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
______
______
A(0) (FID)
A(0) (FID)
T2
T2
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.
NMR Relaxation:
BPP model for liquids
T1
T2 increases monotonically
with molecular mobility
log T1
log T2
τc=1/ω0
liquid
High T, low M
T2
viscous
log(τc)
∝1/T(K)
solid
Low T, high M
T2
T1(>T2)
T2
Expression for Diffusional Echo Attenuation:
Two diffusing Species
A(2τ, X)
= f1exp[ − γ 2D1X] + (1- f1) exp[ − γ 2D 2 X ]
A(2τ,0)
where
X = δ 2 G 2 ( Δ − δ / 3) + G ⋅ G o (..... )
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
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)
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)].
Trace diffusion coefficients and kF for linear and starbranched PI in CCl4 [ reproduced from 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 randomflight chain molecules of star architecture yielded, for
theta solutions, a result independent of M to a first (largeM) approximation:
h(F) = F0.5 [ 2 – F + 20.5 (F – 1) ] -1 .
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 RG3,
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 solventcontaining 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 Mhost0,
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).
Extrapolated PGSE trace diffusion coefficients of 14 plasticizer
species in PVC at 125.5oC.
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.
Spaghetti-meatball model of penetrant diffusion
local host anisotropy
vs.
diffusant anisometry
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)
Diffusion in Melts and Binary Blends
Di (T, M1, M2, v1) = A
x
exp (-Ea / RT)
x
1 / Mi
x
exp [-Bd / f(T,M*)]
constant over T, Mi, and v1
“true” thermal activation
Rouse (diffusant i = 1 or 2)
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.
-3.5
n-alkanes 8 – 60 carbons
-4.0
log (D, cm^2/s)
-4.5
-5.0
T (deg. C) A
30.5 0.077
50.5 0.088
70.5 0.092
90.5 0.102
110.5 0.107
130.5 0.109
150.5 0.115
170.5 0.122
-5.5
-6.0
Finf
0.126
0.134
0.149
0.160
0.174
0.191
0.206
0.221
-6.5
2.0
2.2
2.4
2.6
2.8
log(M, Da)
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)].
3.0
Contour
length
fluctuation:
any effect on D?
Entanglements
(“tube”)
“Strict”
Reptation
(Constraint release
not shown here)
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
-3
150.5 deg. C
-4
log D (cm^2 / s)
-5
-6
-7
-8
-9
-10
2.0
2.5
3.0
3.5
log M (Da)
4.0
4.5
5.0
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) [ Mc2 / 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) [ Mc02 Mi / Mhost3 ] ,
or better:
DCR(i) = α Drep(i,Mc0) [ Mc01.5 Mi / Mhost2.5 ] ,
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
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)
T2 decay
NR devulcanized
Ultrasound
Devulcanization
(Recycling)
three-component fit
inversion
1000
70.5 deg. C
PDMS
Vulcanizate
unfilled
Component T2 (ms)
T2(long)
Virgin
melt
100
T2(med)
10
T2(short)
1
0
20
40
60
Extracted Sol (%)
80
100
100
PDMS
Virgin ----
Proton T2 Fractions (%)
80
F(short)
60
40
F(med)
F(sol)
20
------
F(long)
------
0
0
10
20
Extracted Sol (%)
30
unfilled
PDMS filled
vulcanized
devulcanized
diffusional echo
attenuation
(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
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)
Bicontinuous
Micrcomposite:
HEMA/MMA
Surfactant, water
DMPA photo-initiator
EDGMA linker
Water Diffusion
-4.4
T[diff] (ms)
lo g (D fast, cm ^ 2/s)
-4.5
8
13
30
400
200
100
-4.6
-4.7
1000
-4.8
(+/- 0.025)
^
-4.9
Wt.% Aqueous Phase
-5.0
20
30
40
50
60
70
80
90
100
Surfactant Diffusion
-6.0
-6.1
lo g (D s lo w , c m ^ 2 /s )
-6.2 T[diff] (ms)
8
-6.3
13
-6.4
30
200
100
-6.5
-6.6
400
-6.7
-6.8
(+/- 0.05)
-6.9
Wt.% Aqueous Phase
-7.0
30
40
50
60
70
80
90
100
Permeability of Composites to Water
16
Reduced Permeability
14
assuming 1) free interstitial water diffusion
2) Do = D(water w. surfactant, no network)
12
10
8
6
4
2
Wt.% Aqueous Phase
0
30
40
50
60
70
80
90
100
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 selfassembling morphologies, are of current interest.