uso de la teoría de lifshitz-slyosov-wagner para la predicción

8/19/2019 Uso de La Teoría de Lifshitz-slyosov-wagner Para La Predicción

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847DECEMBER 2015, VOL. 40 Nº 12 0378-1844/14/07/468-08 $ 3.00/0

KEYWORDS / LSW / Coalescence / Drop Size / Emulsion / Ostwald / Ripening /Recibido: 09/02/2015. Modicado: 26/10/2015. Aceptado: 28/10/2015.

Introduction

According to the Laplaceequation (Evans andWennerström, 1994), the inter -nal pressure of a drop of oil

suspended in water is directly propor tional to its interfacialtension (γ ), and inversely pro-

por tional to the radius of thedrop (R i). The excess pressurecauses a difference betweenthe chemical potential of themolecules of oil inside thedrop and the ones belonging toan unbounded bulk oil phase.This difference (Δµ) is equalto (Kabalnov, 1991):

Δµ =2γ V

M

R i

(1)

where VM: molar volume of theoil. According to Eq. 1, theexcess chemical potential is

pos itive, which means that a particle will be always dissol-ving when it is in contact withan aqueous phase. Furthermore,the derivative of the excesschemical potential is negative:

dΔµ

dR = -

2γ VM

R i

2 (2)

This means that an ensemble

of particles of different sizes

cannot be in equilibrium witheach other (Kabalnov, 2001).As a result, larger particlesgrow at the expense of smaller

particles by exchanging mole-cules of oil through the aque-

ous solution; a process referredto as Ostwald ripening. Thetheories of ripening start fromthe Kelvin equation:

C R i( )= C∞ exp

1

R i

⎛

⎝ ⎜⎞

⎠ ⎟ 2γ V

m

R T

⎛

⎝ ⎜

⎞

⎠ ⎟ =

C∞ exp αR

i

⎛

⎝ ⎜⎞

⎠ ⎟ ≈ C∞ 1+

αR

i

⎛

⎝ ⎜⎞

⎠ ⎟

(3)

where : universal gas cons-tant, T: absolute temperature,R i: radius of a drop, and C(∞):solubility of the oil moleculesin the presence of a planar oil/water (R i=∞). Thus, accordingto Eq. 3, the difference betwe-en the aqueous solubility of aslab of oil in contact with wa-ter (C(∞)) and the one of adrop of oil submerged in water(C(R i)), depends on the quo-tient between the capillarylength of the oil (a) defined as

α= 2γ VM

R T (4)

and the radius of the particle.According to the theory of

Lifthitz, Slyosov, and Wagner

the dispersion is equal to itsnumber average radius (R a):

R c= R

a=

1

NT

R k

k

∑ (7)

According to our simula-tions (Urbina-Villalba et al .,2009b, 2012; Urbina-Villalba,2014) the stationary regimeresults from two opposite pro-cesses: the exchange of oilmolecules alone, which leadsto a decrease of the averageradius, and the elimination of

pa rti cles (by dis solu tion orcoalescence), which favors theincrease of the average radius).The result is a saw-tooth vari-ation of R a superimposed tothe average slope predicted by

Eq. 6. This phenomena and itsrelation to the shape of thedrop size distribution was re-cently studied experimentally

by Nazarzadeh et al . (2013).Integration of Eq. 6 between

t0 and t leads to

R c

3t( ) =

VOR

t - t0

⎡⎣ ⎤⎦ + R c

3t0

( ) (8)

where t0 is the initial time ofthe measurements. Eq. 8 is

invariably used for the

SUMMARY

A novel theoretical expression for the estimation of the temporal variation of the drop size of an oil-in-water (o/w) na-noemulsion subject to occulation, coalescence and Ostwaldripening is proposed. It is based on the experimental evaluationof the mixed occulation-coalescence rate. The predictions of

the theory are contrasted with experimental results correspond-ing to a set of dodecane-in-water nanoemulsions stabilized with

sodium dodecylsulfate. A satisfactory agreement is found when-ever the size-dependence of the aggregates is conveniently rep-resented.

USE OF THE LIFSHITZ-SLYOSOV-WAGNER THEORY FOR THE

PREDICTION OF THE DROP SIZE OF AN EMULSION SUBJECT TO

FLOCCULATION AND COALESCENCE Kareem Rahn-Chique and German Urbina-Villalba

Kareem Rahn-Chique. Doctor ofSciences in Chemistry,Universidad Central deVenezuela (UCV). AssociateResearch Professional, Instituto

Venezolano de InvestigacionesCientíficas (IVIC). Venezuela.e-mail: [email protected]

German Urbina-Villalba. Doctorof Sciences in Chemistry,

UCV, Venezuela. Researcher,IVIC, Venezuela. Address:Laboratorio de Fisicoquímicade Coloides, Centro deEstudios Interdisciplinarios de

la Física, IVIC. CarreteraPanamericana, Km. 11, EstadoMiranda, Venezuela. e-mail:[email protected]

(LSW theory; Lifshitz andSlesov, 1959; Lifshitz andSlyosov, 1961; Wagner, 1961),the radius of a particle chan-ges with time according to

dR i

dt= Dm

R i

Δ - αR

i

⎛ ⎝ ⎜

⎞ ⎠ ⎟

(5)

where Dm: diffusion coefficientof the oil, and Δ: supersatura-tion of the solution nearby thedrop, Δ= C(R i)-C(∞). For eachvalue of Δ there exists a criti-cal radius (R c =a/Δ) at which a

pa rticle (or a monodi sp er seensamble of suspended parti-cles) is in equilibrium with thesolution (dR i/dt=0). Otherwise,the drop either grows (R i>R c,dR i/dt>0) or dissolves (R i<R c,

dR i/dt<0). Eventually, a ‘statio-nary regime’ is attained, cha-racterized by a self-similardrop size distribution. At thistime, the ripening rate (VOR )can be quantified in terms of alinear increase of the cube ofthe critical radius (R c) of thecolloid as a function of time:

VOR

= dR c

3dt =

4αDm

C ∞( ) 9 (6)

Finsy (2004) demonstrated

that the critical radius (R c) of



848 DECEMBER 2015, VOL. 40 Nº 12

USO DE LA TEORÍA DE LIFSHITZ-SLYOSOV-WAGNER PARA LA PREDICCIÓNDEL TAMAÑO DE GOTA DE UNA EMULSIÓN SUJETA A FLOCULACIÓN Y COALESCENCIAKareem Rahn-Chique y German Urbina-Villalba

RESUMEN

dicciones de la teoría se contrastan con resultados experimen-tales correspondientes a un conjunto de nanoemulsiones de do-decano en agua estabilizadas con dodecil sulfato de sodio. Seencuentra un acuerdo satisfactorio siempre que la dependenciade tamaño de los agregados sea convenientemente representada.

Se propone una nueva expresión para la estimación de la va-riación temporal del tamaño de gota de una nanoemulsión deaceite-en-agua (o/w) sujeta a oculación, coalescencia y madu-ración de Ostwald. La expresión está basada en la evaluaciónexperimental de la tasa de oculación y coalescencia. Las pre-

USO CORRECTO DA EXPRESSÃO LIFSHITZ-SLYOSOV-WAGNER PARA OVCÁLCULO DO RAIOMÉDIO DE UMA EMULSÃO ÓLEO-EM-ÁGUA (O /A) SUJEITA A FLOCULAÇÃO E COALESCÊNCIAKareem Rahn-Chique e German Urbina-Villalba

RESUMO

se contrastam com resultados experimentais correspondentes aum conjunto de nanoemulsões de dodecano/água estabilizadascom dodecil sulfato de sódio. Encontra-se um acordo satisfató-rio sempre que a dependência de tamanho dos agregados sejaconvenientemente representada.

Propõe-se uma nova expressão para a estimação da variaçãotemporal do tamanho de gota de uma nanoemulsão de óleo-em--água (o/a) sujeita a oculação , coalescência e maturação deOstwald. A expressão está baseada na avaliação experimentalda taxa de oculação e coalescência. As predições da teoria

experimental evaluation ofthe Ostwald ripening rate (seefor example Kabalnov et al .,1990; Taylor, 1995; Weisset al ., 1999; Izquierdo et al .,2002; Tadros et al ., 2004;Sole et al ., 2006, 2012;

Na zarzadeh et al ., 2013).However, the average radiusweighted in volume, area, in-tensity or mass is commonlyused for this purpose.Moreover, the vast majority

of the equipment available forstatic and dynamic light scat-tering measures the hydrody-namic radius of the ‘parti -cles’. Hence, they are unableto distinguish between theradius of a big drop and thehydrodynamic radius of anaggregate of drops. This cre-ates an uncertainty regardingthe effect of flocculation andcoalescence on the increaseof the average radius of theemulsion. If aggregation oc-curs, the slope dR

c

3/ dt will

not be the product of ripeningalone, and yet it will be usu-ally contrasted with Eq. 6.

Average Radius of anEmulsion Subject toFlocculation andCoalescence

Recently, a novel theoreticalexpression for the turbidity of

an emulsion (t) as a functionof time was deduced (Rahn-Chique et al ., 2012a, b, c).According to this equation,and in the absence of mixedaggregates (Rahn-Chiqueet al ., 2012a), the scattering oflight results from the originaldrops of the emulsion (prima-ry drops), aggregates of prima-ry drops, and larger spherical(secondary) drops:

τ= n1σ

1 +

nk

k=2

k max

∑ xaσ

k,a+ 1 -x

a( )σk,s⎡⎣

⎤⎦

(9)

where σk,a and σk,s representthe optical cross sections of anaggregate of size k and that ofa spherical drop of the samevolume. According to recentstudies (Mendoza et al ., 2015),xa is related to the non-globu-larity of the aggregates. Theterm inside the brackets standsfor the average cross section

of an aggregate of size k, andnk is the number density ofaggregates of size k existingin the dispersion at time t

(Smoluchowski, 1917):

nk t( )=

n0 k

FCn0t( )

k-1

1+k FCn0t( )

k+1 (10)

In Eq. 10 n0 is the totalnumber of aggregates at time

t= 0 n0= n

k t =0( )∑( ) and k FC

is an average aggregation-co-alescence rate. The values ofk FC and xa are obtained fittingEq. 9 to the experimental vari-ation of the turbidity as a func-tion of time.

If the average radius of anemulsion is solely the result offlocculation it can be calculatedas

R a = R FC =

nk

n

⎡

⎣⎢

⎤

⎦⎥k =1

k max

∑ R k (11)

where R k : average radius of anaggregate composed of k pri-mary particles, and n: totalnumber of aggregates per unitvolume. The term in parenthe-sis corresponds to the probabi-lity of occurrence of an aggre-gate of size k at a given time.The value of n is equal to

n(t)=k=1

k max

∑ nk t( ) =

n0

1+ k FCn0t

(12)

where k FC= k F in the absence ofcoalescence. If an emulsion issubject to both flocculation andcoalescence (FC), the averagesize of the aggregates resultsfrom the contributions of the‘true’ aggregates of the parti-cles, and the bigger drops resul-ting from coalescence.According to our simulations(Urbina-Villalba et al ., 2005,

2009a) Eq. 12 also applies tothe mixed process of floccula-tion and coalescence whenevercoalescence is much faster thanflocculation.

Every time Eq. 9 is valid,Eq. 11 could be recast in theform

R FC=

n1

n

⎡

⎣⎢

⎤

⎦⎥R 1 +

nk

n

⎡

⎣

⎢⎤

⎦

⎥k=2

k max

∑ xaR

k,a + 1−x

a( )R k,s⎡⎣

⎤⎦

where: R k,a: average radius ofthe aggregates with k primary

particles, and R k,s: radius of adrop resulting from the coales-

cence of k primary particles

R k,s

= k 3

R 0( ). Only in the

case in which R k,a≈R k,s Eq. 13

can be easily evaluated as

R FC=

n1

n

⎡

⎣⎢

⎤

⎦⎥R 1+

nk

n⎡⎣⎢ ⎤

⎦⎥

k=2

k max

∑ R k,a

xa+ 1 -x

a( )⎡⎣ ⎤⎦=

nk

n

⎡

⎣⎢

⎤

⎦⎥

k=1

k max

∑ R k

In general, the dependence ofR k,a on k is unknown due tothe variety of conformationsavailable for each aggregatesize. Alternative approximate



849DECEMBER 2015, VOL. 40 Nº 12

expressions can be formulated based on Eq. 9 and the connec-tion between the optical crosssection of a spherical particle (σ)and its radius (σ=Qs πR 2) aswhere Q is the scattering coe-fficient (Gregory, 2009). Thevalue of Q for a spherical par -

ticle, Qs, can be estimatedusing the program Mieplot(w w w . p h i l i p l a v e n . c o m ) .Alternatively, a k-dependenceof the average radius of an ag-gregate composed of k primarydrops of size R 0 can be extra-

polated based on the variationof the average radius of thecoalescing drops (R k,s=k 1/3R 0):

R FC

= n1R

0+

nk

k=2

k max

∑ xak

m+ 1-x

a( )k n⎡

⎣ ⎤

⎦R 0

(16)

where m and n are rationalnumbers. Hence: it is possibleto calculate the variation ofthe average radius of an emul-sion due to flocculation andcoalescence, if the value of k FC can be obtained by adjustingEq. 9 to the experimental data.

Number Average Radius ofan Emulsion Subject toFlocculation, Coalescenceand Ostwald Ripening

From the previous sectionsit is clear that once the valueof k FC has been evaluated bymeans of Eq. 9, equations 14,15 or 16 can be employed to

pred ict the effect of f loccula-tion and coalescence on thetemporal variation of the aver -age radius. Instead, Eq 8 can

be used to predict the sole in-fluence of Ostwald ripeningaccording to LSW. In fact, thechange of the average numberof drops (nd) due to Ostwald

ripening has the same mathe-matical structure of Eq. 12(Weers 1999; Urbina-Villalbaet al ., 2014):

nd =

n0

1+ k OR

n0t

(17)

where

k OR =16 παD

mC∞

27 ϕ (18)

where φ is the volume fractionof oil. Eq. 17 is similar toEq. 12 and, therefore, the evalu-ation of the rate constant bymeans of the change in thenumber of aggregates as a func-tion of time does not identifythe process of destabilization.

It is evident that floccula-tion, coalescence and ripeningare not independent processes.This situation is clearly illus-trated in the algorithm pro-

posed De Smet et al . (1997) tosimulate the process ofOstwald ripening. Startingfrom Fick’s law, using theKelvin’s equation, and assum-ing that the capillary length ofthe oil is substantially lowerthan the radii of the drops(a<<R i), these authors demon-

strated that the number ofmolecules of a drop of oil (i)suspended in water (m i),changes in time according to

dmi

dt= 4πD

mC∞ α

R i

t( )R

ct( )

-1

⎛

⎝ ⎜

⎞

⎠ ⎟

According to Eq. 19 thedrops can increase or decreasetheir volume depending on thequotient between their particu-lar radius and the critical radi-us of the emulsion. The

amount of molecules trans-ferred to/from a particle depends on the referred quotient.Hence, if the average (critical)radius of the emulsion increas-es due to a mechanism differ -ent from Ostwald ripening, the

process of ripening should also be affected. As previously not-ed, the simulations suggestthat the average radius of theemulsion only increases due toripening when the total num-

be r of dro ps de cr ea se s bycomplete dissolution. Hence, if

the average radius of theemulsion increases faster than

predicted by LSW, the numberof drops whose size (R i(t)) fall

be low the cr it ical ra dius attime t (R c(t)) increases.Consequently, the number ofdrops subject to dissolutioncan be substantially largerthan the one predicted byLSW. This should promote a

further increase in the averageradius, and a ripening rate thatmight be substantially largerthan predicted by Eq. 6.

In this paper we only con-sider the simplest scenario inwhich Oswald ripening occursindependently of flocculationand coalescence. In this case theaverage radius of the emulsionat time t should be equal to

R t( ) = R t0

( ) +

ΔR ( )OR

+ ΔR ( )FC

(20)

Here the symbol (ΔR) p standsfor the change of the averageradius due to process ‘p’:

ΔR ( ) p

= R p

t( ) - R p

t0( ) (21)

We had previously deducedan explicit expression for floc-culation and coalescence (FC)which allows the evaluation of

(ΔR)FC (Eqs. 11, and 14-16). Inthe case of Ostwald ripening(OR), Eq. 8 leads to

R OR

t( )= VOR

t - t0

⎡⎣ ⎤⎦+R OR

3t

0( )3 (22)

Notice that

R t0

( )= R FC

t0

( )=

R OR

t0

( )= R 0

(23)

hence

R t( )= R 0+

VOR t - t0⎡⎣ ⎤⎦+R 03

3

- R 0( )+ R

FCt( )- R

0( ) (24)

R t( ) = VOR

t - t0

⎡⎣ ⎤⎦+R 0

33 +

R FC

t( )- R 0

(25)

Eq. 25 has the correct limit(R 0) for t= t0. Moreover, it isthe most accurate expressionthat can be derived with theclassical t heories, assum ingthat FC and OR occur simulta-neously but independently.

An interesting expressionresults if Eq. 6 is integrated

between t-Δt and t, and it isassumed that the process offlocculation and coalescenceoccurs much faster than theone of Ostwald ripening

R OR

3t-Δt( )= R

FC

3t-Δt( )( ) :

R c

t( ) = VOR Δt+R

FC

3t-Δt( )3 (26)

Using Eq. 26 recursively inorder to reproduce the totaltime elapsed (see Eq. A.6 inAppendix), and assuming thatthe FC contribution is inde-

pendent from the one of OR:

R a

t( )= R FC

t( )+R c

t( ) (27)

one obtains a very good appro-ximation to all the experimen-tal data of the systems with the

stronger variation of the avera-ge radius. However, whileEq. 26 is a reasonable approxi-mation to the critical radius,Eq. 27 is incorrect, since itobliterates the fact that there isonly one average radius for thesystem (R a(t) = R c (t)), and the-refore, Eq. 26 already containsthe effects of both OR and FC(see Appendix).

Experimental Details

Materials

N- do deca ne (C12; Merck98%) was eluted twicethrough an alumina column

prior to use. Sodium dodecyl-sulfate (SDS; Merck) was re-crystallized from ethanol twotimes. Sodium chloride (NaCl;Merck 99.5%), and isopenta-nol (iP; Scharlau Chemie99%) were used as received.Millipore’s Simplicity waterwas employed (conductivity<1μS·cm-1 at 25ºC).

Dispersion preparation andcharacterization

An equilibrated system ofwater + liquid crystal + oilwith 10% wt SDS, 8% wt

NaCl, 6.5% wt iP and a weightfraction of oil f o= 0.80 (f o +f w= 1, f= 0.84) was suddenlydiluted with water at constantstirring until the final condi-tions were attained: 5% wtSDS, 3.3% wt iP, and f o= 0.38(ϕ= 0.44). This procedure al-

lowed the synthesis of mothernano-emulsions (MN) with anaverage radius of 72.5nm. Anappropriate aliquot of MN wasthen diluted with a suitableaqueous solution containingSDS, NaCl and iP (W/SDS/

NaCl/iP) in order to obtain sys-tems with f o= 0.35, 0.30, 0.25,0.20, 0.15 and 0.10, and saltconcentrations of 2 and 4% wt

R FC

= n1 σ

1 Q

sπ( )

1/2

+ nk

k=2

k max

∑ xa σ

k,a Q

a π( )

1/2

+ 1-xa( ) σk,s

Qsπ( )

1/2⎡⎣⎢

⎤⎦⎥

(15)




NaCl. To avoid the risk of per -turbation and/or contaminationof a unique sample, each ofthese emulsions was dividedinto 15 vials and stored at25°C. These vials were usedlater to study the evolution ofthe emulsion during 6h. Thewhole procedure was repeatedthrice with independently-pre-

pared mother nano-emulsions.

In all cases, the concentrationof SDS and iP was kept fixedat 5% wt and 3.3% wt, respec-tively. An additional set ofemulsions was prepared leavinga set of MN to evolve in timeuntil their average radiusreached R ~500nm (mothermacro-emulsion: MM). As be-fore, aliquots of MM were di-luted with W/SDS/NaCl/iPsolution in order to preparesystems with the same physico-chemical conditions of the na-no-emulsions, but with a higher

particle ra dius. The averagesize of the dispersions, andtheir drop size distribution(DSD) were measured using aLS 230 (Beckman-Coulter).Out of the total of 24 systems,6 representative emulsions wereselected to illustrate in thiswork the behavior of the dropsize (Table I).

Evaluation of k FC

At specific times an aliquot

was taken from one bottle ofthe set of vials correspondingto each system. The aliquotwas diluted with an aqueoussolution of SDS in order toreach a volume fraction of oilequal to f = 10 -4 ([SDS]= 8mM). Then, the value of theturbidity was measured using aTurner spectrophotometer

(Fisher Scientific) at l= 800nm(Rahn-Chique et al ., 2012a, c).This procedure was repeatedfor 6h. When the whole set ofmeasurements was complete,Eq. 9 was fitted to the experi-mental data of the turbidity(τ=230Abs) as a function oftime using Mathematica 8.0.1.0.

The optical cross sections ofthe aggregates used in Eq. 9

are valid whenever

CRGD

= 4πR/λ( ) mr

-1( )<<1 (28)

where λ: wavelength of lightin the liquid medium, and mr the relative refractive index

bet ween the particle and thesolvent. In the case of a dode-cane/water emulsion (m r =1.07), the values of CRGD cor -responding to radii R= 50, 60,70, 80, 100 and 500nm are0.07, 0.09, 0.10, 0.12, 0.15 and0.75, respectively. These valuesare reasonably low, guaran-teeing errors in the cross sec-tions lower than 10% withinthe Rayleigh-Gans-Debye ap-

proximation (Kerker, 1969).The values of k FC, t0,teo (theo-

retical starting time of the ag-gregation process), and xa weredirectly obtained from the fit-ting of Eq. 9 to the experimen-tal data. The theoretical valueof the radius is a parameter ofthe calculation which can besystematically varied (R teo =

R exp ±δ) to optimize the fitting,and guarantee a value of t0,teo

close to the experiment. Theerrors bars were calculatedusing the procedure describedin (Rahn-Chique et al ., 2012a,c). The effect of buoyancyduring these measurements isknown to be negligible (Cruz-Barrios et al ., 2014).

For the eval-uation of k FC ,aliquots of theconcent ra t edemulsion were

taken periodi-cally, and dilut-ed. Hence, thenumber of par -ticles per unitvolume used inthe determina-tion of k FC cor -responds to adilute (d) sys-tem (k FC,d) and

not to the actual, concentrated(c) emulsion (k FC,c) understudy. Simple arithmetic showsthat k FC,d has to be multiplied

by the dilution factor in orderto obtain

k FC,c.

If an aliquot of volume Vc isremoved from a concentratedemulsion with an aggregatedensity nc, and diluted withaqueous solution until reaching

a final volume Vd, the newaggregate density nd fulfillsthe relationship

nd

t( ) Vd

= nc

t( ) Vc

(29)

and this allows the definitionof a dilution factor f d :

nd

t( ) = nc

t( )V

c

Vd

⎛

⎝ ⎜⎞

⎠ ⎟ = n

ct( )

1

f d

(30)

f d= n

ct( ) n

dt( ) (31)

Using the expression ofSmoluchowski for a total num-

ber of aggregates at time t:

n0,d

1+k FC,d

n0, dt=

1

f d

⎛

⎝ ⎜⎞

⎠ ⎟ n

0,c

1+k FC,c

n0,ct

(32)

the following equality isobtained:

n0,d

1+k FC,c

n0, ct( ) =

1

f d

⎛ ⎝ ⎜

⎞ ⎠ ⎟ n0, c

1+k FC,d

n0,dt( )

(33)

but for any time t, and in par -

ticular for t= 0:

f d

= nc

0( ) nd

0( ) = n0, c

n0, d

(34)

and therefore,

k FC, c

n0, c

= k FC, d

n0, d

(35)

which means that the value of

k FC obtained from the turbiditymeasurements of the dilutesystems has to be divided bythe dilution factor in order toget the rate of flocculation andcoalescence of the actual con-centrated emulsions:

k FC,c

= k FC,d

n0, d

n0, c

= k FC,d

1

f d

(36)

Prediction of the averageradius

The variation of the averageradius of the emulsions as afunction of time was estimatedunder different premises:1) An order-of-magnitude esti-mation of the mixed floccula-tion/coalescence rate was obtai-ned using the experimental ra-

dius of the emulsions to compu-te an approximate number ofdrops n= f/V1 (where 4/3πR 3).A rough Smoluchoswki’s rate(k S) was calculated from thefitting of this data to Eq. 12supposing that the number ofaggregates was equal to thenumber of drops. Following, R a was computed using Eq. 11, as-

suming R k,a

≈ R k,s

= k 3

R 0.

2) Only Ostwald ripening oc-curs. The LSW radius was evaluated using Eq. 8. Thevalue of VOR (2.6×10-28m3·s-1) was estimated from Eqs. 4 and 6 using γ = 1.1mN·m -1, C(∞)= 5.4×10-9cm3·cm-3, VM =2.3×10-4m3·mol-1, Dm = 5.4×10-

10 m2·s-1 (Sakai et al ., 2002).

3) Only flocculation and coa-lescence occur. The proceduredescribed under k FC evaluationwas used to calculate k FC. Thevalue of R a was estimatedusing Eq. 14.

4) Flocculation, coalescence

and Ostwald ripening occur.The procedure described underk FC evaluation was used to cal-culate k FC. Following, Eq. 25was employed to estimate R a.

Results and Discussion

Depending on the chemicalnature (ionic or non-ionic) ofthe surfactant and the methodof preparation of the emulsion(low vs high energy), the

polyd ispe rs it y (measured interms of the coefficient of

variation) of a nanoemulsion(%CV= SD×100/R ex p, whereSD is the standard deviation ofthe drop size distribution, andR exp is the experimental aver -age radius) can be high. Whenthe present MN and MM sys-tems are diluted with a salinesolution, the polidispersityreaches values between 37 and77% in only a few minutes. If

TABLE ICOMPOSITION OF THE EMULSIONS

STUDIED. WEIGHT FRACTION OF OIL

(f o), SALT CONCENTRATION, INITIALRADIUS, AND PARTICLE DENSITY OF THE CONCENTRATED (n0,c) AND DILUTE (n0,d) SYSTEMS

ID f o % NaCl R 0,exp (nm) n0,c (m-3) n0,d (m-3)

A 0.35 2 74 2.5×1020 5.9×1016

B 0.30 4 156 2.3×1019 6.3×1015

C 0.25 2 165 1.6×1019 5.3×1015

D 0.25 4 204 8.6×1018 2.8×1015

E 0.30 2 505 6.8×1017 1.9×1014

F 0.25 2 560 4.2×1017 1.4×1014




this polydispersity is translatedinto the standard deviation ofthe radius, very large error

bars result (Fig ure 1). Theseerrors are conveniently obliter -ated in the scientific literatureeither by evaluating the repro-ducibility (precision) of themeasurements, or by referringto the polydispersity indexonly. Large error bars compli-

cate the comparison with thetheory. In the absence of a

bett er cr ite rion, we cont ra stour theoretical approximationswith the average value of themeasurements despite themagnitude of the error bars.

The theoretical approxima-tions used in this paper as-sume that the destabilization

processes occur simultaneously but independently of each oth-er. Moreover, only the process-es of flocculation, coalescenceand Ostwald ripening are in-cluded. The emulsions weregently shaken before each

measurement to include thelarger drops possibly locatedwithin the cream.

As previously discussed,the destabilization processesaffect each other. For exam-

ple, co ales ce nc e in f luen ce sOstwald ripening by constant-ly changing the average radi-us of the emulsion. Hence, theexperimental drop size is ex-

pected to show a more pro-nounced time dependencethan the one predicted by thetheoretical approximations.Conversely, if one of the pro-cesses does not occur in agiven sample, the experimen-tal curve should lie below thetheoretical approximationwhich takes it into account.Moreover, only Ostwald rip-ening is able to decrease theaverage size (whenever thecomplete dissolution of thesmaller particles does not oc-cur). Otherwise the averageradius always increases.

The variation of the dropsize is a sound measurementof the overall stability of thesystems. In all cases studiedthe drop size increased as afunction of time. A few sys-tems were selected to illustratethis dependence. Three typicalsituations were found:

1) The average radius of the

emulsion was substantially hi-gher than the predictions ofEq. 25 using Eq. 14 to appro-ximate R FC. This is illustrated

by systems A and B of TablesI and II (curve FCOR inFigures 1a, b).

2) Only flocculation and coa-lescence (FC) occur without asensible contribution of ripe-ning (systems C and D, Figures1c, d). In this case, the forecastof Ostwald ripening (OR) ac-cording to LSW is too low, and

the prediction of Eq. 25, FCORcoincides with the one of FC.Both FCOR and FC lie veryclose to the experimental data.

3) Only ripening takes place(systems E and F, Figures 1e,f). The curves corresponding toFC and FCOR largely surpassthe experimental data. Instead,the LSW prediction reproducesvery well the experimental me-asurements. In fact, only in thesystems in which the initialaverage radius is large (sho-

wing the slowest variation ofthe average radius with time)the prevalence of Ostwald ripe-ning is observed.

Within the period of timeconsidered, the standard con-tribution of Ostwald ripeningto the drop size is very small.Hence, the FCOR rates arevery similar to the ones of FC.If only OR occurs, any other

theoretical approximation tothe average radius should lieabove the experimental curve.If FC mostly contributes, it isdifficult to appraise the effectof Ostwald, since the rates ofFC and FCOR are similar.Accordingly, there are systemsin which only Ostwald ripen-ing might prevail (Figures 1e,f), and others in which FC

predominates (Figures 1c, d).However, Figures 1a and b

show an unexpected behavior:the experimental curve liesabove the theoretical approxi-mation for FCOR. This might

be the result of a synergisticeffect resulting from the mutu-al influence of the differentdestabilization phenomena. Itmight also be the result of thelack of accuracy of the theo-retical equations used. In par -ticular, it could be the conse-quence of our inability to esti-mate the average radius of anaggregate properly, since inthese evaluations it was as-sumed that R

k,a ≈ R

k,s= k

3R

0.

Figure 2 illustrates the ef -

fect of the initial time of mea-surement, the k-dependence ofthe cross section on the aver -age radius of the aggregates(Eq. 16), and the magnitude ofQ (Q= Qs= Qa) in Eq. 15, onthe predictions of Eq. 25 forsystems A and B. It is clearthat in these cases, the aggre-

gates formed appear to bemore extended than globular(R k,a = k mR 0, m ≈ 0.75). As aconsequence, the optical scat-tering coefficient that fits theexperimental data lies approx-imately between one tenth (Q=0.0025) and one third (Q=0.11) of the values predicted

by Mie theory (0.022 and 0.30,respectively) for spherical

Figure 1. Comparison between the experimental data and the predictionsof average radius as a function of time for systems A to F (Table I). Thevalues of the mixed flocculation/coalescence rates are given in Table II.S, OR, FC, and FCOR, stand for the predictions of the theory forSmoluchowski (Eq. 12), Ostwald ripening according to LSW (Eq. 8),

flocculation and coalescence (Eq. 14, with R k,a

≈ R k,s

= k 3

R 0), and all

processes combined (Eq. 25).

TABLE IIPARAMETERS OBTAINED FROM THE FITTING

OF Ec. 9 TO THE EXPERIMENTAL DATA. THE ERRORS LISTED RESULT FROM THE AVERAGE

OF THE ERRORS OF THREE INDEPENDENTMEASUREMENTS (RAHN-CHIQUE, 2012a)

ID R 0,teo (nm) xa k FC,d (m3·s-1) k FC,c (m3·s-1) t0 (s)

A 90 0.51 ±0.18 (9.1 ±1.6)×10-21 (2.3 ±0.4)×10-24 54.9B 141 0.15 ±0.16 (1.3 ±0.2)×10-18 (2.8 ±0.4)×10-22 18.6C 173 0.97 ±0.04 (4 ±6)×10-20 (1.5 ±2.1)×10-23 39.9D 181 0.33 ±0.04 (2.0 ±0.2)×10-18 (7.0 ±0.7)×10-22 14.5E 520 1.5 ±0.2 (2 ±12)×10-20 (4 ±30)×10-24 26.6F 424 1.6 ±0.2 (1.6 ±1.1)×10-20 (1.1 ±0.7)×10-23 8.1




drops with the same initialaverage radius. Thus, thediscrepancy between the pre-

dictions of Eq. 25 and theexperiment in the most un-stable (concentrated) systemscould possibly be adjudicat-ed to the inexact representa-tion of the average radius ofthe floccules.

It is remarkable that the useof Eq. 27 produces theoreticalcurves that lie just above theexperimental points in thosesystems in which the FCOR

pr ed ic tion fa il s to do so(Figures 1a and b). However,the FC contribution is counted

twice in this approximation,once through the estimation ofthe critical radius (Eq. 26) andonce –explicitly– in Eq. 27.The coincidence between the

predict ions of Eq. 27 and theexperiment indicates that ei-ther the critical radius envis-aged by Eq. 8 is too low, orR FC should be twice as high ascurrently estimated. If this is

correct, the discrepancies ob-served in Figure 1 will rathercorrespond to a synergistic

effect due to the simultaneousoccurrence of the processes ofdestabilization.

Conclusion

A novel methodology ableto predict the variation of theaverage radius of an emulsionas a function of time duringa period of –at least– 6h is

proposed. The procedure al-lows discriminating the rela-tive importance of the pro -cesses of flocculation/coales-

cence with respect toOstwald ripening on the tem-

pora l change of the ave rageradius of the emulsions. Asatisfactory ag reement be-tween theory and experimentis found in most cases, butthe reliability of the methoddepends on the soundness ofthe theoretical expressionused to represent the

variation of the average radi-us of the clusters with thenumber of individual drops.

REFERENCES

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Figure 2. Predictions of Eq. 25 for systems A and B of Figure 1 usingdifferent approximations for R FC: a.1) and b.1) changing the initial timeof measurement, a.2) and b.2) using different values of m for Eq. 16,and a.3) and b.3) adjusting the value of Q (Q= Q s= Qa) in Eq. 15.




Appendix A

If it is assumed that

dR

dt=

dR OR

dt

⎛

⎝ ⎜⎞

⎠ ⎟ +

dR FC

dt

⎛

⎝ ⎜⎞

⎠ ⎟ (A.1)

and

VOR

= dR 3

dt= 3R 2 dR

dt= 4αD

mC ∞( ) 9 (A.2)

the second term on the right hand side of Eq. A.1 can be integrated, but the first term leads to an unsolvable integral due to our lack of

knowledge on the variation of the radius with t:

dR

dt∫ = V

OR

dt

R 2

t( )∫ (A.3)

If on the other hand, it is supposed that

dR 3

dt=

dR OR

3

dt

⎛

⎝ ⎜

⎞

⎠ ⎟ +

dR FC

3

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

(A.4)

integration yields:

R 3

t( ) - R 3

t0

( )= VOR

t - t0

⎡⎣ ⎤⎦ + R FC

3t( )- R

FC

3t

0( ) (A.5)

and simplifying,

R 3

t( ) = VOR

t - t0

⎡⎣ ⎤⎦ + R FC

3t( ) (A.6)

This approximation leads to the correct limit for t= t0. Moreover, the

same ansatz leads to

3R 2 dR

dt= 3R

OR

2 dR

OR

dt

⎛

⎝ ⎜⎞

⎠ ⎟ + 3R

FC

2 dR

FC

dt

⎛

⎝ ⎜⎞

⎠ ⎟ (A.7)

or what is equivalent:

dR

dt=

R OR

2

R 2

dR OR

dt

⎛

⎝ ⎜⎞

⎠ ⎟ +

R FC

2

R 2

dR FC

dt

⎛

⎝ ⎜⎞

⎠ ⎟ (A.8)

Thus, the total first derivative of the radius is a linear combination

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uso de la teoría de lifshitz-slyosov-wagner para la predicción

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