Sedimentation Equilibrium of Mixtures of Charged Colloids

Sedimentation Equilibrium of Mixtures ofCharged ColloidsJos Zwanikken 1Supervisor: dr. R. van Roij29th June 20051 j.w.zwanikken@phys.uu.nl

Contents1 Introduction 11.1 Sedimentation Equilibrium of Mixtures of Charged Colloids . . 11.2 Sedimentation equilibrium of mixtures of charged colloids . . . 22 Overview of required statistical mechanics 52.1 Thermodynamics and Legendre transforms . . . . . . . . . . . 52.2 Phase space distributions and averages . . . . . . . . . . . . . 62.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . 72.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Mean Field Approximation and Local Density Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . 83 Sedimentation of a monodisperse system 113.1 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Donnan-method . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Poisson-Boltzmann-method . . . . . . . . . . . . . . . . . . . 134 Sedimentation of a polydisperse mixture 154.1 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 The functional and minimum conditions . . . . . . . . . . . . 164.3 Binary systems . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 Ternary systems . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Polydisperse systems . . . . . . . . . . . . . . . . . . . . . . . 254.6 Explanation of phenomena . . . . . . . . . . . . . . . . . . . . 274.6.1 The influence of the diameter . . . . . . . . . . . . . . 304.6.2 The influence of the salt . . . . . . . . . . . . . . . . . 314.6.3 The layer width . . . . . . . . . . . . . . . . . . . . . . 314.6.4 The slope model . . . . . . . . . . . . . . . . . . . . . 324.6.5 Donnan-method for polydisperse cases . . . . . . . . . 384.6.6 The colloidal Brazil nut effect . . . . . . . . . . . . . . 40iii

5 Charge Regulation in the Low Density Regime 435.1 Regulation, the next challenge . . . . . . . . . . . . . . . . . . 435.1.1 The extension on the PB theory . . . . . . . . . . . . . 445.1.2 Numerical results . . . . . . . . . . . . . . . . . . . . . 455.2 Ideal distributions . . . . . . . . . . . . . . . . . . . . . . . . . 466 The C++ code 497 Conclusions and Outlook 51A Article 53A.1 abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54A.4 Segregation in binary and ternary mixtures . . . . . . . . . . . 56A.5 Polydisperse Mixtures . . . . . . . . . . . . . . . . . . . . . . 59A.6 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . 60A.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 62B Important length scales 63B.1 The Debye screening length κ −1 . . . . . . . . . . . . . . . . . 63B.2 The gravitational length . . . . . . . . . . . . . . . . . . . . . 66B.3 The colloidal diameter, σ . . . . . . . . . . . . . . . . . . . . . 67C The block model 69Index of frequently used symbols and names 71Acknowledgements 73iv

WarningThis is not an ordinary document. It can seriously bore an ignorant reader,or blunt a lyric mind. Even readers that are already familiar with the subjectcan get lost between old facts and new, unclear formulations. Thereforethe reader is requested to read with carefulness and is informed about thedocument he/she/it is staring at. May this warning prevent you from anyunwanted consequences, or at least hide them.This document can be compared with a small encyclopedia. Many topicsare explained in a formal way, although not in alphabetical order and notas clearly listed (resembling the subject of the story, the form is like its content).Some pages contain more information than others and some are moreaccessible. Regrettably these pages cannot be indicated here, since their accessibilityis strongly dependent on the reader. Some parts however can beskipped by physicists. These parts are written in italic script. Non-physicistsare even advised to read those sections, since they are popularly written anddo not require full knowledge of classical statistical mechanics and thermodynamics.. .What? Classical-Statistical-Mechanics?(. . . see section 1.2 for a description). Only this warning is advised to allreaders. Instead of a clear introduction and a labyrinth-like story, all fuzzinessis put here to make the structure and the story transparent.Why not make everything clear?Unfortunately, some unclarity remains because of entropic reasons. Thesereasons will be explained later. First I shall give you an overview of howthis document can be used. Chapter A is a nice chapter for experts. Muchunderstanding is needed of soft condensed matter, the reader is assumed toknow many definitions, and it offers a high density of information. ChapterA summarizes a big part of the document and skips the long road betweenthe basic thermodynamic laws and the end results.This chapter, together with chapters 1.2 and 4.6.6 may be the nicest partsfor readers that are totally unfamiliar with statistical physics, because of itsreadability together with the not so unimportant information... Do you likev

programming?Do I like programming?Then you should copy chapter 6. It offers you an opportunity to check thenumerical results in this document and to play with the programme, or evento optimize it.Many other chapters can be used to refresh the mind, like an encyclopedia,to recheck definitions or to gain some more insight in the origin of the finalresults.The reader will repeatedly be warned against unimportant information, advisedto skip chapters, but the reader should feel free to defy my understandingof importance.At last, and again: the form of this document is like its content.Rectangular?Seeking balance between opposing tendencies, and finding equilibrium, althoughseveral tensions and strong deviations from well known examplesremain. The structure will be diverse, with the permission of entropy, andlayered in density of information, suspense, entertainment, and many otherproperties too many to name.Let’s start!I hope you can enjoy reading now.vi

SummaryThis thesis is about sedimentation equilibrium of polydisperse colloidal suspensionsin the gravity field of the earth. The first part of the documentfocusses on the physical background of the subject. From a quick overviewof some elementary thermodynamics, classical statistical mechanics, densityfunctional theory, the Bjerrum length λ B , the Debye screening length κ −1 ,and the gravitational length L will be introduced. These length scales willprove useful in describing the following colloidal systems.The first colloidal systems under consideration are monodisperse. By usingan analytical method, and a more consistent Poisson-Boltzmann theory, thenumber densities and electrostatic potential of the systems are obtained numerically.In the limit of low salinity a phenomenon is observed known as theentropic lift [8], colloids forming highly non-barometric density profiles. Thetheory will be expanded to describe polydisperse systems of charge stabilizedcolloids. In the same low salinity limit the entropic lift is again observed, aswell as segregation between colloidal species. The so-called Brazil nut effectis predicted by the theory. The layering is found to be dependent on themass per charge of the colloidal particles, as well as the system height. Afterthe consideration of binary, ternary and polydisperse systems (N = 10,and N = 21), two analytical models follow, that approximate the theory formulti-component systems. Monte-Carlo simulation data show almost perfectagreement with the numerical calculations, based on the theory.The last colloidal systems to be discussed contain charge regulating colloids.Three different methods determine the equilibrium distributions, and theexpectation value of the colloidal charge as a function of height. Still, manyexamples of systems remain to be explained, systems that violate primaryassumptions. In the region of high packing fractions, of non-spherical particleshape, dipolar colloids, the theory lacks answers. Some possible answers andadditional ideas are discussed in the outlook.One of the last chapters describes the C++ program that was used for thenumerical calculations. The summarizing article concludes the thesis, givingan overview of the main results.vii

Additional information, often written in italic script, and sheep are includedto improve quantum mechanical, and especially relativistic thoughts, that arelacking in the theory. They can be skipped by observers that do not wishstories from other observers, considering them contaminations, untrue, justimproper, or anything else, or can be skipped for any other reason, since theyare not the official part of the story. I just considered them indispensable.viii

Chapter 1Introduction1.1 Sedimentation Equilibrium of Mixturesof Charged ColloidsThis thesis is not an attempt to gain a Platonic idea by careful inductionand comparisons between diversities. Instead of unifying and generalisingexisting theories, we will go the other way. Some good old ideas, known bythe names Thermodynamics, classical statistical mechanics, classical electrostaticsand Newton’s gravity 1 , will be taken as solid laws, with which we willtry to understand complex systems. Systems that contain groups of different,interacting objects, in equilibrium.The objects we consider are colloids and salt-ions. Colloids are particleswith a size between 10nm and 1µm. These ’big’ particles will be suspendedin a solvent, where they can act like a gas. They interact with gravity andother particles. By the influence of gravity, the colloids sediment, while themuch lighter salt-ions do not. By the influence of their charge, the colloidsrepel each other and interact with the salt-ions. Our main goal will be todetermine the equilibrium state of the system, by looking at the distributionof the particles in space, and their ’influence’, the electric field. We will seethat the balance will lead to structures, very different from ’ordinary’ idealgases.Still, the feeling of understanding often leads to wild analogies, and generalisations,although it was not initially intended. Something shown by manyspecialists. Care was taken to keep these ideas from the thesis.1 a 0 th order approximation1

1.2 Sedimentation equilibrium of mixtures ofcharged colloids 2I know the word ’of’. . .What is a colloid?!I’m glad you ask that question so soon. It is very important to know what acolloid is before reading about suspensions of them.Suspensions?That is what I mean. Wait, first the colloid. . . The physicist uses the wordcolloid for an object of a specific size. Size is the only property that makessomething a colloid and another object not. To become a colloid an objectmust be small. Readers, fingers, letters, books are all too big to be calledcolloids. On the other hand, objects may not be too small either. At leastnot smaller than 10 nm, that is 0.00000001 meter. Their maximum size isabout 1µm, or 0.000001m.What’s so special about this size? Why is the word ’colloid’ invented?Colloids are very special. They live between two size-ordering worlds, themicroscopic and the macroscopic one.Size-ordering worlds?Physicists like to make orderings in mind. They make a difference betweenbig things and very small things. This imaginary collection I call ’world’. Letme show you around. The macroscopic world categorizes all things biggerthan 1µm. Since it it very common to speak of these sizes in everyday life,the objects with these sizes have long been observed and described already.Like running boars, the sun, the moon, clouds, apples and many, many otherthings. The surviving physical theories that describe the laws of this worldare therefore called ’classical’ mechanics. In the other world, the microscopicworld, these laws proved unsatisfactory for many physicists. All objects thatare described in this world are not measurable by eye nor ruler, ever heardabout ’atoms’ or ’electrons’? They have a size smaller than 10 nm. Nota typical size one experiences during childhood. These objects are ratherdescribed by ’quantum’ mechanics, and may behave magically for classicalpeople. And since 1905 many of these objects have been moving freely andrandomly in an ever growing chaos called ’entropy’.Sounds scary.Big (=macroscopic) groups of these objects however, can be described withclassical statistical mechanics, or thermodynamics.And the colloids?Well, they live in between. That is why the physicist often uses the word2 Chapters in italic script can be skipped by physicists2

mesoscopic, in which one could recognize the Greek word ’mesos’, meaning’middle’. The special thing about mesoscopic objects is that physicistsdare to describe them with classical theories, while they also give them someproperties of microscopic objects, in the sense that they also consider the’entropy’. So they don’t necessarily need scary, magical quantum theoriesin order to get magical predictions. Hence, these objects deserve an extraname: ’colloids’.Ok. I see. And can you tell me about these ’magical predictions’?Are you ready for it? Maybe I can impress you even more with facts ratherthan predictions. Did you ever see stiff gel become liquid by shaking, andimmediately freeze at rest? Or have you ever pulled something liquid by amagnet? Did you ever see a bouncing ball becoming a puddle after it stoppedbouncing? Heard about liquid crystals?Things you buy at the ’Expo’.If you know how to play with the properties of liquids you can make billionsin paint, transport and communication industries. Zillions. And I don’tmean liquids.Nice.Well, if that does not impress you, maybe the colloid itself can win yourrespect. Colloidal societies can become very complex, since they know manykinds of interactions, tendencies and other influences that complicate equilibrium.I’m waiting.By the way, are there many colloids?In fact everything can become a colloid if you have a huge hammer. As Isaid, it must be small enough, that’s all. However, you can already find manycolloids in nature, the things in our blood, in plants, think of viruses, thenumerous building blocks of life, milk. But also in paint, in flat screens, orclay, ... everywhere!You are talking a lot about fluids, are colloids always liquids?No, no, the colloids in these examples are not the liquid. They are oftenconsidered in a liquid, because they do many interesting things in it. By theway, this is why the word suspension is used. They ’hang’ in a fluid. And ifyou’ve got many different colloids in a fluid. . .Like in your blood?Exactly! Then the physicist calls the fluid polydisperse. It is the opposite ofmonodisperse.When you don’t have different colloids in a fluid. . .When there is only one type of colloids in a fluid. Now take a glass ofwater and put a bunch of different colloids in it and give them a charge.Like your hair in winter, make them ’static’, make them repel each other,3

make them spark. . . Wait, wait! I’ve got it! Then you’ve got a polydisperse.. . suspension. . . of charged colloids!Exactly. You speak the language. Now, if you wait for several days or weeks,or years, you will see the colloids sinking slowly, until they reach the bottomor just stop sinking.They just stop? While they are in the air?Yes, well, in the water you mean. They want to go down to the bottom,but not all together. By entropy, a certain love or longing for chaos. Justbelieve me. What I want to say is that this sinking of the colloids is calledsedimentation, and when they all stop sinking they’ve reached equilibrium.Sedimentation equilibrium of mixtures of charged colloids?Sedimentation equilibrium of mixtures of charged colloids! You’re great!My brain hurts.4

Chapter 2Overview of required statisticalmechanics2.1 Thermodynamics and Legendre transformsThe first and second law of thermodynamics provide the relationsdE = T dS − P dV + ∑ αµ α dN α (2.1)dS ≥ 0 (2.2)between the internal energy E = E(S, V, {N α }) and the entropy S of aclosed system of volume V and N α particles of species α. The intensivefield variables T , P and {µ α }, where T is the temperature, P the pressureand {µ α } the chemical potential, are conjugate to the extensive variables S,V and N α . The entropy S is experimentally hard to control, and can bereplaced by its conjugate field variable T by a Legendre transfomation:F (T, V, {N α }) = E(S, V, {N α }) − T S (2.3)dF = −SdT − P dV + ∑ αµ α dN α (2.4)where F is called the Helmholtz free energy. If one is still left with fieldvariables that are difficult to handle in experiments, more potentials can befound with Legendre transformations. The Gibbs free energy, or free enthalpyisG(T, P, {N α }) = F (T, V, {N α }) + P V (2.5)5

dG = −SdT − V dP + ∑ αµ α dN α , (2.6)and the grand potential is defined byΩ(T, V, {µ α }) = F (T, V, {N α }) − ∑ αµ α dN α (2.7)dΩ = −SdT − P dV − ∑ αN α dµ α . (2.8)E, F , G and Ω are associated with the micro-canonical, canonical, isothermalisobaricand grand canonical ensembles respectively. A direct consequence ofthe second law of thermodynamics is that the potentials reach a minimumfor thermodynamic equilibrium. For example, for arbitrary internal changesthe free energy satisfies(dF ) T,V,{Nα} ≤ 0. (2.9)2.2 Phase space distributions and averagesSystems of many classical particles can be described in phase space, whereeach microscopic state is characterized by one point Γ ≡ (r N , p N ), withr = (r 1 , r 2 , ..., r d ) the coordinates, p = (p 1 , p 2 , ..., p d ) the conjugate momenta,d the dimension of the coordinates and momenta, and N the total number ofparticles. Time evolution of the system can be seen as a motion of the phasepoint along a path, following from the Hamiltonian H(r N , p N ), that can bewritten asN∑H(r N , p N p 2 i) =2m + Φ(pN , r N ), (2.10)i=1where m is the mass of the particles, and Φ the potential energy, that usuallyonly depends on r N . An ensemble is an arbitrary large collection of macroscopicallyidentical systems, that only differ by their position in phase space.The Liouville equation∂f(Γ, t)∂t+ ∂∂Γ · ( ˙Γf(Γ, t))= 0 (2.11)governs the time evolution of f(Γ, t), the probability density that describesthe phase point distribution of the ensemble. A combination with the Hamiltonequations yields∂f(Γ, t)∂t= − ˙Γ ·∂f(Γ, t)∂Γ6= {H, f(Γ, t)}, (2.12)

with F [ρ] the intrinsic Helmholtz free energy functional obeying the Euler-Lagrange equationδ[ ∫]∣∣∣ρ(r)=ρeqF [ρ(r)]− drρ(r)ψ(r) = δF [ρ eq(r)]+φ(r)−µ = 0 (2.20)δρ(r)(r) δρ eq (r)Hohenberg, Kohn and Mermin proved two important theorems:Theorem 1: The intrinsic free energy functional F [ρ] is a unique functionalof the one-particle density ρ(r), i.e. for a given intermolecular potential energyV N , F [ρ] has the same functional form, whatever the external potentialφ(r).So, only one external potential φ(r) can be associated with a given ρ(r).Theorem 2: The auxiliary functional∫Ω V ext[˜ρ] = F [˜ρ] + dr˜ρ(r)(V ext (r) − µ) (2.21)reaches its minimum when the trial density ˜ρ(r) coincides with the equilibriumdensity ρ(r),δΩ V ext[˜ρ(r)]∣ = 0 (2.22)δ ˜ρ(r) ∣˜ρ(r)=ρ(r)and the minimum of Ω V ext coincides with the equilibrium grand potential Ω(defined in 2.7, and the first line of this section).For the proof see [16]. This formalism and its concepts are also referredto as density functional theory or ’DFT’.2.3.2 Mean Field Approximation and Local DensityApproximationThe mean field approximation is an approximation scheme, often usedfor Coulombic systems. For these systems the pair interaction ν(r) is writtenas ν 0 (r, r ′ ) + w(r, r ′ ). By viewing the long range-part w(r, r ′ ) as a perturbation,ν λ (r, r ′ ) = ν 0 (r, r ′ ) + λw(r, r ′ ). (2.23)In this thesis we use w to be a perturbation on the ideal gas, ν 0 (r, r ′ ) = 0).By introducing the two particle densityρ (2)λ (r, r′ ) = 〈ˆρ(r)ˆρ(r ′ )〉 λ − ρ(r)δ(r − r ′ ) (2.24)=δF2δν λ (r, r ′ ) , (2.25)8

with ν λ = ν 0 +λw, the free energy functional can be written, after integrationof equation (2.25),F [ρ] = F 0 [ρ] + 1 2∫ 10dλdrdr ′ ρ (2)λ (r, r′ )w(r, r ′ ), (2.26)where F 0 [ρ] is the free energy functional of the reference system with ν = ν 0 .Substitution of the two particle density by the pair correlation functionh (2) (r, r ′ ) = ρ(2) (r, r ′ )ρ(r)ρ(r ′ ) − 1 (2.27)splits the free energy functional into three partsF [ρ] = F 0 [ρ] + 1 ∫ 1 ∫ ∫dλ dr dr ′ ρ(r)w(r, r ′ )ρ(r ′ )2 0+ 1 ∫ 1 ∫ ∫dλ dr dr ′ h (2)λ2(r, r′ )ρ(r)w(r, r ′ )ρ(r ′ ), (2.28)0where the last part (2.28) is the correlation term of the free energy. Themean field approximation is made by neglecting this correlation term.Stated less formally and taking the electrostatic potential as an example:by using the MFA, one considers the long-range pair potential as a selfconsistentexternal field for which∫φ(r) = dr ′ Q(r ′ )|r − r ′ | , (2.29)where Q(r) is the total charge density.VThe local density approximation , also known as LDA, is based on theassumption that macroscopic thermodynamics applies locally. The intrinsicHelmholtz free energy may then be written as∫F [ρ] = drf(ρ(r)) (2.30)where the free energy density f(ρ) satisfiesVF (N, V, T )f(ρ) = , (2.31)Vwith ρ = N/V the homogeneous density. For systems with long-range interactions(like Coulomb interactions), the LDA cannot be expected to hold.For example, hydrostatic equilibrium (eq.density approximation.B.16) is derived using a local9

Chapter 3Sedimentation of amonodisperse system3.1 The systemUsing the theory of last chapter, we will turn now to a more complicatedsystem: charged colloidal spheres in an ionic solution in the gravity field ofthe earth. The colloidal spheres have a charge −Ze, gravitational length L,and a diameter σ, and their number density as a function of height is denotedby ρ(x). Instead of the bare mass, the buoyant mass of the colloid is takenin L = k B T/mg, dependent on the density of the incompressible solvent.The solvent further provides a dielectric constant ɛ and temperature T . Themicro-ions have a charge ±e, and have yet-unknown number densities ρ ± (x).In addition we will describe the micro-ions grand canonically, the colloidalspheres canonically. Such a situation can be realised by putting the solutionin osmotic contact with a salt reservoir, from which the colloids are separatedby a semipermeable membrane, while the co- and counter-ions can flow freely.The equilibrium properties of such a system can be determined with twodifferent methods. We will call them the Donnan-method and the Poisson-Boltzmann-method inspired by [8], which treats both methods profoundly.Subsections 3.2 and 3.3 follow the steps of [8].3.2 Donnan-methodThe osmotic equilibrium with the reservoir is also called Donnan equilibrium,named after the person who described this system first. In this way thechemical potential of the ions in the suspension can be fixed, since it mustbe equal to that of the reservoir. The chemical potential of the ions in the11

eservoir is:µ res± = k B T ln(ρ s Λ 3 ±), (3.1)where there ions are treated as an ideal gas of density ρ s , and Λ 3 ± is the(irrelevant) thermal wavelength of the ions. Since the reservoir is chargeneutral the macroscopic electric potential in the reservoir can be set to zero.In the suspension this potential can be expected to be nonzero, because ofthe colloids and ρ ± ≠ ρ s . Therefore, the chemical potential of the ions in thesuspension is denoted by:µ sus± = k B T ln(ρ ± Λ 3 ±) ± eψ, (3.2)where ψ is this electric (Donnan) potential. By demanding that equation(3.1) is equal to equation (3.2) one finds:µ res± = µ sus± → ρ ± = ρ s exp[∓φ], (3.3)where we use the dimensionless Donnan potential φ ≡k B. Next we demandglobal charge neutrality, such that the colloidal charge density Zρ isTcompensated by the ionic charge:which results in combination with equation (3.3):Zρ = ρ + − ρ − , (3.4)− sinh φ = Zρ2ρ s≡ y. (3.5)Now consider the osmotic equation of state in the ideal-gas approximation:P = k B T (ρ c + ρ + + ρ − − 2ρ s ). Its dimensionless form is:eψP ∗ ==P Z2ρ s k B T = y + Z(cosh φ − 1) = y + Z(√ 1 + y 2 − 1) (3.6)⎧⎨ y, y ≪ Z −1 ;Zy 2 /2, Z −1 ≪ y ≪ 1;(3.7)⎩(Z + 1)y, y ≫ 1.Although this system is not a one component system, we will use ’hydrostaticequilibrium’ from section B.2, where P (ρ) is now the osmotic equation ofstate. From dP/dρ = k B T dP ∗ /dy, we get:⎧1⎨ exp(− x−x 2), x > x Z L 2;y(x) = 1 − x−x 1ZL⎩, x 1 < y < x 2 ;(3.8)exp(−x ), x < x (Z+1)L 1.12

The integration constants are found by imposing continuity at the crossoversbetween the regimes. The constants x 1 and x 2 are defined such that y(x 1 ) = 1and y(x 2 ) = Z −1 , and satisfy x 2 − x 1 = (Z − 1)L. For φ(x) we find thefollowing solutions, remembering − sinh φ = y:⎧⎨ − 1 exp(− x−x 2), x > x Z L 2;φ(x) = −1 + x−x 1ZL⎩, x 1 < x < x 2 ;− ln 2 + exp(− x−x 1), x < x (Z+1)L 1.(3.9)where we used sinh φ ≈ φ for |φ| ≪ 1, and sinh φ ≈ − 1 exp(−φ) for φ ≪ −1.2More explanation can be found in ref.[8]. The most important points are theconditions (charge neutrality, hydrostatic and Donnan equilibrium, ideal gas)and the results, equations (3.8) and (3.9). This ’Donnan-approach’ predictsstrong deviations from the barometric height distribution of the density. Thesolution for y(x) and φ(x) is divided into three regions. The region where y 2ρ s , the barometric distributionis corrected with L → (Z + 1)L. However, this last region should be takenthe least serious of all, since we used the ideal gas approximation. The greatbenefit of this method is that it is a fast, clear and straightforward way toget an idea of the equilibrium properties. Unfortunately, the Donnan methodmakes use of local charge neutrality, thanks to the combination of equation(B.16) with equation (3.7). This leads to an internal inconsistency, sincethe electric field, φ ′ (x) needs a local violation of charge neutrality. Also, thesolution for φ(x) is discontinuous. In the next subsection these problems willbe solved, at least numerically.3.3 Poisson-Boltzmann-methodAnother way to determine the equilibrium properties of this suspension iscalled the Poisson-Boltzmann-method, named after the famous equation thatis the key to our solution. The first steps follow the Donnan-method, untilthe inconsistent osmotic equation of state, P [ρ]. It will be corrected withthe so-called Maxwell stress tensor τ to allow for a local net charge. Thistensor is given by τ mn = (ɛ/4π)[−E m E n − B m B n + 1 2 δ mn(E 2 + B 2 )], and theintegral of τ over a closed volume equals the force on the enclosed volume[19]. Here, ɛ is the dielectric constant, E and B are the electric and magneticfield components, and δ mn is the Kronecker delta, respectively. In our case,a system in equilibrium, with a symmetry in the plane perpendicular to the13

x-axis, only τ xx = −(ɛ/8π)E 2 x is non-vanishing. The total force Π = P + τ xxtogether with E x = −ψ ′ (x) becomesβΠ(ρ c (x), φ(x)) = ρ c (x) + 2ρ s (cosh φ(x) − 1) − 18πλ B(φ ′ (x)) 2 . (3.10)We again use hydrostatic equilibrium (B.16), with P replaced by ΠdβΠ(ρ c (x), φ(x))dx= ρ c ′ (x) + 2ρ s φ ′ (x) sinh φ(x) − 14πλ Bφ ′ (x)φ ′′ (x) = − ρ c(x)L .To solve the force balance, we use the Poisson-Boltzmann equation 1 .(3.11)φ ′′ (x) = κ 2 sinh φ(x) + 4πλ B Zρ c (x). (3.12)The differential equation now simplifies todβΠ(ρ c (x), φ(x))ρ ′ c (x) − Zρ c (x) = − ρ c(x)dxL . (3.13)After integration of equation (3.13) the Boltzmann distributionρ c (x) = ρ 0 exp[− x + Zφ(x)] (3.14)Lis found, where ρ 0 is determined by the total amount of colloids in the suspension.Equation (3.14) is a direct generalization of the barometric distribution(B.17) to include electrostatic interactions. While (B.17) is an explicit formfor the density profile, (3.14) is an equation for two unknown profiles ρ c (x)and φ(x), and needs at least a second equation to be solved, the Poisson-Boltzmann equation (3.12).The great benefit of the Poisson-Boltzmann-method is that it does not containthe inconsistencies of the Donnan-method. Only global charge neutralityis demanded, and the solutions of ρ c (x) and φ(x) are not restricted to specifieddensity or spatial regimes (with discontinuous jumps). The disadvantageis that profiles can only be found numerically. Some properties however canstill be seen analytically, but that will remain a secret until the next chapter.. .1 combination of Poisson’s equation (B.7) with the Boltzmann distributions of the salt,equation (B.6)14

Chapter 4Sedimentation of a polydispersemixtureAt last! I thought you’d never begin. . .Let’s proceed. It is getting interesting now! We’ve had theouverture’, we’ve been introduced to the main characters, their backgroundand the scene. We saw ions, charged and Brownian, only interested in theirentropy and in neutrality. We saw lazy colloids, impenetrable and hard, tornbetween the same feelings as the salt, obsessed by chaos and the ’other side’,and still a preference of direction because of their heavy nature. And hidden,though omnipresent, the medium, firing the Brownians with its temperature,and tempering the Coulombic desires with its dielectric cloak, the backgroundthat lightens our colloids. They came all together. . . and found balance. Thecommunity of colloids was homogeneous and of one kind. Things are stirring,new intentions, new groups, and all for the good of course! But will theyfind it? What will happen if two families meet. . .or 20 families!!4.1 The systemInspired by earlier examples, we are now going to observe suspensions thatcontain multiple colloidal spheres of different kinds. They can be distinguishedby their charge, mass or diameter, or combinations of them. Theability to adapt their charge, called charge regulation, will follow later. Thatwill complicate the situation even more.The colloidal components (’families’) in the system are numbered by theindex i, n is the total number of components. The colloids have a charge−Z i e, gravitational length L i and diameter σ i . The colloid number density15

is denoted by ρ i . The salt ions have a charge ±e and number density ρ ± .The medium is responsible for a Bjerrum length λ B , the height of the systemis H. The suspension is in Donnan-equilibrium with a neutral salt reservoir,with a total salt concentration of 2ρ s . The system must sound veryfamiliar, after section 3. With the named section in mind, we can approachthe system in two ways, using either the Donnan-method or the Poisson-Boltzmann-method. The first method proved to be, in the one componentcase, a fast and clear way to obtain an explicit form for the number density.For mixtures, this method did not prove very successful, therefore we willstart with a PB 1 -theory. The system parameters that will be standard in thefollowing sections are denoted in table 4.1 below.Table 4.1, standard system parametersParameter Number Units Commentσ i 150 nm colloidal diameterλ B 2.3 nm ethanol at room temperatureH 20 cm hand size systemρ s 3 µM interesting region (see 3.2)¯η tot 0.005 low packing fraction4.2 The functional and minimum conditionsSince the Donnan-approach did not prove very successful to us in solving theequilibrium properties of the system, neither numerically nor analytically,another more fundamental approach is chosen. The potential and densitieswill now follow from a mean field grand potential functionalβΩ[{˜ρ(x)}] = ∑++n∑i=1n∑i=1− ∑α=+,−1 =Poisson-Boltzmann∫ H0∫ H0∫ H0α=+,−∫ H0dx ˜ρ α (x)(ln ˜ρ α (x)Λ 3 α − 1) +dx ˜ρ i (x)(ln ˜ρ i (x)Λ 3 i − 1)dx x L i˜ρ i (x) + λ B2dx βµ α ˜ρ α (x) −∫ H0n∑i=1dx Q(x)φ(x)∫ H0dx βµ i ˜ρ i (x) (4.1)16

with Q(x) = ρ ˜ + (x) − ρ ˜ − (x) − ∑ N∫ i=1 Z i ˜ρ i (x) is the dimensionless total chargedensity, and φ(x) = −2πλ B dx ′ Q(x ′ )|x − x ′ | the dimensionless potential,equivalent to (B.1)-(B.4). This functional will be the equilibrium grand potentialfor the equilibrium distribution ρ i,α (x) that satisfyδΩ[ ρ˜i,α (x)]∣= 0 (4.2)δ ˜ρ i,α (x)∣∣˜ρi,α (x)=ρ i,α (x)with ρ i (x) and ρ α (x) the equilibrium densities of the components, micro-ionsrespectively.The first two terms on the right hand side of equation 4.1 are ideal gascontributions, for the colloids and the salt ions respectively. The third termrepresents the external potential of the earth’s gravity field, interacting withthe colloids, but not with the much lighter co-ions. The fourth term accountsfor the coulomb interactions between all charges. For simplicity a mean fieldapproximation is made. In this way the particles behave like an ideal gas insome self-consistent electric potential φ(x).By applying condition(4.2) one finds the following equations for the unknownρ ± and ρ i (x):ln ρ ± (x)Λ 3 ± − βµ ± ± φ(x) = 0,with µ ± = kT ln ρ s Λ 3 sln ρ i (x)Λ 3 i − βµ i − Z i φ(x) + x L i= 0, with µ i = kT ln a i Λ 3 i ,with a i ≡ 1 Λ 3 ie βµ iis called the activity, of fugacity of colloidal component i.Thanks to the salt reservoir, we can fix the chemical potential of the saltµ ± = µ s , and the equilibrium density profiles are found to satisfy:ρ i (x) = a i exp ( − x L i+ Z i φ(x) ) (4.3)ρ ± (x) = ρ s exp ( ∓ φ(x) ) (4.4)Now, by observing that d 2 φ(x)/dx 2 = −4πλ B Q(x), and using the solutions(4.3) and (4.4) we arrive at the Poisson-Boltzmann-equation:d 2dx 2 φ(x) = κ2 sinh φ(x) + 4πλ Bn∑Z i ρ i (x) (4.5)with κ −1 the Debye screening length of the reservoir. The suspension isglobally charge neutral, which is provided by the applied Neumann conditionsφ ′ (x) = 0 at the boundaries of the suspension (x = 0 and x = H). From (4.3)and (4.5) we can solve φ(x) and ρ i (x) numerically on an x-grid, i.e. by using17i=1

numbers for the parameters Z i , L i , σ i , ρ s , λ B and a mean density ¯ρ i to fix a i .By using an iteration method, shown and discussed in chapter 6, a common,four-year-old computer can calculate the solution for a polydisperse systemof n = 20 components and H = 20cm in a minute. We will now explore somenumerical results, before we continue the analytical path for which we needassumptions.4.3 Binary systemsWe will start with a binary system, i.e. a system with n = 2 colloidalcomponents. The reason for this choice is that, considering these systems,the complexity is still reduced, while several multi-component phenomenacan already be examined. The next examples will show if the componentswill form layers, if they will order, and how their profiles are influenced whenthey are put together.After careful study three binary systems were selected for comparison. Thegravitational length are fixed at L 1 = 1mm, L 2 = 2mm and Z 1 = 250. Thecharge number Z 2 , however, will be varied, in case (a) Z 2 = 125, in case (b)Z = 500, in case (c) Z = 1000. The calculated density profiles are shownbelow in figure 4.1, in terms of the so-called packing fraction, a dimensionlessdensity that expresses the ratio of the volume that is occupied by the colloids,η i (x) = πσ3 i6 ρ i(x). (4.6)The particles of component 2 are twice as heavy as the particles of component1, and on the basis of their gravitational length, component 2 is expected tobe on the bottom. However, due to their charge ratio, other situations mayoccur as well, as is shown. Figure 4.1a shows a strong segregation of thecomponents, not a mixture of barometric distributions like our atmosphere.Although the heavier component 2 is below component 1, the situation isreversed in figure 4.1c. Apparently independent of mass, the colloidal componentscan live together as well in figure 4.1b.How can this be?We cannot see this directly from theory, the ’Donnan-method 2 ’ needed toomany assumptions and the ’PB-method 3 ’ could not be solved analytically.An explanation can still be found after an examination of φ(x), or equivalently,the salt profiles, and related, the electric field (see figure 4.2)2 see section 3.23 see section 3.318

20.02η i(a)0.02(b)0.01η i0.012100 5 10 15 20x (cm)00 5 10 15 20x (cm)10.021(c)η i0.01200 5 10 15 20x (cm)Figure 4.1: Case (a): Density profiles with Z = (250, 125). The heavier component2 is on the bottom. Case (b): Density profiles with Z = (250, 500).The components do not segregate. Case (c): Density profiles with Z =(250, 1000). The lighter component 1 is on the bottom.The crossovers in the electric field and the cusps in the salt profiles caneasily be related to the density profiles from theory. The Donnan-methodpredicted that linear profiles go together with linear potentials, i.e. constantelectric fields: −Zρ/2ρ s = sinh φ ≈ φ and ⃗ E(x) = −(k B T φ ′ (x)/e, 0, 0). Thepredicted slope of the density profiles was (4.35): ρ ′ (x) = −2ρ s /Z 2 L, andhence, the electric field isE x = k BTeφ′ (x) ≈ − k BTeZρ ′ (x)2ρ s= mgZe . (4.7)Note that the resulting electric force on the colloids exactly cancels the gravitationalforce, ZeE x = mg.Comparable conclusions follow from the PB-method. Using the solution (4.3)for ρ i (x), the maximum of ρ i (x) is found to be at x i , wheredρ i (x)dx(∣ = − 1 + Z i φ ′ (x))ρ i (x)) ∣ ∣∣x=xi = 0. (4.8)x=xiL i19

2Case (a)4.5Case (a)ρ +E (mV/cm)1.510.5ρ (µM)Case (b)31.5Case (c)ρ −0 5 10 15 20x (cm)00 5 10 15 20x (cm)Figure 4.2: The electric field of the three cases (pointing upwards), and thesalt profiles of case (a) (inset). Case (b) has an almost constant electric field,while case (a) and (c) both show a sharp crossover at the height where theircomponents form an interface. At this same height the salt profiles show acusp, at least on the cm length scale.Since ρ i (x i ) ≠ 0 at its maximum, we haveφ ′ (x i ) = 1 ⇒ E = mgZ i L i Ze(4.9)So, if ρ i (x) has a maximum at 0 < x i < H, the electric field must beE x (x i ) = mg/Ze. For many models using hard-core particles this maximumwill always occur, but since we’re considering our components as continuousfields, components can have their (local) maximum at the boundaries. Therelation between φ and ρ in the linear regimes becomes visible when weexpand (4.3):[ ( xρ i (x) = a i 1+ − +Z i φ(x) ) + 1 ( x − +Z i φ(x) ) (2 (− x +O +Z i φ(x) ) ) 3].L i 2 L i L i(4.10)For ρ i (x) to be (approximately) linear, φ(x) must satisfyφ(x) =x −ɛ x + c, (4.11)Z i L i Z i L i20

where ɛ ≪ L i /H is an unknown, but small constant, and c is a constant tobe determined by boundary conditions. This yieldsE x (x) = − k BT φ ′ (x)e= k BTe1 − ɛZ i L i= mg (1 − ɛ). (4.12)ZeFor the realistic case, L i ≪ H this will approximately be E ≈ mg/Ze. Thelinearity of the salt profiles can also be related to the theory, sinceρ ′ ± (x) = 2ρ s φ ′ (x)e ∓φ(x) 1 x≈ 2ρ s + O( ), (4.13)Z i L i (Z i L i )2where the constant c is neglected, since it does not influence linearity.The colloidal Brazil nut effect The colloidal Brazil nut effect is the namefor a phenomenon where smaller components move below larger components.The words ’smaller’ and ’larger’ can apply to size as well as mass. Thenomenclature stems from the field of granular matter, but was recently alsocoined in a colloidal context.We will now have a closer look at this effect. What is the origin, the drivingmechanism en when does it occur. Inspired by earlier work by Löwen andEsztermann [15], we will examine suspensions following the recipe of [15].We will calculate a series of binary mixtures, where Z 2 , the charge number ofthe colloidal particles of component 2, will (again) be the varying parameter.Tabular 4.3, simulation parameters of [15]Parameter Number Units Nameσ 1 = σ 2 150 nm colloidal diameterL 1 10 σ grav. length of comp. 120L 2 σ id. of component 23Z 1 15 charge number of comp. 1H 1000 cm height1λ B σ Bjerrum length128π¯η 1 = ¯η 2 6 10−4 mean packing fractionThese parameters were originally chosen for Monte Carlo simulations byLöwen and Esztermann. Z 2 takes values between 15 and 45. We performeda series of 40 calculations and compared them with the simulation data. Themean height h i of component i, defined ash i =∫ H0 dx x ρ i(x)∫ H0 dx ρ i(x) , (4.14)21

is plotted as a function of Z 2 /Z 1 in figure 4.4. The large difference betweenh 1 and h 2 indicates a strong segregation (see figure 4.3), whereas a smalldifference means that the regions overlap almost completely.0.01a1.50.0080.006i=1ρ salt(µM)10.5ρ +η i ρ −0.00400 200 400 600 800 1000X/σ0.002i=200 200 400 600 800 1000X/σFigure 4.3: Colloidal packing fractions and micro-ion distribution (inset).The co-ion distribution cannot be distinguished from the horizontal axis. Thesuspension is therefore considered deionized. For this specific case Z 2 /Z 1 = 3,a strong segregation takes place as can also be seen from the significantdifference in h i , i.e. say h 2 /h 1 > 2. Compare with figure 4.4 for an indication.From figure 4.4 the colloidal Brazil nut effect is clearly visible for Z 2 /Z 1 ≥1, 6. Figure 4.3 shows the density profiles of such a case. Nevertheless, it is aphenomenon that can only be seen for specific parameters. Not only need thecondition for the mass per charge hold Z heavier L heavier > Z lighter L lighter , themean charge density of the colloids need also be bigger than the mean chargedensity of the co-ions ∑ i ¯ρ i > ¯ρ − for a clear separation. In other words, it isdesirable to have a very low concentration of salt in the reservoir ρ s ≈ 0 tosee the effect. In Monte Carlo simulations this condition is easily met, evenpreferred, since the inclusion of additional co-ions slows down the calculations.The results of [15] were obtained from simulations without co-ions.From our theory (4.2) we cannot consider this situation, strictly speaking,since the salt is seen as a grand-canonical ensemble. A concentration ρ s > 0is needed to provide global charge neutrality, but also includes the existenceof co-ions in the solution. We approached their situation by choosing a verylow ρ s , in order to get a very low co-ion concentration ¯ρ − ≪ 10 −3 · ¯ρ + (seeinset figure 4.3). Although the methods differ, since the simulations do notmake a mean field approximation (sec. 2.3.2), nor an ideal gas assumption,the results seem to coincide rather well. We conclude from this agreementthat the volume effects of the colloids can be neglected, at least for these lowdensities, since the simulations include them and our theory not.22

300bi=2h i/σ200100i=101 1.5 2 2.5 3Z 2/Z 1Figure 4.4: Mean height h i of the two species for several values Z 2 , showinga density inversion for Z 2 /Z 1 > 1.6 (where h 2 > h 1 , the heavier particlesof type 2 are on top of the lighter particles of type 1). The curves arepredictions of the present PB theory (sec. 4.2), the symbols are simulationdata of Esztermann and Löwen [15].4.4 Ternary systemsWe want more proof. More pictures.The statements (4.7)-(4.13) can be checked for the three binary mixtureswe already considered, but why not mix them all together? The result isa ternary mixture with Z = (125, 250, 1000) and L = (1, 2, 1)mm. Thesuspension will be adjusted such that the mixture is equimolar with a meanpacking fraction ¯η i = 0.005.Figure 4.5 shows an example for which the theoretical predictions of theelectric field (equation (4.12)) and the slopes of the density profiles (equation(4.35)) match with the numerical calculations (equations (4.3) and (4.5)).Another example shows the agreement with simulation data. The standardparameters 4.1 are not very suitable for simulations. Especially the lengthscales are chosen to be smaller. The diameter of the particles is still taken tobe σ = 150nm, the system height H is 100σ the gravitational length is L =(4, 6, 6)σ, and the colloidal charge is Z = (50, 20, 10). The salt concentration23

0.1i=320a0.080.06η i0 5 10 15 20ρ salt(µM)15100.04i=25X (cm)0.02i=100 5 10 15 20X (cm)210 −6bm 3g/Z 3eE (mV/cm)1.510.5Q/2ρ ssinhφ(x)−10 −6 00 5 10 15 20X (cm)m 2g/Z 2em 1g/Z 1e00 5 10 15 20X (cm)Figure 4.5: Density profiles (a) and the electric field profile (b) of anequimolar ternary mixture of equally-sized colloids with charges Z i =(1000, 250, 125) and gravitational lengths L i = (1, 2, 1)mm with total packingfractions ¯η i = 0.005 , with monovalent ionic strength ρ s = 10µM in thereservoir. Almost complete segregation takes place into well-defined layers ofpure components, with a slope η i(x) ′ = πρ s σ 3 /3Zi 2 L i and a constant electricfield m i g/eZ i in the layer with component i, as denoted by the dashed linesin (a) and (b), respectively. The inset in (b) shows a dimensionless measurefor local charge neutrality.24

in the calculations is chosen to be un-physically low (ρ s < 1pM), to imitatethe coion-less situation of the simulations.0.030.02η i0.0100 50 100x (σ)Figure 4.6: An agreement with Monte Carlo simulation data (wiggly lines)by [22]. The diameter of the particles is taken to be σ = 150nm, the systemheight H is 100σ the gravitational length is L = (4, 6, 6)σ, and the colloidalcharge is Z = (50, 20, 10). The salt concentration in the calculations is chosento be un-physically low (ρ s < 1pM), to imitate the coion-less situation of thesimulations.4.5 Polydisperse systemsEven more exciting examples. . .Now we will compare two polydisperse systems, example 1 and example 2.The fixed system parameters can be found in table 4.1. The two examplescan be discerned by the mass per charge ratio Z i L i of the particles.Example 1, figure 4.7, shows a system of 21 components of equal mass, suchthat L i = 2mm. The colloidal charge number Z i however varies between 150and 350 according to a Gaussian function:¯η i = ¯η mean · exp[− (Z i − Z mean ) 22∆ 2 ] (4.15)¯η mean is a normalization constant that provides the total mean packing fractionto be 0.005, ∆ ≈ 50 is the standard deviation, and Z mean = 250 themean charge number.25

Z 16=2993a0.6210 3 η iZ 11=2500.40.20150 200 250 300 350Z i1Z 14=27900 5 10 15 2010 3 η iX (cm)Figure 4.7: Example 1: Density profiles of a 21-component colloidal suspensionwith a Gaussian charge distribution as illustrated in the inset anddescribed by (4.15). The components have the same gravitational lengthsL i = 2mm.Example 2, figure 4.8, has the same charge distribution as example 1. Thegravitational length is scaled with the colloidal charge, L i ∝ Z − 3 2i . Thisexample describes a system with different particles, that have a constantmass and charge density, L i /σi 3 = L j /σj 3 resp. Z i /σi 2 = Z j /σj 2 . Except forthe gravitational lengths L i , the system is identical to example 1, to makethem comparable.The results of example 1 show layering of components, with the highestcharge on top. The results of example 2 do not show a layering. This canbe explained from the difference in mass per charge of the components. Inexample 1 the ratio isZ i L i= Z i, (4.16)Z j L j Z jwhereas in example 2√Z i L i Zj= . (4.17)Z j L j Z iTherefore, the variation of the mass per charge in example 2 is much smaller,as we can see in figure 4.8. The total packing fractions are very close to the26

packing fraction of a monodisperse system with colloids, having Z = 250 andL = 2mm, as is shown in the inset of figure 4.8. One could conclude from the 2examples that a small distribution of charge and/or mass, does not influencethe equilibrium properties, except for the distribution of components. Inexperiments the distributions can have a width of about 5%, compared tothe width of 20% here.1.51b11010 3 η tot 25010 3 0 5 10 15 20η i X (cm)150.500 5 10 15 20X (cm)Figure 4.8: Example 2: Density profiles of a 21-component colloidal suspensionwith a Gaussian charge distribution and a distibution for the gravitationallength L i = ( Z 0 2L 0 . The inset shows a comparison between thetotal packing fractions of examples 1 and 2 and a monodisperse system with{Z, L} = {250, 2mm}.Z i) 34.6 Explanation of phenomenaConsidering the results of chapters 4.3 - 4.5, we immediately see two majordifferences with ideal gases,• the highly non-barometric profiles,• the segregation of colloidal components.27

0.02210.83E x(mV/cm)0.60.40.0140.256078η i0 20 40 60 80 100x (cm)91000 20 40 60 80 100x (cm)Figure 4.9: Packing fractions of a suspension with 10 different types of colloids.The charge number Z n = n · 250, while the gravitational lengthL n = 0.1. The other parameters of the system can be found in tab. 4.1,except for the height H = 100cm, instead of 20cm. The inset shows theelectric field strength.The origin lies in the presence of Coulomb forces. Systems of charge neutralgases only need to find a balance between entropy and gravitational energy.Systems of charged particles need to find a balance with electrostatic energyas well, before reaching equilibrium. We will explain this balance in a slightlydifferent way.Non-barometric profiles The colloids prefer a barometric distribution.The salt prefers a homogeneous distribution. Charge neutrality prefers thecolloids to be mixed with sufficient counter-ions. A compromise is made bylifting the colloids, and shifting co-ions up and counter-ions down, leaving stilla (small) net electric field in the fluid. Depending on the salt concentration,the colloids can seriously be lifted in a situation with very few micro-ions.This lifting effect is caused by an electric field, generated by a thin layer ofcharge at the bottom and top of the distribution. Salty suspensions, likeseawater, do not show these effects, because only a small fraction of thecounter-ions can screen all colloids, resulting in (almost) barometric profiles28

0.0412130.034E x(mV/cm)0.80.60.40.2η i0 5 10 15 200.02506x (cm)70.01810900 5 10 15 20x (cm)Figure 4.10: Packing fractions of a suspension with 10 different types ofcolloids. The system has the same properties as in the case of figure 4.9,except for the system height H = 20cm, instead of 100cm. The colloids,having less space to distribute, favor overlapping regions instead of narrowlayers. The inset shows the electric field strength.of the colloids. In almost salt-less suspensions this would cost too muchentropy. A smart balance is found by creating a condenser by the salt, andsometimes even by some colloids (see fig. 4.1).Segregation A suspension containing identical colloids and only a littlesalt has a homogeneous electric field in equilibrium, as we saw many timesbefore, and explained above. If a second component, with a lower mass percharge, would be added, these colloids would float to the top of the first componentdue to the stronger interaction with the electric field. A componentof higher mass per charge would sink. This can be seen as the origin of thelayering.The next sections provide some additional information about these two effects,and about the influence of the parameters.29

4.6.1 The influence of the diameterThe volume of the colloids plays no part in the theory of section 4.2. Changingthe diameter σ i will therefore not lead to different solutions for the densityprofiles and potential. The origin of the independency of size lies in the densityregion for which the theory applies. Since the density is low, the colloidsare considered an ideal gas, interacting as point particles, although having aspecific size σ. As already shown in figures 4.4 and 4.6, this approximationis not that bad, when comparing to simulations that do include size. Althoughthe density profiles ρ i (x) and the potential φ(x) do not change, thedimensionless packing fraction η can have spectacular solutions, though. Intheory, every colloidal family with charge number Z, gravitational length Land diameter σ, can be positioned at a desirable height, by adding ’invisible’small components of other colloids. Also the dimensionless density distributioncan be adjusted as well, by adding the right components. This maysound remarkable, but it is nothing more than a magic trick. The diameterof the particles is just a scaling factor for the density profiles. By choosing avery low diameter for some components it looks like if the others ’float’ in thesuspension. An example, figure 4.11, shows the deceitful packing fraction.0.02110.80.012E x(mV/cm)0.60.40.2η i0 20 40 60 80 100304x (cm)500 20 40 60 80 100x (cm)Figure 4.11: An example of a 10 component system. Components 1 - 5 havea particle radius σ big = 1500nm. Components 6 - 10 cannot be seen, becausetheir radius σ small is too small: σ small ≪ 0.01σ big . The number density isidentical to the number density of figure 4.9. The packing fraction though,looks spectacular. The diameter of the particles is just a scaling factor of thepacking fractions in our theory.It remains a question for experimentalists how big they can make their particlediameters in order to get results like figure 4.11. For our theoreticalobservations though, this factor does not play a role until we are going to30

observe higher densities, where excluded volume becomes important.4.6.2 The influence of the saltIn all sections the salt concentration is chosen to be either ρ s = 3µM, orρ s =< 0.1nM. The first value is a very convenient concentration to showinteresting phenomena like the ’colloidal Brazil nut effect’ (4.3), the layeringof components, linear profiles and many other phenomena discussed sofar. The latter concentration(s), ρ s < 0.1nM, shows the effects somewhatstronger, but cannot easily be realised in experiments since e.g. water alreadyhas a minimum concentration of 0.1µM of ions. It is chosen however,to imitate a system of colloids in the presence of counter-ions only, as oftenused in simulations or in anorganic solvents.Two examples will show an effect we already knew from theory. By increasedsalt concentrations the density distributions collapse, while the electric fielddisappears, see figure 4.12. As well, the segregation disappears, as shownfor a binary system in figure 4.13. While decreasing the salt concentrationthe mean heights of the components tend to a constant. This constant isdetermined by the system height H. The mean hight of component i in thehigh salt limit is equal to the gravitational length h i = L i , as it should be(equal to a barometric distribution). The region where ρ s < 1 2 Z2 ρ, wherenon-barometric profiles are expected, is below ρ s < 3 · 10mM.4.6.3 The layer widthThis section is a consideration of the layer width of the profiles, i.e. thelength or height over which the colloids of a certain species extend. It will bedenoted by ξ i . Most arguments will be qualitative, since an exact knowledgeof ξ i requires analytical solutions for the density profiles or, equivalently, thepotential. Most attention will be paid to the so-called linear regimes, i.e.where Z −1 ≪ y ≪ 1, with y = Zρ/2ρ s (see equations (3.7)).The width of the regime where the density profiles decay linearly decreaseswith the number of components, as could already be seen by comparingmonodisperse with binary suspensions. If one proceeds to add components,these linear regimes disappear at a certain point, due to entropy. The numberof components that can be added until the linear profiles break down, dependson the height of the suspension and the mass per charge of the components.This loose argument cannot be made rigorous, though from a physical point ofview this can still be understood: as explained, layering occurs at sufficientlow salt concentrations. As an effect of the layering, the lower colloidalcomponents are compressed by the weight of the higher components (see31

0.01ρ s=10 −3 Mρ s=10 −5 Mρ s=10 −6 Mρ s=10 −7 Mη0.00500 5 10 15 20x [cm]Figure 4.12: Imitation of figure 3 from [8]. Shown are the predicted packingfraction profiles η(x) = (π/6)ρ(x)σ 3 , in a sample of hight H = 20 cm, with anaverage packing fraction of 0.0005 for several salt concentrations ρ s . Theyare calculated by the Poisson-Boltzmann-method. The colloidal charge isZ = 200, the Bjerrum length λ B = 2.3nm (ethanol at room temperature),the gravitational length L = 2mm, and the colloidal diameter σ = 150nmfigures 4.14 and 4.15). If the total distribution reaches the top of the system,this effect will become stronger by entropy. In other words, by additionthe components would be restricted to increasingly narrow layers, which isenergetically expensive, due to the low entropy. Instead, the componentsform layers that overlap each other forming gaussian like density profiles,rather than linear ones. Related to the density, the electric field will not be astepwise function, but a smooth decaying function with height. Figure 4.16clarifies these arguments (earlier described in figures 4.9 and 4.10).4.6.4 The slope modelWhen the computers fail, it is always nice to have a method that can be doneby hand.The following model approximates the numerical results based on the Poisson-32

15i=110h i(cm)5i=200 2 4 6 8 10− 10 logρ sFigure 4.13: The mean height as a function of the salt concentration. Systemparameters are the standard ones (tab. 4.1), The colloidal charge numberZ 1 = 2Z 2 = 500 and the gravitational length is L 1 = 2L 2 = 2mm.20001500Z100050000 L 2 4 6 8 10h (cm)Figure 4.14: Mean height of a monodisperse suspension, as a function of thecharge number. For the fixed parameters see tab.4.1. Further, the gravitationallength L = 1mm, and Z varies between 0 and 1800. For Z = 0 themean height of the barometric distribution is found, h = L. The point wherethe colloids reach the top of the suspension, when h ≈ 7, the dependence isno longer linear, but goes asymptotically to 1 H = 10cm. The total width of2the distribution is about 3h, valid for linear density profiles and 3h < H.Boltzmann theory. For specific conditions it offers analytical expressionsfor the packing fractions and layer widths. For polydisperse mixtures thesewidths are found by a recursion relation. The assumptions that are neededare• The density profiles are linear, or equivalently, the electric field is con-33

0.03108642η i 0.01500 5 10 150.03108642η i 0.01500 5 10 15η 2/η 1η 1=20η 20 5 10 15 20η 2/η 1η 1=η 2h ih i00 5 10 15 20x (cm)0x (cm)Figure 4.15: The addition of a second component. Charge numbers areZ 1 = 1Z 2 2 = 250, gravitational lengths are L 1 = 1L 2 2 = 1mm. The meanpacking fraction of component 1 is fixed, ¯η 1 = 0.005, while the mean packingfraction of component 2 varies between 0.1¯η 1 < ¯η 2 < 10¯η 1 . The dotted lineis the packing fraction of component 1 in the absence of component 2. Theinsets show the mean heights of both components and the total mean height(see also eq. A.5).0.0223E x(mV/cm)10.80.60.4η 0.2i 0.01 4050 20 40 60 80 100x (cm)67891000 20 40 60 80 100x (cm)0.040.03123E x(mV/cm)0.80.60.440.2η i 0.02500 5 10 15 206x (cm)70.018109100 5 10 15 20x (cm)Figure 4.16: Packing fractions of a suspension with 10 different types ofcolloids. The charge number Z n = n · 250, while the gravitational lengthL n = 1mm. The other parameters of the system can be found in tab. 4.1,except for the left figure, height H = 100cm, instead of 20cm. The insetshows the electric field strength.stant.• Overlap of regimes can be neglected, segregation is strong. That is,34

2ρ s ≪ Z 2 i ρ i , the suspension height H is larger than the layer width ξ,and the charge per mass differs strongly per component Z i L i ≪ Z j L jfor all i < j < n, with n the number of components.For simplicity, consider an equimolar mixture, with mean packing fraction¯η i = ¯η. Call x i the height at which two components i and i + 1 form aninterface. Number the components from highest to lowest mass per charge,i.e. Z 1 L 1 < Z 2 L 2 < . . . < Z n L n . The form of the packing fractions is, byassumption,η i (x) = b i − c i x (4.18)where the slope c i = ρ s πσ 3 /3Z 2 i L i of η i is calculated from equation (3.8).Force balance (see equation (3.10)) dictatesβΠ(x i ) = limx↑xiρ i (x) + 2ρ s (cosh φ(x) − 1) − 18πλ B(φ ′ (x)) 2= limx↓xiρ i+1 (x) + 2ρ s (cosh φ(x) − 1) − 18πλ B(φ ′ (x)) 2 . (4.19)Now we use that the potential is continuous, whereas the electric field makesa jump (use equation (4.12)):φ ′ (x) ≈ 1Z i L i, for i ∈ I i , (4.20)where the approximation is justified by the assumptions, and I i is the intervalon which η i is non-zero. From equation (4.19) we now obtain:ρ i (x i ) − 1 ( 1) 2= ρi+1 (x i ) − 1 (1) 2(4.21)8πλ B Z i L i 8πλ B Z i+1 L i+1The jumps in the packing fraction are therefore((η i (x i ) − η i+1 (x i ) =σ3 1) 2 (1) 2 )− ≡ t i . (4.22)48λ B Z i L i Z i+1 L i+1Where t n is just a convenient abbreviation, called the jump constant. If oneη i (x) can be found, all packing fractions can be found by recursion. Thecomponent with the lowest mass per charge is expected on top, labelled byn. For b n the following relation applies:1H∫ xn +ξ nx ndx()b n − c n (x − x n ) = ¯η, (4.23)where ξ n is the layer width of component n. The constant b n is now foundto be:b n = √ 2H ¯ηc n . (4.24)35

The next constant that can be solved is b n−1 . From the relation between b n−1and b n , using equation (4.22),where ξ n−1 can be found by the normalisation1H∫ xnx n −ξ n−1dxb n−1 = b n + t n + c n−1 ξ n−1 (4.25)()b n + t n − c n−1 (x − x n ) = ¯η (4.26)we find an expression for b n−1 and thus, for all b i . By repeating the stepsfrom equation (4.24) we find expressions for all constants b n and layer widthsξ i√2Hρ s¯ηπση(x i ) = b n =3+ (b3ZnL 2 n+1 + t n+1 ) 2 (4.27)nξ i = 3Z2 i L iρ s πσ 3 (b i − b i+1 − t i+1 ) (4.28)Figure 4.17 shows a comparison with the full numerical solutions. The dottedlines represent the approximation. The left figure shows an approximationof figure 4.1. The right figure shows the packing fractions of a systemwith colloidal charge number Z = (200, 250, 300), and gravitational lengthL = (0.4, 0.7, 1)mm, and the usual system parameters (see tab. 4.1). Althoughthe slope model calculates the equilibrium properties in an instant,and can even be done by hand, the Poisson-Boltzmann calculations are consideredmore reliable, and do not need the condition of pure layers. Especiallythis last condition is not often met for the systems considered in this thesis.An additional problem with the slope model is the jump constant. Multiplecomparisons with PB-calculations show that it is indeed proportional tot n ∝ ((Z N L n ) −2 − (Z n−1 L n−1 ) −2 ), but the proportionality constant provednot always equal to σ 3 /48λ B .36

0.03(a)0.02η i0.0100 5 10 15 20x (cm)(b)0.02η i00 5 10 15 20Figure 4.17: A comparison between the slope model and the numerical solutionsbased on the Poisson-Boltzmann model. The dotted lines representthe results of the slope model. Figure (a) shows an approximation of figure4.1. Figure (b) shows the packing fractions of a system with colloidal chargenumber Z = (200, 250, 300), and gravitational length L = (0.4, 0.7, 1)mm,and the usual system parameters (see tab. 4.1).37

4.6.5 Donnan-method for polydisperse casesThe micro-ions are considered to be an ideal gas. The micro-ion profilesare determined by chemical equilibrium, and the assumption of a Donnanpotentialψ = k B T φ/e in the suspension,Global charge neutrality is now dictated byρ ± = ρ s exp[±φ] (4.29)n∑Z i ρ i = ρ + − ρ − (4.30)i=1where n is the number of colloidal components in the suspension. Next, thedimensionless densities y i are obtained from (4.29) and (4.30),− sinh φ =n∑i=1Z i ρ i2ρ s≡n∑y i . (4.31)i=1The osmotic equation of state in the ideal gas approximation is P/k B T =∑ ni=1 ρ i + ρ + + ρ − − 2ρ s or in dimensionless formP ∗ ==Pn∑ y i= + cosh φ − 1 (4.32)2k B T ρ s Zi=1 i⎧ ∑ n y⎨i∑i=1 Z i,ni=1 y i ≪ Zmax;−11⎩(∑ n2 i=1 y i) 2 , Z −1min ≪ ∑ n∑ i=1 y i ≪ 1; (4.33)n y ii=1 Z i+ y i , y ≫ 1,slightly different from equation (3.6). We used Z max and Z min for the highestand lowest charge number resp. in the system. The approximations haveeven more differences: the three regimes of (4.33) do not cover all possibledensities, and the dimensionless osmotic pressure P ∗ is expressed as a sumof densities, which makes it impossible to obtain the densities for separatecomponents, when combined with a force balance 4 of the formβn∑i=1dP (ρ i (x))dx= −n∑i=1ρ i (x)L i. (4.34)With some additional assumptions however, solutions can be found. If oneassumes that the colloidal components will form pure layers, the layers can4 Even more, this force balance is not properly defined, while the equation for hydrostaticequilibrium (B.16) is38

e considered monodisperse, and the solutions (3.8) for each component arefound. With an additional condition that the salt concentration is low enoughto obtain linear profiles, and the assumption that ρ tot (x) is convex the followingsolution is found: From (3.8) and Z −1i ≪ y i ≪ 1ρ i ′ (x) = 2ρ sy i (x)Z i= − 2ρ sZi 2L , x ∈ I i , (4.35)iwhere the interval I i has at most, the width of the layer, I i < (Z i − 1)L i .And by the ’convexity assumption’ρ i ′ (x) ≥ ρ j ′ (x ′ ) ⇔ x ≤ x ′ for x ∈ I i , x ′ ∈ I j . (4.36)An example is shown in figure 4.181Artist’s impression0.500 0.5 1Figure 4.18: 4 components. An artist’s impression of the density profiles,following from the ’Donnan-method’. Several assumptions were needed forthe impression, like a convex ρ and the formation of pure layers. The slopesdecrease with height, while the intervals increase.Still, these speculations are based on a desire to solve equation (4.32), possiblyat the cost of experimental accessability, and does not prove a solid theoreticalbasis.The inspired reader may ponder about the differential equationsthat follow from equations (4.33) and (4.34):−n∑i=1y i (x)L i=⎧⎨⎩∑ n 1+Z ii=1 Z iy ′ i (x)∑ ni=1 y i(x)y ′ i (x)∑ Ni=1 y i ≪ Z −1∑ n 1i=1 Z iy ′ i (x) = y i ≫ 1.39max;Z −1min ≪ ∑ Ni=1 y i ≪ 1;(4.37)

and try to fill the holes, or try to solve for n 3 regions, where each density istreated seperately, for example.4.6.6 The colloidal Brazil nut effectReaders that do not like informal language can skip this section and go tothe next.Why?Because of you, and the poor farmer with his heavy cargo.What farmer? Where?Somewhere far from here. He was a happy, but poor man. No time he hadto enjoy the summer, to smell the lovely spring. He had to work all day aslong as the sun gave him its light. And when he went to sleep, he dreamt ofnuts, huge bags, loads, storehouses, seas and oceans of nuts! That’s what hegrew. . .Now he is on his way to the market. His cart can sure use some new loyalwheels, that aren’t weary of the bumpy road. With the help of six smallhands, the load is freed from its last ditch and reaches the market. Lo! Fromthe opened bags, a wonder clouds the place. ”What happened?!” the farmershouts ”My nuts, they’ve grown! No wait! They’ve moved! They put themselvesinto order! The biggest first, the smallest last, and smaller follow thebigger!”. With shaking knees he does not know whom to thank. It wasn’tMother Earth, she likes the opposite order. It was certainly not Entropius,he likes no order at all. And the nuts themselves. . .That day he did not sell his famous ’mixed-nuts’, and began to think aboutthe miracle. He decided to buy a large coat from the money he just earned,the biggest he could find. He put it on and stuffed it with bags of hay andfeathers, until it fitted. His daily work was terrible with all these clothes,and the sun made it even worse. He took a rest in the shadow of a tree.”Good Gracious, you are sweaty, sir!” a well-mannered traveller said ”Areyou alright? Please take something to drink, and join me in this carriage. Icannot let you walk any further.” And so they went to the city. The farmerdrank something he didn’t know but still felt hot. The holes in the roadseemed to dear him far less than the well-mannered traveller. When theyreached the city, the poor traveller asked the carriage to halt for a minuteand ran into a hotel. The farmer did not hesitate to take a breath. ”Are youhere for the conference?” a man with golden ropes and stripes asked him.”Is it shady there?” the farmer asked back. The man with the golden stripeslaughed and said:”Follow me, sir.” The farmer entered a huge hall with chandeliers,champagne, chatting men with clothes he’d never seen before, there40

was lovely cool air, and lobsters! He sure felt hungry. While he started tostuff his mouth, he steadily removed more feathers and stealthy hay. Theyweren’t needed anymore. When there was nothing left he felt sleepy, and wasbrought to the biggest bed of the hotel. That night he dreamt about one nut,one huge nut, like a mountain, a planet, like. . . something that was immenseand empty, though still containing all there is. The sun did not wake himthat morning, though pointed with a small beam at a set of clothes. ”Fancy!”the farmer said, while he looked into the mirror. Although they wereeven larger than the coat he wore before, they fitted perfectly. Just outsidethe room he nearly bumped into the manager of the hotel. ”O, I’m sorry,sir. Ah, was the suite to your liking?” he asked. The farmer looked content,and was invited that evening to the house of the manager. A banquet withwine, roast turkey, chestnuts, other nuts he’d never tasted before, he enjoyedeverything. The maire of the city was somewhat impolite and asked wherethe remarkable citizen had his house. The farmer pointed towards the north,a line exactly from where they were standing to his field, met in between bythe royal palace. When he was delivered that night at the gate, by the mairehimself, the guards brought him to the palace hall. ”Who is this late guest?”the king asked, a huge, fat, king-sized king. ”Daddy, I cannot sleep, I donot want to merry prince Peanut!” his huge, fat daughter said while comingdown the royal stairs.Need I tell any more? The farmer and princess Macadamia got married.Soon, the neighboring countries bowed for the new great prince, and he becameking of a stable and ordered kingdom, of mythic proportions. As lazyand ever-growing rulers they ate happily ever after. Some sources say hisname was ’Buddha’. . . well, do you think the farmer cared?Moral: Be big and you will be carried.41

Chapter 5Charge Regulation in the LowDensity RegimeAgain, a new chapter is going to disturb the equilibrium we found. Thesediment of charged particles, with differing properties, is now shaken again,and is given new degrees of freedom with the magic pen of the theorist. Thecolloidal particles so far had a fixed charge. In the suspensions we are goingto describe the colloids have the ability to adapt, to change their charge.This phenomenon is called charge regulation and complicates the systems wehave discussed so far considerably. The next sections show several attemptsto describe sedimentation of regulating colloids. The first attempt describesmultiple polydisperse systems of colloids of fixed charge. The systems onlydiffer in the charge distribution of the colloids. By comparing the sedimentswe will see what charge distribution has the lowest energy, or in other words,what distribution is favored by the colloids. This will be seen as the equilibriumdensity of regulating colloids that can exchange charge for free withother colloids, but not with the solvent.The next section is based on an article written by Biesheuvel [9]. The model isbased on a monodisperse system, where the colloidal charge number dependson the potential, by the so-called Langmuir isotherm.5.1 Regulation, the next challengeThe first model that we build to describe regulating systems is an extension ofthe PB theory of 4.2, and considers a monodisperse colloidal suspension, witha potential-dependent charge. This dependence is governed by the Langmuirisotherm (5.1.1).43

5.1.1 The extension on the PB theoryWith the Langmuir isotherm we can calculate the expectation value of thecolloidal charge, dependent on the potential. To derive it the following modelcan be used. Consider the colloidal particle to have Z 0 sites at which positivemonovalent micro-ions can be adsorbed. The colloid can exchange particleswith the surrounding medium. Adsorbing a counter-ion costs chemical potentialµ, but releases an energy −ɛ. For this simple situation the grandpartition sum isΞ =∑Z 0n=0and hence, the grand potentialZ 0 !n!(Z 0 − n)! e−βµn e −βɛ(Z 0−n) = (e −βµ + e −βɛ ) Z 0, (5.1)βΩ = − ln Ξ = Z 0 βµ − Z 0 ln(1 + e β(µ−ɛ) ) (5.2)The expectation value of the number of appended particles isso the charge of the particle is〈N〉 = − ∂Ω∂µ = Z 0, (5.3)1 + eβ(µ−ɛ) 〈Z〉 = Z 0 − 〈N〉 =Z 0. (5.4)1 + eβ(ɛ−µ) Now, in order to obtain the identical expression as in [9], we must make theidentification,βµ → φ(x) (5.5)βɛ → φ N , (5.6)where φ N is the Nernst potential, and φ(x) the self-consistent dimensionlesspotential from the PB theory. We now extend the PB theory by assigning acolloidal charge to every component that is a function of φ(x) byZ(φ(x)) =Z 01 + e φ N −φ(x) . (5.7)In fact this adds an inconsistency to the theory, since the entropy considersthe components to be consisting of identical colloids, whereas they actuallyhave different charges. Also, the association 5.5 is not well understood. Fornow we are only interested in the results of this extension.44

5.1.2 Numerical resultsThe example we used for the extended theory is a binary system. We usedthe standard parameters (tab.4.1). For the maximum colloidal charge 1 wechose Z (0)1 = 250 and Z (0)2 = 500, for the gravitational length we chose L 1 =2L 2 = 2mm. The Nernst potential needs also be chosen. From the form ofZ(φ(x)) (eq. (5.7)) we expect the most spectacular results for φ N ≈ φ Donnan(of the order of φ(x)), so we took φ N = 1φ 2 Donnan. The results are shown infigure 5.1.0.030.02i=2E (mV/cm)321(a)300250i=2(b)0Z (φ(x))η i0 5 10 15 20c (cm)0.01i=1200150i=100 5 10 15 20x (cm)1000 5 10 15 20x (cm)Figure 5.1: Density profiles, electric field (a) and colloidal charge (b) as afunction of the height. In this example φ N = 1φ 2 Donnan. The maximumcolloidal charge Z (0)1 = 500 and Z (0)2 = 250, the gravitational length L 1 =2L 2 = 2mm. A comparison is made with a system of colloids of fixed charge,that is identical except for the charge number Z 1 = 2Z 2 = 250 (dashed lines).Discussion Physically this model still needs some questions to answer.The functional form of the colloidal charge number Z(φ(x)) is not fully understood.Also, we cannot fully explain the behaviour of the colloids. Thecharge of the colloids increases with height, as the potential decreases withheight. This has an effect, comparable to the charge-fixed polydisperse systemsin figures 4.9 and 4.7: causing more convex density distributions instead of linear ones. The next chapter will make additional considerationsabout the convexity of distributions.1 number of ionisable groups on the colloid45

5.2 Ideal distributionsCharge regulating systems have an additional degree of freedom to lowerthe energy, when compared to systems containing colloids of fixed charge.Therefore the equilibrium free energy of a charge fixed system is expected tobe an upper bound of a regulating system. A regulating system will now beimitated by a series of calculations of polydisperse systems (n=21) of colloidsof fixed charge. The total charge of the colloids∑21Q c =i=1∫ H0dx Z i ρ i (x), (5.8)is kept fixed, while the distribution is changed. The system with the lowestmean height is considered the equilibrium distribution of a regulating systemwith total colloid charge Q c , and free exchange of charge between the colloids(only). Actually we should consider the system with the lowest grandpotential, see discussion below. The sedimentation is pictured in figure 5.2.The distribution is chosen to be Gaussian.Z mean=250, 21 components, λ B=2.3nm, κ −1 =98nm, D=150, L=1mm, η mean=0.0050.020.015Center of mass4.53.50 1 2 3η tot 0.01Gauβwidth/Z meanequimolar case0.00500 5 10 15 20X (cm)Figure 5.2:046

Discussion The mean height of the system (inset in figure 5.1) decreasesas the width of the charge distribution increases, and goes asymptotically tothe mean height of the equimolar distribution. For a model of colloids thatcan freely exchange charge with other colloids, and not with the solvent, theequilibrium distribution is expected to be the equimolar distribution, sinceit has the lowest mean height. Actually the grand potential of the systemshould be considered, since the equilibrium distribution minimises the grandpotential functional, and not the mean height. However, if the system withthe lowest mean height, the system with equimolar distribution, would have ahigher grand potential than a system with an other distribution, componentscan by proper mixing effectively change the distribution to the more favoureddistribution. This is a point that needs further investigation and explanation,as will hopefully follow in future.47

Chapter 6The C++ codeComputer programmes are renowned for their readability. I fear this one isno exception. It did not have this form when it worked for the first time, onthe contrary. As a clear and simple version made by my supervisor, I wasallowed to use it. But I let it grow, steadily, to prepare it for bigger tasks. Itbecame more capable in solving problems, and when parts lost their purpose,it sometimes kept them as a trophy. I know it may not be the most elegant,accurate or efficient programme you have ever seen, but I think it is. In thebeginning it could not cope with two components, but now it can calculatesuspensions of up to 99 components, at least, or do series of calculations.The parameters are easily changed in the highlighted area, and one can evenmake distributions of the parameters in an equal facile manner. If one wantsto save time, do no try to understand every step, or read it chronologically,just believe the comments. From now on, everyone can calculate profiles ofpolydisperse suspensions of charged colloids. The secrets of the C++ codewill be revealed:The program calculates numerical solutions forandρ i (x) = a i exp[− x L i+ Z i φ(x)] (6.1)d 2dx 2 φ(x) = κ2 sinh φ(x) + 4πλ Bn∑Z i ρ i (x) (6.2)in the following way. At first, the parameters i, Z i , L i , σ i , λ B , ρ s (in κ), mustbe∫chosen. The parameter a i is determined by a normalization that provides1 dxρi (x) = ¯ρHi , where ¯ρ i must be chosen. Also, two other parameters,φ tol and φ reltol , can be adjusted that determine the desired precision of thesolution. For Z i , L i and σ i a distribution can be chosen, and even a function49i=1

Z i (x) that converges, at least in the case whereZ i (x) =Z (i)01 + e φ i−φ(x) . (6.3)Then the program starts an iteration process on a non-equidistant x-grid.Near the bottom and top, the grid points are a distance of the order ofthe Debye screening length κ −1 apart from each other, and in between thedistance is of the order of H/200. From homogeneous densities ρ i (x) = ¯ρ iand potential φ(x) = φ d 1 a new density ˜ρ i (x) is calculated with the use ofequation (6.1). The old density ρ i (x) is now shifted towards ˜ρ i (x) with afactor α = 0.001, and normalized:˜ρ i (x)ρ i (x) → (1 − α)ρ i (x) + α¯ρ i ∫ Hdx ˜ρ 0 i(x) . (6.4)The new φ(x) is now calculated by (6.2) where the second derivative is approximatedbyφ ′′ (x) ≈φ(x + a) + φ(x − a) − 2φ(x)a 2 , (6.5)for a small, but finite. This process is repeated until the solution remainsstable between certain boundaries, set at the beginning. If this solution isreached or if the iterations reach their set maximum, the solutions are writtento data files. Other properties of the system, like the salt profiles ρ ± (x), theelectric field E x (x), the total charge distribution Q(x), the centers of massh i , the colloidal packing fractions η i (x), and total packing fraction η tot (x) arecalculated from ρ i (x) and φ(x).In the region where ρ s is of the order of 1 2 Z2 ¯ρ tot , convergence is often a problem.The program has a special option for this region. The first iterationshould start with a salt concentration that converges desirably. When thesolution is found, the iteration starts again with a lowered salt concentration,and with the solutions found for ρ i (x) and φ(x). The solutions to be foundare much closer to these results then to ρ i (x) = ¯ρ i and φ(x) = φ d , whichimproves the convergence.The iteration process and its solutions still contain a bit of black magic andrequires some experience. About uniqueness, speed of convergence and reliabilityI do not wish to speak.1 d stands for Donnan, φ d is defined by equation (4.31)50

Chapter 7Conclusions and OutlookWithin the framework of density functional theory we derived a Poisson-Boltzmann theory to describe the equilibrium of diluted systems of chargedcolloids under sedimentation. The numerical solutions for monodisperse(n=1) cases show strong deviations from the barometric distribution in thelow salt limit, as was already shown in experiments [11]. For binary (n=2)and ternary (n=3) systems the solutions show segregation of components aswell. The salt entropy is seen to be the driving mechanism behind theseeffects. The colloids are lifted by an electric field that cancels gravity. Sincecomponents of different mass per charge cannot be lifted by the same electricfield, the components order according to their mass per charge and formlayers. Although the ordering still lacks experimental confirmation, the numericalsolutions agree very well with Monte Carlo simulations [15, 22]. Forpolydisperse systems (n=10,n=21), the same effects are predicted by the theory,although the segregation is in many cases weaker than in the binary andternary examples.An extension of the theory to charge regulating systems can be made byconsidering the colloidal charge to be dependent on the electrostatic potential,according to the Langmuir isotherm. The numerical results from thisextension do not show the linear density profiles, and constant electric fieldsas calculated from the theory describing charge stabilised colloids. Althoughthe solutions show the same phenomena, the profiles decrease more stronglywith height. This can be understood by comparing monodisperse systemsof charge regulating colloids with polydisperse systems of charge stabilisedcolloids.The exact dependence of the charge still lacks a proper derivation, and thenumerical calculations are not confirmed by simulation data or experiments,making the extension less underpinned than the PB theory for charge stabilisedsystems.51

Other extensions are planned to include the high density regime, where packingeffects and excluded volume are important. For spherical particles theCarnahan-Starling approximation is thought to be suitable, for long rodsthe virial expansion, but this will be worked on in future. Charge regulation,strongly confined fluids, van der Waals forces. . . colloidal suspensionscan be made increasingly complicated, and their diversity will not be easilyexplained by a (simple) theory.The author is delighted to have the possibility to continue this work, participatingin the research on colloidal systems. For now, the reader will haveto wait. New problems, and new answers will follow in future.52

Appendix AArticleA.1 abstractWe theoretically study sedimentation-diffusion equilibrium of dilute binary,ternary, and polydisperse mixtures of colloidal particles with different buoyantmasses and/or charges. We focus on the low-salt regime, where theentropy of the screening ions drives spontaneous charge separation and theformation of an inhomogeneous macroscopic electric field. The resulting electricforce lifts the colloids against gravity, yielding highly nonbarometric andeven nonmonotonic colloidal density profiles. The most profound effect is thephenomenon of segregation into layers of colloids with equal mass-per-charge,including the possibility that heavy colloidal species float onto lighter ones.A.2 IntroductionSedimentation in a suspension of colloidal particles is of profound fundamentalimportance, and has been studied in detail for a long time. For instance,in 1910 Perrin determined the Boltzmann constant k B (and from this Avogadro’snumber) by comparing the measured equilibrium density profile ofa dilute suspension with the theoretically predicted barometric law [1], andmore recently the full hard-sphere equation of state was determined accuratelyfrom a single measurement of the sedimentation profile of a densesuspension of colloidal hard spheres [2, 3]. In rather dilute suspensions ofcharged colloids, however, strong deviations from the barometric distributionhave recently been theoretically predicted [4, 5, 6, 7, 8, 9], experimentally observed[10, 11, 12, 13], and simulated [14], at least in the regime of extremelylow salinity. The most striking phenomenon is that the distribution of colloidsextends to much higher altitudes than is to be expected on the basis of53

their buoyant mass. The force that lifts the colloids against gravity is providedby an electric field that is induced by spontaneous charge separationover macroscopic distances. Although this phenomenon costs energy, the systemgains entropy because of a more homogeneous distribution of salt ions(and colloids). The same entropy-induced lifting force was recently found tobe responsible for a so-called Brazil-nut effect in binary mixtures of chargedcolloidal particles under low-salt conditions: the heavy particles can resideat higher altitudes than the lighter ones [15]. In this Letter we extend thetheoretical study of sedimentation of charged colloids to binary, ternary, andpolydisperse mixtures, for which we calculate colloidal density profiles andthe entropy-induced electric field selfconsistently within a Poisson-Boltzmannlike density functional theory.A.3 TheoryWe consider an n-component suspension of negatively charged colloidal spheresin a salt solution in the Earth’s gravity field. We label the colloidal speciesby i = 1, 2, · · · , n, and denote the electric charge, diameter, and buoyantmass for species i by −Z i e, σ i , and m i , respectively, where e is the protoncharge. We imagine the suspension to be in thermal and osmotic contactwith a reservoir that contains a solvent with temperature T , dielectric constantɛ, monovalent ions (charge ±e) at a concentration 2ρ s , such that theinverse Debye screening length is κ = (8πλ B ρ s ) 1/2 with λ B = e 2 /ɛk B T theBjerrum length. The ions are assumed to be massless point particles, andthe solvent mass density is taken into account by considering the buoyantinstead of the actual colloidal masses according to Archimedes. The volumeof the suspension is V = AH, with A the (macroscopic) horizontal area andH the vertical height of the solvent meniscus above the bottom of the system.The gravitational acceleration g is in the negative vertical direction. We areinterested in the equilibrium colloidal density profiles ρ i (x) as a function ofthe altitude x above the bottom at x = 0 and below the meniscus at x = H.The ion density profiles are denoted by ρ ± (x). In order to calculate theseprofiles we employ the framework of density functional theory [16], where theequilibrium density profiles follow from the minimisation of a grand potentialfunctional Ω[{ρ i }, ρ ± ] with respect to all the profiles. Here we employ themean-field free energy functional that consists of entropic, gravitational, and54

electrostatic contributions,Ωk B T A = ∑ α=±n∑+i=1∫ H0∫ H0dxρ α (x) ( ln ρ α(x)ρ s− 1 ) +dx x L iρ i (x) − 2πλ B2∫ H0dxn∑∫ Hi=1 0∫ H0dxρ i (x) ( ln ρ i(x)a i− 1 )dx ′ |x − x ′ |Q(x)Q(x ′ (A.1) ),where we introduced the fugacity (or activity) a i and the gravitational lengthL i = k B T/m i g of species i, and the local charge density Q(x) = ρ + (x) −ρ − (x)− ∑ ni=1 Z iρ i (x). The electrostatic term follows from an in-plane integrationof the three-dimensional Coulomb law, 2π ∫ ∞dRR/ √ |x − x0 ′ | 2 + R 2 =−2π|x−x ′ | with R the in-plane distance, where we ignored an irrelevant integrationconstant. The functional (A.1) only couples the total charge densityat different heights in a mean-field fashion, i.e. electric double layers andmany other correlation effects are not taken into account. In fact one easilychecks that the thermodynamics of the system reduces in the absenceof gravity to an (n + 2)-component ideal-gas mixture, since then Q(x) ≡ 0because of translational invariance and charge neutrality. In the presence ofgravity, however, interesting structures already appear at this relatively lowlevel of sophistication. The Euler-Lagrange equations δΩ/δρ α (x) = 0 andδΩ/δρ i (x) = 0 can be cast in the formρ ± (x) = ρ s exp[∓φ(x)]; (A.2)ρ i (x) = a i exp[−x/L i + Z i φ(x)]; (i = 1, 2, · · · , n) (A.3)where φ(x) = −2πλ B∫ H0 dx′ Q(x ′ )|x − x ′ | is the dimensionless electrostaticpotential gauged such that it is zero in the reservoir where Q(x) ≡ 0. Itturns out to be convenient to rewrite it with Eq.(A.2) in differential form asthe Poisson-Boltzmann equationd 2 φ(x)n∑= −4πλdx 2 B Q(x) = κ 2 sinh φ(x) + 4πλ B Z i ρ i (x).i=1(A.4)Together with the boundary conditions φ ′ (0) = φ ′ (H) = 0 (where a primedenotes a derivative w.r.t. x) the Eqs.(A.3) and (A.4) form a closed set ofn + 1 equations that can in principle be solved to yield ρ i (x) and φ(x) for agiven thermodynamic state. In an experimental situation, however, the totalpacking fraction ¯η i = (π/6)N i σ 3 i /V of species i is usually fixed instead of thefugacity a i , with N i the number of colloids of species i. This conversion caneasily be accomplished here by regarding the fugacities a i as normalizationconstants that take values such that ¯η i = (1/H) ∫ H0 dxη i(x), with η i (x) =55

(π/6)ρ i (x)σi3 the local packing fraction of species i. Note that the hard-corediameters σ i do not appear at all in the Euler-Lagrange equations, since thefunctional (A.1) ignores the hard-core part of the (direct) correlations; weuse for all colloidal species that σ i = σ = 150nm only to be able to convertdensities to physically reasonable packing fractions. Once the profiles φ(x)and ρ i (x) are known, then ρ ± (x) follows from Eq.(A.2), and from this thetotal charge density Q(x). The magnitude of the electric field is given byE(x) = k B T φ ′ (x)/e.Unfortunately one cannot solve Eqs.(A.3) and (A.4) analytically. However,its one-dimensional character allows for a rather straightforward numericalsolution on a grid of heights, although some care must be taken in dealingwith the widely different length scales H (say of the order of centimeters) andκ −1 (at most of the order of microns) in realistic cases. In all our calculationswe use a non-equidistant grid that resolves the length scale κ −1 close tox = 0, H and fractions of mm’s in between. Typically we use 600 grid pointsfor a system with a meniscus at H = 20cm. Our iterative scheme to solveEqs.(A.3) and (A.4) on the grid takes typically a few seconds on a desktopPC for n = 1, and a minute for n = 21.Before discussing our numerical results, we wish to point out that the presenttheory reduces, for n = 1, to the treatment of Ref.[8]. For instance, the threeregimes for the colloidal density profile (barometric, linear, and exponentialwith a large decay length) follow within the assumption of local chargeneutrality Q(x) = 0, i.e. − sinh φ(x) = Z 1 ρ 1 (x)/2ρ s ≡ y 1 (x). In the linearregime, where 1/Z 1 < y 1 (x) < 1, the potential is then from Eq.(A.4) linearin x, with a slope φ ′ (x) = 1/Z 1 L 1 that is proportional to the electricfield that lifts the colloids against gravity [8]. This one-component resultalready hints at an important complication for mixtures (n ≥ 2): since generallyZ i L i ≠ Z j L j a single electric field strength cannot lift all the colloidalspecies simultaneously. On this basis one could therefore already expect segregationin mixtures, such that colloids with the same value Z i L i (i.e. thesame charge-per-mass) are found at the same height. This is indeed whatour numerical results will show below.A.4 Segregation in binary and ternary mixturesWe start our numerical investigation with a class of binary mixtures (n = 2)of equal-sized light (i = 1) and heavy (i = 2) colloidal particles of variouscharge ratios. We choose the system parameters identical to those of the56

Monte Carlo simulations of figure 3 of Ref.[15]: σ = 150nm, H = 1000σ,λ B = σ/128, L 1 = 3L 2 2 = 10σ, ¯η 1 = ¯η 2 = (π/6) × 10 −4 , Z 1 = 15, and Z 2varies between 15 and 45. The simulated system does not contain addedsalt but only counterions, which we can represent within our theory by theextremely low reservoir salt concentration ρ s = 1nM, which we checked tobe low enough to be in the zero-added-salt limit as regards the colloidalprofiles. In Fig.1(a) we show, for Z 2 = 45, the density profiles of the twocolloidal species, as well as that of the counterions (and the coions) in theinset. We observe profound colloidal segregation into two layers, with theheavy species floating on top of the lighter ones. The counterions are seen tobe distributed throughout the whole volume, i.e. much more homogeneouslythan when all colloids would have been barometrically distributed in a thinlayer of thickness L i just above the bottom (since in that case the net ioncharge would have been located in that same thin layer). The resulting gainof ion entropy is the driving force for the formation of the electric field thatpushes the heavy (highly charged) colloids to high altitudes against gravity[8, 14]. In Fig.1(b) we show the mean height h i of species i, defined ash i =∫ H0 dx x ρ i(x)∫ H0 dx ρ i(x) , (A.5)as a function of Z 2 . We replot the Monte Carlo simulation results of figure3 of Ref.[15] (symbols), together with the predictions for h i that follow fromthe present theory (continuous curves). Given that there is not a single fitparameter involved, the agreement is remarkable, certainly when comparedwith the theoretical analysis on the basis exponentially decaying density profilesas in Ref.[15]. Fig.1 shows that the heavy particles are on top of thelighter ones, h 2 > h 1 , provided Z 2 /Z 1 ≃ 1.6. For barometric profiles onewould find that h i = L i , but we see in all cases that h i ≫ L i due to the lifteffect of the induced electric field. The good agreement between theory andsimulation in Fig.1(b) also indicates that hard-core effects (which are takeninto account in the simulations but not in the theory) are not so relevant, atleast not in the parameter regime studied here.The next system we study is a ternary system (n = 3) in a solvent characterisedby λ B = 2.3nm (ethanol at room temperature) with a reservoirsalt concentration ρ s = 10µM and meniscus height H = 20cm. The colloidalcharges and gravitational lengths are Z i = (1000, 250, 125) and L i =(1, 2, 1)mm, respectively, and the system is equimolar with ¯η i = 0.005 for allthree species. Fig. 2 shows the density profiles as predicted by the presenttheory in (a), as well as the electric field in (b). We find almost perfect segregationinto three layers, such that the mean heights satisfy h 3 < h 2 < h 1 .57

0.010.008ρ +0.006 i=10.5hη i/σi ρ −00.0040 200 400 600 800 1000X/σ0.002aρ salt(µM)1.51i=200 200 400 600 800 1000X/σ3002001000bi=2i=11 1.5 2 2.5 3Z 2/Z 1Figure A.1: (a) Colloidal density profiles and counterion distribution (inset)for a deionised binary mixture of equal-sized light (i = 1) and heavy (i = 2)colloidal spheres (diameter σ = 150nm), with meniscus height H = 1000σ,gravitational lengths L 1 = 10σ and L 2 = 20σ/3, average packing fractions¯η 1 = ¯η 2 = (π/6) × 10 −4 , colloidal charges Z 1 = 15 and Z 2 = 45, and Bjerrumlength λ B = σ/128. (b) Mean height h i of the two species for the samesystem as in (a) except now for a range of Z 2 , showing a density inversion(h 2 > h 1 , the heavy particles floating on top of light ones) for Z 2 /Z 1 > 1.6.The curves are predictions of the present theory, the symbols are simulationdata of Ref.[15].Note that this ordering coincides with the ordering Z 3 L 3 < Z 2 L 2 < Z 1 L 1 ,and not with the corresponding ordering of L i which one would expect onthe basis of a barometric distribution. In other words, this system segregatesaccording to mass-per-charge instead of the more usual ordering accordingto mass: the colloids with the largest mass-per-charge are found at the bottom.In the present case this implies that the lightest colloids (species 2) arefound in a layer in between the equally heavy species 1 and 3. On the basisof the one-component theory of Ref.[8] one would expect a linearly decayingdensity profile of species i in the layer where species i is dominant, aswell as a linear electrostatic potential, such that η ′ i(x) = πρ s σ 3 /3Z 2 i L i andφ ′ (x) = 1/Z i L i , corresponding to an electric field strength m i g/Z i e in thislayer. These values for the density gradients and the electric field are indicatedin Fig.2(a) and (b), respectively, and are in good agreement with thenumerical results. The result for the electric field can also be easily obtainedanalytically for a mixture provided one assumes that segregation takes place,such that ρ i (x) takes a maximum at some height x ∗ in the layer of speciesi. From ρ ′ i(x ∗ ) = 0 one obtains from Eq.(A.3) that φ ′ (x ∗ ) = 1/Z i L i . The58

0.10.08100.06η i 50 5 10 15 200.04i=2X (cm)0.02i=3ρ salt(µM)2015i=100 5 10 15 20X (cm)aE (mV/cm)21.510.50Q/2ρ ssinhφ(x)10 −6−10 −6 00 5 10 15 20X (cm)0 5 10 15 20X (cm)m 3g/Z 3ebm 2g/Z 2em 1g/Z 1eFigure A.2: Density profiles (a) and the electric field profile (b) of anequimolar ternary mixture of equally-sized colloids with charges Z i =(1000, 250, 125) and gravitational lengths L i = (1, 2, 1)mm with total packingfractions ¯η i = 0.005 in a 20cm ethanol suspension at room temperature,with monovalent ionic strength ρ s = 10µM in the reservoir. Almost completesegregation takes place into well-defined layers of pure components, with aslope η i(x) ′ = πρ s σ 3 /3Zi 2 L i and a constant electric field m i g/eZ i in the layerwith component i, as denoted by the dashed lines in (a) and (b), respectively.The inset in (b) shows a dimensionless measure for local charge neutrality(see main text).inset of Fig.2(b) shows the ratio of the total charge density Q(x) and the ioncharge density 2ρ s sinh φ(x), which is such that |Q(x)/2ρ s sinh φ(x)| ≪ 1 forall x except close to x = 0 and x = H where it is ∼ 1. This indicates that thesystem, with its inhomogeneous electric field, is yet essentially locally chargeneutral (but not exactly, and not at all at the boundaries), suggesting thata description on the basis of hydrostatic equilibrium within a local densityapproximation should be rather accurate [17].A.5 Polydisperse MixturesWe now extend our study to polydisperse mixtures, where one could expectsegregation into many layers on the basis of the results for two and threecomponents. We mimic the polydispersity by considering a system of n = 21components, with Z i distributed as a Gaussian with average of 250 and astandard deviation of 50. This distribution is shown in the inset of Fig. 3(a),where the vertical axis (¯η i ) is proportional to the relative frequency of speciesi in the sample. We consider two distributions for L i : (A) L i = 2mm for all59

species, and (B) L i = 2 × (250/Z i ) 3/2 mm. Case B mimics the situation forspheres of different size but the same mass density and surface charge density,such that Z i is proportional to the surface area and L i to the inverse volumeof species i, i.e. L 2 i Zi3 is a constant independent if i. The density profiles,numerically obtained by solving the Eqs.(A.3) and (A.4) for H = 20cm,λ B = 2.3nm, ρ s = 3µM, and ¯η tot = ∑ 21i=1 ¯η i = 0.005, are shown in Fig.3 forall 21 components. Case A shows profound lifting and layering, where theordering is again determined by mass-per-charge as illustrated by the threedashed curves (for Z i = 250, 279, and 299) showing that the colloids in thehigh-charge wing of the distribution reside at high altitudes. Fig.3(b) showsthe density profiles for case B, which does exhibit lifting, but hardly anylayering, and no density inversion at all. This is completely consistent withthe picture that the ordering is determined by Z i L i , which for case B is suchthat Z i L i /Z j L j = √ Z j /Z i , i.e. the highly-charged particle are expected atthe bottom while the relative spread in Z i L i is relatively small compared tocase A, where Z i L i /Z j L j = Z i /Z j . The inset of (b) shows the total packingfraction profiles η tot (x) = ∑ 21i=1 η i(x) of both case A and B together with theone-component profile (n = 1) with Z 1 = 250 and L 1 = 2mm at ¯η 1 = 0.005.Perhaps surprisingly there is hardly any distinction between the pure systemand case B, whereas there is a small difference with case A. These profilesshow that the main distinction between these polydisperse systems and theunderlying one-component one concerns the layering phenomenon (providedZ i L i varies sufficiently for all the species), and not the total distribution of thecolloids. Perhaps this fractionation effect could be exploited experimentallyto purify a polydisperse mixture.A.6 Conclusions and discussionWe have studied sedimentation equilibrium of n-component systems of chargedcolloidal particles at low salinity by minimising a Poisson-Boltzmann-likedensity functional w.r.t. density profiles on a one dimensional grid of heights.For n = 1 the theory reduces to the one-component studies as presented inRef.[8, 14], and for n = 2 we quantitatively reproduce the simulation resultsof Ref.[15], where density inversion was found. These effects are caused by aself-consistent electric field that lifts the higher charged (heavy) particles tohigher altitudes than the lower charged (lighter) colloids. We show that thelayering of the colloids according to mass-per-charge can persist for ternary(n = 3) as well as for polydisperse (here n = 21) mixtures. Given the goodaccount that the present theory gives for simulations [14] and experiments[11, 12, 13] of one-component systems, and for the simulations of binary60

Z 16 =29932a10 3 η iZ 11 =2500.60.40.20150 200 250 300 350Z i10 3 η i0 5 10 15 201.5bA1010 3 η totB51010 3 0 5 10 15 20η i X (cm)151Z 14 =2790.500 5 10 15 20X (cm)00 5 10 15 20X (cm)Figure A.3: Density profiles of a 21-component colloidal suspension with aGaussian charge distribution as illustrated in the inset of (a), at total packingfraction 0.005, reservoir salt concentration ρ s = 3µM, and Bjerrum lengthλ B = 2.3nm. The gravitational lengths are as in case A (see main text) in(a), and case B in (b). In (b) the solid curves are for Z i > 250, and thedashed ones for Z i ≤ 250. The inset of (b) shows the total packing fractionprofile for case A and B, together with that of the underlying one-componentsystem (see main text).systems [15], it is tempting to argue that the theory is also (qualitatively)reliable for ternary or polydisperse systems, i.e. the predicted segregationand layering should be experimentally observable. One should bare in mind,however, that the present theory ignores the hard-core of the colloidal particles,and is therefore expected to break down at higher packing fractions(say η > 0.1) . It also ignores effects due to charge renormalisation, whichbecomes relevant when Z i λ B /σ i ≫ 1. Work on extending the theory in thesedirections is in progress [18, 17].All results presented in this paper were obtained with the zero-field boundaryconditions φ ′ (0) = φ ′ (H) = 0. We checked explicitly, however, thatother boundary conditions that respect global charge neutrality, such asφ ′ (0) = φ ′ (H) = eE ext /k B T (describing a suspension in a homogeneous externalelectric field E ext ) or φ(0) = φ(H) and φ ′ (0) = φ ′ (H) (describing ashort-circuited bottom and meniscus) give indistinguishable density profiles,except in two layers of thickness ∼ κ −1 ≪ 10µm in the vicinity of the bottomand the meniscus. This insensitivity to the boundary conditions is not surprising:the whole phenomenology in these systems is driven by the entropyof the microscopic ions, i.e. a bulk contribution to the grand potential thatshould dominate any boundary (surface) contribution.61

A.7 AcknowledgementsIt is a pleasure to thank Maarten Biesheuvel for interesting discussions. Thiswork is part of the research programme of the ”Stichting voor FundamenteelOnderzoek der Materie (FOM)”, which is financially supported by the ”Nederlandseorganisatie voor Wetenschappelijk Onderzoek (NWO)”.62

Appendix BImportant length scalesIn the following sections systems are considered that are a prelude to colloidalsystems. Often they are easier to describe, because they only includeone type of interaction, have simple geometries and less objects. They provideuseful parameters, and insights for the more complex cases, the colloidalsuspensions. Some length scales will be used frequently in this thesis, andare explained in the following sections using the MFA, LDA (see 2.3.2) respectively:B.1 The Debye screening length κ −1 and the Bjerrum length λ B , a considerationof a charged plane in an ionic solution.B.2 The gravitational length, an ideal gas in the gravity field of the earth.B.3 The colloidal diameter.B.1 The Debye screening length κ −1This section is about κ −1 , or as it is called, the (bare) Debye screening length.It will turn out to be a convenient parameter for the theory in the previouschapters. When colloidal particles are suspended in some polar medium,they will often release or acquire micro-ions, and acquire some surface charge.This will have effect on the micro-ion density in the solvent. Equally chargedmicro-ions, called co-ions, tend to move away from the colloids, whereas oppositelycharged micro-ions, called counter-ions, are attracted to the colloids.These ions will form some distribution after balance is found between theelectrostatic forces and entropy. The counter-ions form what can be calleda layer around the colloids, that screens the colloidal charge. The typicalwidth of the layer will turn out to be κ −1 . Hence, κ −1 was called screening63

length by Peter Debye. In dense colloidal systems this length depends on thecolloid density as well, and is denoted as ¯κ as discussed in ??.To derive κ and λ B the following example is considered: a charged planarsurface in an ionic solution. A plane, at x = 0, is surrounded by monovalentions within some medium. The plane has a homogeneous surface charge σ.The ions, of charge ±e, are considered as point particles and have numberdensities ρ ± (x). They will be called ’the salt’. The medium only providesa dielectric constant ɛ, temperature T and asymptotic salt concentrationlim x→∞ ρ ± (x) = ρ s .The free energy functional of this system is as follows:F [ρ + (r), ρ − (r)] = F id [ρ + (r)] + F id [ρ − (r)] + F C [ρ + (r), ρ − (r)]∫∫= k B T dr ρ + (r)(ln ρ + (r)Λ 3 + − 1) + dr ρ − (r)(ln ρ − (r)Λ 3 − − 1)VV∫+ e2drdr ′ (ρ + (r) − ρ − (r) + σδ(r))(ρ + (r ′ ) − ρ − (r ′ ) + σδ(r ′ ))(B.1)2ɛ|r − r ′ |Vwhere F C stands for the mean-field coulomb energy, and Λ 3 ± the deBrogliewavelength. For the derivation see chapter 2. Since this example has translationalsymmetry in the plane of the surface this functional can be simplifiedin the following form:A −1 F [ρ + (x), ρ − (x)] = k B T ∑−πe2ɛα=+,1∫ ∞0∫ ∞0dx ρ α (x)(ln ρ α (x)Λ 3 α − 1)dxdx ′ Q(x)Q(x ′ )|x − x ′ |(B.2)where A is the area of the surface, eQ(x) = ρ + (x) − ρ − (x) + σδ(x) is calledthe total charge distribution, and for F C we used that:∫∫dr ′ 1H ∫ R∫V |r − r ′ | = 2π dx ′ 1Hdr r √0 0 |x − x′ | 2 + r = 2 −2π|x−x′ |+ dx ′ R0(B.3)where R ≫ H. Take R to infinity: R is a constant, so not of any importancefor our functional. Next, we introduce a new symbol:∫ ∞φ(x) ≡ −2πλ B dx ′ Q(x ′ )|x − x ′ |0called ’the dimensionless electrostatic potential’. It must be zero in infinitywhere Q(x) = 0, from global charge neutrality. A convenient abbreviationis the ’Bjerrum length’ λ B = . It can be seen as the distance at whiche2k B T ɛ64

two unit charges have a coulomb energy equal to k B T . It will appear manytimes in the theory to come.The free energy functional is now more conveniently expressed as:βF [ρ + (x), ρ − (x)] = ∑ ∫ ∞dx ρ α (x)(ln ρ α (x)Λα 3 − 1) + 1 ∫ ∞dx Q(x)φ(x)2α=+,10(B.4)where β = (k B T ) −1 and keeping the (irrelevant) A −1 in mind. By minimizingthis functional with respect to ρ ± we find the equilibrium free energy of thesystem and the equilibrium profiles of the ions:δF [ρ + (x), ρ − (x)]δρ ± (x)= ln ρ ± (x)Λ 3 ± ± φ(x) = βµ ± . (B.5)With βµ ± = ln ρ s Λ 3 ±, the densities that minimise F are found to be:ρ ± (x) = ρ s exp[∓φ(x)]ρ s is the salt concentration at infinity, where Q(x) = 0. Note now thatd 2 φ(x)dx 2∫ ∞= −2πλ B dx ′ Q(x ′ ) d2dx |x − 2 x′ | = −4πλ B Q(x)00(B.6)(B.7)is equal to Poissons’ equation. Together with (B.6) this results in the Poisson-Boltzmann equation:d 2 φ(x)dx 2 = −4πλ B (ρ s exp[−φ(x)] − ρ s exp[φ(x)] + σδ(x))= 8ρ s πλ B sinh φ(x) − 4πλ B σδ(x) (B.8)From now on we will call κ 2 ≡ 8ρ s πλ B . The meaning of κ 2 will follow, still.Although we can solve equation (B.8) analytically in this example, it will notbe possible in later examples. We make the approximation:d 2 φ(x)dx 2 ≈ κ 2 φ(x) − 4πλ B σδ(x) (B.9)For this approximation the dimensionless potential must be small, φ(x) ≪ 1,so take T high enough, a high ɛ is also helpful. The general solution of φ(x)is then:φ(x) = Ae κx + Be −κxFrom the boundary conditions lim x→∞ φ(x) = 0 and φ ′ (0) = −4πλ B σ, Amust be zero, and B = κσ2ρ s. The profiles ρ ± (x) become from the linearizedPoisson-Boltzmann equation:ρ ± (x) = ρ s (1 ∓ φ(x)) = ρ s (1 ∓ κσ e −κx )(B.10)2ρ s65

One quick check shows ∫ ∞dx Q(x) = −σ, which reassures global charge0neutrality. From equation (B.10) the meaning of κ becomes obvious. Thetypical decay length of ρ ± (x) is κ −1 as is shown in figure (B.1) below.ρ −ρ sρ +κσ/2ρ sφ0 1 2 3 4κxFigure B.1: Density profiles ρ ± (x) of the ions near the charged plane at x = 0as well as the dimensionless potential (dashed line). Typical decay length isequal to κ −1 , the Debye screening length.B.2 The gravitational lengthThis section is about the gravitational length. This parameter is widelyused in models that consider sedimentation of gases in gravitational fields.The grand potential functional for a system of uncharged colloids under theexternal potential of the Earth’s gravity field is of the formβΩ[ρ] = βF [ρ] +∫ H0dx ρ(x)( x ρ(x) − βµ),L (B.11)with x/L the external potential due to gravity. By the variational principlewe find the Euler-Lagrange equation, using an LDA (section 2.3.2)βf ′ (ρ) + x − βµ = 0.L (B.12)66

The derivative to x results inAgain we use the LDA inf ′′ (ρ)ρ ′ (x) = −mg.(B.13)∂F (N, V, T )p = − = −f(ρ) + ρf ′ (ρ),∂V(B.14)sodpdρ = ρf ′′ (ρ).(B.15)Substitution of equation B.15 into B.13 leads to the equationdP (ρ(x))= −mgρ(x), (B.16)dxcalled hydrostatic equilibrium, and describes the relation between the pressureof the bulk fluid P (ρ) and the number density ρ at sedimentation equilibrium.The height above the surface is denoted by x, the mass of the particles by m,and the gravitational acceleration by g. For an ideal gas, P (ρ) = k B T ρ, thesolution of equation (B.16) is the so-called barometric height distribution:ρ(x) = ρ 0 exp(− x L ),(B.17)where ρ 0 is determined by the total number of particles in the system, andthe gravitational length.L = k B T/mg(B.18)B.3 The colloidal diameter, σIn this thesis the particles interact as point particles, as an ideal gas. Thediameter of the colloids will only be a scaling factor for the packing fractionη = πσ 3 ρ/6. In this way, the size can be used to check if the ideal gas approximationis applicable. For packing fractions higher than η > 0.1, packingeffects are expected to become important. For the theory, the diameter willtherefore play a minor role, although it can lead to confusing results whencomponents have strongly different diameters (section 4.6.1). For the MonteCarlo simulations the size does matter. Comparisons between theory andsimulations will show that for diluted systems (η < 0.05), packing effects arevery small.The diameter will be taken σ = 150nm as standard.67

Appendix CThe block modelThe model described in the section 4.6.4 can be simplified by a crude approximationof the form of the packing fraction. Although the expressionsfollowing from this model are even less reliable than the slope model, it canbe used to determine the order of the layer widths quickly, and by hand. Thecondition that the total width ξ should be smaller than the total height His not needed either.The equations (4.27) and (4.28) become easier to obtain by assuming thepacking fraction to be constant η i (x) = b i . Then the electric field needs noapproximation, since by ρ ′ i(x) = ddx a i exp(−x/L + Zφ(x)) = 0,φ ′ (x) = 1Z i L i, for i ∈ I i . (C.1)From the equation of state the same solution is found as in equations (4.22),but now also((η i (x) − η j (x) = b i − b j =σ3 1) 2 ( 1) 2 )− . (C.2)48λ B Z i L i Z j L jbecause η i (x) is constant. Only one b i is needed to solve all b’s by recursion.For ξ < H, the value of b n is taken to be b n = √ 8ρ s σ 3 H ¯η/3Z 2 nL n (equation(4.24)), so(b i = η(x i ) = σ3 1( ) 2 1)− ( ) 2 + b n48λ B Z i L i Z n L nξ i = ¯η b i(C.3)In the case where the colloids reach the top of the system, ξ = H, theexpressions are slightly different. Two conditions then determine the layer69

widths ξ i and block heights b i :n∑ξ i = Hi=1n∑b i ξ i = H ¯ηi=1(C.4)(C.5)Figure C.1 shows the full numerical solutions of a ternary system, with standardparameters as in tab. 4.1, charge number Z = (3000, 2500, 2000) andgravitational length L = (4, 3, 2)mm, together with the approximation withthe block model.0.010.008η i0.0060.0040.00200 5 10 15 20x (cm)Figure C.1: The full numerical solutions of a ternary system, with standardparameters as in tab. 4.1, charge number Z = (3000, 2500, 2000) and gravitationallength L = (4, 3, 2)mm, together with the approximation with theblock modelConclusion The block model is not accurate. It needs a fit parameter forthe jump constant t i . Moreover, the crossovers do not seem to be proportionalto ((Z N L n ) −2 − (Z n−1 L n−1 ) −2 ). A reason can be given by the lengths overwhich the crossovers take place, about 4 cm per crossover. A second reasonis the fact that the slopes of the profiles are not taken into account. Theslope model is therefore more accurate, and does not provide (much) morework.70

Index of frequently usedsymbols and namesBjerrum length? Dimensionless density? κ −1 ?!Here they are:symbol name relation to other symbols section of explanationLgravitational lengthk B TmgB.2λ B Bjerrum lengthe 2k B T ɛB.1κ −1 Debye screening length κ 2 = 8ρ s πλ B = 8ρ sπe 2k B T ɛB.1σ colloidal diameter B.3Z colloidal charge numbere elementary charge e1β inverse thermal energyk B TN number of particles 2µ chemical potential 2V (system) volume 2p pressure 2T temperature 2S entropy 2ˆρ microsc. number dens.∑ Ni=1 δ(r − r i) 2.3ρ number density 〈ˆρ〉 2.3η packing fraction πσ 3 ρ/6 B.2ψ electrostatic potential 3.2eψφ dimensionless potentialk B T3.2E(x) electric field ψ ′ (x) 3.2F intrinsic Helholtz free en. 2Ω grand potential 2k B Tmg71

AcknowledgementsI am very grateful to a lot of people for many reasons. For strategic reasons Ishall not mention all, and not all reasons. Among many important teacherswere:My supervisor, René, for introducing me to what I found to be a very excitingtopic in physics, for introducing me to many interesting conferences,talks, articles, and for all the support.Josefien and Stoppelenburg family. For the sheep. And for mind stirring concertsand communication. Thanks to my brother Olaf for the questions:”Watheb je eraan?” and:”Wat word je ermee?”. They proved to be two of the mostchallenging questions during my study. Thanks to Elli Zwart, for helping mewith the understanding of equilibrium, (force) balance, consciousness of systems,and the (un)covering of phenomena. Jan Roest, for all (colloidal)discussions. Wim and Marga Hilhorst, for all the puzzles and meatballs.My parents who did great efforts to make this possible. My father Felixwho was incredible, as my brave mother Nel, always being supportive andshowing an unbounded trust.73

Bibliography[1] J. Perrin, (1913), Les Atomes (Paris: Alcan).[2] R. Piazza, T. Bellini, V. Degiorgio, Phys. Rev. Lett. 71, 4267 (1993).[3] M.A. Rutgers, J-Z Xue, W.B. Russel, and P.M. Chaikin, Phys. Rev. B53, 5043 (1996).[4] T. Biben and J.-P. Hansen, J. Phys.:Cond. Matt. 6, A345 (1994).[5] J.-P. Simonin, J. Phys. Chem. 99, 1577 (1995).[6] G. Téllez and T. Biben, Eur. Phys. J. E 2, 137 (2000).[7] H. Löwen, J. Phys.: Cond. Matt. 10, L479 (1998).[8] R. van Roij, J. Phys.: Condens. Matter 15, S3569 (2003).[9] M. Biesheuvel, J. Phys.:Cond. Matt. 16, L499 (2004)[10] A.P. Philipse and G.H. Koenderink, Adv. in Colloid and Interface Science100-102, 613 (2003).[11] M. Rasa and A.P. Philipse, Nature 429, 857 (2004).[12] C.P. Royall, R. van Roij, and A. van Blaaderen, J.Phys.:Cond. Matt.17, 2315 (2005).[13] M. Rasa, B. Erné, B. Zoetekouw, R. van Roij, and A.P. Philipse,J.Phys.:Cond. Matter. 17, 2293 (2005).[14] A.P. Hynninen, R. van Roij, and M. Dijkstra, Europhys. Lett. 65, 719(2004).[15] A. Esztermann and H. Löwen, Europhys. Lett, 68, 120 (2004).[16] R. Evans, Adv. Phys. 28, 143 (1979).75

[17] A. Torres and R. van Roij, to be published.[18] M. Biesheuvel, private communications and preprint.[19] H.C.Ohanian, Classical Electrodynamics (Boston, MA: Allyn and Bacon)1988[20] A. Cuetos Menendez, private communications and article, in preparation.[21] R. van Roij, Soft Condensed Matter Theory, course book 2004[22] J.P.Hansen, Statistical Mechanics of Fluid Interfaces, course book 200376