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Langmuir 2009, 25, 34253434
Diffusion of an Ionic Drug in Micellar Aqueous Solutions
Huixiang Zhang†,‡ and Onofrio Annunziata*,†
Department of Chemistry, Texas Christian UniV
ersity, Fort Worth, Texas 76129, and Alcon Research Ltd.,
ReceiV
ed NoV
ember 4, 2008. ReV
ised Manuscript ReceiV
ed December 18, 2008
Supramolecular carriers such as micelles can be used to noncovalently bind drug molecules for pharmaceutical
applications. However, these carriers can fundamentally affect diffusionbased drug transport due to hostguestcoupled diffusion. We report a ternary interdiffusion study on an ionic drug in aqueous micellar solutions. Specifically,highprecision Rayleigh interferometry was used to determine the four multicomponent diffusion coefficients for thepotassium naproxenatetyloxapolwater ternary system at 25 °C and pH 7. In addition, we have measured drugsolubility as a function of tyloxapol concentration. These measurements were used to characterize drugsurfactantthermodynamic interactions using the twophase partitioning model. Furthermore, we propose a novel model onhostguest coupled diffusion that includes counterions. We show that quantitative agreement between model andexperimental diffusion results can be achieved if the effect of micelle solvation on transport parameters is includedin the model. This work represents an essential addition to our previous diffusion study on a nonionic drug and providesguidance for the development of accurate models of drug diffusionbased controlled release in the presence of nanocarriers.
Introduction
corresponding molar fluxes. The four
Dij (with
i,
j ) 1,2) are the
Supramolecular systems such as micelles, liposomes, and other
ternary diffusion coefficients. Maindiffusion coefficients,
D11
nanoparticles are valuable tools in the chemical and pharma
and
D22, describe the flux of a solute due to its own concentration
ceutical fields because they can be used to reversibly bind drug
gradient, while crossdiffusion coefficients,
D12 and
D21, are
compounds, thereby enabling controlled release and targeted
responsible for the flux of a solute due to the concentration gradient
delivery. They also reduce toxicity, enhance bioavailability, and
improve stability of therapeutic agents.13
Several interdiffusion studies have been reported in relation
Interdiffusion (or mutualdiffusion) coefficients of drug com
to hostguest systems forming 1:1 complexes. The most relevant
pounds are crucial parameters used for modeling, predicting, and
cases involve binding of small molecules to cyclodextrines.10,11
designing drug release from delivery devices (gels or other porous
One important aspect of these investigations is the observation
materials) and other processes such as transport across mem
of large negative values of the crossdiffusion coefficient
branes.48 However, in the presence of supramolecular systems,
responsible for the flux of guest molecules from low to high
drug diffusion becomes coupled to that of the hosting particle.913
cyclodextrine concentration.1517 However, in many cases, host
The description of drughost diffusion transport in solution
particles may bind more than one guest molecule.3 The most
requires the use of Fick’s first law extended to ternary systems:14
common example is represented by micellar systems.1 Clearly,diffusion studies on these systems represent an essential addition

J )
D ∇
C +
D ∇
C
to those performed on 1:1 hostguest complexes.
We note that accurate selfdiffusion coefficients for drug and
J )
D ∇
C +
D ∇
C
surfactant molecules in solution have been obtained by pulsed
gradient spinecho NMR (PGSENMR). The dependence of
1 and
C2 are the molar concentrations of the two solutes,
drug(1) and host system(2), respectively, and
J
selfdiffusion coefficients on system composition has been used
to determine micellization parameters and drugmicelle bind
* Corresponding author. Phone: (817) 2576215. Fax: (817) 2575851.
ing.1821 However, selfdiffusion coefficients cannot generally
replace interdiffusion coefficients when describing transport
processes in the presence of concentration gradients. This is
especially true when considering ionic species and chemical
(1) Malmsten, M.
Surfactants and Polymers in Drug DeliV
ery; Marcel Dekker:
association.22,23 Furthermore, selfdiffusion studies on multi
(2) Batrakova, E. V.; Kabanov, A. V.
J. Controlled Release 2008,
130, 98–
(3) Li, C.; Wallace, S.
AdV
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Langmuir 2003,
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3426
Langmuir, Vol. 25, No. 6, 2009
component systems yield no information on crossdiffusion
in targeting and controlledrelease applications because drug
binding strength to the host particle can be tuned by physico
Several interdiffusion studies have been reported on surfactant
chemical changes of their surrounding environment.35,36 Hence,
multicomponent systems.9,2428 These investigations have mainly
investigating diffusion of an ionic drug in the presence of micelles
focused on the formation of mixed micelles in aqueous
represents an essential addition to our previous diffusion study
solutions.2426 In relation to micellar solubilization, few studies
have been reported on aqueous solutions of
nalcohols and sodium
We report measurements of the four diffusion coefficients
dodecylsulfate.27,28 In relation to drug compounds, we have
for the naproxentyloxapolwater ternary system at 25 °C
recently reported a diffusion study on drug molecules in micellar
and pH 7.3 by Rayleigh interferometry. Naproxen (
S(+)
aqueous solutions.9 Specifically, using Rayleigh interferometry,
2(6methoxy2naphthyl) propionic acid, p
K )
we have determined the four diffusion coefficients for the
nonsteroidal antiinflammatory drug with analgesic and
hydrocortisonetyloxapolwater ternary system at 25 °C, where
antipyretic properties.3639 At the experimental pH, naproxen
hydrocortisone is a nonionic drug and tyloxapol is a nonionic
exists predominantly in its anionic form. In our investigation,
the drug is a potassium salt. We include solubility measurements
Tyloxapol, which is a commercially available surfactant at a
for potassium naproxenate as a function of tyloxapol concentra
relatively low cost, is essentially an oligomer of octoxynol 9
tion. We then introduce a novel model on hostguest coupled
(Triton X100) mostly used in marketed ophthalmic products
diffusion that includes counterion effects. We show that
and as a mucolytic agent for treating pulmonary diseases.2932
quantitative agreement can be obtained between model and
Tyloxapol has a critical micellar concentration (cmc) of 0.038
behavior of the four experimental diffusion coefficients if the
g/L in water at 25 °C.29 This cmc value is much lower than that
effect of micelle hydration is taken into account.
of Triton X100. Hence, the presence of free surfactant can be
Materials and Methods
neglected with respect to micellar surfactant for concentrationsof the order of 1 g/L or higher. We note that tyloxapol micelles
Materials. Naproxen was purchased from TCI America (Portland,
are spherical with a diameter of 7 nm, and their size and shape
OR). Potassium hydroxide and tyloxapol (SigmaUltra grade) were
do not change significantly for concentration as high as 10% by
purchased from Sigma Chemical Co. (St. Louis, MO). Glacial aceticacid and acetonitrile were purchased from EMScience and EMD
weight according to cryotransmission electron microscopy.30
Chemicals Inc. (Gibbstown, NJ), respectively. Materials were used
We also point out that tyloxapol hydrophilic groups are
as received from the manufacturers. The molecular weights for
poly(ethylene glycol) chains, a chemical motif often encountered
naproxen and tyloxapol were taken to be 230.26 and 4500 g mol1,
in supramolecular systems of pharmaceutical relevance.33 All of
respectively. Deionized water was passed through a fourstage
these features make tyloxapol micelles a model supramolecular
Millipore filter system to provide highpurity water for all of the
system for hostguest physicochemical studies relevant to
experiments. Stock solutions of tyloxapolwater and naproxenwater
were made by weight to 0.1 mg. To prepare potassium naproxenate,
For the previously investigated hydrocortisonetyloxapolwater
the pH of the naproxen stock solution was increased to pH ≈ 7 using
system, the determined diffusion coefficients were examined
KOH. Precise masses of stock solutions were added to flasks anddiluted with pure water to reach the final target concentrations of
using a drugmicelle coupleddiffusion model based on drug
the solutions used for the diffusion experiments. All final solutions
partitioning between the aqueous and micellar pseudophases.
used for diffusion and solubility measurements displayed pH values
Drug partitioning was characterized by measuring the solubility
within the range 7.3 ( 0.3. At these pH values, the neutral form of
of hydrocortisone as a function of tyloxapol concentration. A
naproxen has a concentration of 0.1% or lower based on p
K )
quantitative agreement between the experimental behavior of
drug diffusion coefficients,
D11 and
D12, and a model based on
Density Measurements. Molar concentrations of the solutions
dependence of drug solubility on surfactant concentration was
were obtained from density. All density measurements were made
obtained. One important result of this investigation is that
at 25.00 °C with a computerinterfaced MettlerPaar DMA40 density
hydrocortisone diffusion is not only modulated by its binding to
meter, thermostatted with water from a large, wellregulated ((1
the slowly diffusing micelles but also because of the presence
Solubility Measurements. Solid naproxen compound was added
in excess to tyloxapolwater solutions in glass vials, and pH was
In this article, we extend our interdiffusion studies to the case
adjusted to pH 7.3 using KOH. The obtained heterogeneous samples
of ionic drugs. Drugs with ionic structure are frequently
were continuously agitated for 10 days in a regulated water bath at
encountered in pharmaceutical applications.34 Furthermore, they
25.0 ( 0.1 °C. Aliquots of the suspensions were then passed through
can be also generated from nonionic drugs in situ by a pH change
0.2
µm filters (Millipore) and, if necessary, diluted with the HLPC
under physiological conditions. This feature is very important
mobile phase (see below) so that the final drug concentration wasaround 0.1 mg/mL. The drug concentration of the properly dilutedsamples was then measured using HPLC (Waters Alliance 2695)
(23) Annunziata, O.; D’Errico, G.; Ortona, O.; Paduano, L.; Vitagliano, V.
J.
Colloid Interface Sci. 1999,
216, 16–24.
equipped with a UV detector (Waters model 2487). A Waters
(24) Halvorsen, H. C.; Leaist, D. G.
Phys. Chem. Chem. Phys. 2004,
6, 3515–
Symmetry C18 column (size: 4.6 150 mm) was employed with a
mobile phase consisting of a 39.7/59.5/0.008 (v/v/v) mixture of
(25) Leaist, D. G.; MacEwan, K.
J. Phys. Chem. B 2001,
105, 690–695.
acetonitrile/water/glacial acetic acid with a flow rate of 1.2 mL/min.
(26) Castaldi, M.; Constatino, L.; Ortona, O.; Paduano, L.; Vitagliano, V.
Langmuir 1998,
14, 5994–5998.
Chromatograms were obtained at 254 nm.
(27) Leaist, D. G.
Can. J. Chem. 1990,
68, 33–35.
Rayleigh Interferometry. Diffusion measurements on naproxen
(28) Leaist, D. G.; Hao, L.
J. Chem. Soc., Faraday Trans. 1995,
91, 2837–
(1)tyloxapol(2)water(0) ternary systems and corresponding binary
aqueous systems were made with the highprecision Gosting
(29) Scott, H.
J. Colloid Interface Sci. 1998,
205, 496–502.
(30) Regev, O.; Zana, R.
J. Colloid Interface Sci. 1999,
210, 8–17.
(31) Westesen, K.
Int. J. Pharm. 1994,
102, 91–100.
(35) Choi, S.U.; Bui, T.; Ho, R. J. Y.
J. Pharm. Sci. 2007,
97, 931–943.
(32) Westesen, K.; Koch, M. H. J.
Int. J. Pharm. 1994,
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Int. J. Pharm. 1996,
129, 13–20.
(33) Jain, A.; Jain, S. K.
Crit. ReV
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25, 403–
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Int. J. Pharm. 1995,
126, 95–102.
(38) Cirri, M.; Maestrelli, F.; Corti, G.; Furlanetto, S.; Mura, P.
J. Pharm.
(34) Stahl, P. H., Wermuth, C. G., Eds.
Handbook of Pharmaceutical Salts:
Biomed. Anal. 2006,
42, 126–131.
Properties, Selection, and Use; Verlag Helvetica Chimica Acta: Zu¨rich, 2002.
(39) Chowhan, Z. T.
J. Pharm. Sci. 1978,
67, 1257–1260.
Diffusion of an Ionic Drug in Micellar Aqueous Solutions
Langmuir, Vol. 25, No. 6, 2009 3427
where
K is the partitioning constant,
C(M)
molar concentrations in the micellar and water pseudophases,respectively, and
CD is the free drug molar concentration in thetotal volume. We note that
C(W)
in pure water. Drug solubility,
S1, is the sum of two contributions:
C ) 0
S1(1 
φ) (free drug) and
CD
φ )
K S1
φ (bound drug).
We therefore obtain the following linear relation:9
S )
S0[1 + (
K  1)
φ]
We note that eq 3 applies to both neutral and ionic drugs and
Figure 1. Solubility of naproxen in tyloxapolwater mixtures as a
electroneutrality is not assumed to hold for the pseudophases.
function of tyloxapol volume fraction at 25 °C and pH 7. The solid curve
We fit our solubility data to eq 3 and obtain:
S0 )
is a linear fit through the data using eq 3.
and
K ) 25 ( 1 at 25 °C and pH 7.3. The obtained value of
Kwill be used to compare the experimental diffusion results with
diffusiometer operated in its Rayleigh interferometric mode.14,4043
A comprehensive description of the Gosting diffusiometer can be
Ternary Diffusion Coefficients. The interdiffusion coefficients
found in ref 43 and references therein.
Diffusion experiments were accomplished by setting up a sharp
in eqs 1a,b can be described relative to different reference
horizontal interface (free boundary) between a bottom and a top
frames.48 Diffusion measurements yield, to an excellent ap
solution with different composition in a vertical diffusion channel.
proximation, diffusion coefficients relative to the volumefixed
A necessary condition for eliminating convection is that the fluid
frame. Here, the fluxes of the components of a ternary system
density must decrease from bottom to top along the diffusion channel
at any point during the experiment.44 For each ternary solution
“2”, and “0” denotes the drug, surfactant, and solvent components,
composition, at least four diffusion experiments were performed at
respectively, and the subscript “
V” appended outside the
virtually the same average concentration of naproxen,
C1, and
parentheses identifies the volumefixed frame. Hence, the
tyloxapol,
C2. To obtain the four diffusion coefficients, experiments
measured diffusion coefficients will be denoted as (
D
must be performed with different values of the ratio ∆
C
∆
C2), where ∆
Ci is the difference in concentration of solute
i betweenthe bottom and top sides of the initial diffusion boundary. Details
The four interdiffusion coefficients for the naproxen
on the method and individual diffusion experiments are given as
tyloxapolwater ternary system were determined as a function
Supporting Information. We note that experiments with ∆
C
of tyloxapol concentration at 25 °C and pH 7.3. The naproxen
concentration was kept constant at
C ) 6 mM. Our results are
2) ≈ 1 displayed doublediffusive convection due to dynamic
gravitational instability,4446 which arises at the interface (see
Supporting Information). Hence, the experiments used for the
0 in Figure 2a is the diffusion coefficient, (
D1)
V, for the
determination of ternary diffusion coefficients were performed away
drugwater binary system. At infinite dilution, (
D
where
D( is the meanionic tracer diffusion coefficient. Thiscoefficient is related to the tracer diffusion coefficients of the
naproxenate anion,
DD, and the potassium cation,
DK, through
Drug Solubility. Figure 1 shows solubility of potassium
the NernstHartley equation:
D( ) 2
DD
DK/(
DD
our experimental drug concentration is low, the obtained diffusion
1, as a function of volume fraction,
φ, of
tyloxapol(2) at 25 °C and pH 7.3. Surfactant volume fractions
value can be assumed to be equal to
D(. Because
DK
is the tyloxapol partial molar volume.9 Drug solubility,
S
NernstHartley equation. Comparison between
DD and (
D1)
V
shows that diffusion of ionic drugs can be significantly faster
φ within the experimental error up to surfactant
volume fractions as high as 0.10. This result demonstrates that
than that predicted from its tracer diffusion value. In other words,
naproxenate anions bind to tyloxapol micelles. Binding can be
counterion diffusion generates an electrostatic dragging effect
quantitatively characterized by employing a twophase partition
ing model.35,47 Within this model, drug molecules are assumed
The drug maindiffusion coefficient (
D11)
V in Figure 2a
to partition between the micellefree aqueous pseudophase (free
decreases as the surfactant concentration increases. This behavior,
drug) and the micellar pseudophase (bound drug). This partition
which can be related to the formation of drugmicelle com
ing equilibrium is described by the following idealdilute
plexes,9 is qualitatively consistent with our solubility results.
The relation between (
D11)
V and
K for the case of ionic drugswill be given in the following section.
In Figure 2b, we show the surfactant maindiffusion coefficient,
(40) Miller, D. G.; Albright; J. G. In
Measurement of the Transport Properties
of Fluids: Experimental Thermodynamics; Wakeham, W. A., Nagashima, A.,
(
D22)
V. For comparison, we include the corresponding surfactant
Sengers, J. V., Eds.; Blackwell Scientific Publications: Oxford, 1991; Vol.
3, pp
binary values, (
D2)
V (dashed curve).9 We can see that (
D2)
V slightly
(41) Annunziata, O.; Buzatu, D.; Albright, J. G.
Langmuir 2005,
21, 12085–
increases with increasing surfactant concentration. This behavior
can be attributed to steric repulsive interactions between micelles.9
(42) Zhang, H.; Annunziata, O.
J. Phys. Chem. B 2008,
112, 3633–3643.
The ternary values have been found to be 58% higher than the
(43) Albright, J. G.; Annunziata, O.; Miller, D. G.; Paduano, L.; Pearlstein,
A. J.
J. Am. Chem. Soc. 1999,
121, 3256–3266.
(44) Miller, D. G.; Vitagliano, V.
J. Phys. Chem. 1986,
90, 1706–1717.
(48) Kirkwood, J. G.; Baldwin, R. L.; Dunlop, P. J.; Gosting, L. J.; Kegeles,
(45) Huppert, E. H.; Hallworth, M. A.
J. Phys. Chem. 1984,
88, 2902–2905.
G.
J. Chem. Phys. 1960,
33, 1505–1513.
(46) Vitagliano, V.
Pure Appl. Chem. 1991,
63, 1441–1448.
(49) Robinson, R. A.; Stokes, R. H.
Electrolyte Solutions; Academic Press:
(47) Stilbs, P.
J. Colloid Interface Sci. 1982,
87, 385–394.
3428
Langmuir, Vol. 25, No. 6, 2009
Figure 2. Ternary interdiffusion coefficients for the naproxen(1)tyloxapol(2)water system as a function of tyloxapol volume fraction at 25 °C
and pH 7.3. (A) Naproxen maindiffusion coefficient (
D11)
V. (B) Tyloxapol maindiffusion coefficient (
D22)
V; the dashed curve represents the corresponding
binary interdiffusion coefficients for the tyloxapolwater system calculated using (
D
DM (1 + 17.13
φ) / (1 + 14.34
φ), where
DM
× 109 m2 s1 is the tracer diffusion coefficient of tyloxapol micelles in water. Binary diffusion data are reported in ref 9. (C) Naproxen crossdiffusioncoefficient (
D12)
V. (D) Tyloxapol crossdiffusion coefficient (
D21)
V. Solid curves are weighted fits through the data.
corresponding binary ones. This change, which has not been
counterions. Hence, a net diffusion of surfactant component occurs
observed in the case of the hydrocortisonetyloxapolwater
from high to low drug concentration. We will discuss this
system, can be related to a druginduced charge on the micelles
electrostatic effect in the following section.
and counterions. We will discuss this electrostatic effect in thefollowing section.
Diffusion Model
The drug crossdiffusion coefficient (
D12)
V in Figure 2c is
negative. This result has been obtained also in the case of the
General Diffusion Equations. Multicomponent interdiffusion
hydrocortisonetyloxapolwater system, and it has been gener
coefficients,
Dij, are combinations of thermodynamic factors and
ally observed for hostguest systems. A direct comparison
fundamental transport coefficients. Hence, to obtain theoretical
between the hydrocortisone and naproxen systems can be
expressions for
Dij, we need to model both thermodynamic and
performed by considering the ratio (
D
transport properties of the system. Although diffusion coefficients
12)
V/
C1, because
Dij (with
i *
j) is directly proportional to
C
are obtained in the volumefixed frame, the relation of diffusion
i.14 We find that (
D12)
V/
C1
ranges from 3 to 4 × 108 m2 s1 M1 for the naproxen case
to thermodynamics is simpler in the solventfixed frame for which
and from 3 to 7 × 108 m2 s1 M1 for the hydrocortisone
0.5052 Here, the subscript “0” appended outside the
case9 in the same range of tyloxapol concentrations. Thus, the
parentheses identifies the solventfixed frame. The corresponding
two sets values have similar magnitude. The negative sign of
diffusion coefficients will be denoted as (
Dij)0 (with
i,
j ) 1,2).
The theoretical (
Dij)
V values can be calculated from the
12)
V can be understood by considering a concentration gradient
of micelles in the presence of a uniform concentration of the
corresponding (
Dij)0, provided that the
Vi are known.
drug component. In these conditions, the concentration of free
According to nonequilibrium thermodynamics, diffusion for
drug molecules increases from high to low micelle concentration.
a ternary system can be described using the following linear
The resulting gradient will induce drug diffusion from low to
high micelle concentration on a time scale shorter than thatrequired for dissipating the concentration gradient of the slowly
(
J ) ) (
L ) ∇
µ + (
L ) ∇
µ
diffusing micelles. As in the case of (
D11)
V, we will discuss the
(
J ) ) (
L ) ∇
µ + (
L ) ∇
µ
12)
V to the partitioning constant,
K, in the following
where
µi is the chemical potential of the
ith component, and (
Lij)0
The surfactant crossdiffusion coefficient (
D21)
V in Figure 2d
are the solventframe Onsager transport coefficients. These
significantly increases with
C2. The positive value of (
D21)
V/
C2
coefficients satisfy the Onsager reciprocal relation (ORR): (
L12)0
can be attributed to a druginduced charge on the micelles andcan be understood by considering a concentration gradient of
(50) Miller, D. G.; Vitagliano, V.; Sartorio, R.
J. Phys. Chem. 1986,
90, 1509–
drug component in the presence of a uniform concentration of
(51) Dunlop, P. J.; Gosting, L. J.
J. Phys. Chem. 1959,
63, 86–93.
micelles. In these conditions, both free drug and micelledrug
(52) Woolf, L. A.; Miller, D. G.; Gosting, L. J.
J. Am. Chem. Soc. 1962,
84,
complexes will be electrostatically dragged by the faster potassium
Diffusion of an Ionic Drug in Micellar Aqueous Solutions
Langmuir, Vol. 25, No. 6, 2009 3429
) (
L21)0.53,54 We can use eqs 1a,b and 4a,b to relate the solvent
be used to calculate the average number of drug species, 〈
i〉, and
fixed diffusion coefficients and Onsager transport coefficients
counterions, 〈
j〉, bound to each micelle by:
(
D ) ) (
L )
µ + (
L )
µ
(
D ) ) (
L )
µ + (
L )
µ
(
C /
m) 1 + (
K  1)
φ
(
D ) ) (
L )
µ + (
L )
µ
(
D ) ) (
L )
µ + (
L )
µ
ij ≡ (
∂µi/
∂Cj)
T,
p,
C
k,
k*
j,
T is the temperature, and
p is the
(
C /
m) 1 + (
Kτ  1)
φ
pressure.5054 We will now derive expressions for the thermo
ij, and the Onsager coefficients (
Lij)0.
Thermodynamic Factors. We consider a drug(1)surfactant
To obtain expressions for the four thermodynamic factors in eqs
(2)water(0) ternary system at constant temperature. The drug
5ad, the following chemicalpotential expressions are hypoth
component is ionic, and the surfactant component is neutral. The
composition of this system is characterized by the drug andsurfactant molar concentrations,
C1 and
C2, respectively. We
µ )
µ0 +
RT ln
C(W) +
RT ln
C(W)
neglect the concentration of the free surfactant because the
tyloxapol cmc is significantly lower than our experimental
C
mµ )
mµ +
RT ln
C +
RT ln
y(
φ)
values. Micelles are assumed to be monodisperse with aggregation
number,
m, and molar concentration,
C
where
µ and
µ are the standard chemical potentials,
C
Solubilization of ionic drug molecules into micelles is related
molar concentration of the free micelles (MD
iK
j with
i )
j )
to the formation of a wide range of micelledrug complexes and
0), and
R is the idealgas constant. Equation 9a is consistent with
generally includes counterion binding. We will denote the generic
the twophase partitioning model. On the other hand, eq 9b is
an addition to the twophase model and characterizes the
iK
j, where M, D, and K identify micelle,
drug, and counterion, respectively. For anionic drugs, micellar
translational entropy of the micelles, which is the driving force
complexes display a net charge equal to
j 
i. Concentrations
for their diffusion. The micelle activity coefficient,
y(
φ), describing
of individual species are related to the component concentrations
the deviation from idealdilute solution, is assumed to be not
The free micelle concentration in eq 9b cannot be directly
determined from the twophase partitioning model. Its deter
mination requires knowledge of the distribution function,
f(
i,
j),of the MD
iK
j species so that
CMD
f(
i,
j) (
C2/
m). However, if
drugmicelle binding is assumed to be independent of
i +
j, we
can assume that
f(
i,
j) is given by the bivariate Poisson distribution
where
CD,
CK, and
CMD
function
f(
i,
j) ) (
e〈
i+
j〉/(
i +
j)!)〈
i +
j〉(
i+
j).55 For this special case,
drug, free counterion, and drugmicelle complexes, respectively.
CM can be determined provided that 〈
i〉 and 〈
j〉 are known:
These concentrations can be determined if the equilibrium
constants for the formation the drugmicelle complexes are
C )
f(0, 0)(
C /
m) )
e i〉
e j〉(
C /
m)
known. This complicated chemicalequilibrium problem can be
simplified using the twophase partitioning model19,47 based on
The concentrations
CD and
CK in eq 9a can be related to
C1
eq 2. To take into account binding of counterions to micelles,
using eqs 2 and 7, while the concentration
CM in eq 9b can be
we also introduce the following partitioning equilibrium condition:
related to
C2 using eqs 8a,b and 10. We can therefore rewrite theexpressions of the solute chemical potentials in the following
where
τ is the drugcounterion partitioning constant, and
C(M)
are the counterion molar concentrations in the micellar
and water pseudophases, respectively. The value of
τ ) 0
(
µ 
µ )/
RT ) 1 ln
corresponds to the case of no counterion binding to micelles. On
the other hand, the value of
τ ) 1 corresponds to the case of
counterion and drug binding strengths to micelles being identical.
In this latter case, the net micelle charge is zero. It can be easily
Expressions for the thermodynamic factors are then extracted
shown that the product
Kτ represents the partitioning constant
for the counterions between the aqueous and micellar pseudophases.
We note that counterion binding is likely to occur within the
hydrophilic domain of the micelles. Because the volume of themicellar hydrophilic domain is directly proportional to the total
C (
µ /
RT) ) 
φ
partitioning remains valid. Because
K and
τ are related to
CD and
C (
µ /
RT) ) 
φ(
K (see eqs 2 and 7), the values of these binding constants can
(53) Onsager, L.
Phys. ReV
. 1931,
38, 2265–2279.
(55) Brownlee, K. A.
Statistical Theory and Methodology in Science and
(54) Miller, D. G.
J. Phys. Chem. 1959,
63, 570–578.
Engineering, 2nd ed.; John Wiley & Sons: New York, 1965.
3430
Langmuir, Vol. 25, No. 6, 2009
the resulting expressions into eqs 14a,b, we obtain two equations
+ [
K(
K1)
φ2 +
Kτ(
Kτ1)
φ2 ]
C1
that can be directly compared to eqs 4a,b. This comparison allows
(1 
φ +
Kτφ)2
C2
us to obtain the following expressions for the (
L
≡ 1 + (d ln
y/d ln
C2) is the thermodynamic factor of
(
L ) ) (
L )  2
C [(
ν/
m)
V
low surfactant concentrations. We observe that the expression
of
µ21 based on the Poisson distribution is equal to (1 
φ)
µ12
[1 
C [(
ν/
m)
V
1/
C2)
µ11. This result could be also derived from the
thermodynamic relation between the four
µ
) 0.9,54 This is consistent with the twophase partitioning model,
which assumes that drug molecules do not affect the volume of
[1 
C (
ν/
m)
V
both pseudophases. This approximation is reasonable at low drug
We will now derive expressions for the (
L
(1) the freesolvent fluxes of individual solvated species in solution
Onsager Transport Coefficients. The determination of the
are uncoupled, (2) the diffusion coefficient of each species is
solventframe Onsager transport coefficients, (
Lij)0, requires the
constant and equal to the corresponding tracer diffusion coefficient
identification of the actual diffusing species in solution. Typically
in water, and (3) the diffusion coefficient of the micelledrug
coupled diffusion between these species is assumed to be
complexes is equal to that of the free micelles. We can therefore
negligible, and models are then constructed to obtain expressions
Recently, it has been experimentally observed that the solvent

J )
C D ∇
µ˜ /
RT
12)0 is negative for poly(ethylene glycol), a
hydrophilic macromolecule, in aqueous salt solutions.56 This

J )
C D ∇
µ˜ /
RT
result can be explained by considering the role of solute solvation.
Indeed, the actual diffusing solute species are solvated, and their
/
RT with
i,
j ) 0, 1, 2, .
diffusion behavior should be described with respect to the free
solvent reference frame, where (
J0ˆ)0ˆ ) 0 with the subscript “0ˆ”denoting the free solvent.57 Because surfactants with polyethylene
where
DD,
DK, and
DM are the tracer diffusion coefficients of free
oxide head groups are significantly hydrated, we will include the
drug anion, free counterion, and micelle complexes, respectively.
effect of micelle solvation in our model.
In eqs 17ac,
JD,
JK, and
JMD are the freesolvent frame fluxes
The actual thermodynamic driving forces for diffusion
of the individual species (where we have omitted frame notation
described in the freesolvent reference frame are the chemical
for simplicity), and
µ˜D,
µ˜K, and
µ˜MD
potentials for hydrated solutes,
µˆ
binding, the chemical potential of the hydrated surfactant is
µˆ2
To determine the relations between the (
Lij)0ˆ and the tracer
(
ν/
m)
µ0, where
ν is the number of solvent molecules
diffusion coefficients, we need to link the fluxes and the
bound to the micelle and
µ0 is the water chemical potential. We
electrochemicalpotential gradients of the species to the fluxes
shall neglect the contribution of drug and counterion hydration
and the chemicalpotential gradients of the components. Fluxes
because (1) it is expected to be significantly smaller than that
of individual species are linked to those of the components through
of the micelles and (2) it considerably increases the number of
the following mass balances based on eqs 6a,b:
variables because the hydration state of these species will changeupon binding to the micelles. We will therefore set
µˆ )
(
J ) )
J + ∑ ∑
iJ
Linear laws in the freesolvent reference frame are:57
where (
Lij)0ˆ are the corresponding Onsager transport coefficients
Electroneutrality and chemicalequilibrium conditions allow us
that satisfy the ORR: (
L12)0ˆ ) (
L21)0ˆ. The relation of (
Lij)0 to (
Lij)0ˆ
can be obtained by considering the following relations for the
results can be extended to the corresponding gradients:
(
J ) ) (
J ) 
C [(
ν/
m)
V
∇
µ˜ + ∇
µ˜ ) ∇
µˆ
(
J ) ) [1 
C (
ν/
m)
V
m ∇
µˆ with
i,
j ) 0, 1, 2, .
(
Ji)0ˆ  (
Ci/
C0)(
J0)0ˆ, and the following
relations for the chemicalpotential gradients:
If we insert eqs 17ac and eq 19b into eq 18a, we obtain an
equation that relates ∇
µ˜D and ∇
µ˜K to ∇
µˆ2. This equation together
with eq 19a allows us to obtain the following expressions for
ˆ ) [1 
C (
ν/
m)
V
where we have applied the GibbsDuhem equation to elimi
nate ∇
µ0. By inserting eqs 15a,b into eqs 13a,b and then inserting
[
C D + (
C /
m)
D (〈
j2 〉 〈
ij 〉 )] ∇
µˆ 
C D (〈
i 〉 〈
j 〉 ) ∇
µˆ
(56) Tan, C.; Albright, J. G.; Annunziata, O.
J. Phys. Chem. B 2008,
112,
[
C D +
C D + (
C /
m)
D (〈
i2 〉 + 〈
j2 〉 2〈
ij 〉 )]
(57) Annunziata, O.
J. Phys. Chem. B 2008,
112, 11968–11975.
Diffusion of an Ionic Drug in Micellar Aqueous Solutions
Langmuir, Vol. 25, No. 6, 2009 3431
[
C D + (
C /
m)
D (〈
i2 〉 〈
ij 〉 )] ∇
µˆ 
C D (〈
j 〉 〈
i 〉 ) ∇
µˆ
2(1 
φ)2
D D + 2
Kφ[(1 
φ)(
τD +
D ) +
KτφD ]
D
(1 
φ){[1 + (
Kτ  1)
φ]
D + [1 + (
K  1)
φ]
D } +
Kφ[(1 +
τ)(1 
φ) + 2
Kτφ]
D
[
C D +
C D + (
C /
m)
D (〈
i2 〉 + 〈
j2 〉 2〈
ij 〉 )]
In the case of nonionic drugs, eq 25 reduces to (
D
where we have used the definition 〈
x〉 ≡ ∑
j ) 0
x f(
i,
j). The
M]/(1 
φ +
Kφ) is the self
corresponding expression for ∇
µ˜
diffusion coefficient of the nonionic drug in the presence of
eqs 20a,b into eq 19b. We are now in position to obtain expressions
˜ D is a weighed average between
DD and
for the (
Ji)0ˆ (with
i ) 1,2) as a function of the ∇
µˆ
i. Comparison
DM. We can also consider eq 25 in the limiting case of
τ ) 0
with eqs 13a,b yield the following expressions for the (
Lij)0ˆ:
(no counterions binding). In this case, we obtain the NernstHartleyequation: (
D
C D C D + (
C /
m)
D [
C D 〈
j2 〉 +
C D 〈
i2 〉 + (
C /
m)
D (〈
i2 〉 〈
j2 〉 〈
ij〉2)]
counterions, we conclude that (
D11)
V
RT[
C D +
C D + (
C /
m)
D (〈
i2 〉 + 〈
j2 〉 2〈
ij 〉 )]
counterions exert an electrostatic dragging effect on the slower
drug ions to preserve electroneutrality. Finally, we note that
mand
ν have no effect on (
D11)
V in this limit.
To examine the other three interdiffusion coefficients, we shall
C D 〈
j 〉 +
C D 〈
i 〉 + (
C /
m)
D [〈
i 〉 (〈
j2 〉 〈
ij 〉 ) + 〈
j 〉 (〈
i2 〉 〈
ij 〉 )]
consider the limit of infinite dilution with respect to both
C
RT[
C D +
C D + (
C /
m)
D (〈
i2 〉 + 〈
j2 〉 2〈
ij 〉 )]
2. In the case of (
D12)
V, we obtain:
12
V ) 
D([
K(1 +
τ)  2] +
C D +
C D + (
C /
m)
D [〈
i2 〉 〈
i〉2 + 〈
j2 〉 〈
j〉2  2(〈
ij 〉 〈
i 〉 〈
j 〉 )]
RT[
C D +
C D + (
C /
m)
D (〈
i2 〉 + 〈
j2 〉 2〈
ij 〉 )]
The values of 〈
i〉 and 〈
j〉 can be determined from eqs 8a,b providedthat
K and
τ are known. However, the determination of 〈
i2〉, 〈
j2〉,
In eq 26, the second term contributes marginally to the value of
and 〈
ij〉 requires a further assumption on
f(
i,
j). Although the
(
D12)
V/
C1, because
DM is small as compared to
D( ) 2
DD
DK/(
DD
values of 〈
i〉 and 〈
j〉 are related to each other, we can assume that
+
DK). Indeed, we can approximately write: (
D12)
V/
C1 ≈ 
D(
j  〈
j〉 for the counterion does not correlate with
i  〈
i〉 for the
V2
K(1 +
τ), where we have also assumed that
K . 2. We conclude
drug. Within this assumption,
f(
i,
j) becomes the product of two
that also this coefficient is not very sensitive to the values of
m
independent Poisson distribution functions:55
and
ν. This crossterm is predicted to be negative and directly
proportional to
K for both ionic and nonionic drugs. In other
〈
i +
j〉(
i+
j) ) (
e 〈
i〉
i
words, due to drugmicelle binding, a concentration gradient
of micelle induces a flux of drug component from low to high
Using the mathematical properties of independent Poisson
micelle concentration. Our experimental results on both potassium
distribution functions, we can determine 〈
i2〉, 〈
j2〉, and 〈
ij〉 from
naproxenate and hydrocortisone are in agreement with the
〈
i〉 and 〈
j〉 according to:
〈
i2 〉 ) 〈
i 〉 + 〈
i〉2
The limiting expression of (
D21)
V is
〈
j2 〉 ) 〈
j 〉 + 〈
j〉2
〈
ij 〉 ) 〈
i 〉 〈
j〉
Diffusion Coefficients. The values of (
L
ij)0 and
µij/
RT can be
determined provided that
K,
τ,
m,
ν,
D
D,
DK, and
DM are known.
We can then calculate (
Dij)0 using eqs 5ad. Finally, the (
Dij)0can be converted into (
D
ij)
V using previously reported equations
21)
V/
φ is directly proportional to
m and
DM. This
coefficient is also proportional to a difference between two terms.
The first term, which is associated with druginduced micelle
(
D ) ) (
D ) 
C (
φ/
C )(
D )
becomes zero when
τ ) 1 corresponding to neutral micelles. The
second term is associated with micelle solvation. The relative
contributions of these two terms depend on the values of
τ and
ν/
m. If
τ ) 0 and
ν ) 0, eq 27 reduces to (
D
21)
V/
φ )
K(
DK
DK). We therefore conclude that the sign of (
D21)
V
The explicit expressions for the (
Dij)
V are cumbersome. Thus, to
strongly depends on the sign of
D 
gain physical insight on the behavior of the diffusion coefficients,
qualitative agreement with our experimental results and explana
we shall consider simplified expressions obtained by considering
limit conditions. We further notice that this model describesdiffusion of nonionic drugs9 in the limit of
τ ) 1 and
D )
Finally, we examine (
D22)
V. The effect of ionic drugs can be
11)
V, we consider the limit of infinite dilution
described by considering the limiting expression: (
D22)
V
3432
Langmuir, Vol. 25, No. 6, 2009
an ethoxy group,59 and each tyloxapol consists of ∼70 ethoxy
groups, we obtain
ν/
m ≈ 280 and
ν ≈ 3400. Estimation of
τ isdifficult. This quantity is expected to depend on the chemical
(1 
K 1)
D  (1 
K 1
τ 1)
τD  (1 
τ)
D
nature of the surfactant hydrophilic groups and charge distribution
on the micelle. Thus, we examine our model by varying the
value of
τ from zero to one.
The surfactant nonideality term, (
φ) in eq 12d, is expected
0 (
K +
Kτ  2)] (28) to be close to unity at our experimental low values of
φ.
Nonetheless, we have estimated it from the experimental binary
In eq 28, R is directly proportional to
m and a sum of two terms.
The first term is associated with druginduced micelle charge
and vanishes when
τ ) 1. The second term in eq 28 is associated
with micelle solvation. Interestingly, micelle solvation has
(1 
φ)[1 
C (
ν/
m)
V
22)
V and (
D21)
V. If
Kτ . 1 and
DM
the first term becomes
K2(1 
τ)(
D 
Equation 29 was obtained by assuming that (
L22)0ˆ )
C2
DM for
indicates that the effect of ionic drugs on the surfactant main
the binary tyloxapolwater system, consistent with our diffusion
diffusion coefficient depends on the sign of
D 
model. We have also used (
D2)
V
(1 
φ) [(
L22)0/
C2]
4d and 24d) and converted (
L22)0ˆ into (
L22)0 using eq 16c. We
DK. Small values of
τ imply that micelle diffusion is
enhanced by drug binding. However, as
τ increases,
D
can be calculated provided that
ν/
m is known.
To evaluate whether micelle solvation can be invoked to explain
K. This implies that micelle diffusion is hindered
in these conditions. This behavior can be physically understood
our experimental diffusion results, we initially compute (
Dij)
V by
by considering that a micelle concentration gradient at constant
setting
ν ) 0 and changing
τ. Our experimental results and
theoretical predictions are shown in Figure 3.
1 generates a concentration gradient of free drug anions and
counterions. These gradients generate a net flux of drug component
In Figure 3, we note that our experimental (
Dij)
V values with
toward the micelles as discussed above. Drug crossdiffusion
φ < 0.005 display some discrepancy from those at a higher
φ.
can be driven either by drug ions or by counterions, depending
This small deviation encountered at low surfactant concentrations
can be attributed to a relatively large drug load of micelles,
D/
DK and the corresponding ratio in
which may have a small effect on the micellization process itself.
D/
∂CK)
C . We obtain (
∂C
1/
τ from differentiation of eqs 8a,b in the limit of small
φ. If
τ
Furthermore, we also note that the precision of diffusion
) 0, the concentration of counterions is uniform and drug diffusion
measurements reduces as the solute concentration decreases.9
toward the micelles is driven by the concentration gradient of
Thus, we give more relevance to our results with
φ > 0.005.
drug anions. This diffusion process drives a net negative charge
For (
D11)
V and (
D12)
V, our model is in good quantitative
toward the micelles. The corresponding electric field drives the
agreement with the experimental results if we set
τ ) 0.4 ( 0.2.
negatively charged micelles in the direction opposite of that of
Numerical analysis shows that the behavior of these two diffusion
drug diffusion and equal to that of micelle diffusion. The net
coefficients significantly depends on
K. This implies that our
K
result is an enhancement of micelle diffusion. However, as
τ
value extracted from solubility measurements characterizes the
increases, concentration gradients of both drug anions and
behavior of (
D11)
V and (
D12)
V quite well. Similar conclusions
counterions are present. Because
D >
the micelles becomes driven by the concentration gradient of
For (
D21)
V, our model is in good quantitative agreement with
counterions if
τ is large enough. This diffusion process drives
the experimental results if
τ ) 0.5 ( 0.1. However, a good
a net positive charge toward the micelles. The corresponding
quantitative agreement for (
D22)
V can be obtained only if
τ < 0.2.
electric field drives the negatively charged micelles in the same
Although
τ < 0.2 may still give acceptable predictions for (
D11)
V
direction as that of drug diffusion. Hence, the net result
and (
D12)
V, it predicts (
D21)
V values 100% larger than the
corresponds to a reduction of micelle diffusion.
experimental data (see Figure 3d). Furthermore, our calculationshows that (
D22)
V is lower than (
D2)
V if
τ > 0.2. On the otherhand, we experimentally obtain the opposite behavior. We have
Discussion
examined whether this discrepancy can be related to inaccurate
In this section, we quantitatively compare our results with the
estimations of
m. However, eqs 2528 indicate that (1)
m has
proposed diffusion model. We set
D )
a small effect on (
D11)
V and (
D12)
V; (2) (
D21)
V is directly
0.58 × 109 m2 s1 from our drugwater diffusion data,
proportional to
m; and (3) a change in
m has no effect on the
0.0694 × 109 m2 s1 from our tyloxapolwater
sign of R. Numerical examination on the (
Dij)
V general expressions
diffusion data previously reported.9 We then set
K ) 25 according
confirms our conclusions. Thus, a change in
m does not account
to our solubility results. The tyloxapol micelle aggregation number
for the observed discrepancy between (
D21)
V and (
D22)
V.
is set to
m ) 12. This value was estimated from the micelle
We now examine the role of micelle solvation. Equations 27
hydrodynamic volume and micelle hydration.9 It corresponds to
and 28 show that micelle solvation has an opposite effect on the
∼90 octylphenolethoxylate monomers inside one micelle and
behavior of (
D21)
V and (
D22)
V. As
ν increases, (
D22)
V/(
D2)
V
is comparable with the aggregation number of ∼100 for the
increases, while (
D21)
V/
φ decreases. Thus,
ν can be used to improve
octylphenolethoxylate surfactant.58 According to our diffusion
the agreement between the model and the experimental results.
model,
m is expected to have a significant effect only on the
By varying both
τ and
ν, we find that the best agreement is
behavior of (
D21)
V and (
D22)
V. The number of water molecules
obtained for all four diffusion coefficients when
τ ) 0.27 and
bound to a micelle can be estimated from the chemical properties
ν ) 5000. The results are shown in Figure 4. Our results with
of the hydrophilic ethoxy groups of the surfactant. Because it is
τ ) 0.27 and
ν ) 3400 estimated from the hydration of ethoxy
known that there are about four water molecules associated with
groups are also included in the same figure. We can see that the
(58) Tummino, P. J.; Gafni, A.
Biophys. J. 1993,
64, 1580–1587.
(59) Nilsson, P. G.; Lindman, B.
J. Phys. Chem. 1983,
87, 4756–4761.
Diffusion of an Ionic Drug in Micellar Aqueous Solutions
Langmuir, Vol. 25, No. 6, 2009 3433
Figure 3. Ternary diffusion ratios for the naproxen(1)tyloxapol(2)water system ((
D11)
V/(
D1)
V, A; (
D22)
V/(
D2)
V, B; (
D12)
V/[
C1(
D1)
V], C; (
D21)
V/
[
C2
DM], D). The dashed curves represent the model predictions for
K ) 25 and
ν ) 0. The numbers associated with each curve identify the
corresponding values of
τ.
Figure 4. Ternary diffusion ratios for the naproxen(1)tyloxapol(2)water system ((
D11)
V/(
D1)
V, A; (
D22)
V/(
D2)
V, B; (
D12)
V/[
C1(
D1)
V], C; (
D21)
V/
[
C2
DM], D). The curves represent the model predictions for
K ) 25 and
τ ) 0.27. The solid curves, long dashed curves, and short dashed curves
were obtained setting
ν ) 5000,
ν ) 3400, and
ν ) 0, respectively.
experimental behavior of both (
D21)
V and (
D22)
V is reproduced
of our assumption, we calculate the average number of bound
fairly well also for
ν ) 3400. It is expected that neglecting
drug, 〈
i〉, within the experimental range of micelle volume fraction
electrostatic nonideality effects in the model may account for the
using eq 8a. As
φ increases from 0.0018 to 0.018, 〈
i〉 decreases
from 6.9 to 5.0. Using
m ) 12 and component molar masses,
We note that our proposed diffusion model assumes that drug
we estimate that the drug contribution to the micelle mass
binding has no effect on
DM. However, drug binding may affect
(ignoring the contribution of solvation) is 23%. For globular
DM by changing the size of micelles. To examine the accuracy
particles such as micelles, the estimated increase in mass is
3434
Langmuir, Vol. 25, No. 6, 2009
expected to reduce the corresponding value of
DM by less than
Conclusions
1%. We therefore conclude that the assumption of
DM constant
To quantitatively understand the experimental data of the four
interdiffusion coefficients, we have built a diffusion model based
Finally, we discuss the obtained value of
τ. Using eqs 8a,b,
on drugmicelle binding, counterion effects, and micelle
we can use
τ ) 0.27 to calculate the degree of counterion binding,
solvation for a drugmicellewater ternary system. We remark
〈
j〉/〈
i〉. Within the experimental range of micelle volume fraction,
that diffusionbased transport of ionic drugs is relatively fast due
〈
j〉/〈
i〉 varies from 0.28 to 0.35. For ionic surfactants, it has been
to the presence of counterions. Because (
D11)
V decreases as the
experimentally and theoretically found that the degree of
surfactant concentration increases, micellar systems can be used
counterion binding for the corresponding micelles is significantly
to bind drug molecules, thereby reducing their diffusion in a
higher and ranges from 0.5 to 0.8.20,60 However, in the case of
controllable fashion. Moreover, because (
D12)
V is negative, a
ionicnonionic mixed micelles, it has been shown that 〈
j〉/〈
i〉
concentration gradient of micelles may be used as a tool to further
steadily decreases approaching zero as the contribution of neutral
reduce drug diffusion rate from high to low micelle concentration.
surfactant to the micelle increases.6163 Our drugloaded tyloxapol
This work provides guidance for the development of models for
system is better described as a mixed micelle. Furthermore,
controlled drug release in the presence of nanocarriers based on
because there are about 90 neutral head groups in a tyloxapol
multicomponent diffusion coefficients.
micelle, the ratio of naproxenate anions to tyloxapol head groupsis quite small within our experimental range. Thus, the obtained
Acknowledgment. We are in debt with Prof. John G. Albright
small value of
τ is qualitatively consistent with previous
for his assistance with the Gosting diffusiometer. This work was
experimental and theoretical studies on mixed micelles.
partially supported by TCU Research and Creative Activity Funds.
Supporting Information Available: Interferometric diffusion
(60) Srinivasan, V.; Blankschtein, D.
Langmuir 2003,
19, 9932–9945.
(61) Hall, D. G.; Price, T. J.
J. Chem. Soc., Faraday Trans. 1984,
80, 1193–
data; convective flow induced by drug diffusion. This material is available
free of charge via the Internet at http://pubs.acs.org.
(62) Akisada, H.
J. Colloid Interface Sci. 2001,
240, 323–334.
(63) Goldsipe, A.; Blankschtein, D.
Langmuir 2005,
21, 9850–9865.
Source: http://geo1.tcu.edu/annunziata/2009_LANG_01.pdf
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