Journal of Colloid and Interface Science 316 (2007) 984–995www.elsevier.com/locate/jcis

A new approach to the phenomena at the interfaces of finely dispersedsystems

Aleksandar M. Spasic a,∗, Mihailo P. Lazarevic b

a Department of Chemical Engineering, Institute for Technology of Nuclear and Other Mineral Raw Materials, 86 Franchet d’Esperey St.,P.O. Box 390, 11000 Belgrade, Serbia

b Faculty of Mechanical Engineering, University of Belgrade, 16 Kraljice Marije St., 11120 Belgrade 35, Serbia

Received 20 April 2007; accepted 23 July 2007

Available online 28 August 2007

Abstract

A new idea has been applied for the elucidation of the electron and momentum transfer phenomena, at both rigid and deformable interfaces, infinely (micro-, nano-, atto-) dispersed systems. The electroviscoelastic behavior of, e.g., liquid/liquid interfaces (emulsions and double emulsions),is based on three forms of “instabilities”; these are rigid, elastic, and plastic. The events are understood as interactions between the internal(immanent) and external (incident) periodical physical fields. Since the events at the interfaces of finely dispersed systems must be consideredat the molecular, atomic, and/or entities level it is inevitable to introduce the electron transfer phenomenon beside the classical heat, mass, andmomentum transfer phenomena commonly used in chemical engineering. Therefore, an entity can be defined as the smallest indivisible elementof matter that is related to the particular transfer phenomena. Hence, the entity can be either differential element of mass/demon, ion, phonon asquanta of acoustic energy, infon as quanta of information, photon, and electron. Three possible mathematical formalisms have been derived anddiscussed related to this physical formalism, i.e., to the developed theory of electroviscoelasticity. The first is the stretching tensor model, wherethe normal and tangential forces are considered, only in mathematical formalism, regardless of their origin (mechanical and/or electrical). Thesecond is the classical integer-order van der Pol derivative model. Finally, the third model comprises an effort to generalize the previous van derPol differential equations, both linear and nonlinear, where the ordinary time derivatives and integrals are replaced by corresponding fractional-order time derivatives and integrals of order p < 2 (p = n − δ, n = 1,2, δ � 1). In order to justify and corroborate a more general approach theobtained calculated results were compared to those experimentally measured using the representative liquid/liquid system.© 2007 Elsevier Inc. All rights reserved.

Keywords: Electroviscoelasticity; Electron transfer phenomena; Electrified liquid/liquid interfaces; Fractional-order model; Finely dispersed systems; Numericalevaluation; Predictor–corrector method; Asymptotic expansion; Emulsions; Double emulsions

1. Introduction

A number of theories that describe the behavior of liq-uid/liquid interfaces have been developed and applied to variousdispersed systems, e.g., Stokes, Reiner–Rivelin, Ercsen, Ein-stein, Smolichowski, and Kinch. A new, recently developedtheory of electroviscoelasticity describes the behavior of elec-trified liquid/liquid interfaces in finely dispersed systems, andis based on a new constitutive model of liquids [1]. The behav-

* Corresponding author. Fax: +381 11 3691 583.E-mail addresses: a.spasic@itnms.ac.yu (A.M. Spasic),

mlazarevic@mas.bg.ac.yu (M.P. Lazarevic).

0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2007.07.051

ior of, e.g., liquid/liquid interfaces, is based on three forms of“instabilities”; these are rigid, elastic, and plastic. The eventsare understood as interactions between the internal (immanent)and the external (incident) periodical physical fields. Thus,hydrodynamic and electrodynamic motions are considered inthe presence of both potential (elastic forces) and nonpotential(resistance forces) fields. The elastic forces are gravitational,buoyancy, and electrostatic/electrodynamic (Lorentz), and theresistance forces are continuum resistance/viscosity and electri-cal resistance/impedance. According to this model liquid/liquiddroplet or droplet–film structure is considered as a macro-scopic system with internal structure determined by the waythe molecules (ions) are tuned (structured) into the primary

A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995 985

components of a cluster configuration. All these microelementsof the primary structure can be considered as electromechan-ical oscillators assembled into groups, so that excitation byan external physical field may cause oscillations at the reso-nant/characteristic frequency of the system itself (coupling atthe characteristic frequency). Three possible mathematical for-malisms have been considered related to this physical formal-ism, i.e., to the theory of electroviscoelasticity; the first is thestretching tensor model, the second is the integer-order van derPol derivative model, and the third is the fractional-order vander Pol derivative model [2]. The third model presents an effortto generalize the previous van der Pol equation using fractionalintegro-differential operators (fractional calculus). Fractionalderivatives provide an excellent instrument for the descriptionof memory and hereditary properties of various materials andprocesses, including more degrees of freedom in the model [3].

Central and new parts of presentation are analytical solu-tions using the Riemann–Liouville and Caputo definitions offractional operators to obtain a nonhomogeneous solution ofa nonlinear/linearized equation of the fractional-order van derPol model. In particular the cases of low fractionality are dis-cussed [4], i.e., when the order of fractional derivative p slightlydeviates from an integer value n (p = n − δ, n = 1,2, δ � 1).Also, the nonhomogeneous solutions, using numerical approxi-mation of Caputo-type derivatives, are evaluated; i.e., a numeri-cal algorithm is proposed, using a predictor–corrector approachfor calculating corresponding nonhomogeneous solutions ofnonlinear differential equations of the van der Pol type.

2. General—classification of finely dispersed systems

2.1. Classification based on scales [1]—macro- andmicroscale

Classical chemical engineering has been intensively devel-oped during the last century. Theoretical backgrounds of mass,momentum, energy balances, and equilibrium states are com-monly used as well as chemical thermodynamics and kinetics.Physical and mathematical formalisms are related to heat, mass,and momentum transfer phenomena as well as on homogeneousand heterogeneous catalysis. Entire object models, continuummodels, and constrained continuum models are frequently usedfor the description of the events, and for equipment designing.The usual, principal equipment consist of reactors, tanks, andcolumns. Output is, generally, demonstrated as conventionalproducts, precision products, chemistry (solutions), and bio-chemistry.

2.2. Nanoscale

Molecular engineering nowadays still suffers substantial de-velopment. Beside heat, mass, and momentum transfer phe-nomena, commonly used in classical chemical engineering, itis necessary to introduce the electron transfer phenomenon. De-scription of the events is based on molecular mechanics, molec-ular orbits, and electrodynamics. Principal tools and equipmentare microreactors, membrane systems, microanalytical sensors,

Table 1A new classification of finely dispersed systems

DM, DP Gas Liquid Solid

Gas Plasma D Foam D Metal RLiquid Fluosol/fog D Emulsion D Vesicle DSolid Fluosol/smoke R Suspension R Dispersoide R

DP, dispersed phase; DM, dispersion medium; D, deformable interfaces; R,rigid interfaces.

and microelectronic devices. Output is, generally, demonstratedas molecules, chemistry (solutions), and biochemistry.

2.3. Attoscale

Attoengineering already more than a whole century is in per-manent and almost infinite development. The theoretical back-ground is related to surface physics and chemistry, quantumand wave mechanics, and quantum electrodynamics. Discreetmodels and constrained discreet models are convenient for de-scription of related events. Tools and equipment are nano- andattodispersions and beams (demons, ions, phonons, infons, pho-tons, and electrons), ultrathin films and membranes, fullerenesand bucky tubules, Langmuir–Blodget systems, molecular ma-chines, nanoelectronic devices, various beam generators. Out-put is, generally, demonstrated as finely dispersed particles(plasma, fluosol-fog, fluosol-smoke, foam, emulsion, suspen-sion, metal, vesicle, dispersoide).

2.4. Classification based on entities

An entity can be defined as the smallest indivisible elementof matter that is related to the particular transfer phenomena.Hence, the entity can be a differential element of mass/demon,an ion, a phonon as quantum of acoustic energy, an infon asquantum of information, a photon, or an electron, the first philo-sophical breakpoint [1,5].

A possible approach is proposed to the general formulationof the links between the basic characteristics, levels of approxi-mation and levels of abstraction related to the existence of finelydispersed systems/DS [1,5]. First, for simpler and easier phys-ical and mathematical modeling, it is convenient to introducethe terms: homoaggregate (phases in the same state of aggrega-tion/HOA) and heteroaggregate (phases in a more than one stateof aggregation/HEA). Now the matrix presentation of finelydispersed systems is given by

(1)[(DS)ij = (HOA)δij + (HEA)τ ij

],

where i and j refer to the particular finely dispersed sys-tem position; i.e., when i = j , then diagonal positions cor-respond to the homoaggregate finely dispersed systems (plas-mas, emulsions, and dispersoids, respectively), and when i �= j ,then tangential positions correspond to the heteroaggregate sys-tems (fluosols/fog, fluosols/smoke, foam, suspension, metal,and vesicle, respectively). Furthermore, the interfaces may bedeformable D and rigid R that are presented in Table 1.

Now, related to the levels of abstraction and approxima-tion it is possible to distinguish continuum models (the phases

986 A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995

considered as a continuum, i.e., without discontinuities in-side the entire phase, homogeneous, and isotropic) and dis-crete models (the phases considered according to the Born–Oppenheimer approximation: entities and nucleus/CTE mo-tions are considered separately). Continuum models are con-venient for microscale description (entire object models), e.g.,conventional products, precision products, chemistry/solutions,biochemistry, while discrete models are convenient for eithernanoscale description (molecular mechanics, molecular orbits),e.g., chemistry/solutions, biochemistry, molecular engineering,and or attoscale description (quantum electrodynamics), e.g.,molecular engineering, attoengineering. Since the interfaces infinely dispersed systems are very developed it is sensible toconsider the discrete models approach for description of relatedevents [5]. For easier understanding it is convenient to consult,among others, e.g., [6–19].

2.5. Hierarchy of entities

Fig. 1a shows a stereographic projection/mapping from aRiemann sphere, i.e., Fig. 1b shows a “hierarchy” of enti-ties, which must be understood as a limit value of the ra-tio u0/Z [withdrawn from magnetic Reynolds criteria (Rem =4πlGu0/c

2), where the conductivity G is expressed as a recip-rocal of viscosity/impedance Z (G = 1/Z), l is the path length

that an entity “overrides,” u0 is the characteristic velocity, andc is the velocity of light].

In general S corresponds to the slow system/superfluid andF corresponds to the fast system/superconductor; now, it ispossible to propose that all real dynamic systems are situ-ated between these limits. Also, it seems sensible to con-sider the further structure of entities, the second philosophi-cal breakpoint; e.g., the basic entity can be understood as anenergetic ellipsoid shown in Fig. 1c (based on both quantum-mechanical/microscopic and electrodynamic/macroscopic prin-ciples, Schrodinger, Lorentz, Dirac) [14,19].

3. Particular—liquid/liquid interfaces

3.1. Physical formalism–structure–mechanism–dynamics

If the liquid/liquid interface, e.g., emulsion or double emul-sion, is taken as a central and representative (i = j = 2) finelydispersed system it is possible to propose a theory of electrovis-coelasticity [1,2,5,6,20–24] based on a new constitutive modelof liquids. The principles of conservation of mass, momentum,energy, and charge are used to define the state of a real fluidsystem quantitatively. In addition to the conservation equations,which are insufficient to define the system uniquely, statementson the material behavior are also required; these statements are

Fig. 1. (a) The first philosophical breakpoint—a stereographic projection/mapping from a Riemann sphere. (b) The hierarchy of entities, correlation viscos-ity/impedance characteristic velocity. S, slow/demon (superfluid), and F, fast/electron (superconductor). (c) The second philosophical breakpoint—entity as anenergetic ellipsoid (at the same time macroscopic and microscopic); CTE, center of total energy, motions (translation, rotation, vibration, precession, angle rota-tion). From Ref. [1], p. 20, courtesy of CRC Press/Taylor & Francis.

A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995 987

termed constitutive relations, e.g., Fourier’s law, Fick’s law, andOhm’s law.

Now, the droplet or droplet–film structure is considered asa macroscopic system with internal structure determined bythe way the molecules (ions) are tuned (structured) into theprimary components of a cluster configuration. At first, dur-ing the droplet formation and/or destruction periods, it maybe assumed that the electrical analogue consists of a num-ber of serial equivalent circuits; after rearrangement or cou-pling at resonant/characteristic frequency a probable equiva-lent circuit is shown in Figs. 2a and 2b. Electrical analogueFig. 2a consists of passive elements (R, L, and C), and anactive element (emitter-coupled oscillator W). Further on theemitter-coupled oscillator is represented by the equivalent cir-cuit as shown in Fig. 2b. Fig. 2c shows the electrical (oscil-lators) and/or mechanical (structural volumes Vj ) analogueswhen they are coupled to each other, e.g., in the droplet. Hence,the droplet consists of a finite number of structural volumes orspaces/electromechanical oscillators (clusters) Vj , a finite num-ber of excluded surface volumes or interspaces Vs , and a finitenumber of excluded bulk volumes or interspaces Vb . Further-more, the interoscillator/cluster distance or internal separationSi represents the equilibrium of all forces involved (electrosta-tic, solvation, van der Waals, and steric) [1,2,5,6]. The externalseparation Se is introduced as a permitted distance when thedroplet is in interaction with any external periodical physicalfield. The rigid droplet boundary R presents a form of dropletinstability when all forces involved are in equilibrium. Never-theless, two-way disturbance spreading (propagation or trans-fer) of entities occurs, either by a tunneling mechanism (low en-ergy dissipation and occurrence probability) or by an inductionmechanism (medium or high energy dissipation and occurrenceprobability). The elastic droplet boundary E represents a formof droplet instability when equilibrium of all forces involved isdisturbed by the action of any external periodical physical field,but the droplet still exists as a dispersed phase. In the region be-tween the rigid and the elastic droplet boundaries, a reversibledisturbance spreading occurs with or without hysteresis. Afterthe elastic droplet boundary, the plastic form of droplet insta-bility takes place, electromechanical oscillators/clusters do notexist any more, and then the beams of entities or attoclustersappear. Attoclusters are the entities that appear in the attodis-persed systems. In this region one-way propagation of entitiesoccurs.

3.2. Mathematical formalisms—stretching tensor model

Now, using the presented propositions and electromechan-ical analogies, an approach to non-Newtonian behaviors andto electroviscoelasticity is introduced. When the droplet isstopped, e.g., at an inclined plate in the lamellar coallescer,and when the droplet is in the state of “forced” levitation, thenthe volume forces balance each other, and the surface forcesalone will be considered [1,2,6,25,26]. It is assumed that thesurface forces are, for the general case that includes the elec-troviscoelastic fluids, composed of interaction terms expressed

Fig. 2. A graphical interpretation of the structural model: (a) the electricaland mechanical analogue of the microcollective/cluster, (b) the equivalent cir-cuit for the emitter-coupled oscillator, (c) the macrocollective—a schematiccross-section of the droplet and its characteristics (Vj , structural vol-umes/clusters; Vs , excluded surface volumes/interspaces; Vb , excluded bulkvolumes/interspaces; Si , internal separation/interoscillator distance; Se , exter-nal separation; R, rigidity droplet boundary, elasticity droplet boundary). FromRef. [1], p. 374, courtesy of CRC Press/Taylor & Francis.

by

(2)dF is = T ij dAj ,

where the tensor T ij is given by

(3)T ij = −α0δij + α1δ

ij + α2ζij + α3ζ

ikζ

kj ,

and T ij is composed of two tensors, δij is the Kronecker sym-bol, and ζ ij is the stretching tensor. In the first and secondisotropic tensors α0 = α0(ρ,U) and α1(ρ,U) are the potentio-static pressures, where U represents hydrostatic or electrostaticpotential. Now, the general equilibrium condition may be de-rived from Eq. (3), and may be expressed by

(4)τ = −α0 + α1 + α2(σd) + α3(

σd)

2(α2 + α3),

where τ is the tangential stress [26]. Note that for dispersedsystems consisting of, or behaving as, Newtonian fluids, α3 =α3(ρ,U) is equal to zero.

The processes of formation/destruction of the droplet ordroplet–film structure are nonlinear. Furthermore, the viscositycoefficients μi (i = 0,1,2), where each consists of bulk, shear,and tensile components, when correlated to the tangential ten-

988 A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995

sions of mechanical origin τv can be written as

(5)τv = μ0du

dx+ μ1

d2u

dx2+ μ2

(du

dx

)2

,

where u is the velocity and x is one of the space coordinates.Using the electrical analog, the impedance coefficients Zi

(i = 0,1,2), where each consists of ohm, capacitive, and induc-tive components, will be correlated with the tangential tensionsof electrical origin τe, as

(6)τe = Z0dφe

dt+ Z1

d2φe

dt2+ Z2

(dφe

dt

)2

,

where φe is the electron flux density, and t is the time coordi-nate.

More detailed discussion about derivation of these equationscan be found in Refs. [6,9,25,26].

3.3. Van der Pol integer-order derivative model

Here, postulated assumptions for an electrical analogue[1,9,26,27] are:

1. The droplet is a macrosystem (collective of particles) con-sisting of structural elements that may be considered aselectromechanical oscillators.

2. Droplets as microcollectives undergo tuning or couplingprocesses, and so build the droplet as a macrocollective.

3. The external physical fields (temperature, ultrasonic, elec-tromagnetic, or any other periodic) cause the excitation ofa macrosystem through the excitation of microsystems atthe resonant/characteristic frequency, where elastic and/orplastic deformations may occur.

Hence, the study of the electromechanical oscillators isbased on electromechanical and electrodynamic principles. Atfirst, during the droplet formation it is possible that the serialanalog circuits are more probable, but later, as a consequenceof tuning and coupling processes, the parallel circuitry becomesdominant. Also, since the transfer of entities by tunneling (al-though with low energy dissipation) is much less probable itis sensible to consider the transfer of entities by induction(medium or high energy dissipation). Fig. 3 presents the resul-tant equivalent electrical circuit, rearranged under the influenceof an applied physical field, such as an antenna output circuit[1,2,26].

A nonlinear integral–differential equation of the van der Poltype represents the initial electromagnetic oscillation

(7)CdU

dt+

(U

R− αU

)+ γU3 + 1

L

∫U dt = 0,

where U is the overall potential difference at the junction pointof the spherical capacitor C and the plate, L is the inductancecaused by potential difference, and R is the ohm resistance (re-sistance of the energy transformation, electromagnetic into themechanical or damping resistance), t is the time; α and γ are

Fig. 3. The structural model of electroviscoelasticity: (a) the rigid droplet,(b) the incident periodical physical field, (c) the equivalent electrical circuitWd represents the emitter-coupled oscillator, Cd, Ld, and Rd are capacitive, in-ductive, and resistive elements of the equivalent electrical circuit, respectively.Subscript d is related to the particular diameter of the droplet under considera-tion. From Ref. [27], p. 854, courtesy of Marcel Dekker, Inc.

constants determining the linear and nonlinear parts of the char-acteristic current and potential curves. U0, the primary steady-state solution of this equation, is a sinusoid of frequency closeto ω0 = 1/(LC)0.5 and amplitude A0 = [(α − 1/R)/3γ /4]0.5.

The noise in this system, due to linear amplification of thesource noise (the electromagnetic force assumed to be the inci-dent external force, which initiates the mechanical disturbance),causes the oscillations of the “continuum” particle (moleculesurrounding the droplet or droplet–film structure), which canbe represented by the particular integral

(8)CdU

dt+

(1

R− α

)U + γU3 + 1

L

∫U dt = −2An cosωt,

where ω is the frequency of the incident oscillations.Finally, considering the droplet or droplet–film structure for-

mation, “breathing,” and/or destruction processes, and takinginto account all the noise frequency components, which are in-cluded in the driving force, the corresponding equation is givenby

(9)CdU

dt+

(1

R− α

)U + 1

L

∫U dt + γU3

= i(t) = 1

2π

∞∫−∞

exp(jωt)An(ω)dω,

where i(t) is the noise current and An(ω) is the spectral distri-bution of the noise current as a function of frequency [1,2].

In the case of nonlinear oscillators, however, the problemof determining of the noise output is complicated by the factthat the output is fed back into the system, thus modifying in

A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995 989

a complicated manner the effective noise input [1,2]. The noiseoutput appears as an induced anisotropic effect.

3.4. Van der Pol fractional-order derivative model—linearized

In an effort to generalize Eqs. (7) and (8) the ordinary timederivatives and integrals are now replaced with correspond-ing fractional-order time derivatives and integrals [1,2]. Here,the capacitive and inductive elements, using fractional-orderp ∈ (0,1), enable formation of the fractional differential equa-tion, i.e., more flexible or general model of liquid/liquid inter-faces behavior. Now, a differ-integral form using the Riemann–Liouville definition is given by

0Dpt

[U(t)

] = dpU

dtp= 1

Γ (1 − p)

d

dt

t∫0

U(τ)

(t − τ)pdτ,

0 < p < 1,

0D−pt

[U(t)

] = 1

Γ (p)

t∫0

U(τ)

(t − τ)1−pdτ,

(10)p > 0.

So, in that way one can obtain a linear fractional differentialequation with zero initial conditions as follows:

(11)C0Dpt

[U(t)

] +(

1

R− α

)U + 1

L 0D

−pt

[U(t)

] = i(t).

Further evaluation and calculation related to the solu-tions of the linearized, both homogeneous and nonhomoge-neous, fractional integro-differential equations are presented inAppendix A.

Now again, the initial electromagnetic oscillation is rep-resented by the differential equation, Eq. (7), and when thenonlinear terms are omitted and/or canceled the first step, i.e.,homogeneous solution, may be obtained using numerical cal-culation derived from the Grunwald definition [24], as seen inFig. 4.

The calculation has been done for the following parameters:α = 0.9995; U0 = 15 mV; p = 0.95; T = 0.001 s.

Further on, considering Eq. (8), nonnomogeneous solutionis obtained, and presented in Fig. 5.

The obtained result appears as a band because the input (cos)is of the fractional order, and output is in a damped oscillatorymode, of high frequencies!

The calculation has been done for the parameters [wherethe derivative of cos (ω0t ) is fractional too, of the order r]:α = 0.95; U0 = 15 mV; p = 0.95; T = 0.01 s; An = 0,05 nm;cos(ω0t) = dr/dt r (sinω0t).

3.5. Van der Pol fractional-order derivative model: Nonlinearcase

Nonlinear fractional differential equations have receivedrather less attention in the literature, partly because many ofthe model equations proposed have been linear. Here, a nonlin-ear homogeneous (i(t) = 0) differential equation (7) of the van

Fig. 4. Calculated solution of the linearized Eq. (8), homogeneous solution;α = 0.9995; U0 = 15 mV; p = 0.95; T = 0.001 s.

Fig. 5. Calculated solution of the linearized Eq. (9), nonhomogeneous so-lution; α = 0.95; U0 = 15 mV; p = 0.95; T = 0.01 s; An = 0.05 nm;cos(ω0t) = dr /dtr (sinω0t).

der Pol type which represents the droplet or droplet–film struc-ture formation is considered. Also, one can obtain an equivalentnonlinear problem applying differentiation of Eq. (7), such as

(12)Cd2U

dt2+

(1

R− α + 3γU2

)dU

dt+ 1

LU = 0.

In an effort to generalize the previous equation the ordinarytime derivative and integral are now replaced with correspond-ing fractional-order time derivative and integral of order p, ortaking into account the Caputo definition (see Appendix B) onecan obtain the fractional-order van der Pol equation:

c0D

2pt

U(t) = − 1

C

(1

R− α

)c0D

ptU(t)

(13)− 3γ

CU(t)2c

0DptU(t) − 1

CLU(t).

990 A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995

(a)

(b)

Fig. 6. Calculated solution of the nonlinear Eq. (B.21) homogeneous solution;(a) α = 8 × 10−7, U0 = 8 mV, p = 1.2, T = 0.004 s, γ = 3 × 10−3; (b) detail.

Further evaluation and calculation related to the solu-tions of the nonlinear, both homogeneous and nonhomoge-neous, fractional integro-differential equations are presented inAppendix B.

Hence, the initial electromagnetic oscillation is representedby the nonlinear fractional differential equation (11); homoge-neous solution may be obtained using numerical calculationof the Caputo derivative and PECE algorithm (Appendix B,(B.21)–(B.24)), as seen in Fig. 6. The calculation has been donefor the following parameters:

α = 8 × 10−7, U0 = 8 mV, p = 1.2,

T = 0.004 s, γ = 3 × 10−3.

Finally, the particular incident electromagnetic oscillationis represented by the nonlinear fractional differential equa-tion (13); a nonhomogeneous solution may be obtained using

(a)

(b)

Fig. 7. Calculated solution of the nonlinear Eq. (B.23) nonhomogeneous solu-tion; (a) α = 8 × 10−7, U0 = 8 mV, p = 1.2, T = 0.004 s, γ = 3 × 10−3,ω1 = 1.6 × 108 Hz, A = 0.1 nm; (b) detail.

numerical calculation of the Caputo derivative and the PECEalgorithm (Appendix B, (B.21)–(B.24)), as seen in Fig. 7. Thecalculation has been done for the following parameters:

α = 8 × 10−7, U0 = 8 mV, p = 1.2,

T = 0.004 s, γ = 3 × 10−3, ω1 = 1.6 × 108 Hz,

A = 1 × 10−10 m.

4. Experimental confirmation

The presented theoretical predictions including both phys-ical and mathematical formalisms have been experimentallycorroborated by means of electrical interfacial potential (EIP)measurements, and by means of nuclear magnetic resonancespectroscopy (NMR). The obtained experimental results werein fair agreement with the postulated theory [1,2,5,6,9,20–39].

A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995 991

Fig. 8. Measured variations of the electrical interfacial potential (EIP) with timefor the system phosphoric acid/D2EHPA–TOPO–kerosene at a spherical inter-face.

4.1. Description of the system

The heavy liquid was 5.6 M phosphoric acid, and the lightliquid was the synergistic mixture of 0.5 M di(2-ethylhexyl)phosphoric acid (D2EHPA) and 0.125 M tri-n-octylphosphineoxide (TOPO) in dearomatized kerosene [1,5,6,25,27].

4.2. EIP measurements

A method and apparatus were developed to voltametricallymonitor the change of electrical interfacial potential appearingduring the formation of the electric double layer (EDL) whilethe two-phase contact occurs [1,25,27]. Measurements of theEIP have been performed during the processes of formation andtransition of the electroviscoelastic sphere into the rigid sphereas shown in Figs. 8 and 9 [1,25,27].

Fig. 8 shows the measured change in EIP appearing duringthe introduction of the heavy-phase droplet into the light-phasecontinuum [27]. It can be seen in the figure that an interfacialjump potential peak appears during the formation of the EDL.Thereafter, the EIP decreases to a constant value. The loweringof the EIP in absolute value during the flow is due to the par-ticipation of cations that form the dense part of the EDL. Theanions are the counterions in the diffuse part. Redistribution ofthese anions and cations between the region close to the surfaceand the surface layers of the heavy-phase define the kinetics ofthe EIP [1,27].

Fig. 9 shows the measured spontaneous oscillations of theEIP during the “breathing” period. After the EIP jump, whichis in the millivolt-millisecond scale, the EIP continues to os-cillate in the parts of millivolt-minute range. Its damped oscil-latory mode is (probably) due to the hydrodynamic instabilityof the interfacial surface, as a consequence of the local gra-dients of interfacial tension and density in mutual saturationprocesses of liquids [1,27]. Another relevant interpretation ofthe EIP spontaneous oscillations may be expressed as follows:the electroviscoelastic sphere undergoes transformation into therigid sphere. This transformation can be understood as a mem-ory storage process.

Fig. 9. Measured spontaneous oscillations of the EIP during the “breathing”period; transformation of the electroviscoelastic sphere into the rigid sphere.

Fig. 10. 31P NMR spectrum of D2EHPA–TOPO–kerosene sample (phosphoricacid standard solution, sweep width of 15,000 Hz); the peak at 7 ppm corre-sponds to D2EHPA, and that at 62 ppm corresponds to TOPO.

4.3. 31P NMR measurements

In order to determine the resonant (characteristic) frequencya NMR spectrometer was used as a reactor for the energeticanalysis. The impedance Z at the resonant frequency ω0 isequal to the resistance R. The resonant frequency of the electro-mechanical oscillator can be considered as some characteristicfrequency within the vibrorotational spectrum of the molecu-lar complex that builds the droplet film structure [27]. Fig. 10shows 31P NMR spectra of the examined droplet–film structure.

All experiments were performed and all spectra acquired ona Bruker MSL 400 spectrometer with a 9.395 T magnet and at a31P frequency of 161.924 MHz. The transmitter was set at res-onance frequency with the phosphoric acid standard solution,and a sweep width of 15,000 Hz was employed. The swept re-gion corresponded to the range between −10 and 90 ppm.

5. Conclusion

Recently a number of authors have demonstrated applica-tions of fractional calculus in various fields, such as physics,chemistry, and engineering [3,4,40]; also a few works dealingwith the application of this mathematical tool in signal process-

992 A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995

ing, anomalous diffusion [41], and control theory [3] were pub-lished. Concerning a continuous-time modeling the fractionalcalculus may be of great interest, e.g., in a problems related tothe viscoelasticity [42,43], electrochemical processes [44,45],polymer chemistry, heat, mass, momentum, and electron trans-fer phenomena [1,2,46]. The main reason for the success of thetheory in these cases is that these new fractional-order mod-els are more accurate than integer-order models; i.e., there aremore degrees of freedom in the fractional-order model. Further-more, fractional derivatives provide an excellent instrument forthe description of a memory and hereditary properties of vari-ous materials and processes due to the existence of a “memory”term in a model. This memory term ensures the history andits impact to the present and future. Fractional-order modelshave an unlimited memory compared to the integer-order mod-els that have a limited memory. Based on these arguments it wasnecessary to apply a fractional-order calculus approach to therecently developed theory of electroviscoelasticity [1]. The ob-tained results were in much better agreement with experimentthan the results obtained using previously developed models,where a classical approach was applied. Hence, both the phys-ical and the mathematical formalisms were generalized, and aconsequence was more accurate and deeper elucidation of thephenomena at the interfaces of finely dispersed systems.

Acknowledgments

This work was supported by the Ministry of Science andEnvironmental Protection of Republic of Serbia as a Fundamen-tal Research Project 142034. Fruitful consultations, comments,discussions, and suggestions have been held and receivedfrom Professors J. Jaric, M. Plavsic, M. Mitrovic, D.N. Krstic,J. Prochazka, H.-J. Bart, A.V. Delgado, H. Oshima, A.T. Hub-bard, J.P. Hsu, S. Tseng, and A. Saboni!

Appendix A

Using the Laplace transform leads to

G(s) = U(s)

i(s)= 1

Csp + 1/Ls−p + (1/R − α)

(A.1)= sp

Cs2p + (1/R − α)sp + 1/L

or

G(s) = spG3(s), G3(s) = 1

as2p + bsp + c,

(A.2)a = C, b = (1/R − α), c = 1/L.

Further, G3(s) in the form

G3(s) = 1

as2p + bsp + c= 1

c

cs−p

as2p−p + b

1

1 + cs−p

as2p−p+b

(A.3)= 1

c

∞∑k=0

(−1)k(

c

a

)k+1s−pk−p

(s2p−p + b/a)k+1.

The term-by-term inversion, based on the general expansiontheorem for the Laplace transform [28], and using the two-parameter function of the Mittag–Leffler type which is defined

by the series expansion:

(A.4)Eα,β(z) =∞∑

k=0

zk

Γ (αk + β)(α,β > 0).

The Mittag–Leffler function is a generalization of exponen-tial function ez and the exponential function is a particular caseof the Mittag–Leffler function. Here is the relationship given in

E1,1(z) =∞∑

k=0

zk

Γ (k + 1)=

∞∑k=0

zk

k! = ez.

(A.5)E1,m(z) = 1

zm−1

{ez −

m−2∑k=0

zk

k!

}.

The one-parameter function of the Mittag–Leffler type is

(A.6)Eα(z) =∞∑

k=0

zk

Γ (αk + 1).

The Laplace transform of the Mittag–Leffler function in twoparameters is

(A.7)

∞∫0

e−t tβ−1Eα,β(ztα)dt = 1

1 − z(|z| < 1).

And a pair of Laplace transforms of the function tαk+β−1 ×E

(k)α,β(±ztα) is

∞∫0

e−pt tαk+β−1E(k)α,β(±atα)dt = k!pα−β

(pα ∓ a)k+1

(A.8)(Re(p) > |a|1/α

).

Finally, Eq. (8) is

G3(t) = 1

a

∞∑k=0

(−1)k

k!(

c

a

)k

t2p(k+1)−1E(k)2p−p,2p+pk

×(

−b

at2p−p

)

= 1

a

∞∑k=0

(−1)k

k!(

c

a

)k

t2p(k+1)−1E(k)p,2p+pk

(−b

atp

),

(A.9)

where Eλ,μ(z) is the Mittag–Leffler function in two parametersand its kth derivative is given by

E(k)λ,μ(t) = dk

dtkEλ,μ(t) =

∞∑j=0

(j + k)!t jj !Γ (λj + λk + μ)

,

(A.10)k = 0,1,2, . . . .

Inverse Laplace transform of G(s) is fractional Green’sfunction,

(A.11)G(t) = 0Dpt

[G3(t)

] = dpG3(t)

dtp,

A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995 993

where the fractional derivative of G3(t), (9) is evaluated withthe help of

∞∫0

e−st0D

pt f (t)dt = spF (s) −

n−1∑k=0

sk0D

p−k−1t f (t)|t=0.

(A.12)

The solution of the initial-value problem for the ordinaryfractional linear differential equation with constant coefficientsusing Green’s function is presented. The consideration takeninto account for the given differential equation under homoge-neous initial conditions is bk = 0, k = 1,2, . . . , n:

aDσnt y(t) +

n−1∑k=1

pk(t)aDσn−kt y(t) + pn(t)y(t) = f (t),

(A.13)σk =k∑

j=1

αj , 0 � αj � 1, j = 1,2, . . . , n.

The analytical solution of the given problem takes the form

(A.14)y(t) =t∫

0

G(t, τ )f (τ )dτ,

where G(t, τ ) is known as Green’s function of Eq. (11); for thefractional differential equations with constant coefficients thisfunction is

(A.15)G(t, τ ) = G(t − τ)

and in such case the Green function can be obtained by theLaplace transform method. Therefore, the solution of linearfractional differential equations with constant coefficients re-duces to finding the fractional Green function. Finally, for in-homogeneous initial conditions solution has the form:

y(t) =n∑

k=1

bkψk(t) +t∫

0

G(t − τ)f (τ)dτ,

(A.16)bk = [0D

σk−1t y(t)

]t=0.

Lastly, this allows an explicit representation of the solution:

(A.17)U(t) =t∫

0

G(t − τ)i(τ )dτ.

Appendix B

Now, one may convert previous equations into an equivalentsystem of equations of low order. Let

(B.1)x1(t) = U(t), x2(t) = c0D

ptU(t), p ∈ Q,

where c0D

ptU(t) denotes Caputo definition of fractional deriva-

tive which is given by

c0D

pt

[U(t)

] = dpU

dtp= 1

Γ (n − p)

t∫U(n)(τ )

(t − τ)n−p−1dτ,

0

(B.2)n − 1 < p < n, U(n)(τ ) = dnU/dτn.

In that way, introducing vector x(t) = (x1, x2)T one can get in

condensed form,

c0D

ptx(t) =

[0 1

−1/LC −(1/R − α)/C

]{x1(t)

x2(t)

}

(B.3)+[

0 00 −3γ x2

1(t)/C

]{x1(t)

x2(t)

},

or

(B.4)c0D

ptx(t) = Ax(t) + B

(x1(t)

)x(t).

It is easily observed that the previous case is one general casefor this nonlinear problem which can be obtained in the form

(B.5)c0D

ptx(t) = f

(t, x(t)

)subject to the initial conditions

(B.6)xk(0) ={

x(kp)

0 , kp ∈ N00, else.

Hence, the problem of finding a unique continuous solutionof (B.5) and (B.6) is omitted (for more details, see [28,29]).Now a case of low fractionality is considered [4]; i.e., the orderof fractional derivative p slightly deviates from an integer valuen (p = n − δ, n = 1,2, δ � 1).

c0D

2−δt

U(t) = − 1

C

(1

R− α

)c0D

1−δt

U(t)

(B.7)− 3γ

CU(t)2c

0D1−δt

U(t) − 1

CLU(t).

Introducing Caputo fractional derivative in the form

0Dpt f (t) = f (n)(0)tn−p

Γ (n − p + 1)

+ 1

Γ (n − p + 1)

t∫0

f (n+1)(τ )(t − τ)n−p dτ,

(B.8)n − 1 < p � n,

one can obtain

0Dn−δt f (t) = f (n)(0)tδ

Γ (1 + δ)

+ 1

Γ (1 + δ)

t∫0

f (n+1)(τ )(t − τ)δ dτ,

(B.9)n = 1,2.

First, considering the case δt � 1 and using the expansion

(B.10)1

Γ (1 + ε)(t − τ)ε = 1

Γ (1 + ε)eε ln(t−τ)

it yields the fractional derivative in a form of the perturbationto the second derivative and first derivative,

0D2−δt f (t) = f (2)(t) + δ

(f (2)(0) ln t + γf (2)(t)

994 A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995

(B.11)+t∫

0

f (3)(τ ) ln(t − τ)dτ

)+ · · · ,

0D1−δt f (t) = f (1)(t) + δ

(f (1)(0) ln t + γf (1)(t)

(B.12)+t∫

0

f (1)(τ ) ln(t − τ)dτ

)+ · · · ,

where τ < t , γ = 0.5772156649 . . . , is a constant. As a result,when the limit δ → 0 the correct expansion is

(B.13)limδ→0

0Dn−δt f (t) = f (n)(t), n = 1,2,

and one can obtain

(B.14)

∞∫0

e−st C0 D

p

tf (t)dt = spF (s) −

n−1∑k=0

sp−k−1f (k)(0).

Now, the asymptotic representation of fractional derivative, us-ing the Laplace transform and its inversion, expansion δt 1,can be formally obtained as follows.

0Dpt f (t) = −f (0)t−p

Γ (1 − p)+ −f ′(0)t1−p

Γ (2 − p)

+∞∑

k=0

F (k)(0)t−p−k−1

Γ (−p − k)k! ,

(B.15)1 < p < 2 (t → ∞)

0Dpt f (t) = −f (0)t−p

Γ (1 − p)+

∞∑k=0

F (k)(0)t−p−k−1

Γ (−p − k)k! ,

(B.16)0 < p < 1 (t → ∞),

where F(s) is a Laplace transform of f (t). Taking p = n − δ ityields

0D2−δt f (t) ≈ −f (0)t−2+δ

Γ (−1 + δ)− f ′(0)t−1+δ

Γ (δ)

(B.17)≈ δf (0)t−2+δ − δf ′(0)t−1+δ (δt 1),

(B.18)0D1−δt f (t) ≈ −f (0)t−1+δ

Γ (−1 + δ)≈ δf (0)t−1+δ (δt 1).

After changing the fractional derivatives with (B.17) and(B.18), Eq. (B.7) becomes

−3γ

CU2(t)

(δU(0)t−1+δ

) − 1

CLU(t)

= δU(0)t−2+δ − δU ′(0)t−1+δ + 1

C

(1

R− α

)(δU(0)t−1+δ

).

(B.19)

In condensed form

(B.20)a1(t)U2(t) + a2(t)U(t) = a0(t),

where

a1(t) = −3γ (δU(0)t−1+δ

), a2 = − 1

,

C LC

a0(t) = δt−1+δ(U(0)t−1 − U ′(0)

)+ 1

C

(1

R− α

)(δU(0)t−1+δ

).

Solving this quadratic equation one can see the asymptoticbehavior U(t) = U(δ, t−1+δ, t−2+δ) when t → ∞.

B.1. Numerical method—predictor–corrector algorithm

A numerical algorithm that solves Caputo-type fractionaldifferential equations is listed below, (0 < p < 2,p �= 1). So,one may introduce the uniformly distributed grid points for thetime interval [0, t = X]tk = kh, k = 0,1,2, . . . ,N : h = X/N .

B.1.1. One stepPredict using the quadrature weights (derived from a product

rectangular rule) [28],

xPh (N) =

[p]∑m=0

Xm

l! x(m)

0+ +[

hp

Γ (1 + p)

]N−1∑k=0

bk,Nf (tk, xk),

(B.21)

where

(B.22)bk,N = (N − k)p − (N − k − 1)p.

B.1.2. Two stepEvaluate f

(X,xP

N = xPh (N)

).

B.1.3. Three stepCorrect with following expression where the quadrature

weights (derived from the product trapezoidal rule)

xh(N) =[p]∑

m=0

Xm

l! x(m)

0+ +[

hp

Γ (2 + p)

]

(B.23)×(

N−1∑k=0

ck,Nf (tk, xk) + cN,Nf(X,xP

N

)),

where

ck,N =

⎧⎪⎨⎪⎩

(1 + p)Np − N1+p + (N − 1)1+p, k = 0,

(N − k + 1)1+p − 2(N − k)1+p + (N − k − 1)1+p,

0 < k < N,

1, k = N,(B.24)

B.1.4. Four stepReevaluate f (X,xN = xh(N)) and saving it as f (tN , xN),

which is then used in the next integration step. Finally, it is saidto be of the PECE (Predict, Evaluate, Correct, Evaluate) typebecause, in this implementation start run by calculating the pre-dictor in Eq. (B.22), evaluate f (X,xP

N = xPh (N)), and then use

this to calculate the corrector in Eq. (B.24), and finally evaluatef (tN , xN).

A.M. Spasic, M.P. Lazarevic / Journal of Colloid and Interface Science 316 (2007) 984–995 995

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