Self-diffusion near the percolation threshold in reverse microemulsions
Brian Antalek, Antony J. Williams, and John Texter
Analytical Technology Division, Eastman Kodak Company, Rochester, New York 14650
~Received 7 May 1996!
Self-diffusion measurements of reverse water–acrylamide–sodium bis~2-ethylhexyl!sulfosuccinate–toluene
microemulsions have identified a distinct increase in water proton diffusion above the percolation threshold in
cosurfactant chemical potential. This increase is assigned to water transport through fractal aggregates and
clusters. Above threshold, increasing apparent partitioning of water and cosurfactant into the continuous
pseudophase yields an order parameter for estimating percolating cluster volume, and shows that cosurfactant
preferentially segregates into these clusters. @S1063-651X~96!50612-4#
PACS number~s!: 82.70.Kj, 05.70.2a, 05.60.1w, 47.20.Ma
Reverse microemulsions of water suspended as nanodrop-
lets in a continuous pseudophase of water immiscible sol-
vent, where these nanodroplets are stabilized by a monolayer
of surfactant, are of continuing physical and chemical inter-
est @1#. A particularly important transport property of interest
in these systems is electrical conductivity, and the increase
of conductivity above percolation thresholds in temperature,
volume fraction, or other field variable.
One of the key structural features associated with reverse
microemulsions, besides the details of the droplet composi-
tion, diameter, and the interfacial arrangement of surfactant
between the water and oil, is the interaction of such particles
in forming clusters and aggregates, and the relation of such
aggregation to percolation. In addition, a fascinating and yet
unresolved issue is the mechanism of transformation of re-
verse microemulsions into bicontinuous microemulsions
having low to zero mean curvature ~so-called sponge phase
or middle phase microemulsions!. Controversy persists @2–4#
in experimentally distinguishing reverse and bicontinuous
microemulsions, and whether percolation itself indicates a
transformation to sponge phase bicontinuity @5–7# or the for-
mation of particle clusters @8,9#. Chen, Chang, and Strey @10#
have mapped out a cohesive picture that leads from reverse
microemulsion droplets to droplet clusters and aggregates to
bicontinuous connectivity. Percolation in reverse microemul-
sions has been described as arising from increased transport
from particle aggregation and clustering @11,12# On the other
hand, percolation is sometimes taken as de facto proof of
sponge phase bicontinuity. The differences in these views is
more than semantic, because reverse microemulsions and
clusters of nanodroplets retain high average curvature, while
the average curvature in sponge phase bicontinuous micro-
emulsions is low to vanishingly small. However, experimen-
tally distinguishing these models has remained elusive.
Self-diffusion measurements by NMR of microemulsions
have proven very useful in establishing transitions from oil-
in-water to bicontinuous to water-in-oil isotropic phases
@13#. This approach is particularly useful, because the diffu-
sivity of each chemical component may be monitored more
or less simultaneously. Sponge phase bicontinuity in micro-
emulsions is inferred on the basis of the magnitude of self-
diffusion coefficients @14,15#. In this paper we show that
diffusion coefficients of intermediate magnitude arise natu-
rally in reverse microemulsions as a result of partitioning
between continuous and discontinuous pseudophases. We
further show that the detailed consideration of such partition-
ing provides means to quantify diffusivity arising from per-
colation, and to quantify the volume ~fraction! of percolating
cluster aggregates.
Electrochemical studies @16,17# have revealed a threshold
phenomenon in Faradaic electron transfer involving electro-
active species included in the aqueous phase of certain mi-
croemulsions. The redox chemistry appears to switch on
when the cosurfactant chemical potential is raised above a
certain critical threshold. This phenomenon was reminiscent
of the percolation in electrical conductivity reported for a
similar system @18#, and was subsequently confirmed to co-
incide with the percolation threshold @19#. These Faradaic
and conductivity thresholds are illustrated in Fig. 1 for re-
verse microemulsions of water in toluene, stabilized by the
surfactant AOT ~sodium bis@2-ethylhexyl#sulfosuccinate!.
Acrylamide is a cosurfactant @20# in this system, and its
FIG. 1. Peak current ~ ! for oxidation of ferrocyanide and
electrical conductivity ~ ! in water–acrylamide–AOT–
toluene reverse microemulsions, as acrylamide composition is var-
ied. Peak currents are from square wave voltammetry at 10-mm-
diam platinum microelectrodes. Microemulsions were formulated
with 1.84 g AOT, 7.32 g toluene, 0.833 g aqueous 10 mM potas-
sium ferrocyanide ~ ! or 0.833 g water ~ !, and varying
amounts of acrylamide ~0–0.65 g! to cover the range of 0–5 %
~w/w!. Measurements were made at 25 °C.
PHYSICAL REVIEW E DECEMBER 1996VOLUME 54, NUMBER 6
541063-651X/96/54~6!/5913~4!/$10.00 R5913 © 1996 The American Physical Society

chemical potential serves as a field variable in modifying the
microstructure and increasing water solubilization, as well as
inducing percolation.
The onset of percolation in this microemulsion system has
been linked to physical changes within the interface, such as
increased interfacial flexibility @18# and decreased chain
packing order @19,21#. Such changes lead to increased inter-
droplet attraction, coalescence, and cluster formation. In this
paper we further examine microstructure near the percolation
threshold by considering the self-diffusion of all four com-
ponents, water, AOT, toluene, and acrylamide. These data
provide evidence for further distinguishing bicontinuity and
clustering above percolation threshold.
Self-diffusion measurements for each of the four compo-
nents in the microemulsions were determined by pulsed-
gradient spin-echo ~PGSE! NMR spectroscopy. This ap-
proach is based on the method of Stejskal and Tanner @22#
and is derived from the nuclear spin echo concept of Hahn
@23# and Carr and Purcell @24#. The diffusion coefficient is
obtained from the attenuation of the spin echo under the
influence of pulsed magnetic field gradients. Measurements
were made at 2560.5 °C on a Varian Unity 500, narrow bore
~51 mm! spectrometer operating at 499.9 MHz. The spec-
trometer was equipped with a commercially available diffu-
sion system, which includes a 5 mm, 1H-19F, air cooled,
z-gradient probe built by Doty Scientific, a Highland DC
current amplifier for producing current pulses, and a Sorren-
son variable temperature power supply. The current amplifier
was enhanced to provide up to 20 A; this enhancement al-
lowed the probe to reach a gradient of up to 1000 G cm21
with excellent stability.
We employed a stimulated-echo pulse sequence with long
eddy current delay @25#. Eddy currents have been reduced
and amplifier overdrive eliminated by using ramped ~trap-
ezoidal shaped! gradients @26#. Experiments were performed
by varying the gradient strength ~g! and keeping the gradient
width ~d! and all other timing parameters constant. Typi-
cally, a value of 100 ms was used for the diffusion time ~D!.
This value was chosen to insure that the mean displacement
of the molecular center of mass was more than 10 000 Å and
thus much larger than any internal displacements relative to
the center of mass. The relationship between the echo attenu-
ation and self-diffusion coefficient is given by
ln
~
E/E0! 52Dg2g2$d2~D2 ~d/3 !!1 ~e3/30 !2d ~e2/6 ! %,
~1!
where g is the gyromagnetic ratio of the 1H nucleus, E is the
measured signal amplitude E0 is the amplitude with no gra-
dients, and « is the duration of the gradient ramp ~200 ms!.
Half-echoes were acquired for typically 10 values of g and
Fourier transformed. The natural logarithm of the integral
value for the resonance of interest was recorded and plotted
against g2. The value of D was obtained from the slope of a
least-squares fit. The high magnetic field of the spectrometer
resulted in excellent sensitivity and spectral dispersion.
Self-diffusion coefficients obtained from these data are
displayed in Fig. 2 for each of the chemical components as a
function of the acrylamide level. The water, acrylamide, and
AOT data appear to exhibit breakpoints near the percolation
threshold, in the neighborhood of 1.5–2 % acrylamide. It is
noteworthy that the toluene self-diffusion is suppressed only
8% over the illustrated range in composition. This suppres-
sion is due to increasing viscosity and the particle obstruc-
tion. Bulk viscosity increases about three-fold over this com-
position range, from about 1.25 cP to about 4 cP. The water
and acrylamide data initially coincide, below threshold, but
diverge markedly above threshold. The AOT data run paral-
lel to the acrylamide data, but are offset to lower values.
Experiments were performed on the 4% ~w/w! acrylamide
sample to test for a diffusion time ~D! dependence. Varia-
tions of 10–1000 ms in D did not yield any significant varia-
tion in the self-diffusion derived for each of the four chemi-
cal components.
We examine the AOT diffusion in the context of a fast
exchange model, where it is assumed that the AOT ex-
changes between the water swollen droplets of the disperse
pseudophase and the toluene-solvated unimeric state in the
continuous ~toluene! pseudophase. Therefore, the observed
AOT diffusion may be modeled as a mole-fraction weighted
average of the faster diffusing molecule in the continuous
toluene phase ~Dc! and the more slowly diffusing swollen
micelle ~Dmic! @27#,
Dobs5xDc1~12x !Dmic . ~2!
The critical micelle concentration of AOT in toluene is of the
order of that found in benzene @28#, about 431024 to
231023 M, with a geometric average of about 931024 M.
We found the diffusion coefficient of AOT in the molecular
state in toluene at 25 °C to be 1.2931025 cm2 s21. The total
volume concentration of AOT in the microemulsions is of
the order of 0.4 M, so that the mole fraction ~of AOT in the
continuous pseudophase! x>0.002. Essentially all of the
AOT is in the disperse state, and we have, therefore, the
quantiative estimate that Dobs2Dmic<331028 cm2 s21.
Since Dobs for AOT ranges over ~3–6!31027 cm2 s21, we
may take Dmic>Dobs ~DAOT! with about 5–10 % error.
A local viscosity h was derived from the Stokes-Einstein
equation and the assumption that the molecular size ~diam-
FIG. 2. Observed self-diffusion coefficients at 25 °C for toluene
~h!, water ~s!, acrylamide ~j!, and AOT ~d! in the microemul-
sions as a function of acrylamide content. The lines are included as
guides to the eye.
R5914 54BRIAN ANTALEK, ANTONY J. WILLIAMS, AND JOHN TEXTER