Institute for Theoretical Physics

Department of Physics and Astronomy

Spontaneous Emulsification

of Pickering Emulsions without additives

Aikaterini Ioannidou

Supervision by dr. Rene van Roij

August 26, 2010

Contents

1 Introduction 51.1 Colloids, Emulsions and Interfaces . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Pickering Emulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theoretical Background 82.1 Classical Description of the System . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Statistical Description of the System . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 The Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.4 Binary Mixture of Ideal Gases . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Adsorption - Langmuir Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Another Langmuir Isotherm . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Examples of Langmuir Isotherms . . . . . . . . . . . . . . . . . . . . . 14

2.4 Poisson-Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Numerical Solution of PB Equation . . . . . . . . . . . . . . . . . . . 18

2.5 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.1 Examples of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Spontaneous Emulsification 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Theory of Spontaneous Emulsification . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Chemical Reactions and Adsorption . . . . . . . . . . . . . . . . . . . 233.2.2 Responsive Oil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 Methanol Buffering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Numerical Results 284.1 Minimization of the Responsive Oil Model . . . . . . . . . . . . . . . . . . . . 284.2 Considering Only One Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Buffering Methanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4 Comparing with Recent Experiments . . . . . . . . . . . . . . . . . . . . . . . 324.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A Theoretical and Numerical Supplement 39A.1 Dimensionless Form of PB Equation . . . . . . . . . . . . . . . . . . . . . . . 39A.2 Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1

A.3 Definitions of Physical quantities . . . . . . . . . . . . . . . . . . . . . . . . . 41

2

List of Figures

2.1 Langmuir isotherms of the type (2.25) and (2.36), as a function of the surfacepotential φ(a), for several parameter values. . . . . . . . . . . . . . . . . . . . 15

2.2 Electrostatic potential and densities plots, resulting by numerically solving thePB equation of a charged sphere of radius a = 100λB, surrounded by pointions in a cell of radius R = 2a, for screening parameter κa = 3. . . . . . . . . 17

3.1 Area over volume term plotted versus the droplet radius a, for different oilvolume fractions. The minimum occurs at a → ∞ when the system phaseseparate. For small oil volume fractions x, the function drops to zero quickerthan for larger x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Schematic representation of the responsive oil theoretical model. One oildroplet is placed at the center of the cell. The ions produced by the reac-tions (3.2) and (3.3) are given by the plus and minus circles, while the bluecircles represent the neutral methanol molecules. . . . . . . . . . . . . . . . . 23

4.1 Grand potential and charge densities plotted as function of the droplet size.The plots correspond to the parameters of table 4.1. . . . . . . . . . . . . . . 29

4.2 The grand potential is plotted as a function of the droplet size, using the pa-rameters of table 4.1, for different oil volume fractions x and salt concentrationsρs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 The contributions to the grand potential are plotted as function of the dropletsize, considering both reactions and only the reaction (3.2), using the parame-ters of table 4.1. The contributing terms are plotted with dashed lines and thetotal grand potential by solid green line. The black dashed line is the interfaceterm. The red dashed line is the contribution of the ionic electrostatic interac-tion. The blue dashed line is the adsorption term of reaction (3.2), as given byEq. (3.6) and the purple dashed line is the contribution of the reaction (3.3),as given by Eq. (3.8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 The grand potential is plotted with respect to the droplet radius, using theparameters of table (4.1), for several background methanol densities ρm. . . . 32

4.5 Experimental picture of Pickering emulsions without colloidal particles andauxiliary plot of the unknown equilibrium constants Kc, Ka in order to fitthe experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.6 Fitting to the experimental data [2] and the minimization of the grand poten-tial for system parameters of table (4.2). . . . . . . . . . . . . . . . . . . . . . 34

3

4.7 Density plot of the droplet radius with respect to oil volume fraction and saltconcentration ρs and grand potential versus radius a for different Ka values. . 36

4

Chapter 1

Introduction

The making of permanent emulsions has been studied for many years, as it is important forfoods, cosmetics, pharmaceutical preparations, and other products. The breaking of emul-sions is also important in crude oil operations to recover the oil. Several types of emulsionshave been researched for years, thus enhancing theoretical knowledge and industrial appli-cations. The process and the conditions of emulsification are not completely known for alltypes of emulsions, due to the complexity and diversity of the systems. At the present time,there is no theory that seems to apply universally to all emulsions. That is probably becausea number of factors contribute and the relative importance of each varies not only in differenttype emulsions but in the same type emulsion under different circumstances.

In this thesis, we investigate the formation conditions of surfactant free Pickering emul-sions and the parameters that affect their stability. We focus on the equilibrium properties ofPickering emulsions, which are governed by electrostatics, entropic effects and adsorption. Inthis analysis, no dynamic effects are included, although the equilibrium results can possiblydescribe several mesostates of emulsion evolution towards phase separation. Including in themodeling of the Pickering emulsions the attributing interactions, we reach interesting resultsin accordance with recent experimental research [2].

In the following paragraphs, the characteristics of emulsions are discussed. In chapter2, the main theoretical ideas required for the description of the investigated system arepresented. In chapter 3, using the concepts of chapter 2, the theory of the spontaneousformation of Pickering emulsion is analyzed. In chapter 4, the numerical results obtainedfrom the theory of emulsification are presented and discussed.

1.1 Colloids, Emulsions and Interfaces

A colloid or a colloidal dispersion is a mixture that consists of a continuous phase, called thedispersion medium, and a discontinuous phase, called the dispersed phase. The size of theobjects of the dispersed phase is around 5nm−103nm, such that they can perform Brownianmotion in the continuous phase. The objects can be solid particles or liquid droplets, althoughgenerally colloids can be made by any combination of gas, liquid, solid phase. A colloid oftwo or more immiscible liquids is called emulsion. A common example of two immiscibleliquids is oil and water, which can form an emulsion given an energy input, in the form ofshaking, beating or agitation. The provided energy is needed to separate the dispersed phase

5

into smaller globules, thereby increasing the surface area between the two liquids. In thisway more and smaller droplets of the dispersed medium into the continuous one can formed.The question is whether such a droplet state is stable.

The boundary surface between two different phases is called interface, in the case of emul-sions an interface forms between two different liquids. The role of the interface is significantin the case that the area per volume is big, as in colloidal dispersions. After the formationof the emulsion, the droplets of the dispersed phase should be sufficiently small that theywill stay suspended and that there should be a sufficiently viscous film in the interface toprevent coalescing. If the previous conditions are fulfilled, the emulsion may be stable overtime. Nevertheless, stability is not trivial and in most of the cases emulsifying agents areused to increase stability.

The emulsifiers are usually surface active substances, that are adsorbed by the interfaceand form a coating that prevents coalescence of the droplets, for example substance such asegg, honey, mustard. A detergent is also an emulsifier, which reacts with both the oil andwater phase, thus stabilizing the interface, as exemplified by the use of soap to remove fat.Sometimes the dispersed phase can act as its own emulsifier, that is the case of microemul-sions, where the dispersed liquid forms nano droplets. A common example is the addition ofwater in a strong alcoholic anise-based beverage, like ouzo or raki, where a milky oil-in-watermicroemulsion is formed . Another type of stabilized emulsions are the so called Pickeringemulsions, which are stabilized by solid particles, such as magnetite or silica, which are ad-sorbed by the oil-water interface. An example of Pickering emulsion is the homogenized milk,where proteins act as stabilizing particles.

1.2 Pickering Emulsions

Two interesting types of emulsions are microemulsions and Pickering emulsions. In bothcases, the emulsions can be formed spontaneously, by the addition of the ingredients with-out energy input. The stabilization of microemulsion is achieved by surfactants and co-surfactacts, whereas Pickering emulsions are stabilized by solid particles. Microemulsionsconsist of droplets of 5− 100nm, whereas Pickering emulsions form droplets of 10− 1000nm,therefore they are useful for different applications. A significant difference is that microemul-sions reach thermodynamic equilibrium, whereas Pickering emulsions are in general onlykinetically stable, thus the droplet state is metastable. In this thesis, we study two new typesof Pickering emulsions, which both form spontaneously. The first exhibits kinetic stabilityin the absence of additives, and the second is thermodynamically stable with the addition ofcolloidal particles.

The recent experiments [2] on Pickering emulsions involved a specific type of oil andan aqueous solution of a specific type of salt, brought together to spontaneously form anemulsion. After a day the emulsion indeed formed, consisting of mono-disperse droplets[2, 3]. This emulsion without additives was found to be stable for a couple of days, afterwhich the droplets start to coalesce leading to phase separation after a month [2]. If smallcolloidal particles are added, the formed emulsion is turned out to be stable for more thanhalf a year. In addition, if such emulsions of different droplet sizes were mixed, an emulsionwith an intermediate size of droplets was formed, suggesting thermodynamic stability [2, 3].

The theoretical model we develop, includes the essential phenomena that take place in

6

such a Pickering emulsion which we deem to be the electrostatic interaction among ions andcharged oil droplets, adsorption effects in the oil-water interface and the ratio of interfacearea over volume. To model the emulsion, we consider a homogeneous system in equilibrium,which is composed of charged droplets of radius a and ions of positive and negative charge.To treat the interactions between the ions and the charged spheres, as well as these amongthe droplets an the ions, each droplet is assumed to be isolated within a charge neutral cell ofradius R. Each droplet has a charged surface, due to the adsorption of ions into the oil-waterspherical interface. If small colloidal particles are added to the system, they are also adsorbedin the oil-water interface, that effect is not included in this thesis. The colloidal particles areneutral when added, but they can obtain charge by adsorption of positive and negative ions.The adsorption effect is described by the Langmuir isotherm. According to this mechanism,the surface of the adsorbent is uniform and consist of adsorption sites, where only one ioncan bind. The adsorption of small colloidal particles in the oil-water interface decreases thesurface area and favors the formation of droplets of the size of some decades nanometers.

In the next chapters, the theory we developed to describe the system is presented, as wellas the results. We hope the result of this effort may contribute to an understating of suchcomplex systems.

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Chapter 2

Theoretical Background

In this section, several theories are mentioned in order to describe the system of interestin this work. In some cases, a meticulous description is presented, as it is useful for thefollowing calculations. Other times several notions of classical mechanics, statistical physicsand electrodynamics are used without further definition.

2.1 Classical Description of the System

The dynamical state of a classical system of N identical particles is described by a point inthe 6N dimensional phase space Γ, of positions and conjugate momenta. The evolution ofthe system in the phase space is driven by the Hamiltonian H of the system according toLiouville’s theorem. A short description of Liouville’s theorem using Poisson brackets is thefollowing

∂ps∂t

= −ps, H, (2.1)

where ps(r,p) represents the probability density of a microscopic state Γ = (r,p) of thesystem, where r = (r1, r2, ..., rN) and p = (p1,p2, ...,pN) are the position and momentumconfiguration.

The equilibrium state of the system does not depend on time, thus from Liouville’s the-orem, ps, H = 0. Therefore, the probability density of a microscopic state is a constantof motion and may be expressed as a function of other conserved quantities, such as the en-ergy. In the next section, the microscopic statistical description is based on the distributionfunction as a function of the constant of the system such as energy and chemical potential.

2.2 Statistical Description of the System

The system is composed by a large number of particles, of the order of the Avogadro number(≈ 1024), so it can also be treated as a statistical ensemble. Among the usual statisticalensembles, the most interesting ones to describle physical systems at a given temperature arethe canonical and the grand canonical ensemble.

Thermodynamically, the microcanonical ensemble has constant volume V , number ofparticles N and total energy E. The canonical ensemble does not have constant total energy,but it exchanges energy with a heat bath that fixes the temperature T . The grand canonical

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ensemble in addition can also exchange particles with a reservoir of particles, that fixes thechemical potential µ.

Considering the phase space of the system and the distribution function, the three ensem-bles can be described as follows. The distribution function of the microcanonical ensemble isnon zero at the (6N − 1) dimensional surface where the total Hamiltonian equals the totalenergy of the system. At the canonical ensemble, there is a set of such surfaces with energiesE ±∆E, where E is the energy of the heat bath and ∆E the small fluctuations of the sys-tem. In the case of the grand canonical ensemble, the number of particles changes, so we canimagine the same set of surfaces as for the canonical case but the dimension of the surfacesfluctuates (6(N ±∆N)− 1).

By the normalization of the distribution functions, the partition functions arise. The par-tition functions indicate the way that the probabilities are partitioned among the differentmicrostates that result to the same system energy. The importance of partition functionsis the connection between the microscopic and macroscopic theory through the estimationof ensemble average of quantities such as the energy, the number of particles and the en-tropy. Following, we define the partition functions for the canonical and the grand canonicalensemble and connect them to macroscopic quantities.

2.2.1 Canonical Ensemble

The canonical partition function for a system of N identical point particles, in a volume Vat temperature T is defined as

Z(N,V, T ) =1

N !h3N

∫d3r d3p e−βH(r,p), (2.2)

where p = (p1,p2, ...,pN ) and r = (r1, r2, ..., rN ) are the momentum and position configura-tion, respectively and h is Planck constant. The canonical probability that the system is ata microstate of energy H is given by

pc =1

N !h3N

e−βH

Z. (2.3)

Therefore, the average energy is given by

〈E〉 =∫d3r d3p pcH =

1N !h3NZ

∫d3r d3p e−βHH

= − 1N !h3NZ

∂

∂β

∫d3r d3p e−βH = −∂ lnZ

∂β. (2.4)

The entropy is also derived as a function of the partition function

S = −kB∫d3r d3p pc ln pc =

−kBN !h3NZ

∫d3r d3p e−βH ln

e−βH

Z

= −∂ lnZ∂β

+ kB lnZ = kB(logZ + β〈E〉). (2.5)

The Helmholtz free energy F is a thermodynamic potential depending on (V,N, T ), which incase of canonical ensemble are constant. It is also defined from the above relation by

F = 〈E〉 − TS = −kBT lnZ. (2.6)

9

More thermodynamic quantities can be calculated from the free energy such as the pressureand the chemical potential

p = −(∂F

∂V

)N,T

µ =(∂F

∂N

)V,T

. (2.7)

2.2.2 Grand Canonical Ensemble

The grand partition function for a system of chemical potential µ, volume V and at temper-ature T is given as a weighted sum of partition functions with different number of particles

Ξ(µ, V, T ) =∞∑N=0

eβµNZ(N,V, T ). (2.8)

The grand canonical probability that the system of N particles is at a microstate (r,p) ofenergy H(r,p) is

pgc =1

N !h3N

e−βH+βµN

Ξ. (2.9)

The average number of particles is

〈N〉 =∞∑N=0

N

∫d3r d3p pgc

=∞∑N=0

NeβµN

N !h3NΞ

∫d3r d3p e−βH

=∞∑N=0

NeβµNZ

Ξ=

1Ξ

∂

∂βµ

( ∞∑N=0

eβµNZ

)

=∂

∂βµln Ξ. (2.10)

The grand potential Ω is a thermodynamic potential depending on (µ,N, T ), which in caseof the grand canonical ensemble are constant. It is also defined as

Ω = 〈E〉 − TS − µN = F − µN = −kBT ln Ξ. (2.11)

2.2.3 The Ideal Gas

The ideal gas is composed by randomly moving point particle that do not interact with eachother. Even though it is a theoretical concept, it describes satisfactory dilute systems or itcan be used as first approximation. The free energy and the chemical potential of the idealgas is often used in the next chapter, so here we will derive those results.

The canonical partition function for N classical particles in volume V at temperature Twith the interaction potential Φ = 0 is given by Eq. (2.2)

Z =1

N !h3N

N∏i=1

∫d3pi e−β

p2i

2m =V N

N !Λ3N, (2.12)

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wherem is the mass of a particle and Λ is the thermal de Broglie wavelength Λ = h/√

2πmkBT .The free energy is

βF = − lnZ = N

(ln(NΛ3

V

)− 1), (2.13)

where the Stirling approximation lnN ! = N lnN −N was used. The chemical potential is

βµ = ln ρΛ3, (2.14)

where ρ = N/V is the particles density.

2.2.4 Binary Mixture of Ideal Gases

The previous description can be generalized for a system with different species of particles.The partition function for a different species reads

Z(Na, V, T ) =∏a

1Na!Λ3Na

a

∫d3ra e−βΦ(ra), (2.15)

where ra = (r1, r2, ..., rNa). The grand partition function is

Ξ(µa, V, T ) =∏a

∞∑Na=0

eβµaNaZ(Na, V, T )

=∏a

∞∑Na=0

eβµaNa

Na!Λ3Naa

∫d3ra e−βΦ(ra). (2.16)

In the case that all species are treated as ideal gas, the configurational integral of onparticle is just the volume. For a binary mixture of two species of ideal gases these resultsread

Z(Na, V, T ) =V N1

N1!Λ3N11

V N2

N2!Λ3N22

, (2.17)

Ξ(µa, V, T ) =∞∑

N1=0

1N1!

(V eβµ1

Λ31

)N1 ∞∑N2=0

1N2!

(V eβµ2

Λ32

)N2

= expV eβµ1

Λ31

expV eβµ2

Λ32

= expV ρ1Λ3

1

Λ31

expV ρ2Λ3

2

Λ32

= e〈N1〉e〈N2〉. (2.18)

2.3 Adsorption - Langmuir Isotherm

The adsorption of ionized molecules onto a surface is described by the Langmuir isotherm.To derive this, we assume an equilibrium among empty surface sites S, ionized particles P+

and sites filled with an ionized particle SP+. The adsorption reaction is thus written as

S + P+ ↔ SP+. (2.19)

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The number of available sites is called M , while the number of occupied sites is Z ≤ M . Inaddition, only one particle can bind to each site and the binding to one site is independentof the state (neutral or charged) of the other sites.

Under these circumstances, the partition function of the system with Z particles beingabsorbed to M surface sites is

Zp =M !

Z!(M − Z)!e−βFZ , (2.20)

where F is the free energy of an occupied site. The more realistic case, that the number ofoccupied sites fluctuates, is described by the grand partition function, which from Eq. (2.8)is a linear combination of partition functions of different number of particles, such that

Ξ =M∑Z=0

eβµZZp =M∑Z=0

(M

Z

)eβ(µ−F )Z =

(1 + eβ(µ−F )

)M, (2.21)

where µ is the chemical potential.The grand potential Ω of the system of adsorbed particlesis

βΩ = − ln Ξ = −M ln(

1 + eβ(µ−F )). (2.22)

Therefore, the average value of the number of adsorbed ionized molecules is

〈Z〉 = −∂Ω∂µ

=M

1 + eβ(F−µ), (2.23)

which is also called the Langmuir isotherm. For fixed number of sites M and free energy F ,the Langmuir isotherm predicts that 〈Z〉 increases from 〈Z〉 ≈ 0 as µ F , to 〈Z〉 ≈ M forµ F .

The number of occupied sites can be associated with an experimentally measurable quan-tity, the equilibrium constant K of the reaction (2.19)

K =[S][P+][SP+]

=M − 〈Z〉〈Z〉

ρsurf , (2.24)

where [P+] = ρsurf is the local density of ionized particles at the surface. Solving thisequation with respect to 〈Z〉,

〈Z〉 =M

1 + Kρsurf

, (2.25)

yields the Langmuir isotherm in terms of the density of ions P+ near the surface ρsurf , thenumber of available sites M and the equilibrium constant K of the reaction (2.19). Comparingthe two expressions, Eq. (2.23) with Eq. (2.25), we find a relation between the free energyand the equilibrium constant

K

ρsurf= eβ(F−µ) ⇔ K =

eβF

Λ3, (2.26)

where the chemical potential of the ideal gas βµ = ln ρΛ3 (2.14) is used. The free energy isgiven by the partition function Zp of a single particle in the external potential

e−βF = Zp =1

Λ3

∫d3r e−(φ(r)+φsurf ) ⇔ e−βFb =

1Λ3

∫d3r e−φ(r), (2.27)

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where φ(r) is the dimensionless interaction potential between an ion and a site and φsurf thesurface potential. In addition, the binding energy is defined by Fb = F−φsurf and consideredto be in the order of 1 − 10 kBT . Therefore, at room temperature the equilibrium constantis K > 10−4M .

2.3.1 Another Langmuir Isotherm

Another type of Langmuir isotherm is derived for the following equilibrium among emptysurface sites S, ionized particles P−, filled ionized particle sites SP− and neutral moleculesA

S + P− ↔ SP− + A. (2.28)

We consider a two-state surface model of Z1 ions that bind to available sites S, releasing Z1

neutral molecules are released. A background concentration ρm of molecules A is added tothe system, such that the total number of neutral A type molecules is Z2 = Z1 +ρmV , whereV is the volume available to the A molecules.

The partition function of this system, for M surface sites and volume V is given by

Zp =M !

Z1!(M − Z1)!e−βF1Z1

V Z2

Z2!Λ3(Z2)2

, (2.29)

where F1 = F + Ff with F the free energy of an occupied site and Ff the free energy of theformation of a particle A. The thermal length of the neutral molecules is indicated by Λ2.The grand partition function of the system reads

Ξ =M∑

Z1=0

∞∑Z2=0

eβ(µ1+µ2)Z1Zp

=M∑

Z1=0

(M

Z1

)eβ(µr−F1)Z1

V Z2

Z2!Λ3Z22

(2.30)

=(

1 + eβ((µr−F ))M V Z2

Z2!Λ3Z22

, (2.31)

where µ1, µ2 and µr = µ1+µ2 are the chemical potential of SP−, A and their sum respectively.Using the stirling approximation, the grand potential of the system is

βΩ = − ln Ξ = −M ln(

1 + eβ(µr−F1))

+ Z2 lnZ2Λ3

2

V− Z2. (2.32)

Therefore, the average value of the number of absorbed ion and produced neutral moleculesis

〈Z1〉 = − ∂Ω∂µr

=M

1 + eβ(F1−µr), (2.33)

where the partial derivative is with respect to the total chemical potential which characterizesthe reaction at the equilibrium, as

µS + µP− = µSP− + µA = µr. (2.34)

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The dimensionless equilibrium constant of this reaction is

K =[S][P−]

[SP−][A]=

(M − 〈Z1〉)ρsurf〈Z1〉 (ρmV + 〈Z1〉) /V

, (2.35)

which leads to equation

〈Z1〉 =12(−1 +

√1 + xi

)V (ρo + ρm) (2.36)

where xi =4Mρo

V (ρo + ρm)2and ρo =

ρsurfK

(2.37)

This Langmuir not only depends on the density of ions near the surface ρsurf , the number ofsites M and the equilibrium constant K, but also on the background concentration ρm of Amolecules and the volume V .

Comparing the two Langmuir expressions, Eq. (2.23) and (2.36), a relation for the freeenergy and the reaction chemical potential µr is found as

eβ(F1−µr) =12(1 +√

1 + xi)V (ρo + ρm)− 1, (2.38)

depending on the above parameters.

Limit Cases of Langmuir Isotherm Considering the expression (2.36) of the averagenumber of absorbed ions, which depend on the parameter xi, two limiting cases arise forxi 1 and xi 1.

The first case, xi 1, occurs for large droplet radius or small number of sites M . Thesquare root is approximated by Taylor expansion and the following expression is obtained

〈Z1〉 ≈M

1 + ρm

ρo

=M

1 + Kρm

ρsurf

, (2.39)

which is very similar to the Langmuir isotherm (2.25) for a simple adsorption reaction. Therelation suggests that the number of attached anions is tunable by the background concen-tration of A.

The second case, xi 1, occurs for the opposite cases, small droplet radius or large oilvolume fraction. The following expression is obtained

〈Z1〉 ≈√MρoV (2.40)

which, in contrast, is independent of the A concentration ρm. The physics in this case dependsonly on the surface terms.

2.3.2 Examples of Langmuir Isotherms

Firstly, we assume the number of adsorbed ions given by Langmuir isotherm (2.25). Wesubstitute the surface density by the Boltzmann distribution ρsurf = ρse

−φ(a), where ρs isthe bulk ion density and φ(a) is the electrostatic potential at a sphere surface with radius a.Then we have from Eq. (2.25)

〈Z〉 =M

1 + Kρseφ(a)

, (2.41)

14

-4 -2 0 2 4

0.0

0.5

1.0

1.5

2.0

FHaL

Z

KΡs=5& M=2

M=2

KΡs=5

FD

(a) Langmuir Isotherms of type (2.25). The labelindicates the parameters tuned compared to FD dis-tribution, where all are set to one.

-10 -5 0 5 100

1000

2000

3000

4000

5000

FHaL

Z1

ΡsK=0.4Ρm=1

ΡsK=1Ρm=1

ΡsK=0.4Ρm=0

ΡsK=1Ρm=0

(b) Langmuir Isotherms of type (2.36) as a functionof the surface potential Φ(a) for background densityρm = 0 and ρm = 1, and for different ρs/K ratios.

Figure 2.1: Langmuir isotherms of the type (2.25) and (2.36), as a function of the surfacepotential φ(a), for several parameter values.

This Langmuir isotherm is a Fermi-Dirac (FD) distribution, as also shown in Fig. (2.1(a))(red thick line) as a function of φ(a), where the values of parameters M , K, ρs are set to one.The rest three lines show the way the FD distribution changes by tuning the parameters. Thepurple, thick, and dashed dot line shows the change of the FD line by tuning the K/ρs = 5ratio. The black, thick, and dashed line shows the change of the FD line by tuning thenumber of available sites to M = 2. The blue, thin, and dashed dot line shows the change ofFD line by tuning both K/ρs = 5 and M = 2. The higher point depends on M , Z →M forφ(a)→∞. The purple and blue line, with bigger K/ρs ratio than one, decrease quicker.

Secondly, we assume the Langmuir isotherm (2.36) and apply the same substitutions asabove at equation (2.41), therefore it reads

〈Z1〉 =12(−1 +

√1 + xi

)V (ρo + ρm) (2.42)

where xi =4Mρo

V (ρo + ρm)2and ρo =

ρse−φ(a)

K(2.43)

In Fig. (2.1(b)), the number of adsorbed ions is plotted as a function of the surface potential,with background density ρm = 0 and ρm = 1, for two different ρs/K ratios. The dashed lineshave ρs/K = 0.4 ratio, whereas the solid lines have ρs/K = 1 ratio. Adding backgrounddensity ρm = 1, shifts both lines to the right.

15

2.4 Poisson-Boltzmann Equation

We consider a charged sphere of radius a and point ions that screen the charged sphere. Thiscan be described by the Poisson equation

∇2φ(r) = −4πεeQ(r), (2.44)

where φ(r) is the electrostatic potential and eQ(r), is the total charge distribution, such that

Q(r) = ρ(r) + q(r) = ρ+(r)− ρ−(r) + σδ(r − a), (2.45)

where ρ(r) is the average ion charge distribution and q(r) is the external charge distribution,in this case the charged sphere . The average ion charge distribution is given by eρ(r)

ρ(r) = ρ+(r)− ρ−(r)= ρs(g+(r)− g−(r))= ρs(exp(−βqφ(r))− exp(βqφ(r)))= −2ρS sinh(βqφ(r)), (2.46)

where ρ+ and ρ− is the density distribution of positive and negative ions, ρs is the bulkdensity of ions far from the sphere, g+ and g− is the radial distribution function of positiveand negative ions.

The radial distribution function g±(r) describes the way the cation (+) and anion (−)density vary as a function of the distance from a specific particle. For dilute liquids, thecorrelations are due to the potential energy, therefore using the Boltzmann law, g(r) reads

g±(r) = e∓βφ(r). (2.47)

In the case of ideal gas particles, g(r) = 1 for all r. We also mention that g(r →∞) = 1, asφ(r →∞) = 0.

Substituting the charge density to the Poisson equation, the Poisson-Boltzmann (PB)equation is obtained

∇2φ(r) =8πeρsε

sinh(βeφ(r))− 4πeσε

δ(r − a). (2.48)

Using the dimensionless potential φ(r) = βeφ(r), the PB equation takes the form

∇2φ(r) = κ2 sinh(φ(r))− 4πλBσδ(r − a), (2.49)

where λB is the Bjerrum length and κ−1 is the Debye length. The Bjerrum length is definedas the distance at which the electrostatic interaction between two unit charges equals kBT ,

λB =e2

εkBT(2.50)

The Debye length denotes the typical distance at which the ionic charge screens out thesurface charge of a hard sphere,

κ−1 =√

8πλBρs (2.51)

16

where 2ρs is the total ion density.To solve the PB equation, appropriate boundary conditions should be imposed. The

boundary conditions of a charged sphere of radius a are

dφ(r)dr

∣∣∣∣r→∞

= 0 anddφ(r)dr

∣∣∣∣r=a

= −4πλBσ, (2.52)

where λB is the Bjerrum length, σ = Z/(4πa2) the surface charge.The PB equation, as a nonlinear equation, can be linearized and solved analytically.

The linearized solution of PB equation gives satisfactory quantitively results for low chargedsurfaces. For higher charged surface, a numerical solution of the nonlinear PB equationis appropriate. In the following subsection, the dimensionless PB equation (A.1) is solvednumerically. The dimensionless form of PB equation follows, with x = r/a, as

d2φ(x)dx2

+2x

dφ(x)dx

= κ2a2 sinhφ(x), (2.53)

where the dimensionless boundary conditions (2.52) are

dφ(x)dx

∣∣∣∣x=R/a

= 0 anddφ(x)dx

∣∣∣∣x=1

= −4πλBaσ = −λBaZ. (2.54)

1.0 1.2 1.4 1.6 1.8 2.0-2.5

-2.0

-1.5

-1.0

-0.5

0.0

ra

j

(a) The electrostatic potential φ(r) surrounding acharged colloid of radius a = 100λB in a cell of ra-dius R = 2a, in a medium of screening length κa = 3.

1.0 1.2 1.4 1.6 1.8 2.00

2

4

6

8

ra

Ρ

Ρs

Ρ+-Ρ-

Ρ-

Ρ+

(b) The anion ρ−, cation ρ+ and total ρ+ − ρ−densities surrounding a charged colloid of radiusa = 100λB in a cell of radius R = 2a, in a mediumof screening length κa = 3.

Figure 2.2: Electrostatic potential and densities plots, resulting by numerically solving thePB equation of a charged sphere of radius a = 100λB, surrounded by point ions in a cell ofradius R = 2a, for screening parameter κa = 3.

17

2.4.1 Numerical Solution of PB Equation

We consider a charged sphere of radius a in the center of a charged neutral cell of radius R.Using the boundary conditions for a charged sphere (2.54)

dφ(x)dx

∣∣∣∣r=R

= 0 anddφ(x)dx

∣∣∣∣x=1

= −λBaZ (2.55)

and the dimensionless form of PB equation (2.53), the PB equation can be solved numericallyon a radial grid

xi = a+ iR− aN

with i = 0, ..., N , (2.56)

between x = a and x = R with R the cell radius. The numerical method used to solve thePB equation is the Euler method, which can be found at appendix A.2.

Some illustrative results are presented in Fig. (2.2), using as cell radius R/a = 2, sphereradius a/λB = 100, charge Z = −1000 and screening length κa = 3. The slope of the elec-trostatic potential at the cell boundary is zero, it indeed satisfies condition (2.55), indicatingthe charge neutrality of the cell. The positive and negative ionic densities tend to the bulkdensity at the boundary of the cell and the charge density tends to zero, supporting thecharge neutrality of the cell. The ion densities follow from the Boltzmann distribution (2.47)

ρ±(r) = ρse∓φ(r) (2.57)

2.5 Density Functional Theory

The Density Functional Theory (DFT) is used to determine the structure of a many bodysystem using functionals such as the spatially depended electron density. The two mainprinciples of DFT are that

• the equilibrium properties of the system are uniquely determined by a particle den-sity, therefore the many body system of 3N spatial coordinates is reduced to 3 spatialcoordinates through the use of functionals

• defining and minimizing an energy functional for the system, a particle density arises,which is the equilibrium density of the system

A derivation of DFT can be found in sources [6, 7]. Using the notation of [6], we considerthe grand potential functional Ω of a system of N identical particles, interacting with a po-tential Φ(r1, r2, ..., rN ), and an external potential V =

∑Ni=1 Vext(ri). We define the intrinsic

local chemical potential asu(r) = µ− Vext(r), (2.58)

where µ is the chemical potential. Using the functional derivative, the variation of the grandpotential with respect to the intrinsic potential is

δΩ[u]δu(r)

= −ρ0(r). (2.59)

18

where ρ0 is the equilibrium density. The intrinsic Helmholtz free energy can be defined bythe following Legendre transformation

F [ρ0] = Ω[u] +∫dr ρ0(r)u(r) (2.60)

Changing thermodynamic potential and independent function, the minimization condition(2.59) can be expressed as

δF [ρ]δρ(r)

= u(r). (2.61)

Hence, a system with a given external potential is associated with a unique equilibriumdensity and vice versa.

2.5.1 Examples of DFT

The Ideal Gas The functional of the thermodynamic potential is calculated exactly for anideal gas, like in equation (2.13) and also taking into account the spatially varying density.The functional of the intrinsic Helmholtz free energy of an ideal gas is

βFid[ρ] =∫dr ρ(r) (ln ρ(r)Λ3 − 1), (2.62)

by equation (2.60), using also equation (2.14) βµ = ln ρsΛ3, the grand potential functional is

βΩid[ρ] =∫dr ρ(r)

(lnρ(r)ρs− 1), (2.63)

where ρs is the bulk density of ideal particles.

Mean Field Approximation About more interesting cases, where also interactions exist,the functional of the thermodynamic potential cannot be calculated exactly, hence approx-imation methods are applied. We are interested to Coulomb interaction, like a pairwiseinteraction can be approximated by the mean field approximation.

Using the perturbation theory, the interaction v(r, r′) is split into two parts, one withoutlong range interactions v0(r, r′) which is the reference system, and the other is a perturbationto the reference system, containing long range interactions v1(r, r′). Hence, the interactionreads

v(r, r′) = v0(r, r′) + λv1(r, r′), (2.64)

where λ ∈ (0, 1) tunes the perturbation. As the reference system, the ideal gas can be used.The resulting free energy functional can be split into three parts

βF [ρ] = Fid[ρ] +12

∫ 1

0dλ

∫dr∫dr′ρλ(r)v1(r, r′)ρλ(r′)

+12

∫ 1

0dλ

∫dr∫dr′h(2)

λ (r, r′)ρλ(r)v1(r, r′)ρλ(r′), (2.65)

where Fid is the free energy functional of the ideal gas and h(2)λ (r, r′) is the total correlation

function. Within the mean field approximation the correlation term (third) is neglected andthe density is substituted by the reference density ρid(r).

19

Coulomb Interaction We apply the mean field approximation to a system containingion densities ρ± and an external field density q(r), such that the charge density is eq(r). Asreference system, the ideal gas is considered. The long range interaction between the two kindof ions α± is denoted by v1α,α′(r, r′) and between the ions and the external field distributionis denoted by v1α,q(r, r′) = v1q,α(r, r′) and both assumed to obey Coulomb law

v1α,α′(r, r′) =e2

ε

αα′

|r− r′|and v1α,q(r, r′) =

e2

ε

α

|r− r′|. (2.66)

Using the free energy functional (2.65) and Eq. (2.60), the grand potential functional is

βΩ[ρ] = βΩid[ρ] +12

∑α,α′=±

∫Vdr∫Vdr′ρα(r)v1α,α′(r, r′)ρα′(r′)

+∑

α,α′=±

∫Vdr∫Vdr′ρα(r)v1α,q(r, r′)q(r′)

=(2.66)∑α=±

∫Vdr ρα(r)

(lnρα(r)ρs− 1)

+12

∫Vdr Q(r)φ(r), (2.67)

where ρs is the bulk density of ions, the total charge density Q(r) = ρ+(r)−ρ−(r) + q(r) andφ the dimensionless electrostatic potential

φ(r) = λB

∫Vdr′

Q(r′)|r− r′|

, (2.68)

where λB is the Bjerrum lenght, Eq. (2.50). Minimizing the grand potential functional (2.67)with respect to the ion densities ρ±, results to Boltzmann distribution ρ±(r) = ρs exp±φ(r),which in combination with Poisson equation (2.44) leads to Poisson-Boltzmann equation(2.48)

∇2φ(r) = κ2 sinh(φ(r))− 4πλBq(r) (2.69)

Substituting the Boltzmann distributions to the functional (2.67), the equilibrium grandpotential is found

βΩ = 2ρs∫Vdr(

12φ(r) sinhφ(r)− coshφ(r)

)+

12

∫Vdr q(r)φ(r). (2.70)

20

Chapter 3

Spontaneous Emulsification

3.1 Introduction

A Pickering emulsion is an emulsion that is stabilized by solid particles, such as magnetite,colloidal silica etc. The small colloidal particles are adsorbed onto the interface between thetwo phases, usually water and oil. This type of emulsion was named after its discoverer, S.U.Pickering [9]. An example of a Pickering emulsion is homogenized milk, where protein unitsare adsorbed at the surface of milk fat globules and act as a surfactant. Pickering emulsionsusually are metastable, after a while they demix into macroscopic oil and water phases.

Some recent experimental findings suggesting that Pickering emulsion can be thermody-namically stable. This stimulated our interests to develop a theoretical model to describethe formation of this class of emulsions [4, 2]. We briefly mention the experimental resultsand then we introduce a theoretical model to describe thermodynamically stable Pickeringemulsions.

The emulsions were synthesized by pouring together an oil called TPM (3 - (Methacry-loxypropyl) tri-methoxysilane), an aqueous solution of TMAH salt (Tetramethylammoniumhydroxide) and colloidal particles of the size of few nanometers, such as magnetite [4, 2].After a day, an emulsion is formed spontaneously, consisting of water as the continuous phaseand oil droplets as the dispersed phase. The oil droplets are covered by the colloidal nanoparticles, in this way the oil water interface reduces and the formation of finite droplet isfavorable. The droplets are nearly mono-disperse in size. By mixing two systems of differ-ent droplet sizes a new system of intermediate droplet size arises, which suggests that thesystem is thermodynamically stable. The main difference with other Pickering emulsions isthe spontaneous formation and the thermodynamical stability, which contrasts the kineticstability that usually characterizes Pickering emulsions.

The following theory attempts to incorporate the key mechanisms of the formation andstability of emulsions. In section 3.2, we describe the spontaneous formation of emulsionswithout colloidal particles. However, this case is experimentally found to not be stable for along time [2].

21

3.2 Theory of Spontaneous Emulsification

If oil and water are mixed and small oil droplets are formed and dispersed throughout thewater, eventually the droplets will coalesce to decrease the amount of energy in the system.The energy of this the system is dominated by the interfacial term γowAtot, where γow is theoil-water surface tension and Atot the total surface area between oil and water. Denoting thenumber of droplets by Nd, such that Atot = Nd4πa2 with a the droplet radius, one easilychecksthat the surface energy per unit volume scales as

AtotVtot

=NdAdVtot

=NdAdVoilVtotVoil

=NdAdx

NdVd=

3xa, (3.1)

where x = Voil/Vtot is the oil volume fraction, with Voil = NdVd = Nd 4πa3/3. In Fig. 3.1,the interface contribution is plotted as a function of the droplet radius a for different oilvolume fractions. Regardless x, the favorable state is reached at a→∞ where the system ismacroscopically phase separated.

0 20 40 60 80 1000.00

0.02

0.04

0.06

0.08

0.10

a @nmD

AV

@nm

-1D

0.1

0.01

0.001

x

Figure 3.1: Area over volume term plotted versus the droplet radius a, for different oil volumefractions. The minimum occurs at a → ∞ when the system phase separate. For small oilvolume fractions x, the function drops to zero quicker than for larger x.

However, if we assume that ions can be adsorbed onto the oil-water interface, they com-pensate the interface term and hence thermodynamically stable emulsions can be formed.Byadding salt to the water, cations are produced from dissociation. The anions are producedby the hydrolysis and dissociation of the oil. Both can be adsorbed by the oil water interface.The pairwise Coulomb interaction between ions is also considered. The competition amongarea over volume fraction, electrostatics and adsorption effects results to stable droplets offinite size.

22

Figure 3.2: Schematic representation of the responsive oil theoretical model. One oil dropletis placed at the center of the cell. The ions produced by the reactions (3.2) and (3.3) aregiven by the plus and minus circles, while the blue circles represent the neutral methanolmolecules.

3.2.1 Chemical Reactions and Adsorption

During the mixing of the oil TPM (3-(Methacryloxypropyl) trimethoxysilane) with an aque-ous solution of TMAH salt (Tetramethylammonium hydroxide) the following processes takeplace. The alkalic environment produced by the dissociation of the TMAH salt into TMA+

and OH−, enhances the hydrolysis of the TPM oil, which produces negatively charged TPM−molecules of hydrolyzed oil. Both TMA+ cations and TPM− anionic molecules of hydrolyzedoil are amphiphilic and consequently surface active. Therefore, they have a tendency toadsorb onto the oil water interface and lower the interfacial tension.

A simple way to describe these complicated multi-step reactions is by the two followingschematic reactions. The cations TMA+ may be adsorbed at sites S of the surface of the oildroplet, contributing to a positive surface charge. The net reaction is,

TMA+ + S↔ STMA+ (3.2)

The hydrolysis of the TPM oil and the dissociation of the hydrolyzed molecules can jointlybe described by the following equivalent reaction

TPM + OH− ↔ CH3OH + TPM−h (3.3)

23

where h at TPM−h stands for hydralyzed. Note that methanol, CH3OH, is formed which ispresumed to be dissolved in the water phase.

3.2.2 Responsive Oil Model

We consider Nd oil droplets of radius a in a volume V , surrounded by an aqueous ionicsolution. Each droplet is presumed in the center of a charge neutral cell of radius R, whosesize depends on the fixed volume fraction x of the droplets, x = 4πa3N/3V , such that theradius is R = ax−

13 . The droplets have a surface charge eσ due to adsorption and reactions

of oil and ions at the oil-water interface. Symmetry over the polar θ and the azimuthal φangle is assumed, such that we deal with a radially symmetric geometry of a single dropletin a finite spherical cell.

The cations and anions can be absorbed at sites S on the surface of the droplet. Thenumber of available sites per area is M/4πa2 = O(10) nm−2 and they are accessible foranions and cations. The total grand potential per system volume is a physical quantity,which yields important information of the system. By minimizing this quantity with respectto droplet radius, the optimal size of droplets is obtained. If we assume that total grandpotential is Ωtot = NdΩcell, where Nd is the number of droplet and cells, we can write

Ωtot

Vtot=NdΩcell

NdVcell=

Ωcell

Vcell, (3.4)

where Vcell = 4πR3/3 is the volume of the cell. The dimensionless form we consider, isβΩcell/ρsVcell, where ρs the bulk salt concentration in the water phase, we presume the oilto be salt free.

The total grand potential depends on the density profiles ρ±(r) and the total numberof cations Zc and anions Za adsorbed. They determined by the oil volume fraction x andthe droplet radius a though the volume of the cell and the droplet respectively. The grandpotential reads

βΩcell(ρ, Zc, Za) = 4π∑a=±

∫ R

adr r2ρa

(lnρaρs− 1)

+12

4π∫ R

adr r2φ(r)Q(r)

+ 4πa2βγow + βΩ1 + βΩ2, (3.5)

where Q(r) = ρ+(r)−ρ−(r)+σδ(r−a) with σ = (Zc−Za)/4πa2 the surface charge distribu-tion, γow is the interfacial tension and φ(r) is the potential satisfying the Poisson-Boltzmannequation (2.49). From now on, we will use Ω to represent Ωcell. The grand potential term Ω1

is the additional contribution of the cation adsorption (3.2),

βΩ1 = Zc(βFc − βµ)− ln(M

Zc

), (3.6)

where βFc = lnKcΛ3 is the cations binding energy with Kc the equilibrium constant ofreaction (3.2)

Kc =[TMA+][S][STMA+]

, (3.7)

24

and βµ = ln ρsΛ3 is the cation chemical potential with Λ the thermal de Broglie length. Thegrand potential term Ω2 is the additional contribution of the reaction (3.3) and the adsorptionof anions at the oil water interface,

βΩ2 = Za(βFa − βµ)− ln(M

Za

)+ Za(ln

ZaΛ3

Va− 1), (3.8)

where βFa = lnKa is the anions binding energy with Ka the equilibrium constant of reaction(3.3)

Ka =[OH−][TPM]

[CH3OH][TPM−h ]. (3.9)

The volume Va = 4πR3(1−x)/3 is the available volume for the methanol and the third termof Ω2 is the free energy of the methanol, treated as an ideal gas.

If the grand potential is minimized with respect to ion densities ρ±, we have

δβΩδρ±(r)

= lnρ±(r)ρs

± φ(r) = 0, (3.10)

which leads to Boltzmann distributions for ρ±,

ρ±(r) = ρs e∓φ(r), (3.11)

where the potential φ(r) is the solution of the PB equation (2.49) with boundary conditionsφ′(a) = −4πλBσ and φ′(R) = 0. Substituting the densities distributions to Eq. (3.5), gives

βΩ(Zc, Za) = ρs

∫d3r (φ(r) sinhφ(r)− 2 coshφ(r)) + 4πa2βγow

+12

∫d3r φ(r) σ δ(r − a) + βΩ1 + βΩ2. (3.12)

Minimizing the grand potential (3.12) with respect to cation charge number Zc, leads to

δβΩδZc

= lnZc

M − Zc+ φ(a) + βFc − βµ = 0, (3.13)

which can be rewritten into the Langmuir isotherm

Zc =M

1 + Kcρseφ(a)

, (3.14)

with Kc defined in (3.7). Minimizing the grand potential with respect to anion charge numberZa, leads to

δβΩδZa

= lnZa

M − Za− φ(a) + βFa − βµ+ ln

ZaΛ3

Va= 0, (3.15)

which can be written as the following Langmuir isotherm,

Za =12

(−1 +

√1 + ξ

)Vaρo, (3.16)

25

where

ξ =4MVaρo

and ρo =ρse

φ(a)

Ka. (3.17)

Substituting the Langmuir isotherms and approximating the binomial coefficients by theStirling approximation, the final form of the grand potential is

βΩ(Zc, Za) = ρs

∫d3r (φ(r) sinhφ(r)− 2 coshφ(r))

+ 4πa2βγow −12φ(a)(Zc − Za)− Za

+ M ln(

1− ZcM

)+M ln

(1− Za

M

), (3.18)

where φ(r), Zc and Za are the solution of the PB equation and the Langmuir isotherms (3.14)and (3.16), respectively.

The Macroscopic Limit The limit of the total grand potential (3.18) per oil droplet, forthe oil droplet radius a→∞ going to infinity gives

lima→∞

βΩρsVcell

= −2Vcell − Vdrop

Vcell= −2(1− x), (3.19)

which stems from the first line of Eq. (3.18), since the remaining ones drop with 1/a behavior.

3.2.3 Methanol Buffering

In this section, the previous description of water-oil emulsions will be studied with the ad-dition of a background methanol concentration. The extra methanol concentration changesthe reaction rate and consequently the charge of the oil droplets. Therefore, the equilibriumstate of the system changes. The background methanol concentration together with the oilvolume fraction and the salt concentration are parameters that give the ability to tune theequilibrium radius of the droplet and the stability of the system.

To present this system, the same analysis as for the previous section is assumed. Theparts mentioned are the modified ones by the extra methanol in the mixture.

The grand potential term Ω2 related to the anions adsorption and methanol release (3.3),including the background concentration of methanol now becomes

βΩ2 = Za(βFa − βµ)− ln(M

Za

)+(ρmVa + Za)

(ln

(ρmVa + Za)Λ3

Va− 1), (3.20)

where Va = Vcell(1− x) is the available volume for methanol.Minimizing the grand potential with respect to anion charge number Za, now leads to

δβΩδZa

= lnZa

M − Za− φ(a) + βFa − βµ+ ln

(ρmVa + Za)Λ3

Va= 0, (3.21)

26

which can be written as the Langmuir isotherm,

Za =12

(−1 +

√1 + ξ

)Va(ρo + ρm), (3.22)

where

ξ =4Mρo

Va(ρo + ρm)2and ρo =

ρseφ(a)

Ka. (3.23)

Substituting the Langmuir isotherms and approximating the binomial coefficients by theStirling approximation, the final form of the grand potential is

βΩ(Zc, Za) = ρs

∫d3r (φ(r) sinhφ(r)− 2 coshφ(r))

+ 4πa2βγow −12φ(a)(Zc − Za)− ρmVa − Za

+ M ln(

1− ZcM

)+M ln

(1− Za

M

). (3.24)

The Macroscopic Limit The limit of the modified total grand potential (3.24) per oildroplet, for the oil droplet radius a→∞ going to infinity gives

lima→∞

βΩρsVcell

= −2Vcell − Vdrop

Vcell− ρmρs

(1− x) = −(2 +ρmρs

)(1− x). (3.25)

The non zero terms are the first one and the term −ρmVa of Eq. (3.24), the remaining onesdrop with 1/a behavior.

27

Chapter 4

Numerical Results

4.1 Minimization of the Responsive Oil Model

The minimization of the grand potential with respect to droplet radius a is implementednumerically, with the system parameters of table 4.1. For different values of droplet radius a,the Poisson-Boltzmann equation with the following boundary conditions is solved numerically

d2φ(r)dr2

+2r

dφ(r)dr

= κ2a2 sinhφ(r),

φ′(∞) = 0 and φ′(a) = −4πλBaσ = −λBa

(Zc − Za), (4.1)

where φ(r) is the dimensionless potential and κ−1 = 1/√

8πλBρs is the Debye length. As-suming the salt concentration of table 4.1, the screening length is κ−1 = 13.6nm. After theelectrostatic potential is found numerically, the grand potential can be calculated for sev-eral radius values. Therefore, the total grand potential per unit volume can be plotted as afunction of the droplet radius, and the radius that minimizes this quantity is the thermody-namically stable state.

Table 4.1: Parameters

Bjerrum length λB 0.71 nmoil-water interfacial tension γow 8.2 mN/m

oil volume fraction x 0.002ion concentration ρs 0.5 mM

eq. constant oil-cation Kc 10−3 Meq. constant oil-anion Ka 30

In general, the range of the equilibrium constant for cation adsorption and the anionreaction is 10−4M < Kc < 1M and 10−5 < Ka < 105 respectively.

The plot of the grand potential as a function of the droplet radius a is presented at Fig.4.1(a) and the charge densities of the droplet at Fig. 4.1(b), for the parameters of the table(4.1). The optimal size for a spherical droplet is at radius a∗ = 11nm, as seen from theminimum of the red solid line at Fig. 4.1(a). The dashed black line indicates the value of themacroscopic limit, hence the red line is expected to approach monotonically the macroscopic

28

0 20 40 60 80 100-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

a @nmD

ΒW

ΡsV

(a) The dimensionless grand potential per unit vol-ume is plotted versus the droplet radius a. It exhibitsa minimum for finite droplet radius, a = 11nm,hence the emulsion is formed spontaneously by oildroplets of such size dispersed inside the water. Thedashed line indicates the macroscopic limit.

0 20 40 60 80 100

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

a @nmD

Σ@n

m-2D

Σtot

Σa

Σc

(b) The absolute value of the charge densities growin the similar way as the droplet grows. The purpledashed line shows that total charge density at thedroplet surface is zero.

Figure 4.1: Grand potential and charge densities plotted as function of the droplet size. Theplots correspond to the parameters of table 4.1.

0 100 200 300 400 500-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

a @nmD

ΒW

ΡsV

*

*

*

*

*

0.01 0.02

100

250

x

a*@n

mD

0.022

0.017

0.012

0.007

0.002

x

(a) The grand potential is plotted as a function ofthe droplet size for different oil volume fractions.The optimum droplet radius a∗ increases linearlywith the addition of oil, as shown by the inset.

0 20 40 60 80 100 120 140-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

a @nmD

ΒW

ΡsV

*

*

**

*

0.3 0.7

10

30

50

Ρs@mMD

a*

@nm

D

0.7

0.6

0.5

0.4

0.3

Ρs@mMD

(b) The grand potential is plotted with respect tothe droplet size for different salt concentrations. Theoptimal droplet radius a∗ decreases with the additionof salt, as shown by the inset.

Figure 4.2: The grand potential is plotted as a function of the droplet size, using the param-eters of table 4.1, for different oil volume fractions x and salt concentrations ρs.

29

limit. Fig. 4.1(b) shows that |σa| and |σc| charge densities grow in a similar way, such thatσc + σa ≈ 0.

Next, we tune composition parameters such as oil volume fraction x and the salt densityρs, while we keep fixed the other parameters of table 4.1. The resulting plots for Ω(a) areshown at Fig. 4.2(a) and 4.2(b) respectively. For larger oil volume fractions, the minimumof the potential is moved to bigger radius values, therefore less and larger oil droplets arefavorable for the system. In the inset of Fig. 4.2(a), it is shown that the droplet radiusincreases linearly with the increment of oil volume fraction. On the contrary, for largersalt concentrations, the state of more and smaller droplets is optimal. In the inset of Fig.4.2(b), the droplet radius is shown to decreases with the addition of salt. The value of thegrand potential at the minimum radius also decreases with the addition of salt, while themacroscopic limit, indicated by the thin horizontal black dashed line, does not change. Hence,the addition of salt results to more stable systems with smaller and more droplets.

0 20 40 60 80-4

-3

-2

-1

0

1

2

a @nmD

ΒW

ΡsV

ΒW2ΡsV

ΒW1ΡsV

Interface

Ions

Total

(a) Both reactions (3.2) and (3.3) are assumed. Thereaction terms (purple and blue dashed lines) com-pete with the interfacial term (black dashed line),giving a finite minimum for the total grand poten-tial (green solid line).

0 20 40 60 80-4

-3

-2

-1

0

1

2

a @nmD

ΒW

ΡsV

ΒW2ΡsV

ΒW1ΡsV

Interface

Ions

Total

(b) Considering only reaction (3.2), the interfacialterm (black dashed line) dominates. The effect ofthis reaction (blue line) is not enough to compensatethe interface term.

Figure 4.3: The contributions to the grand potential are plotted as function of the dropletsize, considering both reactions and only the reaction (3.2), using the parameters of table4.1. The contributing terms are plotted with dashed lines and the total grand potential bysolid green line. The black dashed line is the interface term. The red dashed line is thecontribution of the ionic electrostatic interaction. The blue dashed line is the adsorptionterm of reaction (3.2), as given by Eq. (3.6) and the purple dashed line is the contributionof the reaction (3.3), as given by Eq. (3.8).

30

4.2 Considering Only One Reaction

The hypothetical case, that only one of the reactions (3.2) and (3.3) takes place, is alsostudied numerically by assuming that the equilibrium constant of one of the reactions goesto infinity.

As we noticed from several numerical attempts for different parameters, the cation ad-sorption (3.2) exhibits a minimum at the macroscopic limit a→∞, as shown in Fig. 4.3(b).Considering only the anion reaction (3.3), it is possilble to find sets of parameters, that giveminima at finite radius, however the minima are less deep than the case that both reactionsoccur. The grand potentials of the two reactions differ only by the term of the free methanol,apparently it is an important term towards the minimization at a finite radius. Assumingboth or one reaction changes the boundary conditions of the PB equation, therefore theresulting electrostatic potential.

The contributions of each term of the grand potential (3.5) are presented in Fig. 4.3. Inthe case that only reaction (3.2) occurs, the leading term is the interface term. The interfaceterm is the black dashed line at both Fig. 4.3(a) and 4.3(b), it decays as 1/a. The red dashedline is the contribution of the electrostatic interaction among the ions. The blue dashed lineis the adsorption term of reaction (3.2), as given by Eq. (3.6). At Fig. 4.3(a), the purpledashed line is the contribution of the reaction (3.3), as given by Eq. (3.8). At Fig. 4.3(b),where we assume only one reaction, the purple line is zero and the total grand potential, givenby the green solid line, goes monotonically to the macroscopic limit. However, in the caseof both reactions Fig. 4.3(a), the reactions contributions to the grand potential compensatethe interfacial term.

4.3 Buffering Methanol

The case of background methanol density 3.2.3, is also solved numerically. Tuning the back-ground methanol the equilibrium of the chemical reactions is shifted and the entropic contri-bution of the methanol molecules in the grand potential Ω2 changes, see Eq. (3.20).

The resulting grand potential Ω(a) is presented at Fig. 4.4, using the parameters of table4.1 for several background methanol densities. Increasing the background methanol density,the minima are deeper and shifted to smaller droplet radius, as also shown by the insets ofthe Fig. 4.4.

At Fig. 4.4(b), the contributing terms to the total grand potential are plotted with dashedlines for background methanol ρm = 0.4mM. The black dashed line is the interface term.The red dashed line is the contribution of the ionic electrostatic interaction. The blue dashedline is the adsorption term of reaction (3.2), as given by Eq. (3.6) and the purple dashed lineis the contribution of the reaction (3.3), as given by Eq. (3.8). The total grand potential isshown by the solid green line. Comparing with Fig. 4.3(a), where the background methanoldensity is zero, we notice that the addition of methanol influences the contribution of bothreaction terms given by Eq. (3.6) and Eq. (3.8).

31

0 20 40 60 80 100-8

-6

-4

-2

0

a @nmD

ΒW

ΡsV

*

*

*

**

0.2 0.6

5

10

Ρm @mMD

a*@n

mD

*

*

*

*

*

0.2 0.6

1

3

Ρm @mMD

ΒD

WΡ

sV

0.8

0.6

0.4

0.2

0

Ρm @mMD

(a) Plot of the grand potential versus the droplet ra-dius for several background methanol densities ρm.The insets show the radius that minimizes the grandpotential a∗ and the difference ∆Ω of the minimumvalue of the grand potential Ω∗ from the macroscopiclimit versus ρm, respectively. The addition of back-ground methanol density results to smaller and morestable droplets.

0 20 40 60 80-6

-4

-2

0

2

a @nmD

ΒW

ΡsV

ΒW2ΡsV

ΒW1ΡsV

Interface

Ions

Total

(b) For ρm = 0.4mM, each contributing terms of thegrand potential is plotted. Each line represents thesame as in Fig. (4.3). Comparing with Fig. (4.3(a)),we notice that the addition of methanol influencesthe contribution of both reaction terms.

Figure 4.4: The grand potential is plotted with respect to the droplet radius, using theparameters of table (4.1), for several background methanol densities ρm.

4.4 Comparing with Recent Experiments

Recently, there have been several experiments about the formation and the stability of Pick-ering emulsions [4, 2]. The experimental findings suggest a new class of thermodynamicallystable meso-emulsions, as indicated by the spontaneous formation and the monodispersedroplet size, which depends on the system parameters.

For a long time, Pickering emulsions have been considered to be thermodynamically unsta-ble. They required supplied energy to compensate the surface energy upon droplet splitting,hence cannot form spontaneously. They can only be kinetically stable, because the oil and wa-ter phases separate again in the long run. However, the new experimental findings of a specifictype of Pickering emulsion [4, 2], consisting of a particular oil TPM (3-(Methacryloxypropyl)tri-methoxysilane), an aqueous solution of TMAH salt (Tetramethylammonium hydroxide)and magnetite particles of size 11nm, show that the emulsion forms spontaneously, withoutshaking or ultrasonication. The thermodynamic stability is proven by mixing two emulsionsof different droplet sizes that spontaneously evolved into an emulsion of intermediate dropletsize. Moreover, the thermodynamic stability is indicated by the monodisperse size of thedroplets, which is determined by the mass ratio of TPM and magnetite colloidal particles.

In this section, we present the results of our theoretical model and compare them with

32

(a) Experimental photos of Pickering emulsionswithout colloidal particles [2]. In the horizontaldirection the TPM oil increases and in the perpen-dicular direction appears the time. After 17 hoursof the preparation the samples turn turbid, how-ever a month later the samples phase separate. Atthe right part of the picture, TEM micrographs ofthe samples containing B) 50 µl, C) 150 µl and D)250 µl TPM.

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*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

0.2 0.4 0.6 0.8

20

40

60

80

100

Kc @MD

Ka

(b) For the system parameters of table (4.2), wefind pairs of equilibrium constants that result tothe experimental measured droplet size. The blackdots are pairs of values for Kc, Ka which resultto minimization radius within the experimental er-ror. The red stars correspond to pairs which resultto the measured experimental value.

Figure 4.5: Experimental picture of Pickering emulsions without colloidal particles and aux-iliary plot of the unknown equilibrium constants Kc, Ka in order to fit the experimentaldata.

the experimental findings of section IV.1 of [2], about emulsions without colloidal particles.According to [2], the description of the chemical processes that take place is summarized atthe following points:

• The TMAH salt shifts the pH towards alkaline conditions 1 hence enhance the hydrolysisof TPM oil and probably also acts as a catalyst in the reaction. However, it also speedsup the self-condensation reaction of the formed silanols 2 into siloxane 3 linkages [12].

• Hydrolyzed TPM molecules can dissociate forming negatively charged surface activemolecules, which can be adsorbed by the oil water interface, therefore lower the inter-facial tension.

The chemical reactions are characterized by the equilibrium constants. From the bibliography1Alkaline conditions characterized by excess of hydroxide OH−, hence pH > 7.2Silanols are compounds containing silicon atoms to which hydroxyl −OH substituents bond directly.3A siloxane is any chemical compound composed of units of the form R2SiO, where R is a hydrogen atom

or a hydrocarbon group. They belong to the wider class of organosilicon compounds.

33

[10], the equilibrium constant for the esterification 4 of silanols is given

Keq =[R3SiOCH3][H2O][R3SiOH][CH3OH]

= 2.5 · 10−2. (4.2)

This value is a good estimate for the hydrolysis of TPM. However, to match with the equi-librium constant of the reaction (3.3), it should be multiplied with the equilibrium constantof the dissociation of the hydrolyzed TPM oil and the equilibrium constant of the adsorptionof the negatively charged molecules at the water oil interface.

Table 4.2: Parameters

Bjerrum length λB 0.71 nmoil-water interfacial tension γow 8.2 mN/m

ion concentration ρs 6mMoil volume fraction x 0.01

**

*

*

0.005 0.01 0.015 0.02 0.025

100

200

300

400

500

600

x = VoilVtot

Radius

@nm

D

* Experimental

- -Kc =0.7 MKa=27.7

-Kc =0.4 MKa=40

Numerical

(a) The experimentally measured droplet radiuswith the experimental error are shown by blackstars, for four different values of oil volume frac-tion. The two lines are obtained by the numeri-cal data of our theoretical model for two differentpairs of equilibrium constants. The radius of thedroplets scales linearly with the TPM oil volumefraction.

0 100 200 300 400 500

-2.04

-2.02

-2.00

-1.98

-1.96

-1.94

-1.92

-1.90

a @nmD

ΒW

ΡsV

0.020

0.015

0.010

0.005

0.002

x

(b) Several curves of the grand potential versus thedroplet radius a, corresponding to oil volume fractionsx of the experimental data. The equilibrium constantsused are Kc = 0.4M and Ka = 40. The droplet sizegrows with the addition of oil and the observed min-ima become less deep. For example, at x = 0.02 themacroscopic limit of the grand potential is −1.98 andthe finite minimum occur at −1.99.

Figure 4.6: Fitting to the experimental data [2] and the minimization of the grand potentialfor system parameters of table (4.2).

4Esterification is the general name for a chemical reaction in which two reactants (typically an alcohol andan acid) form an ester as the reaction product.

34

For the values of the two last equilibrium constants, no estimates can be found in theliterature. For this reason, the equilibrium constants, Kc and Ka, are to be considered asfree parameter within the above mentioned range, section 4.1. To choose the equilibriumconstants for the two reactions, (3.2) and (3.3), random pairs of values are tested for thesystem parameters of table 4.2.

The pairs Kc, Ka for which the resulting minimization radius is close to the measuredone, are shown in Fig. 4.5(b). The black dots are pairs of values for Kc, Ka which result tominimization radius within the experimental error. The red stars correspond to pairs whichresult to the measured experimental value. The experimental measurements of the dropletsizes for different oil volume fractions are shown in Fig. 4.6(a) (black stars with error bars).

The experimental data are obtained by the preparation of samples containing 6mM ofTMAH and different volume fractions of TPM oil, within the range x ∈ [0.005, 0.025]. Aday after the preparation, by gentle mixing, the samples turned turbid and the turbidityincreased by the oil volume fraction. After several days, the emulsions become unstable andfinally phase separate. The experimental results are presented at Fig. 4.5(a). During thestable period of the prepared emulsions, the droplet size was measured. The experimentaldata are shown at Fig. 4.6(a) by black stars with the experimental error. The two lines,shown at Fig. 4.6(a), are obtain by the numerical data of our theoretical model for twodifferent pairs of equilibrium constants. The first pair, represented by the red solid line,was chosen to fit the data point of x = 0.01. The second pair, which results at the purpledashed line, was chosen within the error of the data point of x = 0.01. The experimentaland numerical result of this plot is that the radius of the droplets scales linearly with theTPM oil volume fraction. In Fig. 4.6(b), the grand potential with respect to the dropletradius is plotted for several oil volume fraction, in the range of the experimental data. Thedroplet size and minimum value of the grand potential increases with the oil volume fraction,resulting to less stable systems.

Considering the equilibrium constants Kc,Ka = 0.4M, 40, the dependence of thedroplet size by the oil volume fraction x and the salt density ρs is presented by the densityplots of Fig. 4.7(a). The areas of lighter color depict pairs of x, ρs that result to smalleroil droplets, while the darker areas correspond to larger droplets.

The two important aspects of Pickering meso-emulsions are the spontaneous formationand the thermodynamic stability. The theoretical model describes the spontaneous forma-tion for a broad range of system parameters, where the grand potential is minimized at afinite radius. The experiments confirm the spontaneous formation of emulsion even thoughcolloidal particles are absent. However, emulsions without colloidal particles are not thermo-dynamically stable in the experiments. The chemical effect that results to the destabilizationof the emulsion is probably the self-condensation reaction of silanols into siloxane linkages.The self-condensation reaction is much slower than the TPM hydrolysis, hence it plays arole at a larger time scale. This reaction is absent from our theoretical model, therefore theultimeta phase separation of the system cannot be captured.

A way to effectively include the self condensation reaction is by varying with time theequilibrium constant Ka of reaction (3.3). As shown at Fig. (4.7(b)), the droplets becomebigger increasing the equilibrium constant Ka and the minimum value of the grand potentialgoes to the macroscopic limit. By assuming that the equilibrium constant Ka incorporatesthe self condensation reaction and increases by time, we can see the different droplet sizes inthe passing of time. Unfortunately, the time behavior of intrinsic parameters of the system,

35

0.005 0.010 0.015 0.020 0.025

0.002

0.004

0.006

0.008

0.010

x oil volume fraction

Ρssalt

concentration

@MD

550 nm

5 nm

(a) Density plot of the minimization radius de-pending on oil volume fraction and salt concentra-tion.The range of x and ρs are close the experimen-tal range. Lighter colored pairs of x, ρs result tosmaller radius, while the darker colored pairs tolarger radius.

0 200 400 600 800 1000-2.02

-2.01

-2.00

-1.99

-1.98

-1.97

-1.96

-1.95

a @nmD

ΒW

ΡsV

**

**

**

**

**

40 80 120

250

450

650

Ka

a*@n

mD

130

110

90

70

40

Ka

(b) Several lines of the grand potential versus thedroplet radius a, for different values of the equilib-rium constant Ka. The value of equilibrium con-stant of reaction (3.2) is Kc = 0.4M . The systemparameters are given by table (4.2). The dropletsize grows with the increment of Ka. The horizon-tal dashed line is the macroscopic limit.

Figure 4.7: Density plot of the droplet radius with respect to oil volume fraction and saltconcentration ρs and grand potential versus radius a for different Ka values.

such as the equilibrium constants, is not known.

4.5 Conclusions

The spontaneous formation of an emulsion is a collective effect, it depends on the balanceof the contributing effects. The presence and the intensity of each one of the contributinginteractions can affect the formation and the characteristics of an emulsion, such as thedroplet size and the stability. Including the electrostatic interactions among the ions andbetween the ions and the charged droplets, the adsorption of ions into the oil-water interfaceand the surface tension of the oil droplets, the formation of a thermodynamically oil-wateremulsion is possible within a reasonable range of system parameters.

The favorable state of oil and water would be to not mix in the absence of energy input,thus the surface term would be the leading term. In case of this type of emulsions, the chem-ical reactions, therefore the electrostatic interactions and the adsorption, offer the requiredfree energy to form an emulsion spontaneously. In terms of the grand potential (3.5), theelectrostatic term and the adsorption terms compensate the surface tension term, resultingto an equilibrium system, consisting of oil droplets of tens of nanometers, a meso-emulsion.

In our theoretical model, we included two reactions to represent the chemical processesobserved in the experiments. If the hydrolysis reaction (3.3) is absent, the emulsion is notformed spontaneously. If the reaction (3.2) is absent, the emulsion is formed, however the

36

resulting minimum is less deep than the case of both reactions. Therefore, the adsorptioneffects determine the spontaneous formation of emulsions and their stability.

Tuning the parameters of the theoretical model the characteristics of the emulsions change.First, we tune the composition parameters, such as the oil volume fraction x and the saltconcentration ρs. Increasing the oil volume fraction x, the forming oil droplet are bigger andthe equilibrium grand potential does not change significantly. In comparison, increasing thesalt concentration ρs, the oil droplets are smaller and the minimum of grand potential becomessmaller. Secondly, we add methanol with background concentration ρm, which results tosmaller droplets and smaller grand potential. An interesting remark is that the equilibriumdroplet radius does not change significantly, however the difference of the grand potentialfrom the macroscopic limit at the equilibrium radius decreases significantly, suggesting astabler system.

We also tune the intrinsic parameters of the system, such as the equilibrium constantsKc,Ka of the two reactions. Actually the equilibrium constants are considered to be freeparameters, as there is not sufficient value in the bibliography. It is also possible that theequilibrium constants change through time, as new phenomena take place, such as the selfcondensation reaction. Hence, we tested a broad range of values for the two equilibriumconstants, finding that several pairs of Kc,Ka result to the same size of oil droplets. Tun-ing the equilibrium constants for a given oil and salt composition of the system, we findthat the system can form larger droplets. According to experimental findings, some timeafter the formation of the emulsion, the droplets coalescence and finally the system phaseseparates. Assuming that time dependent equilibrium constants could capture the reactioneffects through time, then the kinetically stable states that the system reaches through time,can be described by this theoretical model.

37

Bibliography

[1] J. Zwanikken, Looking deeper into emulsions and suspensions ionic screening, fractiona-tion and adsorption (2009).

[2] D. Kraft et al., Conditions for equilibrium solid-stabilized emulsions, in preparation.

[3] S. Sacanna et al., Spontaneous Oil-in-Water Emulsification Induced by Charge-StabilizedDispersions of Various Inorganic Colloids (2007).

[4] S. Sacanna et al., Phys. Rev. Lett. 98, 158301 (2007).

[5] P. Pieranski, Phys. Rev. Lett. 45, 569 (1980).

[6] R. van Roij, Soft Condensed Matter Theory, course book (2009).

[7] J.-L. Barrat, and J.-P Hansen, Basic Concepts for Simple and Complex Liquids, Cam-bridge University Press (2003).

[8] F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill InternationalBook Company (1965).

[9] S. U. Pickering, J. Chem. Soc. 91, 2001 (1907)

[10] B. Arkles et al., Factors contributing to the stability of alkoxysilanes in aqueous solution(1991).

[11] S. Melle et al., Langmuir 21, 2158 (2005).

[12] M. Brochier Salon, P. Bayle, M. Abdelmouleh, S.Boufi, M. N. Belgacem Colloids andSurfaces A: Physicochemical and Engineering Aspects (2008).

38

Appendix A

Theoretical and NumericalSupplement

A.1 Dimensionless Form of PB Equation

The dimensionless form of equation reduces the number of parameters and it is easier andsafer to apply approximations by considering a quantity small. The dimensional analysis isapplied on the Poisson-Boltzmann equation,

∇2φ(r) = κ2 sinhφ(r) (A.1)

in spherical coordinates, assuming that the potential φ(~r) is symmetric over the polar θ andthe azimuthal φ angle,

d2φ(r)dr2

+2r

dφ(r)dr

= κ2 sinhφ(r). (A.2)

Consider the dimensionless length x = r/γ, where γ is a characteristic length scale. Thecharacteristic length is such that the derivatives in the new coordinates are of unity orderO(1). The new derivatives are expressed using the chain rule

dφ(r)dr

=dφ(x)dx

dx

dr=

1γ

dφ(x)dx

(A.3)

and similarlyd2φ(r)dr2

=1γ2

d2φ(x)dx2

. (A.4)

Applying the above substitutions to PB equation (A.2) results to

1γ2

d2φ(x)dx2

+2r

1γ

dφ(x)dx

= κ2 sinhφ(x). (A.5)

The dimension of the PB equation (A.1) is [m−2], as can easily be seen from the right handside of the equation (A.2), as the Debye length κ has dimension of inverse length. In order toobtain a dimensionless equation, it should multiplied by a squared length. As characteristic

39

length, the radius a of the charged sphere or the Bjerrum length λB are possible candidates.Here we chose the sphere radius a, therefore

a2

γ2

d2φ(x)dx2

+2r

a2

γ

dφ(x)dx

= κ2a2 sinhφ(x). (A.6)

Since sinhφ and the derivatives are of order O(1) by assumption, then

a2

γ2≈ O(1) such that γ = a. (A.7)

The dimensionless form of PB equation follows

d2φ(x)dx2

+2x

dφ(x)dx

= κ2a2 sinhφ(x). (A.8)

The boundary conditions in case of a charged sphere of radius a are

dφ(r)dr

∣∣∣∣r→∞

= 0 anddφ(r)dr

∣∣∣∣r=a

= −4πλBσ, (A.9)

where λB is the Bjerrum length, σ = Z/(4πa2) the surface charge. The dimensionless bound-ary conditions are

dφ(x)dx

∣∣∣∣r→∞

= 0 anddφ(x)dx

∣∣∣∣x=1

= −4πλBaσ = −λBaZ. (A.10)

A.2 Euler method

A numerical solution to the nonlinear PB equation can be acquired by Euler’ s method. Thisscheme provides a way to approximate the solution φ(r) given the equation and satisfactorynumber of initial conditions.

According to Euler’s method, starting from the initial condition of the function, we canfind the next point by moving in the direction of the derivative, so the iteration schemefollows

f(xn+1) = f(xn) + f ′(xn)∆x (A.11)

where ∆x should be very small, otherwise the approximation gets bad.We will apply the iterative method to PB equation (A.2). Firstly, we consider a grid for

the distance r, r(n+1) = r(n)+∆r, where ∆r is given by the total number of iteration pointsN , ∆r = R/N with R the total interval of interest. Then, we set u(r) the first derivative ofthe potential u(r) = φ′(r), where by one prime we mean the first derivative with respect tor and by two primes the second derivative. Therefore, the PB equation is written

u′(r) = −2ru(r) + sinhφ(r) (A.12)

To approximate the value of the potential for the next grid point, the derivative at the previouspoint should be known, which involves also the value of the potential at the previous point

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as can be seen by equation (A.12), then the two functions u(r), φ(r) are interconnected. Thesystem of equations is

u(rn+1) = u(rn) + u′(rn)∆rφ(rn+1) = φ(rn) + u(rn)∆r (A.13)

if we impose two initial conditions, for the potential and its derivative, the above system canbe solved numerically.

A.3 Definitions of Physical quantities

Table A.1: Definitions of frequently used quantities

λB Bjerrum length λB = e2

εkBT

κ Debye length κ−1 =√

8πλBρ

γ interfacial tension γ = dW/dA

e elementary charge

β inverse thermal energy β = 1kBT

Λ thermal de Broglie length Λ = h√2πmkBT

K equilibrium constant K = [A][B][C][D] , if A+B → C +D

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