iles on theend

ke

by

ehav

Journal of Colloid and Interface Science 293 (2006) 213–221www.elsevier.com/locate/jcis

Sphere to rod transitions in homologous alkylpyridinium salts:A Stauff-Klevens-type equation for the second critical micelle

concentration

A. González-Pérez, L.M. Varela∗, M. García, J.R. Rodríguez

Grupo de Materia Condensada Blanda, Departamento de Física de la Materia Condensada, Universidad de Santiago de Compostela,E-15782 Santiago de Compostela, Spain

Received 21 April 2005; accepted 8 June 2005

Available online 8 August 2005

To the memory of Professor Jan Czapkiewicz, Jagiellonian University, Cracow

Abstract

In the present paper, we analyze the dependence of the second critical micelle concentration (second cmc) of ionic amphiphnumber of atoms in the hydrocarbon molecular chain,nc. A molecular thermodynamic model for the interaction energy between thecaps of rodlike micelles,g(N), is introduced and the linear dependence of this object onnc analyzed, thus leading to a Stauff-Klevens-libehavior of the second cmc. The predictions agree with previously reported data forn-alkyldimethylbenzylammonium chloride (CnBACl)for n-alkylpyridinium chloride (CnPyCl) andn-alkyltrymethylammonium bromide (CnTABr) at 35◦C. These conclusions are reinforcedthe conductivity, density, and ultrasound velocity measurements of the second cmc of several CnPyCl (n = 12, 14, 16) andn-alkylpyridiniumbromides (CnPyBr,n = 12, 14, 15, 16) presented in this paper. 2005 Elsevier Inc. All rights reserved.

Keywords: Ionic amphiphile self-assembly; Micellar growth; Second critical micelle concentration; Molecular thermodynamics; Stauff-Klevens-like bior

ilicor-bee(seee-hy-ncyivergeve an-hic

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1. Introduction

Thermodynamically driven self-assembly of amphiphmoieties into mesomorphic structures is of outmost imptance in many academic and industrial areas, and hasthe object of many studies throughout the past decadese.g., Refs.[1–8] and references therein). Amphiphilic molcules are formed by a hydrophilic polar head and adrophobic tail, consequently showing a striking tendeto minimize their free energies in solution by cooperat[9–11]self-assembly either at interfaces or by forming lamicellar aggregates in the bulk at concentrations abowell-defined concentration called the critical micelle cocentration (cmc)[2]. The size and shape of the mesomorp

* Corresponding author.E-mail address: fmluis@usc.es(L.M. Varela).

0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2005.06.026

n,

equilibrium self-associating structures are determined bybalance of energetic and entropic changes in the mainlytropically driven hydrophobic aggregation process, togewith molecular packing considerations. It took much effto conclude that this polymorphism was essentially dueintraaggregate mechanisms (essentially opposed forcesdrophobic attraction between monomer chains, and resions between surface head groups, and molecular pacconstraints), with interaggregate interactions having a negible effect on the thermodynamic stability of a given sha[3,4].

Depending on the type of surfactant and on the exteconditions (temperature, pressure, concentration, etc.),ious morphologies are accessible to the aggregates ining spherical, globular, rodlike, and even spherical bilay(vesicles). Micelles are generally spherical in the vicinitythe cmc, but other thermodynamically less favorable sha

214 A. González-Pérez et al. / Journal of Colloid and Interface Science 293 (2006) 213–221

on-nnofor-oidular

-

its,ansdy-aseslate

ed, buthe

ionsnlles,rplycen-

qua

llesowticalmi-lles

ex-ing-lar

ondithe

-g-in tosechatbe-bicens

ere-the

ilesvedex-h of

m-sescon-plies

meof

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theer-

tingda-

hen offor a

er

argeione re-ngitsbe a

er-re

are registered in solution due to the molecular packing cstraint that at least one dimension of the aggregate caexceed the monomer chain length in order to avoid themation of an energetically and entropically disfavored vinside the core. This restriction imposes that the molecpacking parameterv/la, wherev andl are the volume andlength of the surfactant tail anda is the area of the hydrophobic core per monomer, is bounded by 1/3 (spheres), 1/2(cylinders), and 1 (bilayers). Beyond the respective limthe aggregates are forced to undergo morphological trformations into shapes compatible with the new thermonamic conditions and the packing constraints, in most cpassing through other globularlike micelles such as proor oblate spheroids and ellipsoids.

For most amphiphile solutions, the previously describgrowing process does not take place in a continuous waytwo well-defined stages are experimentally detected inconcentration range where morphological transformattake place[12,13]. In the lower part of this concentratioregime, all added surfactant aggregates into small micethe ones formed in the first cmc, until a more or less shadefined concentration, called the second micellar contration (second cmc, of molar fractionX2cmc) where theconcentration of small micelles saturates and remains eto X2cmc. In the concentration regimeX > X2cmc all addedsurfactant aggregates into long linear (rod-shaped) miceThe second cmc has an origin similar to that of its lconcentration counterpart, due to the fact that hypothemicelles of intermediate sizes between that of the smallcelles formed in the first cmc and that of rod-shaped miceare not energetically favored. This is indicative of theistence of a large energy barrier in the molecular packspectrum (about 5kBT per micelle for micelles of intermediate sizes)[12] that is responsible for the gap in the micelsize distribution.

Although several thermodynamic studies of the seccmc have been reported during the past decades, from ea phenomenological[1,4,12,14–16]or a molecular thermodynamical[17,18] point of view, several aspects of the agregation process in the sphere to rod transition remabe clarified. This is the case for the dependence of theond cmc on the monomer chain length. It is well known tthe logarithm of the first cmc shows a linear decreasinghavior with the number of carbon atoms in the hydrophochain, constituting the basis of the so-called Stauff-Klevequation[19–21]

(1)lnXcmc= A − Bnc, B > 0,

wherenc is the number of carbon atoms in the monomchain, andA andB are usually treated as adjustable paramters. The above semiphenomenological result applies toformation of self-assembled nanostructures of amphiphheld together by noncovalent interactions, and it has prohighly useful for the understanding of a large amount ofperimental data. In fact, it is a consequence of the growt

t

-

t

l

.

r

-

Chart 1.

the hydrophobicity of the chain with an increase in the nuber of alkyl groups in its molecular structure, which cauthe self-assembly process to start at progressively lowercentrations. Thus, one can reasonably expect that it apalso to the second cmc.

Actually, González-Pérez et al. recently reported soconductivity and density results for the dependencelnX2cmc on the number of carbon atoms in the hydrocarbchain for aqueous solutions ofn-alkyldimethylbenzylam-monium chloride[22], showing that lnX2cmc follows a be-havior similar to that predicted by Eq.(1). A similar behaviorwas found in NMR studies of Lee and Woo[23] for CnPyCland CnTABr at 35◦C. Despite considerable advances inthermodynamical and molecular thermodynamical undstanding of the morphological transitions in self-associamaterials, these results still need a firm theoretical fountion. A contribution to this analysis is the main aim of tpresent paper, together with the experimental verificatiothe obtained results using direct experimental evidencewhole series of alkylpyridinium salts (seeChart 1) and sev-eral results previously reported in the literature[23].

2. Materials and methods

2.1. Materials

Dodecylpyridinium chloride (C12PyCl) was provided byMerck with a purity of 95% and was used without furthpurification. Tetradecylpyridinium chloride (C14PyCl) [24]was obtained by treatment of chlorotetradecane with a lexcess (9 eq) of pyridine at reflux for 100 h. The reactmixture was concentrated under vacuum and the residucrystallized twice from acetone (82% yield) until reachia purity >99%. The compound was characterized byspectrometric data and shown by elemental analysis tomonohydrate. Hexadecylpyridinium chloride (C16PyCl) wasprovided as a monohydrate by Fluka with a purity>98% andit was used without further purification.

Dodecylpyridinium bromide (C12PyBr), kindly donatedby late Prof. J. Czapkiewicz from the Jagiellonian Univsity in Krakow, was synthesized following the procedupreviously described[25] and its purity was>99%. Tetrade-

A. González-Pérez et al. / Journal of Colloid and Interface Science 293 (2006) 213–221 215

-tro-etheri-

n.in-

as a

icsde-g-

ngthanuc-ed.sseea-

An-Thetateers.

-ts.

ms

is

seon-cies

m-

ulk.

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in-f the

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fortheg-

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tes.

ondthe

.-ond

cylpyridinium bromide (C14PyBr)[26] was obtained according to the literature and its purity was established by specscopic analysis to be>99%. Pentadecylpyridinium bromid(C15PyBr) was also donated by Prof. Czapkiewicz fromJagiellonian University in Krakow. Finally, hexadecylpydinium bromide (C16PyBr) was obtained from Aldrich witha purity of 98% and was used without further purificatioResults of potentiometric and thermogravimetric analysisdicated that the air-dried crystals of bromide salts existmonohydrate in all the studied compounds.

2.2. Conductivity measurements

Conductivities were measured using a Kyoto Electronconductimeter (Model CM-117). The cell constants weretermined using KCl solutions following the procedure sugested by Monk[27]. The temperature was controlled usia thermostated bath with temperature variations of less0.1◦C. Appropriate corrections were made for the condtivity of pure water as a function of temperature. Distillwater with conductivity below 3 µS cm−1 at 25◦C was usedThe concentrations of the surfactant solutions are exprein moles per kilogram and were prepared by weight msurements.

2.3. Density and sound velocity measurements

Density and sound velocity were measured with anton Paar DSA 5000 density and sound velocity analyzer.temperature control is attained using a built-in solid-sthermostat made for two integrated platinum thermometThe repeatability is 1× 10−6 gcm−3 for density, 0.01 m s−1

for sound velocity, and 0.001◦C for temperature. The apparatus was calibrated before each series of measuremen

3. Theoretical section

3.1. Thermodynamics of micellization

The size distribution of aggregates in solution in terof the system parameters (T ,P,X1), whereX1 is the mo-lar fraction of free amphiphile monomers in solution,determined by the conventional mass equilibrium,µ(N) =Nµ(1), whereµ(N) is the chemical potential of micelleof aggregation numberN . Thus,µ(1) corresponds to thfree energy of a free monomer in solution. Using the cventional expression for the chemical potential of a spein dilute solution one obtains

(2)XN = NXN1 exp

[−N

(µ0N − µ0

1)

kBT

],

where µ0N is the standard chemical potential of an a

phiphilic monomer in aN -aggregate, and consequentlyµ01

is the standard chemical potential of a monomer in the b

d

Fig. 1. Schematic representation of rodlike micellar aggregates in the srocylinder (a) and generalized spherocylindrical model (b).

The free energy per amphiphile in an aggregate of aggrtion numberN in Tanford’s opposing forces model is giveby [1,5,28]

(3)µ0N = gs + gh + gb = γ a + α

a+ gb,

wheregs = γ a represents the positive contribution of theterfacial tension associated with the residual contacts ochains with water. The contributiongh = α/a is due to therepulsive forces between the polar heads at the surfactheN -aggregate,α being the head-group repulsion paramter andgb stands for the free energy of a monomer insideliquid hydrocarbon core of the aggregate.

The application of Eq.(3) predicts the existence of selassociation finite size spherical aggregates, but it doesaccount for micellar growth. A necessary conditiona chemical potential model to predict the growth ofnanoaggregates is thatµ0

N decreases with the micellar agregation number in a definite manner[1]. Particularly, thecase in which micellar growth is achieved by variation ofamphiphile concentration has been studied using the spocylinder model[1], where the internal packing free enerof a linear micelle of aggregation numberN is

µ0(N) = (N − N0)µ0cyl + N0µ

0sph

(4)= Nµ0cyl + 2δ,

whereµ0cyl is the chemical potential of a monomer in t

cylindrical part of the aggregate andµ0sph its spherical coun

terpart.N0 is the aggregation number of the sphericalgregate, andδ = N0(µ

0sph− µ0

cyl)/2. The standard chemicapotential in the above equation corresponds to an aggreformed up byN − N0 monomers in its cylindrical part anN0/2 in each hemispherical end cap. The spherocylindrbasic structural model is shown inFig. 1a, and it assumethat the free energy of a monomer in the cylindrical pof the aggregate is lower than that in the spherical pcompensating the entropic loss associated with packingamphiphile in a long micelle instead of smaller aggregaHowever, as discussed by Porte et al.[12], the spherocylin-der model is not compatible with the existence of a seccmc as it predicts a continuous, monotonic increase ofaverage aggregation number of the micellar aggregates

As noted by May and Ben-Shaul[13], the necessary condition for the appearance of an abrupt transition (sec

216 A. González-Pérez et al. / Journal of Colloid and Interface Science 293 (2006) 213–221

mi-achnderwndi-ith

and

eirthe

arlyiatedingphellar

ith

ea-m-

alion

themic

ichreen

sion

ticde--for

lily

tionr-ofthen to

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nd

icel-tionrfac-anyhe

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ellarof

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oci-ace.

intri-n be

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pi-i-

her

cmc) is the existence of an internal energy barrier in thecellar size region where the end caps still interact with eother, a gap that does not exist in the simple spherocylimodel. When the micelles are modeled in the way shoin Fig. 1b, there exist thermodynamically vetoed intermeate size micelles in the early stages of micellar growth whigher chemical potentials than those of both smallerlarger ones, because due to packing constraints[29,30] thespherical parts of the rodlike micelle are bulkier than thcylindrical counterparts, with the chemical potential ofmolecules in the junction,µ0

J , larger than eitherµ0cyl or µ0

sph.Thus, an energetic barrier is expected to exist in the estages of the growth of the micellar aggregates assocwith the creation of the two unfavorable end caps, givrise to the existence of a second cmc. This generalized srocylinder model also accounts better for the actual micesize distribution.

The chemical potential of large cylindrical micelles wnon-interacting end caps is governed by Eq.(4). However,for intermediate size micelles for which the asymptotic rsoning is not valid yet, the chemical potential of their aphiphilic molecules must be written as[13]

(5)µ0(N) = Nµ0cyl + 2δ − g(N),

where g(N) accounts for the deviations of the chemicpotential from its asymptotic value due to the interactbetween end caps. Thus,g(N) = 0 for long micelles.µ0

cylcan be set to zero without any loss of generality[13], sothe whole thermodynamics of the system is contained inspherical end-cap terms. The condition of thermodynaequilibrium in Eq.(2) then leads to

(6)XN = NXN1 exp

(Nµ0

1

kBT

)exp

[g(N) − 2δ

kBT

].

Defining the second cmc as the concentration at whthe rodlike micellar concentration equals that of the fmonomers,X2cmc = X1 [7], and using the above equatioone obtains

lnX2cmc= − lnN

N − 1− N

N − 1

µ01

kBT

(7)− 1

N − 1

g(N) − 2δ

kBT.

The analysis of the dependence of the above expreson the number of carbon atoms in the monomer chain,nc,is the main aim of this paper. Evidently, for the asymptocase this analysis is little more than trivial because thependence of 2δ in nc is directly obtainable from the molecular thermodynamics of Tanford’s opposing forces modelspherical micelles in Eq.(3). As pointed out by Israelachviand co-workers[1], the area per monomer is approximateindependent of the chain length, and the bulk contributo the free energy,gb, is linear in this parameter. For spheical micelles, this is the ultimate reason for the validitythe semiempirical Stauff-Klevens equation. However, inregime where the micellar growth begins, the contributio

-

the free energy of interaction of the spherical end caps isidently different from zero, so the answer to the questionthe validity of a Stauff-Klevens-type equation for the secocmc depends crucially on a model forg(N). In the follow-ing, we shall try to obtain such a model.

3.2. Model for g(N)

At concentrations above the second cmc where the mlar growth begins, one can neglect in a first approximaexcluded volume and steric repulsions between the sutant chains in the two spherical end caps together withcontribution of the molecules in the cylindrical part of tmicelle (recall that we can fixµ0

cyl = 0 without loss of gen-erality [13]). In the neighborhood of the second cmc tassumption is particularly appropriate, since no molecis contained in the cylindrical part when micellar growbegins. Thus, the interaction energy between the micend caps is mainly the result of the combined actiontwo contributions: (i) the electrostatic repulsion of the eltric charges of the head groups on the surface of thecaps, and (ii) the interfacial free energy contribution assated with the creation of a new hydrocarbon–water interfWhile the latter is easily mimicked by means of a termγ acylper monomer, whereacyl is the surface area per monomerthe cylindrical part of the micelles, the electrostatic conbution to the interaction energy between the end caps caapproximately calculated as follows.

Let us consider two semispheres of radiusR with a uni-form surface charge densityσ and separated by a distanceL,representing the two spherical end caps of a cylindricalcelle (Fig. 2). As the charges are on the surface of the sphonly, they can be treated as equipotential surfaces similconductors. Thus, the total electrostatic energy in a medof dielectric constantε (the bulk hydrocarbon phase) canwritten as[31]

(8)U = 1

2

2∑i=1

2∑j=1

qiϕij ,

whereqi represents the charge of theith semisphere (q1 =q2) andϕij is the electrostatic potential created by end cai

on end capj , which obviously verifies the symmetry condtion ϕij = ϕji andϕ11 = ϕ22. This potential can be approxmated by the potential generated by the spherej on the axisof spherei (i.e., on the point(0,0,L) in Fig. 2) as

ϕij (L) = σR2

4πε

π/2∫0

2π∫0

sinθ√R2 + L2 − 2LR cosθ

dθ dφ

(9)= σR

ε

[R + L −

√R2 + L2

].

Approximatingϕ11 ≈ ϕ12(L = 0) andϕ11 ≈ ϕ12(L) for thepotential created by distribution 1 on itself and on the otcharge distribution, respectively, one trivially gets

(10)U(L) = qσR [R + 2L −

√R2 + L2

].

2εL

A. González-Pérez et al. / Journal of Colloid and Interface Science 293 (2006) 213–221 217

con-of a

fterof

ss”

aulfthe

r in

ve

e re-

ins,full

g

riv-ing

tivesult,nsid-ince

ends ofe rhsx-y

r inthe. In

th

forsoa-

i-p-rs

er-des,

rfac-

ernsytheted

Fig. 2. Geometric model employed in calculations of the electrostatictribution to the free energy of interaction between spherical end capsrodlike micelle.

For globular micelles, i.e., those formed immediately athe beginning of the micellar growth in the neighborhoodthe second cmcL → 0, and therefore,

(11)U(L) = qσR

2ε

[2− χ

2

],

whereχ = L/R is a measure of the degree of “globularneof the aggregates. Exactly at the second cmcχ = 0, so weget

(12)g(N) = qσR

ε.

The above expression clearly shows thatg(N) is positivein accordance with the predictions of May and Ben-Shfor very small micelles[13]. Thus, in the neighborhood othe second cmc the chemical potential of a monomer inspherical end caps of aN -aggregate is

(13)µ0N = 2δ

N0− qσR

εN0,

where we have used thatN ∼ N0 when the micellar growthbegins. Obviously, the chemical potential of a monomethe interior of a spherical micelle of aggregation numberN0

in the first term on the right-hand side (rhs) of the aboequation is given by the opposing forces model in Eq.(3),and the second term on the rhs contains the effect of thpulsion of the two spherical ends.

In the very same moment when micellar growth begthe radius of the spherical aggregates coincides with themonomer chain length,R = lc, and the spherical packinconstraint still holds[1], so vlc/a0 = 1/3, wherelc is thelength of the fully stretched monomer chain, anda0 is theoptimum area per monomer in the spherical aggregate. Tially, from the above constraint one can get the followrelationships

N0 = 4π

a0l2c ,

q = 2πzβ

a0l2c ,

(14)σ = zβ

a0,

z being the monomer head-group charge, andβ the ion-ization degree of the aggregates which marks the effeccharge of the micelles. As one can see in the above rethe charge density of the spherical end caps can be coered as approximately independent of the chain length, sits only dependence onlc is that of a0 which is currentlyconsidered to be not very strong (see Ref.[1]). Besides, thedependence on this parameter of both the charge of thecaps and the number of monomers in the spherical endthe aggregates is the same. Thus, the second term in thof Eq. (13) exhibits a linear dependence on the fully etended chain length,lc. This same behavior is exhibited bthe bulk term,gb, contributing toδ according to Eq.(3), sothe conclusion is that the chemical potential per monomethe neighborhood of the second cmc varies linearly withmonomer chain length in the model we actually proposefact, combining Eqs.(3), (13), and (14)one gets

(15)µ0N = γ a0 + α

a0+ gb − (zβ)2

2εa0lc.

As evident from the above equationgN = (zβ)2lc/2εa0.According to Tanford[5], at 298.15 K, the extended lengof the surfactant tail is generally a linear function ofnc

and can be evaluated using a contribution of 1.265 Åthe methylene group and 2.765 Å for the methyl groupthat lc = 1.501+ 1.265nc. Thus, for a typical surface area0 ∼ 70 Å2, a relative permittivity of about 20 (typical of liquid alkane micelles), monovalent head groups, andnc ∼ 12and β ∼ 0.3, one getsgN ≈ 4.5kBT . This quantity repre-sents approximately 30% of the�15kBT associated withtypical hydrophobic contributions to the free energy of mcellization per monomer in ionic amphiphiles, and it is aproximately coincident with that of typical energy barriein the molecular packing spectrum (of the order of 5kBT aspreviously noted), so it can explain the splitting of the sphical micelle in the neighborhood of the second cmc. Besicombining Eqs.(7) and (15)one can see that lnX2cmc isgiven by

kBT lnX2cmc≈ µ0N − µ0

1

(16)=(

γ a0 + α

a0

)+ (

gb − µ01

) − (zβ)2

2εa0lc.

The dependence of the surface contributions on the sutant chain length can be neglected sincelc enters their ex-pressions only througha0. On the other hand, the monomfree energyµ0

1 is made up of the sum of the contributioof a hydrophobic termghyd and the electrostatic self-energof the charged head group. However, the contribution oflatter to the free energy of micellization is usually neglec

218 A. González-Pérez et al. / Journal of Colloid and Interface Science 293 (2006) 213–221

then thord

n

iescmcantthence

re-nen-tantluted

af-dthe

at

the

tra-llarences asmsbe-reei-ellecmcithofthefirs

nsed

of

eenrmi-ighumtra-th

ity–the

ast-r-ndd.h-

ta oftheata

rdere oflal-

tivityith

ideed insolid

riticalr fit of

since its change upon transference of a monomer fromdispersed to the aggregated state is much smaller thahydrophobic free energy change. On the other hand, accing to Israelachvili and co-workers,gb − ghyd is a linearfunction of nc, as it is the contribution of the interactiobetween the end caps sincelc is a linear function ofnc ac-cording to the previously noted Tanford result. This implthat the chemical potential per monomer in the secondis a linear function of the number of atoms in the surfactchain, thus leading to a Stauff-Klevens-type behavior forsecond cmc in accordance with the experimental evideexposed in the following.

4. Experimental results and discussion

The molality dependence of the conductivity measuments of aqueous solutions of CnPyBr have been studied iorder to obtain the first and second critical micelle conctrations. An appropriate concentrated solution of surfacwas prepared for each compound and then stepwise diat constant temperature. Conductivities were recordedter thermal equilibration and the conductivity of distillewater was conveniently subtracted before representingconductivity of the amphiphile solutions against molality298.15 K.

The value of critical micelle concentration, cmc, andionization degree of the micelles,β, were estimated fromthe breaks on the specific conductivity, density, and ulsound velocity vs molality plots in the pre- and postmiceregimes, respectively. In both regimes, a linear dependwas obtained and the intercept of both linear sections wasumed to be the cmc according to the conventional Williamethod[32] and the ratio between the slopes above andlow the first cmc can be interpreted as the ionization degof the micelles,β. An additional break was found in the mcellar range and it was interpreted as a second critical micconcentration (second cmc). The value of the secondwas estimated from the intercept of the two, by analogy wits low concentration counterpart. The ionization degreethese different kinds of micelles has been estimated fromratio of the slopes above the second cmc and below thecmc and represented byβt . The results obtained by meaof the conventional Williams method were further confirmby means of the application of the Phillips method[33]. Inthis framework, the cmc is identified with the minimumthe second derivative of the experimental data:

(17)

(d3κ

dm3

)m=cmc

= 0.

The numerical method employed in this paper has bspecially designed to resolve the problem of the detenation of the cmc in weakly associating systems, with hdegrees of preaggregation levels and fluctuations in the nber of associated monomers in the critical micelle concention range. It combines Phillips’ definition of the cmc wi

e-

-

t

-

an analysis of the second derivative of the conductivconcentration data, assumed to be Gaussian, usingRunge-Kutta method and the Levenberg-Marquardt lesquares fitting algorithm[34]. This method allows the detemination of the cmc using a smaller number of points awith greater precision than the conventional Euler metho

Applying both the Williams and the Phillips-based higprecision numerical schemes to direct experimental dathe different techniques, we got compatible results forcmc values of the studied compounds. The conductivity dof CnPyBr, with n = 12, 14, 15, and 16, and CnPyCl, withn = 12, 14, and 16, have been analyzed at 298.15 K in oto find critical aggregation concentrations. As an examplthe experimental results, we show the conductivity vs moity data at 25◦C for C14PyBr and C14PyCl in Fig. 3. In thisfigure one can observe a marked change in the conductrend at a fairly well-defined concentration identified wthe first cmc (0.00277 and 0.00418 mol kg−1 for C14PyBr

Fig. 3. Electrical conductivity of tetradecylpyridinium bromide and chlorat 298.15 K. The arrows denote the second cmc obtained as describthe text. The open circles represent direct experimental data and thelines represent the corresponding linear fits below and above the first cassociation concentration and the dotted line corresponds to the lineathe data obtained above the second cmc.

A. González-Pérez et al. / Journal of Colloid and Interface Science 293 (2006) 213–221 219

a

Table 1First cmc, second cmc,β , andβt as a function of the number of carbon atoms on the alkyl chain length for CnPyBr and CnPyCl

CnPyBr cmc(mol kg−1)

Second cmc(mol kg−1)

β βt cmc(mol kg−1)

cmc(mol kg−1)

12 0.01117 0.05159 0.265 0.2799 0.012[35] 0.00117[36]14 0.00277 0.01673 0.253 0.247 0.0027[35] 0.00282[36]15 0.00136 0.00715 0.275 0.247 0.0013[35] –16 7.2502E–4a 0.00392a 0.312a 0.2594a 0.00064[35] 0.000815[36]

CnPyCl12 0.0162 0.03296 0.491 0.446 0.018[37]14 0.00418 0.00759 0.422 0.391 0.0035[38]16 9.7095E–4 0.00258 0.417 0.368

The numbers in brackets of the values taken from literature represent the corresponding references.a Values reported at 35◦C.

Fig. 4. Electrical conductivity (A), density (B), and ultrasound velocity (C) measurements for dodecylpyridinium bromide at 298.15 K. The symbolsre as inFig. 3.

esedre-

e inse-ther

nedideelf-s, theen-he-

and C14PyCl, respectively). The currently reported valufor the first critical concentrations of these pyridinium-basamphiphiles are in good agreement with those previouslyported by other authors for the first cmc, as one can seTable 1. In this table, the measured cmc’s for the wholeries of compounds treated in this paper are listed, togewith the ionization coefficientsβ, andβt for CnPyBr andCnPyCl, together with other previously cited values[35–38].

One must recall that, despite being treated as a well-deficoncentration, cmc is in fact a concentration range inswhich the changes associated with the formation of sassociation nano- and mesostructures take place. Thumeasurement of the cmc by means of different experimtal techniques usually leads to fairly different values. Tvalues reported for C16PyBr were obtained from the conductivity/molality isotherm at 35◦C in order to work above

220 A. González-Pérez et al. / Journal of Colloid and Interface Science 293 (2006) 213–221

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the Krafft temperature of this compound. Since small dferences in the cmc values at 25 and 35◦C are expected tooccur, we have opted to compare the value of the cmC16PYBr at 35◦C with the cmc’s of the other amphiphilein the alkylpyridinium series at 25◦C in order to test the influence of the chain length for one additional compoundthe alkylpyridinium series.

The arrows inFig. 3 indicate the second cmc’s obtainby means of a combination of the Williams and Phillimethods presented above. At concentrations about anof magnitude above the first cmc, a second structural chis registered in the conductivity/concentration curves thaflect the sphere to a rod micellar transition. However, itmuch more subtle change than its low concentration coterpart, specially for short-chain amphiphiles like C12PyBrand C12PyCl. In order to clarify the value reported for thsecond cmc for these compounds we carried out diffemeasurements of sound velocity against molality at anpropriate range at 25◦C. The results of specific conductivity vs molality, density, and ultrasound velocity vs molalmeasurements are shown inFig. 4 for C12PyBr. As seen inthis representation, the second cmc is marked by a slighcrease of the density below the line for spherical micethat corresponds to a relatively sharp increase in ultrasovelocity in solution, suggesting an structural modificationthe aggregates in the bulk. The values determined appWilliams method to the density and ultrasound velocity teniques are in fair agreement with those calculated applWilliams and Phillips methods to the conductivity resuconfirming the value of the second cmc found for C12PyBrby means of these latter measurements.

Finally, Fig. 5shows the measured values of lnXcmc andlnX2cmc against the number of carbon atoms in the alchain length for CnPyBr and CnPyCl. Despite the existencof a distribution of values of the second cmc associatedthe different experimental techniques, we opted to usevalues of the second cmc obtained by conductimetry sthese are widely recognized to be the most precise. Ascan see, a remarkable linear behavior is found for bothfirst and second cmc’s for the systems of interest in thisper, in reasonable agreement with previously reported re[22,23] and with the theoretical predictions in Section3 ofthis paper.

5. Conclusions

In the present paper we have presented molecular thedynamic considerations to support a Stauff-Klevens-typehavior of the second cmc in the aggregation profile of ioamphiphiles. For this purpose, a model for the interacbetween the spherical end caps of cylindrical micellesbeen introduced and the dependence of this function innumber of carbon atoms in the amphiphile chain analyTaking into account the electrostatic repulsion betweenend caps a linear behavior has been found in the limit of z

r

-

-

Fig. 5. Logarithm of the critical micellization concentration in molar fration units against the number of carbon atoms in the amphiphile chan-alkylpyridinium halides. Solid squares and circles correspond to the msured first and second cmc’s, respectively. Solid lines are the correspolinear fits.

separation between the end caps corresponding to the bning of the micellar growth at the second cmc (sphere totransition). Besides, we have proven the considerabletive effect of the electrostatic repulsion between end cin the early stages of micellar growth as a mechanismovercome the energy barrier in the early stages of the mlar growth process. These predictions are in good agreewith previously reported results and with direct experimenmeasurements for a series of alkylpyridinium halides pformed in this work.

Acknowledgment

The authors thank the financial support of the XuntaGalicia (Project No. PGIDIT03PXIB20601PR).

A. González-Pérez et al. / Journal of Colloid and Interface Science 293 (2006) 213–221 221

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