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DIFFUSION AND POLYMERS: THE PULSED-FIELD-GRADIENT METHOD

Ernst von Meerwall

Departments of Physics, Polymer Science, and Chemistry,

The University of Akron Akron, Ohio 44325

Tutorial Presentation Symposium “NMR Spectroscopy of Polymers” ACS New Orleans, April 6-10, 2008

Outline

I The NMR/PGSE method of measuring self-diffusion spectroscopic non-spectropscopic (wide-line; high gradient) role of spin-spin relaxation II Polymers in solution in small-molecule solvents Infinite dilution Dilute solutions Semidilute solutions Effect of branching (stars) in dilute solutions III Penetrants and diluents in rubbery polymers General principles Free-volume theory Concentration-dependence of diluent diffusion Explicit temperature-dependence Frictional effects not directly related to free volume Effects of diffusant molecular shape, flexibility IV Polymers in polymer melts, blends, and networks Unentangled diffusion Entangled diffusion Definitions and assumptions: strict diffusion Constraint release Transition to entangled diffusion Synthesis: combined approach for arbitrary M, c, and T Entangled binary blends V Selected applications – time permitting Ultrasound devulcanization (environmental) Permeability in bicontinuous microcomposites (biomedical)

NMR T2 and Pulsed-Gradient Spin-Echo Experiment

T2 T2

______A(0) (FID)

______A(0) (FID)

Brownian motion: a reminder Brownian motion of the diffusant does not depend on the presence of a concentration gradient, but causes any existing non-uniform concentration profile c(x,t) to evolve in time. < r2 > = 6 D tdiff , or < x2 > = 2 D tdiff . These relationships may be regarded as definitions of D, commonly referred to as self-diffusion. D will depend on temperature, molecular weight, concentration (in blends and solutions), etc. PGSE measures < x2 > in the laboratory frame of reference, and from it infers D. This microscopic definition becomes the basis of Fick’s first law. In the presence of several distinct molecular species, each will have a separate self-diffusion constant (multicomponent self-diffusion D1, D2, ... etc.) Knowledge of these Di is not tantamount to knowing the system’s mutual diffusion coefficient; the relationship is complex, and any rule of mixture may not apply. Entropic and thermodynamic terms play a role in mutual diffusion; no single theoretical framework for relating mutual diffusion to self-diffusion has been developed to fit all cases.

T1

T2

log(τc)

∝1/T(K)

log T1

log T2

τc=1/ω0

liquid solid

High T, low M Low T, high M

NMR Relaxation: BPP model for liquids

viscous

T2 increases monotonicallywith molecular mobility

T T (>T )T2 2

1 2

Expression for Diffusional Echo Attenuation:Two diffusing Species

A(2 ,X)A(2 ,0)

= f exp[ D X] + (1- f1 1 1ττ

γ γ− −2 22) exp[ ]D X

where

X G G Go= − + ⋅δ δ2 2 3( / ) (..... )Δ

D1 diffusivity of faster-moving species,f1 echo fraction of that species at t=2τ;

D2 diffusivity of slower-moving species,f2 = (1-f1) echo fraction of that species.

Magnetogyric ratio of resonant nuclide (here protons) is γ.Usually Δ = τ . Go is a steady magnetic field gradient during measurement of both A(2τ,G) and A(2τ,0).

PFG / PGSE attributes and capabilities

• No chemical or other external labeling is required to measure D (use nuclear spin, e. g. 1H, 13C, 19F) :

• Measurements yield first-principles, absolute “Self-Diffusion” • Samples are indefinitely reusable (no irreversible changes), e. g. at

different temperatures; samples may be altered between measurements

• T2-weighting can be used to tune out signals from rigid components• Multicomponent self-diffusion is measurable (spectroscopically or by

resolving rate distributions); includes polymer mass dispersity • Anisotropic diffusion is measurable. Macroscopic: reorient sample;

microscopic averages: fit specific geometric model• D(CM) is only reached for large molecules when xrms >> Rg

• Anomalous (segmental) diffusion is measurable: interpret departures from Fickian master curve, hence Dapp = D(t)

• Fully or partly restricted diffusion is measurable (see above)

Outline


Reciprocal diffusion coefficients vs. concentration for 18-armed PI stars in CCl4 [ Fig.3 in Chen Xuexin, Xu Zhongde, E. von Meerwall, N. Seung, N. Hadjichristidis, and L. J. Fetters, Macromolecules 17, 1343 (1984)].

Effect of Branching Several theoretical and experimental investigations have dealt with the diffusion of regular star-branched polymers in dilute solutions. For stars having F equal arms (the linear polymer has F=2 arms), one defines a hydrodynamic ratio h(F), h(F,M) = Do(F=2,M,T) / Do(F,M,T). This ratio should be sensibly temperature-independent. Application of the Kirkwood-Riseman theory† to random-flight chain molecules of star architecture yielded, for theta solutions, a result independent of M to a first (large-M) approximation: h(F) = F0.5 [ 2 – F + 20.5 (F – 1) ] -1 .

Trace diffusion coefficients and kF for linear and star-branched PI in CCl4 [ reproduced from Chen Xuexin, Xu Zhongde, E. von Meerwall, N. Seung, N. Hadjichristidis, and L. J. Fetters, Macromolecules 17, 1343 (1984)].

Semidilute solutions The semidilute solution regime is confined on both sides to a concentration range c* < c < c** , where c* represents an overlap concentration† c* = M / No RG

3, approximately ∝ M Do3 .

Here No represents Avogadro’s number and RG the radius of gyration. The upper bound c** delimits a concentrated solution, that is, the emergence of a space-filling but solvent-containing polymer aggregate or gel, leading to the onset of entanglement constraints, where applicable. However, some recent authors instead locate the entanglement onset at or somewhat higher than c*. In either case, both c* and c** decrease with increasing diffusant molecular mass M. For polymer molecules small enough never to be entangled, the dilute solution regime extends across most of the c range. Only at the highest c is the solution semi-dilute or concentrated; under those conditions diffusion of solvent as well as polymer is adequately described by the free-volume theory (see below). In higher-M polymers the most general case is that of ternary systems, in which monodisperse linear molecules of mass M diffuse through a host consisting of a fully entangled polymer of (usually greater) mass Mhost in a small-molecule solvent, where the overall polymer concentration is c. According to de Gennes††, applying scaling as well as reptational concepts for entangled solutions leads to D(M,c) ∝ M-2 c-K Mhost

0, with K=1.75 (good solvent) and 3 (theta solvent).

Normalized polymer diffusivities for 17 different polymer-solvent systems as a function of reduced concentration. Systems include theta solutions. [ Fig. 4 in V. D. Skirda, V. I. Sundukov, A. I. Maklakov, O. E. Zgadzai, I. Gafurov, and G. I. Vasiljev, Polymer 29, 1294 (1988) ]

Outline

I The NMR/PGSE method of measuring self-diffusion spectroscopic non-spectropscopic (wide-line; high gradient) role of spin-spin relaxation II Polymers in solution in small-molecule solvents Infinite dilution Dilute solutions Semidilute solutions Effect of branching (stars) in dilute solutions III Penetrants, diluents in rubbery polymers General principles Free-volume theory Concentration-dependence of diluent diffusion Explicit temperature-dependence Frictional effects not directly related to free volume Effects of diffusant molecular shape, flexibility IV Polymers in polymer melts, blends, and networks Unentangled diffusion Entangled diffusion Definitions and assumptions: strict diffusion Constraint release Transition to entangled diffusion Synthesis: combined approach for arbitrary M, c, and T Entangled binary blends V Selected applications – time permitting Ultrasound devulcanization (environmental) Permeability in bicontinuous microcomposites (biomedical)

Schematic representation of volume disposition in a rubbery matrix as function of temperature. [ Fig. 2 in J. L. Duda and J. M. and Zielinski, Ch. 3 in “Diffusion in Polymers”, P. Neogi, ed., M. Dekker, N. Y., 1996]. Cohen and Turnbull† proposed what is now the standard form of the observed strong dependence of D on f: D(f) = D’ exp ( - Bd / f ) .

† M. H. Cohen and D. Turnbull, J. Chem. Phys. 31, 1164 (1959).

Hole free volume fraction For most practical purposes, the fractional hole free volume limiting diffusion near or above the glass transition temperature Tg may be approximated by† f(T) = f(Tg) + ∆α (T – Tg) , where ∆α represents the thermal expansivity of only the hole free volume; a good approximation is the difference in specimen thermal expansivity above and below Tg. Typical values for fg = f(Tg) ≈ 0.025, and ∆α ≈ 4.8 x 10-4 K-1, but information about specific polymers is to be preferred whenever available. † see, e. g., F. Bueche, “Physical Properties of Polymers”, Interscience/Wiley, New York, 1962, Ch. 5.

Diffusion of n-alkanes (carbon numbers are indicated) in high-M cis-polyisoprene melts at 51oC as function of concentration. Lines are two-parameter fits of the Fujita-Doolittle equation. Initial slopes (s) are proportional to the difference in fractional free volumes of alkane and rubber; extrapolations to trace concentrations are in accord with the constancy of monomeric friction (to be discussed later). [Fig. 1a in R. D. Ferguson and E. von Meerwall, J. Appl. Polym. Sci. 23, 3657 (1979)]

Effective friction coefficient (ζo = kT / Do ) at 51oC for diffusion of trace concentrations of n-alkanes ( 8 to 36 carbons) dissolved in two uncrosslinked high-M rubbers: styrene(40%)-butadiene rubber (top) and cis-polyisoprene (bottom). Points represent extrapolations to zero concentration using the Fujita-Doolittle equation (see earlier). [ Fig. 2 in R. D. Ferguson and E. von Meerwall, J. Appl. Polym. Sci. 23, 3657 (1979)]

Room-temperature proton PGSE diffusion measurements for 14 plasticizer species dissolved in CCl4 at several small concentrations, extrapolated to zero concentration. E. von Meerwall, D. Skowronski, and A. Hariharan, Macromolecules 24, 2441 (1991).

Diffusion of trace plasticizers in PVC, approximately corrected for molecular mass, plotted as function of the molecules’ minimal transverse diameter ignoring side groups assumed to be mobile enough to evade retardation by host.

Extrapolated PGSE trace diffusion coefficients of 14 plasticizer species in PVC at 125.5oC.

Spaghetti-meatball model of penetrant diffusion

local host anisotropyvs.diffusant anisometry

Outline


Diffusion in Melts and Binary Blends

Di (T, M1, M2, v1) = A constant over T, Mi, and v1 x exp (-Ea / RT) “true” thermal activation x 1 / Mi Rouse (diffusant i = 1 or 2) x exp [-Bd / f(T,M*)] Cohen-Turnbull (host), where the fractional free volume f has the form: f (T,M*) = f (T,∞) + 2 Ve(T) ρ (T,M*) / M* (Bueche)

with ρ(T,M*) = [ 1 / ρ(T,∞ ) + 2 Ve(T) / M* ] -1 , where 1/ M* = v1 / M1 + (1 - v1) / M2 .

Values of A, Ea, ρ(T,∞ ), Ve(T), and f (T,∞) are those for the neat melts (at the same T). The theory applies in the absence of entanglements and generates ideal solution behavior. For melts, one sets v1 = 0.

T (deg. C) A Finf

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

2.0 2.2 2.4 2.6 2.8 3.0

log(M, Da)

log

(D, c

m^2

/s)

30.5 0.077 0.12650.5 0.088 0.13470.5 0.092 0.14990.5 0.102 0.160110.5 0.107 0.174130.5 0.109 0.191150.5 0.115 0.206170.5 0.122 0.221

Diffusion coefficients in the n-alkane series at various temperatures, measured via PGSE. Symbols represent data; each line is a two-parameter fit of theory icomposed of thermal activation, Rouse, Cohen-Turnbull/monomeric friction, and free-volume host effects, with optimized parameters indicated [Fig. 4 in E. von Meerwall, S. Beckman, J. Jang, and W. L. Mattice, J. Chem. Phys. 108, 4299 (1998)].

n-alkanes 8 – 60 carbons

Entanglements(“tube”)

“Strict”Reptation

(Constraint releasenot shown here)

Contourlength fluctuation:any effect on D?

Difference between the M-dependence in a series of polybutadienes in trace concentrations in the M = 44,000 host, and in their own melts. Lines are drawn to guide the eye, and are not fits of any theory. [ Fig. 2 of E. von Meerwall, S. Wang, and S.-Q. Wang, Polymer Prepr. 44/10, 288 (2003) ].

Diffusion of PE melts at 200oC (top) and PS melts at 225oC (bottom) measured via PGSE, corrected for constant free volume, as function of the degree of polymerization. Solid lines represent unmodified Rouse theory (M-exponent -1) in the unentangled regime, and reptation plus constraint release above the entanglement onset. Dashed lines are drawn to guide the eye. [ Fig. 3 of G. Fleischer, Colloid Polym. Sci. 264, 1, (1986) ]

Diffusion of n-Alkanes and PE

-10

-9

-8

-7

-6

-5

-4

-3

2.0 2.5 3.0 3.5 4.0 4.5 5.0

log M (Da)

log

D (c

m^2

/ s)

150.5 deg. C

Entangled diffusion: Constraint Release Enhancement of Dent by constraint release ( rates add ) Dent = Drep + DCR Entangled species 2 dissolved in a light species 1: ( Pearson, et al. ) ( M2 > Mc0 > M1 ) DCR(2) = α Drep(2,Mc) [ Mc

2 / M22 ]

where Mc = Mc (w1), increased from Mc0; add dilute-solution corrections for D2 near w1→ 1 Both species entangled: ( Graessley; Klein ) ( M2 > M1 > Mc0 ) DCR(i) = α Drep(i,Mc0) [ Mc0

2 Mi / Mhost

3 ] , or better: DCR(i) = α Drep(i,Mc0) [ Mc0

1.5 Mi / Mhost

2.5 ] , where the concentration-dependence enters via (extended to w1 > 0: our proposal) Mhost = w1 M1 + ( 1 - w1 ) M2, (no dilute-solution corrections for D1 or D2).

[ Tube dilation is part of DCR even for M1 > Mc0; contour-length fluctuation is not considered ]

Outline


NR devulcanized

UltrasoundDevulcanization(Recycling)T2 decay

inversion

three-component fit

1

10

100

1000

0 20 40 60 80 100Extracted Sol (%)

Com

pone

nt T

2 (m

s)

T2(long)

T2(med)

T2(short)

Vulcanizate

Virginmelt

70.5 deg. C PDMS

unfilled

0

20

40

60

80

100

0 10 20 30Extracted Sol (%)

Pro

ton

T2 F

ract

ions

(%)

Virgin ----

------

------

F(short)

F(med)

F(long)

F(sol)

PDMS

unfilled

PDMS filled

vulcanized devulcanized

diffusional echoattenuation

(T2s network echo no longer contributes)

Estimating the fastest-diffusing (oligomeric) fraction of the sample, FFAST

by correcting the diffusion experiment for differential T2 - weighting:

FFAST = ffast (2τ) fL’(2τ) , with

fL’(2τ) = fL + fM exp[ 2τ (T2L-1 – T2M

-1) ] ,

where ffast is the fast-diffusing echo fraction at t = 2τ, T2M and T2L are the relaxation times of the two more mobile components, and 2τ = 2 x rf pulse spacing (PE: 90o - τ - 90o - τ - echo).

Outline


BicontinuousMicrcomposite:HEMA/MMASurfactant, waterDMPA photo-initiatorEDGMA linker

Water Diffusion

-5.0

-4.9

-4.8

-4.7

-4.6

-4.5

-4.4

20 30 40 50 60 70 80 90 100

Wt.% Aqueous Phase

log

(D fa

st, c

m^2

/s)

13

40030

200100

T[diff] (ms)

1000

^(+/- 0.025)

8

Surfactant Diffusion

-7.0

-6.9

-6.8

-6.7

-6.6

-6.5

-6.4

-6.3

-6.2

-6.1

-6.0

30 40 50 60 70 80 90 100

Wt.% Aqueous Phase

log

(D s

low

, cm

^2/s

)

T[diff] (ms)

13

30

100200

400

(+/- 0.05)

8

Permeability of Composites to Water

0

2

4

6

8

10

12

14

16

30 40 50 60 70 80 90 100

Wt.% Aqueous Phase

Red

uced

Per

mea

bilit

y

assuming 1) free interstitial water diffusion

2) Do = D(water w. surfactant, no network)

Summary

• NMR PGSE diffusion measurements, augmented by T2 , constitute a useful combination of capabilities for characterizing molecular mobilities in polymer systems.

• This kind of information is difficult or impossible to obtain by alternate methods

Perspectives: Diffusion and Polymers

● Detailed comparisons of experiment, theory, and simulation are highly desirable, and informative as to mechanisms.

● Theory and simulation must predict correct dependence of D on M, T, c, architecture, and host effects.

● Theory and simulation should strive to predict absolute diffusion rates, precisely. Current MD comes closest; dynamic MC cannot do so; theories vary greatly.

● The field is wide open; many questions remain. Amorphous and fluid-based nanocomposites, and self-assembling morphologies, are of current interest.

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