 : solvent only v 2 : vol fraction of solute V: vol of soln V e : vol of equiv. sphere c: wt conc n M: mol wt for η s, see handout p92 η rel 1 =η sp = 2.5 cn A V e /M (g/l)(1/mol)(l)/(g/mol) [η sp /c] c 0 = 2.5 N A V e /M = [η] intrinsic viscosity (dl/g) Ch 3-1 Slide 36

37 [η] mol wt V e = (4/3)πR e3 = (4/3)πH 3 R g3 (4/3)πH 3 (M/M 0 ) 3/2 R e = HR g, R g (M/M 0 ) ½ at θ-condition [η] θ = 2.5 N A V e /M (10πN A H 3 /3M 3/2 0 )M ½ [η] θ = K θ M 0.5 at θ-condition in good solvent, R g = α R θ g [η] = α 3 [η] θ = α 3 K θ M 0.5 = λ 3 K θ M (0.5+3Δ) = K M a α = λ M Δ [η] = K M a v Mark-Houwink-Sakurada (MHS) eqn M v ~ viscosity-average mol wt a 0.5 Ch 3-1 Slide 37

38 M v ~ viscosity-average mol wt η sp = Σ(η sp ) i ~ N i moles of M i mol wt = Σ c i [η] i = Σ (N i M i ) (KM ia ) = K ΣN i M 1+a i [η] = [η sp /c] c 0 = K ΣN i M 1+a i /ΣN i M i = K M a v M v = [ΣN i M 1+a i /ΣN i M i ] 1/a 0.8 a at θ-condition when a = 1, M v = M w when a = 1, M v = M n M v close to M w Fig 3.4 Ch 3-1 Slide 38

39 DSV experiment capillary viscometer Poiseulli eqn, Q = V/t = πr 4 P/8ηL η t η/η 0 = t/t 0 Procedure measure t 0, t 1, t at c 0, c 1, c ( 0 ~ solvent) [η] = [η red ] c 0 = [η sp /c] c 0 = [(η rel 1)/c] c 0 = [(η/η 0 1)/c] c 0 = [(t/t 0 1)/c] c 0 or [η] = [η inh ] c 0 = [ln η rel /c] c 0 = [ln (η/η 0 )/c] c 0 = [ln (t/t 0 )/c] c 0 [η] = K M v a Table 3.10 K, a from handbook at the same temp and solvent Cautions: temp control < 0.2 K t 0 > 100 s (laminar) c < 1 g/dl (Newtonian) Fig 3.14 Ch 3-1 Slide 39

40 9. Gel Permeation Chromatography (GPC) size exclusion chromatography separation by size using porous gel substrate Larger molecules elute earlier. Instrumentation injector column(s) detector A chromatogram Fig V R : retention volume V R = t R x flow rate V R ~ size mol wt (M i ) H i ~ amount N i M i Needs calibration Ch 3-1 Slide 40

41 Universal calibration With the same instrument, column, and solvent, the same V R represents the same hydrodynamic volume. [η] M = [2.5 N A V] = K M v a+1 Many polymers fall on the same curve on the [η]m V R plot ~ universal calibration curve Fig 3.23 Procedure of an experiment 1. From the chromatogram, read V Ri and H i (column 1 and 2). 2. Run the same experiment with polystyrene standards. Ch 3-1 Slide 41

42 2 (cont d) PS standard anionically polymerized with known mol wt 3. Draw a calibration curve (M i vs V Ri ) for PS. 4. Read M i (PS) for each V Ri (column 1, sample). Ch 3-1 Slide 42

43 5. M (PS) M (sample) [η] PS M PS = [η] sample M sample K PS M PS a(ps)+1 = K sample M sample a(sample)+1 from handbook from calibration curve column 3 6. N i = H i / M i (column 2 / column 3) 7. Calculate M n, M w, MWD. Are M n and M w obtained absolute? No. Ch 3-1 Slide 43

44 10. Mass spectrometry MS determines mol wt by detecting molecular ion, M + in vapor phase ionization of polymers in gas phase? MALDI-TOF technique soft ionization choice of the matrix critical for not-too-high mol wt useful for (highly) branched polymers Ch 3-1 Slide 44

45 11. Conclusions At theta condition (solvent/temp) A 2 = 0, χ = ½ R g is the same to that of the bulk polymer infinite mol wt fraction just precipitate (poor/good) Absolute and relative methods Table 3.15 Ch 3-1 Slide 45