Numerical Study on the Colloidal Stability of Oxide Nanosheets

Master thesis Inorganic Materials Science University of Twente June 2011

M.W. Baltussen, MSc

Committee Chair: Prof.dr.ing. D.H.A. Blank External: Dr.ir. P.M. Biesheuvel Mentor: Dr.ir. J.E. ten Elshof Members: S. Kumar, MSc

Summary

Consumers require mobile phones and laptops with decreasing dimensions, while increasing computing power. To accommodate these needs, the building blocks, like capacitors, need to decrease in size. The decrease can only be continued when ultrathin high-κ dielectrics with good insulating properties are used. A possibility for these materials are the oxide nanosheets. The nanosheets can be processed using solution-based processes, which leads to low operational costs and the use of inexpensive equipment. To enable the solution based processing, a stable colloidal solution of nanosheets is needed. The colloidal stability of the nanosheets is thus important for the processing of the nanosheets. In this report, the colloidal stability of nanosheets is determined. To determine the colloidal stability, a numerical model is developed. The model is based on the DLVO-theory. Due to some unrealistic elements, the DLVO-theory is adjusted. First of all, in the DLVO-theory the ions are assumed to be point charges. The ions are thus infinitesimal small particles. However, the ions have a finite size. To take this into account the Stern layer is introduced in the model. Furthermore, the DLVO-theory assumes a constant charge at the surface. However, the charge of an oxidic surface depends on the degree of protonation of the oxidic surface. Therefore, the constant charge model is changed to a charge regulation model: the 1-pK model. In this model, the dimensionless potential between two similar oxidic surfaces is determined using the modified DLVO-theory. Following the electrostatic free energy is determined via the indirect route (the integration of the pressure) and the direct route (integration of the dimensionless potential). The electrostatic free energy is combined with the Van der Waals forces to obtain the interaction forces. If these total interaction forces are positive (repulsion is dominant) at any distance the colloid is stable. For the model, some initial settings are determined to ensure reproducibility of the results. The numerical method used in the model is validated using the analytical solution of the DLVO theory for n:n-electrolytes, 1:2-electrolytes and 2:1-electrolytes. Furthermore, the implementation of the charge regulation model is validated using the degree of ionisation of the surfaces and the analytical solution of the DLVO-theory for n:n-electrolytes. The determination of the pressure is validated, using the properties of the pressure. The indirect and the direct method for acidic and amphoteric surfaces is validated using the Linear Superposition Approximation. However, the results of the direct route for n:1-electrolytes (1:n-electrolytes) for positively (negatively) charged amphoteric surfaces could not be validated. The implementation of the Stern layer is validated via several routes. First of all, the changes in the dimensionless potential and the gradient of the dimensionless potential are validated using the definitions of the diffuse layer and the Stern layer. Furthermore, the calculation of the free energy via the indirect route did not change, when the pressure of the diffuse layer is used in the calculations. Therefore, the changes in the direct route due to the implementation of the Stern layer are validated using the indirect route. The changes in the direct route are validated for acidic surfaces. However, the changes could only be validated for amphoteric surface with a |pH − pK| ≥ 6. The numerical results for the silica and aluminium particles do not match the experimental results. This might be caused by the influence of the temperature and the field strength on the relative permittivity and the specific adsorption of ions on the surface. The numerical results for the swelling of the titanium precursor can be simulated very well at medium distances. The

1 2 differences at small and large distances might be explained by the specific adsorption of ions. Furthermore, the difference between the numerical and experimental results at small distances between the layers might be caused due to the fact that the simulations are not yet in the range of the osmotic swelling. Contents

1 Introduction 5

2 Colloidal stability 9 2.1 The theory on colloidal stability ...... 9 2.2 The experimental colloidal stability ...... 14

3 The Poisson-Boltzmann model for isolated surfaces 17 3.1 The governing equations ...... 18 3.2 The numerical simulation ...... 22 3.3 The initial settings ...... 22 3.4 The validation ...... 26 3.5 Discussion ...... 29 3.6 Conclusion ...... 29

4 The Poisson-Boltzmann model for non-isolated surfaces 31 4.1 The governing equations ...... 31 4.2 The numerical simulation ...... 36 4.3 The initial settings ...... 36 4.4 The validation ...... 43 4.5 Discussion ...... 50 4.6 Conclusion ...... 51

5 Simulation results for isolated surfaces 53 5.1 The acidic surface ...... 53 5.2 The amphoteric surface ...... 57 5.3 Conclusion ...... 61

6 Simulation results for non-isolated surfaces 63 6.1 The experiments with silica ...... 63 6.2 The experiments with aluminium ...... 65 6.3 Colloidal solution of titania nanosheets ...... 65 6.4 Discussion ...... 66 6.5 Conclusion ...... 67

7 Conclusions 69

8 Recommendations 71

9 List of symbols 73

A The Matlab scripts for the isolated surface simulations 83 A.1 The input parameters ...... 83 A.2 The outline of the script ...... 83

3 4 CONTENTS

B The Poisson-Boltzmann model for isolated surfaces 87 B.1 The initial settings ...... 87 B.2 The validation ...... 91

C The Matlab scripts for non-isolated surface simulations 93 C.1 Matlab script for a single distance between the surfaces ...... 93 C.2 Matlab script for multiple distances between the surface ...... 96

D The Poisson-Boltzmann model for two non-isolated surfaces 99 D.1 The initial Settings ...... 99 D.2 The validation ...... 103

E Simulation results for isolated surfaces 109 E.1 The acidic surface ...... 109 E.2 The amphoteric surface ...... 111 Chapter 1

Introduction

Small, smaller, smallest... Consumers require mobile phones and laptops with decreasing dimensions, while increasing computing power. To accommodate this need, the elemental electronic building blocks, like capacitors and transistors, need to decrease in size. To satisfy these demands in the future, ultrathin high-κ dielectrics with good insulating properties are necessary. Currently, these materials are mostly produced using vapour deposition and laser ablation. However, these methods are complex processes. Furthermore, high temperature annealing is necessary, which reduces the capacitance and degrades the other materials in the device. Besides, the equipment and operation costs are high due to the ultrahigh-vacuum used in these methods. Therefore, other methods for fabrication of ultrathin high-κ dielectrics are explored [1, 2]. Ultrathin high-κ dielectrics with good insulating properties can be obtained using oxide nanosheets. These oxide nanosheets can be applied to a surface using solution-based processes, which use inexpensive equipment and have low operational costs [1]. Furthermore, the process does not involve high temperatures. Titania nanosheets, as shown in figure 1.1.a, have a relative dielectric constant of 125 even at a thickness of 10 nm, while the leakage current is less than 10−7 A cm−2. Therefore, these nanosheets might be a very promising material for the fabrication of thin film capacitors and transistors [1, 2, 4, 5]. A nanosheet is a new class of nanomaterials. Nanosheets have a 2D morphology. The lateral dimensions of a nanosheet can be up to a micrometer, while the thickness is only about one nanometer [1, 2, 4, 5]. In figure 1.2, the Atomic Force Microscope (AFM) image of a titania nanosheet is shown. The height image in figure 1.2.b shows that the height of the nanosheets is 0.93 ± 0.06 nm, which is comparable to the crystallographic thickness of TiO2 (0.75 nm). The slight difference is probably caused by adsorption of solvent or ions at the nanosheets [2, 4]. The nanosheets are produced from a layered precursor, as shown in figure 1.1.b [3]. The nano-

(a) The titania nanosheet (b) The layered precursor

Figure 1.1: Schematic representation of the layered precursor of the nanosheet and the titania (Ti0.91O2) nanosheet [2, 3].

5 6 CHAPTER 1. INTRODUCTION

(a) The AFM image (b) The height image

Figure 1.2: The Atomic Force Microscope (AFM) image of a Ti0.91O2 nanosheet on a Silica surface. The image is obtained using a SII nanotech E-Sweep AFM in tapping-mode [2].

Figure 1.3: The exfoliation of a layer compound into nanosheets [4]. sheets are clearly visible in the crystal (the hexagonal layers). The layers between the nanosheets are called the interlayers, which in the case of the titania nanosheets consist of potassium, rubi- dium or caesium ions [5]. The nanosheets are obtained via exfoliation of the layered precursor as shown in figure 1.3. In clays, the delamination of the layered precursor is spontaneous due to the weak interactions between the layers in the compound. However, in the layered precursor of oxide nanosheets, the charge density is larger. Therefore, the layered precursor is first hydrated, in which the interlayer is exchanged by protons. Following, the protons between the layers are exchanged by bulky organic ions like tertrabutylammonium cations (TBA+). These bulky compounds will increase the distance between the layers and thereby reduce the electrostatic interactions between the layers. The delamination of the layered precursor can then occur [2, 5, 4]. The colloidal stability of the solution is of importance, because of the production and processing of the nanosheets. Therefore, the colloidal stability of the nanosheets will be determined in this master assignment. In the colloidal suspension, there are two forces acting on the nanosheets: the attractive Van der Waals forces and the repulsive electrostatic forces [6, 7, 8]. Since the 1950’s, the DLVO-theory (Derjaguin-Landau-Verwey-Overbeek-theory) is used for the description of the forces acting in solutions containing colloidal particles [6]. However, the DLVO-theory is unable to explain the reversible potentials obtained in experiments [6]. Therefore, the theory will be adapted to enable an explanation of the reversible potentials. The first of the two adaptions is the use of Stern-Gouy-Chapman model instead of the Gouy-Chapman model, which will both be explained later. The introduction of the Stern layer in this model, will remove the assumption that all ions are point charges [8, 9, 10]. The second adaption is the use of a charge regulation model instead of the constant charge model. In the DLVO-theory, the charge of the examined surface is assumed constant, while the charge depends on the adsorption of protons [6, 9, 11]. The resulting Stern-Gouy-Chapman model with charge regulation boundary condition, will first be used for the determination of the colloidal stability of particles. This will enable a validation of the model. Following, the same model will be used for the determination of the colloidal stability of titania nanosheets. The same model can be used for the study of nanosheets, because studies 7 on clay minerals showed that the behaviour of nanosheets is similar to the behaviour of colloidal particles [4]. The aim of this report is to develop a model which is able to determine the effect of the electrolyte solution, the permittivity of the solvent, the pH and the temperature on the colloidal stability of oxide nanosheets (especially the titania nanosheets made by R. Besselink (2010) [5]). The numerical model is based on the DLVO-theory, which is adjusted for the finite particle size and the charge regulation. This report is divided in several chapters as follows. The first chapter contains a short description of the colloidal stability. Following, the models used for the determination of the colloidal stability will be discussed in detail. The models will also be validated in these chapters. The next chapters contain the results of the numerical simulations. These results will be compared to experimental results, obtained from literature. Finally, some conclusions will be drawn from the simulations and some recommendations will given for further research. 8 CHAPTER 1. INTRODUCTION Chapter 2

Colloidal stability

As discussed in the previous chapter, the stability of this colloid is important for the production and processing of the nanosheets. Therefore, this chapter will focus on the colloidal stability. The colloidal stability can be determined from the Gibbs free energy of the system [12]. If the Gibbs free energy is positive (so repulsion forces are dominant), the colloid is stable and will not flocculate. The Gibbs free energy is the summation of the attractive Van der Waals forces and the repulsive electrostatic force. Since the 1950’s, these forces are combined in the DLVO-theory. Therefore, this chapter starts with a description of the DLVO-theory [6, 10]. In the DLVO-theory, the ions are assumed to be point charges, the adsorption energy to be only electrostatic, no specific interactions between the surface and the ions (so there is no specific adsorption of the ions), a constant charge model and the solvent to be primitive (the solvent is a structureless continuum which affects the distribution only via the dielectric permittivity). The assumptions contain unrealistic elements [8, 10]. The effect of these assumptions will be discussed in this chapter and some refinements on the model will be done. The force over the radius between the colloidal particles, which is related to the Gibbs free energy, can also be determined experimentally. The interaction forces can be determined by for example the Atomic Force Microscope (AFM) and Surface Force Apparatus (SFA). These results can be compared to validate the results of the numerical model. Therefore, the experimental results for the interaction forces for silica and aluminium surfaces are presented in this chapter [6]. The swelling of a layered precursor can also be described thermodynamically using the theory of colloidal stability [13]. The relation between the swelling and the theory of the colloid stability will be discussed in this chapter. Furthermore, the experimentally determined swelling of the titanium precursor is given, to enable verification of the numerical results for nanosheets.

2.1 The theory on colloidal stability

In this paragraph, the theory on colloidal stability will be explained. First of all, the DLVO-theory will be discussed in detail. To do so a detailed description of the charged surface and the adjacent solution are given. The DLVO-theory is developed with some assumptions. However, these assumptions lead to unrealistic elements in the DLVO-theory. The eﬀects of these assumptions will be discussed in this paragraph and the model will be adapted where necessary.

2.1.1 The DLVO-theory

When a charged surface comes into contact with an electrolyte solution, the counter-ions will accumulate at the surface. Besides the attraction to the surface, the Brownian motion and entropy eﬀects also act on the counter-ions, which leads to a attraction toward the bulk of the solution. This leads to an equilibrium in the distribution of the counter-ions, as shown in ﬁgure 2.1.a. The

9 10 CHAPTER 2. COLLOIDAL STABILITY

(a) The distribution of the ions

(b) The concentration proﬁles

Figure 2.1: A schematic representation of the electrical double layer. The grey part of the drawing is the surface, while the white part is the solution. In figure 2.1.a, the distribution of the ions is shown with the surface charges. In figure 2.1.b, the profile of the anions (red line) and the cations (blue line) is shown. In figure 2.1.c, the profile of the potential is shown. concentration of the counter-ions (in figure 2.1 the anions) will gradually decrease to the bulk concentration (c∞), as shown in figure 2.1.b [8, 9, 10]. The co-ions (the cations in figure 2.1) are repelled by the surface. The concentration near the surface is low. In the bulk of the solution the concentration of the co-ions is higher than at the surface, as shown in figure 2.1.a. Figure 2.1.b shows the equilibrium distribution of the co-ions, which is a gradual increase in the concentration of the co-ions. The combination of the charged surface and the adjacent solution, which shows a deviant concentration of the counter- and co-ions (with respect to the bulk concentration), is called the electrical double layer [8, 9, 10]. The concentration of both the counter-ions and co-ions can be described using electrostatic and diffusion theory, which are combined in the Poisson-Boltzmann equation. Besides the distribution of the ions, the average electric potential, ψ, can be determined at any point in the solution. The potential as function of the distance from the surface is shown in figure 2.1.c [8, 9, 10]. However, there are multiple particles in a colloidal solution. The particles can approach each other due to their Brownian motion. The electrical double layers of both surfaces of the particles will then interfere, which effects the distribution of the ions and the potential of both double layers, as shown in figure 2.2. These changes give rise to an increase of the concentration of ions with respect to the bulk concentration. This leads to an osmotic pressure between the surface, which causes the surfaces to repel each other. The energy needed to bring the surfaces together is shown in figure 2.3 by the double layer repulsive force [8, 9, 10]. 2.1. THE THEORY ON COLLOIDAL STABILITY 11

(a) The potential proﬁle of an isolated wall

(b) The potential proﬁle of a non-isolated surface

Figure 2.2: A schematic representation of the potential. In figure 2.2.a the profile of an isolated layer is shown. In figure 2.2.b shows the profile for the non-isolated surface. In this figure is the blue line the profile of the non-isolated surfaces, while the red lines are the potential profiles of the isolated surface.

Besides the repulsive force, there is also an attractive force. The Van der Waals force acts on the atoms of the particles. These forces seem small, but the Van der Waals forces of the different atoms in the particles are additive. Due to the large amount of atoms in the particles, the Van der Waals force is in the same order as the repulsive forces due to the electrical double layer, as shown in figure 2.3.a. The Gibbs free energy of the particles can be determined by the summation of the energy of the Van der Waals force and the energy of the osmotic pressure between the surfaces. This is done at varying distances from the surface. The resulting curve is shown in figure 2.3.a by the solid line [8, 9, 14]. However, there are some unrealistic elements in the DLVO-theory. First of all, other energy effects than the electrostatic adsorption energy, like the counter-ion and solvent interactions, are not taken into account. These energy effects partly compensate each other. Therefore, these effects have little influence on the calculated Gibbs free energy [8]. Besides, the particles are considered to be point charges and the specific interactions between the counter-ions and the surface are neglected. These assumptions can be removed by introducing the Stern layer [8, 9, 10].

2.1.2 The Stern-Gouy-Chapman model In the DLVO-theory, the ions are assumed to be point charges. The ions have infinitesimal radius. However, ions have a finite size. The charge of the ions is considered to be at the center of the ion. Therefore, the charges cannot get closer to the surface than the ion radius, as shown in figure 2.4 [8, 9, 10]. The minimal distance between the ions and the surface is called by the outer-Helmholtz plane. The layer between the outer-Helmholtz plane and the surface is the Stern layer. In this layer, there are no charges present. Therefore, it acts as a molecular condensor. The potential in the Stern layer will drop linearly in the Stern layer, as shown in figure 2.4. The rest of the electrical double layer is called the diffuse layer. This layer acts ideally: the potential and the concentration of the ions can be determined using the Poisson-Boltzmann equation [8, 9, 10]. Besides the effect on the potential, the introduction of the Stern layer also has consequences for 12 CHAPTER 2. COLLOIDAL STABILITY

(a) without Stern layer (b) with Stern layer

Figure 2.3: The course of the Gibbs free energy using the DLVO-theory without and with Stern layer. The ﬁgure shows the energy of the Van der Waals force and the electrostatic force, which are both represented with a dotted line. Furthermore, the net Gibbs free energy is shown by the solid line. The red line in ﬁgure 2.3.b indicates the Stern layer [8].

Figure 2.4: A schematic representation of the potential for an non-isolated surface with Stern layer. 2.1. THE THEORY ON COLLOIDAL STABILITY 13 the determination of the Gibbs free energy. Due to the introduction of the Stern layer, the reference plane for the determination of the repulsion forces has shifted with respect to the reference plane of the determination of the Van der Waals forces, as shown in figure 2.3.b. This leads to a higher repulsion of the surfaces at a certain distance between the planes, which is clear when figure 2.3.b is compared to figure 2.3.a [6, 14]. Besides the size of the ions, the Stern layer can also take into account the specific interactions between ions and the surface. Ions can not only have electrostatic interactions with the surface, but can also form complexes on the surfaces. To take this into account the simple Stern layer concept as discussed above is replaced by a triple layer model; a second plane is introduced between the outer-Helmholtz plane and the surface: the inner-Helmholtz plane. At this plane the complexes are formed. This model works very well for ideally flat surfaces. However, in solids the surface irregularties make it impossible to determine an extra plane between the outer-Helmholtz plane and the surface. Therefore, only the sizes effects will be taken into account [8, 9, 10].

2.1.3 The charge regulation model In the previous paragraphs, the surface charge is assumed to be a constant known value. However, the surface charge of metal (hydr)oxides is determined by the adsorption of protons and hydroxide ions at the surface. The ﬁrst model developed for the determination of the surface charge using the adsorption of protons is the 2-pK model. A surface group can absorb two protons in two consecutive reactions as shown below.

− + MO + H MOH

+ + MOH + H MOH2 This leads to three kinds of surface groups as shown in ﬁgure 2.5.b and 2.5.c [9, 11]. However, only one of these reactions will be active in the normal pH range. Therefore, a new model is created: the 1-pK model. In this model only one reaction will take place, which is given the following reaction. − 1 + + 1 MOH 2 + H MOH 2 The 1-pK model is represented in ﬁgure 2.5.a. The charge of each surface group is determined via 1 Pauling’s bond valance concept. The metal ion (+3) has six bonds with a valance of + 2 . The 1 oxygen has a a valance of -2. With the positive charge of the protons (+1), this results in a - 2 1 charge of the groups containing only one proton (−OH) and a + 2 charge of the −OH2 surface groups [9, 11]. The pK-value, which is the logarithm of the equilibrium constant of the protonation reaction, of these metal (hydr)oxides can be determined from the point of zero charge. At the point of zero charge, there are as many positive sides as negative sides. Therefore, the pH of the point of zero

(a) 1-pK (b) 2-pK model (c) 2-pK model

Figure 2.5: Schematic representation of a metal oxide surface. In ﬁgure 2.5.a the 1-pK model is shown while ﬁgure 2.5.b and 2.5.c show 2-pK models. The numbers in the boxes show the charges of each group [9]. 14 CHAPTER 2. COLLOIDAL STABILITY

Table 2.1: The point of zero charge (pK-values) and the charge density at the surface (N) for silica, aluminium hydroxide and titania surfaces

pK N (nm−2) Reference Silica (SiO2) 7.5 8 [9, 11, 15] Aluminium hydroxide(AlOH) 7.9-10.1 8-10 [6, 11, 16, 17] Titania (TiO2) 4.0-7.5 8 [9, 11, 18, 19, 20, 21] charge is equal to the pK-value [9, 11]. The point of zero charge for aluminium hydroxide and titania are given in table 2.1. The range for the pK-value for titania surfaces is large. The values 4.0 and 7.5 are exceptions: the obtained point of zero charge is mostly in the range 6-6.3. Besides the pK-value, the density of the charged sites, N, should be known. These values are also given in table 2.1. All the metal oxides, which are also called amphoteric materials, can be described using this 1-pK model, except for silica. Silica exhibits in the normal pH-range the following reaction.

− + SiO + H SiOH The sites of silica are thus similar to those in the 2-pK model. However, the silica surface cannot be positively charged in the normal pH-range. The point of zero charge of silica is reached asymptotically. The pK-value is thus not equal to point of zero charge [9, 11]. Both the pK- value and the density of the charged sites of silica are also given in table 2.1. The surface charge can be determined from the concentration of the protons at the surface and the 1-pK model. When the surfaces close up, the potential rises as explained in paragraph 2.1.1. Therefore, the concentration of the protons will increase or decrease depending on the sign of the surface charge. The charge of the surface will then increase.

2.2 The experimental colloidal stability

Theory on the colloidal stability has been explained into detail. The DLVO-theory has been adapted for the ﬁnite size of the ions and the charge regulation at the surface. This model can be used to describe the colloidal stability of the sols. However, to test the validity of the models, the numerical solutions have to be compared to experimental results. These experimental results will be given in this part of the chapter.

2.2.1 Silica en aluminium particles For silica and aluminium, the forces between two surfaces are determined using the AFM and the SFA. The aluminium surface has a single coordinated hydroxide groups. Therefore, these can be modelled using the pK-value of aluminium hydroxide. The results of these measurements are partly presented by P.M. Biesheuvel (2001). All these results are now shown in ﬁgure 2.6. The results are obtained in a NaBr solution with pH of 3 with a SFA [6]. The results for the silica surfaces are obtained using the AFM. The AFM results of silica are obtained by P.M. Biesheuvel. The results are presented in ﬁgure 2.7. These results are measured in a CaCl2 solution. The AFM does not give an absolute distance measurements. Therefore, the results are shifted by 0.8 nm [6].

2.2.2 Titania nanosheets For titania nanosheets, there are no AFM or SFA measurements available. Therefore, another method of validation for the numerical results has to be used. The swelling data obtained by R. Besselink et al. (2011), which are shown in ﬁgure 2.8, can be used for veriﬁcation of the numerical results [5]. 2.2. THE EXPERIMENTAL COLLOIDAL STABILITY 15

(a) 100 mM (b) 10 mM

Figure 2.6: The SFA results for the repulsion between aluminium surfaces in a NaBr solution with a pH of 3. The concentration of NaBr is given in the subscript of the ﬁgures [6].

(a) 10 mM, pH = 4.1 (b) 10 mM, pH = 10.3

Figure 2.7: The AFM results for the repulsion between aluminium surfaces in a CaCl2 solution. The concentration of NaBr and the pH are given in the subscript of the ﬁgures. The distances in these results are shifted by 0.8 nm [6]. 16 CHAPTER 2. COLLOIDAL STABILITY

Figure 2.8: The distance between the titania layers in the swollen precursor. In these experiments, the concentration of the TBA+, in the form of addition of the salt TBAOH, is varied while the pH is kept constant at 12.4-12.7 [5].

The swelling of the layered precursor has 2 stages. In the ﬁrst stage, the swelling is caused by the hydration of the precursor. Water is adsorbed by the layered precursor. The water will form successive layers of water between the layers. The energy of the interaction between the layer is a combination of the hydration energy, the electrostatic repulsion and the Van der Waals attraction [13]. The second stage of swelling is the osmotic swelling. The swelling is caused by the higher concentration of ions between the layers than at inﬁnite distance from the surface. This stage starts when the layers are more than 10 Å apart. The hydration energy is in this stage no longer of importance. The energy of the interaction between the layers is thus only determined by the electrostatic repulsion and the Van der Waals attraction. The energy of the interaction between the layers should be determined varying from the surface. The equilibrium distance between the layers can then be determined from the minimum in this graph [13]. The osmotic swelling can be determined using the adjusted DLVO-theory. The experimental results presented a swelling of 8 nm to 13 nm. This is clearly in the range of osmotic swelling. Therefore, the experimental results can be modelled using the adjusted DLVO-theory of paragraph 2.1. Chapter 3

The Poisson-Boltzmann model for isolated surfaces

As discussed in the previous chapter, the colloidal stability can be determined from the Gibbs free energy of the solution. The Gibbs free energy consists out of the electrical work, the chemical work and the Van der Waals forces. Since the 1950’s, the DLVO-theory is used to determine the Gibbs free energy of a solution [10, 6]. However, the DLVO-theory is not able to explain the reversible minima in the potential, which are obtained for oxide surfaces [6]. Therefore, a model is developed in which the DLVO-theory is adjusted. First of all, the Stern layer is introduced to take into account the finite size of the ions and the discrete surface charges. The introduction of the Stern layer divides the diffuse double layer into two parts. The inner part of the diffuse layer takes into account all the non-ideallities, while the outer layer stays ideal [10]. Besides the changes in the electrochemical potential, the introduction of the Stern layer also introduces a difference in the calculation reference plane for the calculation of the electrostatic forces [14]. Furthermore, the constant charge model, which assumed in the DLVO-theory, is changed to a charge regulation model. In the DLVO-theory, the charge at the surface is assumed constant, but the charge of the surfaces depends on the adsorption of protons and thus the dimensionless potential. These oxide surfaces can be either acidic (silica surfaces) or amphoteric (for example aluminium hydroxide and titania surfaces). The charge of the oxide surfaces depends on the concentration of the protons, the concentration of electrolyte and the pH of the solution [7, 6, 24]. In this chapter the Poisson-Boltzmann model for isolated surfaces is introduced, because the Poisson-Boltzmann equation can be solved analytically for an isolated surface in a electrolyte solutions, with a valency of n:n, 1:2 and 2:1 [10, 23]. These analytical results can be used to verify the numerical results. This will lead to the validation of the numerical method used. Besides the validation of the numerical model, the charge regulation model can be validated, using the analytical results for a n:n-electrolyte solution. The ionisation degree of the oxide surface (for both acidic and amphoteric surfaces) depends on the pH of the solution, the equilibrium constant for the protonation reaction of the surface and the potential at the surface. However, the ionisation degree can also be determined from the Poisson-Boltzmann equation. Comparing these results will enable the validation of the implementation of the charge regulation model [7]. This chapter starts with a description of the governing equations of the model. These equations will then be implemented in a numerical model. Following, to ensure reproducibility the initial settings of the model are determined. Finally, the numerical model and the implementation of the charge regulation model will be validated using analytical solutions. The implementation of the Stern layer will be validated using the definitions of the Stern layer and the diffuse layer.

17 18 CHAPTER 3. THE POISSON-BOLTZMANN MODEL FOR ISOLATED SURFACES

3.1 The governing equations

In this paragraph the governing equations of the model will be discussed. First, the equations for the electrochemical potential in the diﬀuse layer will be derived. Then, the eﬀects of the Stern layer on the governing equations will be discussed. Finally, the charge regulation boundary condition will be explained.

3.1.1 The diffuse layer Electrostatic behaviour can be described using the Maxwell equations. The Gauss law, which is one of these equations, relates the distribution of electric charges to the electric field. Gauss law reduces to the Poisson equation, equation 3.1, if the ions are assumed to be point charges, the ionic adsorptions are assumed to be purely electrostatic and the solvent is assumed to be primitive (which means that the solvent is a structureless continuum which affects the distribution only via the dielectric permittivity) [10, 25]. In this equation, E is the electric field, ψ the electrostatic potential, F the Faraday constant, ε0 the permittivity of vacuum and εr the relative permittivity of the medium. zi is the valency of the ion i and ci is the concentration of ion i.

F X −∇E = ∇2ψ = − (z c ) (3.1) ε ε i i 0 r i In thermodynamic equilibrium, the concentration of the ions can be determined using the Boltz- mann equation (equation 3.2), in which R is the gas constant and T the temperature [12, 6, 11]. The subscript ∞ is used for an inﬁnite distance from the surface.

− ziFψ ci = ci,∞e RT (3.2)

Combining the Poisson and Boltzmann equations results in the Poisson-Boltzmann equation (or the Gouy-Chapman equation), equation 3.5.

2 F X − ziFψ ∇ ψ = − z c e RT (3.3) ε ε i i,∞ 0 r i For an asymmetric electrolyte in a water solution the Gouy-Chapman equation is equal to equation 3.4. In this equation subscript p is used for the positive ions (cations) and subscript m for the negative ions (anions) [10].

z Fψ 2 F − p − zmFψ ∇ ψ = − zpcp,∞e RT + zmcm,∞e RT (3.4) ε0εr Besides the ions caused by the electrolyte, the acidity of the water solution also introduces ions in the solution. It is assumed that both acidic and basic solutions are produced using a 1:1 salt, for example hydrochloric acid (HCl) for acidic solutions and sodium hydroxide (NaOH) for basic solutions. Therefore, the concentration of anions introduced by the acidity in a acidic solution will be equal to the concentration of the protons. Similarly, the concentration of the cations introduced by the acidity in a basic solution will be equal to the concentration of the hydroxide ions. The Gouy-Chapman equation will therefore become equation 3.5. In this equation, the subscript H is the concentration of the protons in an acidic solution and the concentrations of the hydroxide ions in a basic solution.

z Fψ 2 F − p − zmFψ − Fψ Fψ ∇ ψ = − zpcp,∞e RT + zmcm,∞e RT + cH,∞e RT + cH,∞e RT (3.5) ε0εr Now, the dimensionless potential, y, is introduced, which results in equation 3.6. RT F 2 −zpy −zmy −y y ∇ y = − zpcp,∞e + zmcm,∞e + cH,∞e − cH,∞e (3.6) F ε0εr 3.1. THE GOVERNING EQUATIONS 19

This equation can be simpliﬁed using equation 3.7 to 3.9. In these equations, v is the number of ions in which the used salt dissociates and c∞ is the concentration of the undissociated salt. The resulting equation is equation 3.10. zpcp,∞ = zmcm,∞ (3.7)

cp,∞ = vpc∞ (3.8)

cm,∞ = vmc∞ (3.9)

F2c c c 2 ∞ −zpy −zmy H,∞ −y H,∞ y ∇ y = − zpvpe + zmvme + e − e (3.10) RTε0εr c∞ c∞ To decrease the computational time, the equation is integrated to equation 3.12 via equation 3.11 [10].

F2c c c 2 ∞ −zpy −zmy H,∞ −y H,∞ y 2∇y · ∇ y = −2∇y · zpvpe + zmvme + e − e (3.11) RTε0εr c∞ c∞

F2c c c 2 ∞ −zpy −zmy H,∞ −y H,∞ y (∇y) = 2 vpe + vme + e + e + K (3.12) RTε0εr c∞ c∞ In equation 3.12 is K an integration constant, which can be removed when a boundary condition is used. For isolated surfaces the dimensionless potential at inﬁnite distance from the surface is zero. The derivative of the dimensionless potential is also zero at inﬁnite distance from the surface. Therefore, equation 3.12 becomes equation 3.13. s 2 2F c∞ −z y −z y cH,∞ −y ∇y = −sign (σs) vpe p + vme m + e + ··· (3.13) RTε0εr c∞ cH,∞ y cH,∞ cH,∞ ··· + e − vp − vm − − c∞ c∞ c∞

In this equation is sign (σs) the sign of the surface charge density, which is deﬁned by Gauss law (equation 3.14) [10]. σF ∇y = − (3.14) ε0εrRT Equation 3.13 can be rewritten to equation 3.15 using Gauss law.

c 2 −zpy −zmy H,∞ −y σ = 2c∞ε0εrRT vpe + vme + e + ··· (3.15) c∞ cH,∞ y cH,∞ cH,∞ ··· + e −vp − vm − − c∞ c∞ c∞

3.1.2 The Stern layer The introduction of simplest Stern layer, which only takes into account the eﬀect of the ion size, leads to adjustments to the governing equations. In the Stern layer, the gradient of the potential is equal to the gradient at the surface. Therefore, the potential can be determined using equation 3.16. In this equation, d is the thickness of the Stern layer and x the direction of calculation [10].

y − ys F ∇y = = − σs (3.16) x≤d d RTε0εr

At the outer Helmholtz plane, the gradient of the potential will be equal to the gradient at the surface. From the outer Helmholtz plane, the potential will behave ideal. The resulting equation for the gradient in the diﬀuse layer is equation 3.17. 20 CHAPTER 3. THE POISSON-BOLTZMANN MODEL FOR ISOLATED SURFACES

c 2 −zpy(x−d) −zmy(x−d) H,∞ −y(x−d) σ = 2c∞ε0εrRT vpe + vme + e + ··· (3.17) x≥d c∞ cOH−,∞ y(x−d) cH,∞ cH,∞ ··· + e − vp − vm − − c∞ c∞ c∞ This equation is equal to the equation used in the Gouy-Chapman picture. However, the direction of calculation, x, in the Gouy-Chapman picture should be replaced by x − d, to take into account the non-ideal behaviour [10].

3.1.3 The charge regulation model Besides the boundary condition at inﬁnite distance from the surface, another boundary condition is needed to determine the dimensionless potential at all distances. This boundary condition is the potential, the charge or a relation between these at the surface. For constant charge models and constant potential models, this boundary condition is a value for the charge or the potential. For the charge regulation models, which are used in this assignment, this boundary condition is deﬁned by the adsorption of protons at the surface, which means that this boundary condition is a Robin (mixed) boundary condition. The equilibrium of the protonation of the surface is given by the following reaction [10, 11, 26]. S + H SH The equilibrium constant, K, of this reaction is given by equation 3.18. In this equation the subscript s denotes the concentration of a certain molecule at the surface.

[SH] K = s (3.18) [S]s[H]s The concentration of the protons can be determined using the Boltzmann equation, as shown in equation 3.19 [11].

− Fψs [H]s = cH,∞e RT (3.19) When ideal behaviour is assumed, the sites are either empty or they adsorbed a proton. Then equation 3.18 will become equation 3.20, in which θh are the protonated sites. This equation is similar to a Langmuir type of adsorption, but the concentration of protons is the concentration at the surface instead of the bulk concentration [11].

θh K · [H]s = (3.20) 1 − θh For an acidic surface (silica), the negative sites on the surface are protonated to neutral sites as shown by the following reaction equation.

− + SiO + H SiOH This means that the surface charge density is always negative or neutral. The charge of the surface can be described by equation 3.21.

σs = −F · N (1 − θh) (3.21)

In this equation, N is the site density at the surface in mol m−2. Combining equation 3.20 and 3.21 gives the boundary condition for acidic surfaces (equation 3.22) [11].

−F · N σs = (3.22) 1 + K · [H]s 3.2. THE NUMERICAL SIMULATION 21

Start: y(1) = 0

dy via Poisson-Boltzmann y(1) = y(1) +Δy equation

dy via boundary condition

Rdif > threshold dy check

Determination of all y

Simulation finished

Figure 3.1: A schematic representation of the numerical method for the determination of the dimensionless potential. The dimensionless potential at the surface is y(1).

For an amphoteric material (such as aluminium hydroxide), the sites are protonated via the following reaction.

1 + 1 − 2 + 2 AlOH + H AlOH2

According to the reaction, an amphoteric surface has both negative and positive sites. The surface charge density of the amphoteric materials is described by equation 3.23.

1 σ = F · N θ − (3.23) s h 2

The resulting boundary condition for a amphoteric surface is equation 3.24 [11].

1 1 − K · [H]s σs = − FN (3.24) 2 1 + K · [H]s 22 CHAPTER 3. THE POISSON-BOLTZMANN MODEL FOR ISOLATED SURFACES

3.2 The numerical simulation

In this part of the chapter, the numerical simulation model is explained. A more detailed description of the Matlab script used is given in appendix A. The simulations are performed on a Cartesian grid in one direction. The Robin boundary condition is determined using a simple procedure for the determination of the dimensionless potential, which is shown in ﬁgure 3.1. The starting potential at the surface is set to 0. Then the derivative of the potential is determined using the Poisson-Boltzmann equation. (For models including the Stern layer, the gradient of the potential determined using the Poisson-Boltzmann equation should be determined at the outer- Helmholtz plane) The derivative is also determined via the surface charge density (equation 3.14 and 3.22 or 3.24). The relative diﬀerence, Rdif , is calculated via equation 3.25.

∇yPB − ∇yBC Rdif = (3.25) ∇yBC

In this equation is ∇yPB the derivative determined using the Poisson-Boltzmann equation and ∇yBC the derivative determined via the surface charge density. If Rdif is larger than a set maximal relative diﬀerence, a small value, ∆y, is added (subtracted) to y for a positively (negatively) charged surface. (The value of the threshold will be discussed in paragraph 3.3). If the diﬀerence is less than the threshold, the value of y is chosen as the dimensionless potential of the surface. When the dimensionless potential at the surface is known, the values of the other potentials can be determined using a discretisation scheme. In the numerical model, three discretisation schemes can be used: the Euler method, the midpoint method or the third order Runga-Kutta method. The most exact solution of the problem is given by the third order Runga-Kutta method, because of the higher order of this scheme.

3.3 The initial settings

Starting the simulation, some initial settings are deﬁned. These initial settings should be declared for all the simulations. This will ensure the reproducibility of the results and will enable interpretation of the results. In this paragraph the initial settings of the isolated surface simulations will be set. The pK-value used in this simulations for silica is 7.5 and for aluminium hydroxide is 10. The density of the sites, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2 [6, 9, 11].

3.3.1 The step size First of all, the effect of the step size in the distance is determined, for different pH, concentrations and surfaces. A smaller step size will introduce more points for calculation. This will result in a higher accuracy of the numerical method, but will increase the simulation time. The results of these simulations for acidic surfaces (silica) in an acidic environment are presented in figure 3.2.a to 3.2.d. These show that using larger then 0.1 nm as the step size for the isolated surface simulations might lead to large differences between the real solution and the obtained results. When smaller step sizes are used no difference is found between the results. Figure 3.2.e to 3.2.h show that a step size of 0.01 nm is sufficient to obtain results that do not really differ from the results with smaller step sizes for amphoteric surfaces (aluminium hydroxide) in a acidic environment. The effect of the step size on the accuracy of the simulations is also determined in a basic environment. These results are given in Appendix B. These results show that in a basic environment a step size of 0.01 nm is needed for both acidic and amphoteric surfaces. Concluding, a step size of 0.01 nm can be used to obtain reproducible results.

3.3.2 The relative diﬀerence, Rdif

The relative difference, Rdif , is important in the calculation of the dimensionless potential at the surface, as shown in figure 3.1. The effect of this relative difference is determined at different 3.3. THE INITIAL SETTINGS 23

(a) Silica, 0.1 M (b) Silica, 0.1 M, close up

(e) Aluminium hydroxide, 0.1 M (f) Aluminium hydroxide, 0.1 M, close up

(g) Aluminium hydroxide, 1 mM (h) Aluminium hydroxide, 1 mM, close up

Figure 3.2: The effect of the step size on the accuracy of the isolated surface simulations in a acidic environment, pH of 4. The lengths given in the legend are the step sizes. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity is 78. The used relative difference, Rdif , is 0.01% and the ions of water are taken into account. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. 24 CHAPTER 3. THE POISSON-BOLTZMANN MODEL FOR ISOLATED SURFACES

(a) Silica, 0.1 M (b) Silica, 0.1 M, close up

(e) Aluminium hydroxide, 0.1 M (f) Aluminium hydroxide, 0.1 M, close up

(g) Aluminium hydroxide, 1 mM (h) Aluminium hydroxide, 1 mM, close up

Figure 3.3: The effect of the relative difference of the determination of the dimensionless potential, Rdif , at the surface on the accuracy of the isolated surface simulations in a acidic environment, pH of 4. The percentages given in the legend are the different Rdif used. The temperature is 298 K and the relative permittivity is 78. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The step size used is 0.01 nm and the water ions are taken into account. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. 3.3. THE INITIAL SETTINGS 25

(a) Silica, 0.1 M (b) Silica, 1 mM

Figure 3.4: The effect of the ions in the water of the isolated surface simulations in a acidic environment, pH of 4. The blue lines are the simulation without water ions and the red lines are the simulations with water ions. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity is 78. The step sizes are 0.01 nm and the relative difference, Rdif , is 0.01%. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density at the silica surface, N, is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. concentrations, different pH and different surfaces. These results of the simulations in an acidic environment are presented in figure 3.3. The results in basic environment are in Appendix B. Figure 3.3.a to 3.3.d show that at least a relative difference of 0.1 % is needed to obtain similar results for acidic surfaces (silica) in an acidic environment. For amphoteric surfaces (aluminium hydroxide), the relative difference should be less than 1 % to obtain reproducible results. The results presented in the Appendix B show that an relative difference of 1 % is needed to obtain accurate results for both kinds of surfaces in a basic environment. Concluding, to obtain reliable results, the relative difference should be 0.1 % for both acidic and amphoteric surfaces. To ensure reproducibility, the relative difference is set to 0.01 %.

3.3.3 The ions in the water In all the simulations performed above the ions of water are also taken into account. However, this induces a longer simulation time. Therefore, the effect of the ions in the water is determined in this paragraph. The results of the basic environment are shown in Appendix B. Figure 3.4.a and 3.4.c show simulations with a relative high concentration of electrolyte with respect to the concentration of the ions in the water. These simulations show no difference between the simulations with or without the water ions. When the concentration of the electrolyte is decreased (so the concentration of the electrolyte and the concentration of the water ions are in the same order), the ions in water have an effect on the resulting electrochemical potential as shown in figure 3.4.b and 3.4.d. 26 CHAPTER 3. THE POISSON-BOLTZMANN MODEL FOR ISOLATED SURFACES

Also the basic simulations in Appendix B show that the influence on the simulations of the water ions is only found at simulations with low concentrations of electrolytes. However, a small effect of the water ions is obtained for the amphoteric surface at high concentration of electrolyte. Concluding, the ions in the water influence the electrochemical potential. However, the effects are only obtained when the concentration of the electrolyte and the ions of the water are in the same order. Therefore, the ions of the water are taken into account.

3.4 The validation

The previous paragraphs in this chapter describe the Poisson-Boltzmann model for isolated surfaces. However, to ensure the validity of the model some validation simulations are performed. In this paragraph, the implementation of the numerical method and the implementation of the boundary condition are validated using the analytical solution of the DLVO-theory. Finally, the eﬀect of the Stern layer on the potential and the gradient of the potential are validated with the theory described in paragraph 3.1.2.

3.4.1 The numerical method To check the numerical method, the numerical simulation results for the dimensionless potential should be checked with analytical results. The DLVO-theory can be solved analytically, when the eﬀect of the water ions are not taken into account and a n:n, 1:2 or 2:1 electrolyte solution is used [10, 23].

A n:n-electrolyte solution Equation 3.10 will become equation 3.26, when a z:z electrolyte is used and the water ions are not taken into account. F2z2c 2F2z2c ∇2y = − ∞ e−zy − ezy = ∞ sinh(zy) (3.26) RTε0εr RTε0εr This equation can be integrated via the same route as depicted in paragraph 3.1. This leads to equation 3.27 [10]. 4F2c (∇y)2 = ∞ (cosh(zy) − 1) (3.27) RTε0εr Equation 3.27 can be rewritten to equation 3.29 via equation 3.28. s 2 4F c∞ ∇y = −sign (σs) (cosh(zy) − 1) (3.28) RTε0εr s 2 2F c∞ zy ∇y = −sign (σs) · 2 sinh (3.29) RTε0εr 2 Integrating this equation will lead to equation 3.30, which is the analytical solution for the Poisson- Boltzmann equation. In this equation is ys the potential at the surface [10].

q 2 zy zy − 2F c∞ x tanh = tanh s e RTε0εr (3.30) 4 4 The analytical solution and the numerical simulations are obtained using a constant dimensionless potential of 0.2 and a constant charge of 0.11 mC m2. The results of simulations and analytical solution are shown in ﬁgure 3.5. Figure 3.5.a and 3.5.c show only the results of the numerical solution with the third order Runga-Kutta method. This is caused by the overlap of the numerical and the analytical results, as shown in ﬁgure 3.5.b and 3.5.d. Figure 3.5 shows that the numerical simulations are in good comparison with the analytical results for both constant charge and constant dimensionless potential. Therefore, the numerical method is validated for a n:n-electrolyte. 3.4. THE VALIDATION 27

(a) Constant potential (b) Constant potential, Close up

Figure 3.5: The comparison between the analytical of equation 3.30 and the numerical results. The simulations use either the Forward Euler Method, the Midpoint Method or the third order Runga-Kutta Method. The temperature is 298 K, the concentration of the 1:1 electrolyte is 0.1 M, the relative permittivity is 78. The dimensionless potential is 0.2 for the constant potential simulations, while the surface charge is 0.11 mC m2 for the constant charge simulations.

A 1:2-electrolyte or 2:1-electrolyte solution Besides the analytical solution for a n:n-electrolyte solution, the analytical solution for a 1:2- electrolyte or 2:1-electrolyte solution can be determined, when the ions in water are not taken into account. Equation 3.10 can be simpliﬁed to equation 3.31 and equation 3.32 for a 1:2-electrolyte and 2:1-electrolyte solution, respectively [23].

8F2c 1 ∇y = (ey − 1) 1 + 2e−y 2 (3.31) RTε0εr

2 8F c 1 ∇y = e−y − 1 (1 + 2ey) 2 (3.32) RTε0εr These equations can be integrated analytically to equation 3.33 and equation 3.34, for the positive values of the dimensionless potential of 1:2-electrolyte and 2:1-electrolyte solutions, respectively.

3 ln p 1 1 y = − ln tanh2 + tanh−1 √ − (3.33) 2 2 3 2 ! 6p y = ln 1 + (3.34) (p − 1)2 In equation 3.33 and 3.34 is p a direction parameter, which depends on the kind of electrolyte, the concentration of the electrolyte, the permittivity of the solution and the temperature. The 28 CHAPTER 3. THE POISSON-BOLTZMANN MODEL FOR ISOLATED SURFACES

(a) Total (b) Close up

Figure 3.6: The comparison between the analytical results and the numerical results for a 1:2 and 2:1 electrolyte solution. The simulations use the third order Runga-Kutta Method. The temperature is 298 K, the concentration of the 1:1 electrolyte is 0.1 mM, the relative permittivity is 78. The dimensionless potential is 0.2 for the constant potential simulations. function p is equation 3.35 and 3.36 for 1:2-electrolyte and 2:1-electrolyte solutions, respectively [23]. q 1 √ 2 1+2ey 3 8F c x+ √2 tanh−1 2 − √2 tanh−1 √1 p = e RTε0εr 3 3 3 3 (3.35)

√ √ √ q y 8F2c 1 1+2e + 3 3 x+ √ ln √ √ p = e RTε0εr 3 1+2ey− 3 (3.36) The simulations for the 1:2-electrolyte and 2:1-electrolyte solution are performed with a constant dimensionless potential of 0.2. The results of the analytical solution and the numerical simulations are shown in ﬁgure 3.6. The ﬁgure shows that the results of the numerical simulation are similar to the analytical solution. Therefore, the numerical method is also validated for 1:2-electrolyte and 2:1-electrolyte solutions.

The numerical validation For more complicated electrolyte solutions, the analytical integration cannot be performed in terms of normal functions [23]. Therefore, the numerical results cannot be validated for other electrolyte solutions. However, due to the validation with the n:n-electrolyte, 1:2-electrolyte and 2:1-electrolyte solutions, it can be concluded that the numerical model is validated.

3.4.2 The charge regulation model Besides the numerical method, the implementation of the charge regulation method should also be validated. The charge density at the surface depends on the degree of ionisation of the surface, α, via equation 3.37. In this equation is zs the charge sign of the surface groups, which is -1 for acid sites (for example silica) and 1 for amphoteric oxide sites (for example aluminium hydroxide) [7]. σs = zsαN (3.37) Combining this equation with equation 3.28, the ionisation degree depends on the dimensionless potential at the surface, as shown in equation 3.38. In this equation is z the charge of the ions of the n:n electrolyte solution. s 2 2c∞zF p αN = 4c∞z (cosh (ys) − 1) (3.38) εrε0RT 3.5. DISCUSSION 29

The degree of ionisation of the surface however also depends on the surface charge. The ionisation degree for acid and base surfaces (silica like surfaces) is given by equation 3.39, while the ionisation degree for amphoteric surfaces is given by equation 3.40 [7]. 1 α = (3.39) 1 + 10zs(pH−pK)ezsys 1 1 α = − (3.40) 2 1 + 10zs(pK−pH)e−zsys When equation 3.38 is combined with equation 3.39 or 3.40, the ionisation degree and the dimensionless potential at the surface can be determined. These results are compared to the numerically obtained results in ﬁgure 3.7. From the results presented in this ﬁgure, it can be concluded that the results obtained numerically are equal to the results obtained analytically. Therefore, the implementation of the charge regulation model is validated.

3.4.3 The Stern layer Besides the numerical method and the implementation of the charge regulation boundary condition, the implementation of the Stern layer should be validated. As stated in paragraph 3.1.2, the gradient of the potential should be constant in the Stern layer, while ideal behaviour is expected in the diffuse layer. Furthermore, the dimensionless potential should be smooth. To validate the Stern layer, several simulations are performed. The results for simulations in an acidic environment are presented in figure 3.8, while the results for the simulations in a basic environment are presented in appendix B. Figure 3.8.a and 3.8.c show a smooth dimensionless potential. Furthermore, the gradient of the potential is constant within the Stern layer, as shown in figure 3.8.b and 3.8.d. As shown in appendix B, similar results are obtained at high pH. All these simulations are also performed at lower concentrations, which showed similar results. Therefore, the implementation of the Stern layer is validated.

3.5 Discussion

The Stern layer is only validated using the governing equations of the model. To validate this part of the model, another procedure for the calculation of the Stern layer should be determined.

3.6 Conclusion

The Poisson-Boltzmann model for an isolated surface, as described in this chapter, will be used for the simulation of isolated surfaces. The numerical model, the implementation of the boundary condition and the implementation of the Stern layer are validated in this chapter. Furthermore, the initial settings of the model, which will be used in the simulations, are presented in table 3.1.

Table 3.1: Initial settings for the Poisson-Boltzmann model for an isolated surface

Default Step size 1 × 10−11 m The relative diﬀerence, Rdif 0.01 % The ions in water taken into account 30 CHAPTER 3. THE POISSON-BOLTZMANN MODEL FOR ISOLATED SURFACES

(a) The silica surface (b) The aluminium hydroxide surface

Figure 3.7: The validation of the implementation of the charge regulation model. The blue lines are the results of the numerical simulations and the red dots are determined using equation 3.38 and equation 3.39 or 3.40. The concentration of the 1:1 electrolyte solution is 100 mM. The pK is 7.5 and 10 for the silica surface and the aluminium hydroxide surface, respectively. The charge density of both surfaces is 1 nm−2 . The relative permittivity of the water solution is 78.

(a) y, silica (b) ∇y, silica

Figure 3.8: The eﬀect of the introduction of the Stern layer on the dimensionless potential, y, and the gradient of the dimensionless potential, ∇y, in an acidic environment, pH of 4. The red dotted line represents the outer Helmholtz plane, while the blue line represents y or ∇y. The temperature is 298 K, the concentration of the 1:1 electrolyte is 0.1 M and the relative permittivity is 78. The kind of surface used is stated in the name of the ﬁgure. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. The capacity of the Stern layer is 1 F m−2. Chapter 4

The Poisson-Boltzmann model for non-isolated surfaces

In chapter 3, a model is presented for an isolated surface. However, in a colloidal solutions the surface will not be isolated. The particles in a colloidal solution can approach each other due to Brownian motion. The resulting changes in the dimensionless potential, as discussed in chapter 2, cannot be determined analytically. A way to determine the dimensionless potential is simplifying the Poisson-Boltzmann equation [27, 28]. However, to obtain a the dimensionless potential for all distances between two surfaces and in all kinds of solutions a numerical model is needed. In this chapter, this numerical model is described. The model enables the calculation of the dimensionless potential for non-isolated surface simulations. It is based on the DLVO-theory. The DLVO-theory is adjusted in a similar fashion as for an isolated surface, as shown in chapter 3. The constant charge model of the DLVO-theory is replaced by the charge regulation model, in which the amount of protonation of the surface determines the charge of the surface [7, 6, 24]. Furthermore, the Stern layer is introduced to take into account the finite size of the ions [10]. The numerical solution of two non-isolated surfaces will be used for the determination of the colloidal stability of certain surfaces [7, 6, 27]. The electrostatic force can be determined via two routes from the dimensionless potential. First of all, the indirect procedure determines the electrostatic force from the pressure between two plates. The pressure can be determined from the dimensionless potential and the gradient of the dimensionless potential [6]. The second route is the determination of the electrostatic force directly from the dimensionless potential, which is called the direct procedure [7]. The electrostatic force will be combined with the Van Der Waals force to obtain the total interaction force between the surfaces. If the interaction force between the surfaces is positive (repulsion is dominant), the colloid is stable [7, 6, 27]. In this chapter, both routes will be used to determine the electrostatic force. Furthermore, the Van Der Waals force will be calculated. This will be used to determine the interaction force between the surfaces. The chapter will start with the main equations used in the Poisson-Boltzmann model for two non-isolated surfaces. Following, the numerical method used to solve these equations will be explained. The initial settings of the numerical model are determined, to ensure the reproducibility of the simulations. Finally, the model will be validated. The validation is done using properties of the pressure between the surfaces and results of simplifications of the Poisson-Boltzmann equation [7, 29]. Furthermore, the implementation of the Stern layer will be validated using the definitions of the Stern layer and the diffuse layer.

4.1 The governing equations

In this paragraph the governing equations of the model will be discussed. First, the equations for the electrochemical potential in the diﬀuse layer will be derived. Then, the charge regulation boundary condition will be explained. Following, the equations for the determination of the col-

31 32 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES loidal stability will be explained. Finally, the eﬀects of the Stern layer on the governing equations will be discussed.

4.1.1 The dimensionless potential in the diffuse layer For two non-isolated surfaces, Gauss law can be reduced to the Poisson equation, when the ions are assumed to be point charges, only purely electrostatic adsorption is taken into account and the solvent is assumed to be primitive. At thermodynamic equilibrium, the Poisson equation is combined with the Boltzmann distribution, which leads to the Poisson-Boltzmann equation, equation 4.1. In this equation, E is the electric field, ψ the electrostatic potential, F the Faraday constant, ε0 the permittivity of vacuum and εr the relative permittivity of the medium. zi is the charge of the ion i, ci,∞ is the concentration of ion i at infinite distance from the surfaces, R the gas constant and T the temperature.

2 F X − ziFψ ∇ ψ = z c e RT (4.1) ε ε i i,∞ 0 r i For asimilar electrolyte solution, equation 4.1 becomes equation 4.2, when the dimensionless potential, y, is introduced. Besides the ions of the electrolyte, the solution also contains ions caused by the acidity of the solution. To take these into account in equation 4.2, it is assumed that both acidic and basic solutions are produced using a 1:1 salt, for example hydrochloric acid (HCl) for acidic solutions and sodium hydroxide (NaOH) for basic solutions. Therefore, the concentration of anions introduced by the acidity in a acidic solution will be equal to the concentration of the protons. Similarly, the concentration of the cations introduced by the acidity in a basic solution will be equal to the concentration of the hydroxide ions. The concentration of the ions of water is given by cH. F2c c c 2 −zpy −zmy H,∞ −y H,∞ y ∇ y = zpvpe + zmvme + e − e (4.2) RTε0εr c c This equation can be integrated in a similar fashion as depicted in paragraph 3.1. The diﬀerence between the integration in paragraph 3.1 and this equation, is the boundary condition. In a colloidal solution, the surfaces which will approach each other are similar. The gradient of the dimensionless potential will thus be zero halfway the distance between the surfaces. The potential halfway the distance is the unknown value, y0. The integration will then result in equation 4.3.

X cH,∞ σ2 = 2c ε ε RT v e−ziy − v e−ziy0 + e−y + ··· (4.3) ∞ 0 r i i c i ! cH,∞ cH,∞ cH,∞ ··· + ey − e−y0 − ey0 c c c

4.1.2 The charge regulation model Besides the boundary condition midway the surfaces, another boundary condition is needed to determine the dimensionless potential at all distances. This boundary condition is either the potential, the charge or a relation between these at the surface. This boundary condition is similar to the boundary condition at the surface for isolated surfaces. For constant charge models and constant potential models, this boundary condition is a value for the charge or the potential. For the charge regulation models, which are used in this assignment, this boundary condition is deﬁned by the adsorption of protons at the surface, which means that this boundary condition is a Robin (mixed) boundary condition. For acidic and amphoteric oxide surfaces the charge density and the potential are coupled by the reaction of protons with the surface. The charge density of the surface can be described using equation 4.4 for acidic surfaces or equation 4.5 for amphoteric surfaces. Using these equations and 4.1. THE GOVERNING EQUATIONS 33

Gauss law the dimensionless potential at the surface can be determined. In these equations is [H]s the concentration of protons, which can be determined using the Boltzmann distribution, N the density of sites at the surface, which are active and K the equilibrium constant of the reaction of the protons with the surface [11]. −F · N σs = (4.4) 1 + K · [H]s

1 1 − K · [H]s σs = − FN (4.5) 2 1 + K · [H]s

4.1.3 The colloidal stability without Stern layer The dimensionless potential at any distance from the surface can now be determined using the charge regulation model and equation 4.3. To determine the colloidal stability, the free energy between the plates should be determined. The free energy consists of electrostatic forces and Van der Waals forces. First, the electrostatic part will be discussed. The electrostatic forces can be determined by either the indirect procedure or the direct procedure. This electrostatic force will then be combined with the Van der Waals forces to determine the overall interaction force.

The indirect procedure The indirect procedure determines the electrostatic forces from the pressure. The pressure, P, can be calculated from the osmotic pressure and the Maxwell stress, via equation 4.6. If the pressure obtained is positive the surfaces will be repulsive, while a negative pressure indicates an attraction of the surfaces [29].

2 X ε0εr RT P (x) = RT c e−ziy(x) − 1 − ∇y (4.6) i,∞ 2 F i

The result of this calculation should be constant over the total distance between the plates [29]. F The electrostatic force over radius, R , can be determined from the pressure using the Derjaguin approximation, as shown in equation 4.7. The factor 2 should be added when a sphere-plate or a cylinder-cylinder geometry is considered. However, when a sphere-sphere interaction is determined the factor 2 is removed. F (x) x electrostatic = (2) π P(x0)dx0 (4.7) R ˆ∞

The direct procedure The electrostatic force can be determined via the indirect route as shown in the previous paragraph. However, the integration from inﬁnity to the separation distance will induce a long simulation time. Therefore, a direct calculation of the free energy from the dimensionless potential is preferred [7]. The electrostatic free energy of the system, Vsys, can be described by equation 4.8. In this equation Vs is the contribution of the surface and I the ionic strength, which can be determined with equation 4.9. Furthermore, the Debye length, κ, is introduced, which can be calculated with equation 4.10 [7, 30].

x ! X I 2 V (x) = V dS − (c − c ) + (∇y) dx (4.8) sys ˆ s ˆ i i,∞ κ2 S 0 i

1 X I = z2v c (4.9) 2 i i ∞ i F2 1 P z v c κ2 = 2 i i i ∞ (4.10) RTε0εr 34 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES

However, when the electrostatic interaction free energy is determined, the diﬀerence between the interaction free energy of two non-isolated surfaces should be subtracted by the electrostatic free energy of two isolated surfaces. The interaction free energy, Vinteraction, can then be determined by equation 4.11 [7].

x ! X X I 2 V (x) = ∆V − (c − c ) + (∇y) dx + ··· (4.11) interaction s,i ˆ i i,∞ κ2 i=1,2 0 i ∞ ! X I 2 ··· +2 (c − c ) + (∇y ) dx ˆ i,sw i,∞,sw κ2 sw 0 i

In this equation ∆Vs,i is the surface contribution to the electrostatic interaction free energy per surface, which can be calculated for acidic surfaces with equation 4.12 and equation 4.13.

1 − α ∆Vs,i = N · ln (4.12) 1 − α∞

1 α = (4.13) 1 + 10(pH−pK)eys For amphoteric surfaces the equations change to equation 4.14 and equation 4.15 [7].

N (1 + 2α) (1 − 2α) ∆Vs,i = · ln (4.14) 2 (1 + 2α∞) (1 − 2α∞)

1 1 α = − (4.15) 2 1 + 10(pH−pK)eys The interaction free energy cannot be compared directly with experiments. Therefore, the F property R is needed, which can be determined using the Derjaguin approximation, as shown in equation 4.16. The factor 2 in this equation should be added when a sphere-plate or a cylinder- cylinder geometry is considered. However, when a sphere-sphere interaction is determined the factor 2 is removed [7]. F electrostatic (x) = (2) πRTV (x) (4.16) R interaction

The interaction force

The previous two paragraphs shows two procedures to calculate the electrostatic part of the free energy. However, besides the electrochemical contributions, the Van Der Waals force will also F contribute to the forces measured in experiments. Therefore, the property R should be calculated using equation 4.17. In this equation is A the Hamaker constant [6, 31].

F F A (x) = electrostatic (x) − (4.17) R R 6x

4.1.4 The Stern layer The DLVO-theory as explained above does not take into account the finite size of the ions, as discussed in detail in chapter 2. The finite size of the ions can be taken into account by employing the Stern layer. In this assignment the zeroth order Stern layer will be introduced. This leads to changes in the governing equations for the determination of the dimensionless potential, the electrostatic free energy and the interaction force between the plates. The modifications of the equations will be discussed in this paragraph. 4.1. THE GOVERNING EQUATIONS 35

The dimensionless potential For non-isolated surfaces, the eﬀect of the Stern layer on the dimensionless potential is similar as for isolated surfaces. The ions cannot get closer to the surface than a certain distance, d. The distance, d, represents the outer Helmholtz plane. Within the Stern layer, there are no carriers of charge. Therefore, the dimensionless potential has a linear decrease in the Stern layer, as shown in equation 4.18 [10].

y − ys F ∇y = = − σs (4.18) x≤d d RTε0εr In the diffuse layer, the dimensionless potential behaves ideal. The only difference between the DLVO theory and the diffuse layer, is that x should be changed to x-d, as shown in equation 4.19 [10].

c 2 −zpy(x−d) −zmy(x−d) H,∞ −y(x−d) σ = 2c∞ε0εrRT vpe + vme + e + ··· (4.19) x≥d c cH,∞ cH,∞ cH,∞ ··· + ey(x−d) − v e−zpy0(x−d)v e−zmy0(x−d) − e−y0(x−d) − ey0(x−d) c p m c c

The electrostatic force The introduction of the Stern layer also influences the calculation of the electrostatic forces. For the indirect route, there is no differences in the equations for the determination of the electrostatic force. However, the pressure between the surfaces will not be constant within the Stern layer. There is a large increase in the pressure, due to the non-ideal behaviour of the solution in the Stern layer. Therefore, the pressure, used in equation 4.7, should be in the diffuse layer [6]. The changes for the determination of the electrostatic force via the direct route are more drastically. All non-ideallities introduced by the Stern layer are taken into account via the surface free energy. Therefore, the electrostatic free energy can be determined using equation 4.20. Do note, that the integration boundaries change to the boundaries of the diffuse layer [7].

x−d ! X X 2 V (x) = ∆V − (c − c ) + κ2I(∇y) dx+ ··· (4.20) interaction s,i ˆ i i,∞ i=1,2 d i ∞ ! X 2 ··· + 2 (c − c ) + κ2I(∇y ) dx ˆ i,sw i,∞,sw sw d i The Stern layer is taken into account via the surface free energy. The surface free energy is determined using equation 4.21 and 4.22, for acidic and amphoteric surfaces respectively. The ionisation degree of the surface, α, can be determined via the equations given in the previous paragraph [7]. ! 1 − α α2 − α2 (y − y ) ∞ s d ∆Vs,i = N · ln + (4.21) 1 − α∞ 2α ! N (1 + 2α) (1 − 2α) α2 − α2 (y − y ) ∞ s d ∆Vs,i = · ln + (4.22) 2 (1 + 2α∞) (1 − 2α∞) α

The interaction force The introduction of the Stern layer does not only inﬂuence the interaction force via the electrostatic free energy, but also via the Van der Waals forces. This leads to equation 4.23 [14].

F F A (x) = electrostatic (x) − (4.23) R R 6(x + 2dvdW) 36 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES

In this equation, dvdW is the thickness of the Stern layer for the Van der Waals forces. In the equations above, the relative permittivity of the solvent does not change in the Stern layer. Ho- wever, the relative permittivity decreases in the Stern layer [8]. dvdW will thus be smaller than d.

4.2 The numerical simulation

The resulting equations cannot be solved analytically. The governing equations are solved using Matlab scripts. A detailed description of the Matlab scripts is given in appendix C. In this paragraph a short description of the numerical model will be given. The numerical simulation route is schematically represented in figure 4.1. The simulation starts with the simulation of the isolated surface. By performing this simulation, the diffuse contribution of the free energy for isolated surfaces is determined. Furthermore, the dimensionless potential at the wall and halfway the distance between the surface are used as initial guesses for the dimensionless potential at the wall and y0, respectively. When y0 is known, the dimensionless potential at the wall can be determined. This is done by the same route used for the isolated surface simulation. The dimensionless potential at the wall of the isolated surface is used to determine the gradient of the potential via the Poisson-Boltzmann equation and via the 1-pK model. (If the Stern layer is included in the simulation the calculation of the gradient by the Poisson-Boltzmann equation is performed at the outer Helmholtz plane) If the relative difference between the two gradients, as determined by equation 4.24, is more than a set value for Rdif,2, the potential at the surface is increased (or decreased depending on the sign of the charge of the surface) with a certain ∆y. When the relative difference between the two values is less than the threshold, the used potential in the 1-pK model is the surface potential.

∇yPB − ∇yBC Rdif,2 = (4.24) ∇yBC Using this dimensionless surface potential, the dimensionless potential at all distances from the surface can be determined using the Euler, the midpoint or third order Runga Kutta discretisation scheme. This leads to a value of the dimensionless potential midway between the surfaces, y(n). y(n) is compared to the used y0, according to equation 4.25. If the relative difference is more than a certain threshold value for Rdif,3, the value of y0 is changed to new value, by adding a part of the difference between y(n) and y0 to y0. If the difference is less than the set threshold, the dimensionless potentials used in the calculations are chosen to be the dimensionless potentials of the system.

y0 − y (n) Rdif,3 = (4.25) y0 With these dimensionless potentials, the pressure between the surfaces can be determined. When the pressure for multiple distances are performed, the pressure between the surfaces can be integrated using the midpoint method. This leads to the determination of the electrostatic interaction force over the radius via the indirect route. Furthermore, the electrostatic interaction force between the surfaces can be determined using the direct route. The integrations performed in this route are also done using the midpoint discretisation scheme. The electrostatic interaction energy is then combined with the Van der Waals forces to obtain the interaction force over the radius.

4.3 The initial settings

For the non-isolated surface simulations, there are also some initial settings. These need to be deﬁned, to ensure the reproducibility of the results. Furthermore, the initial setting will enable 4.3. THE INITIAL SETTINGS 37

Isolated surface calculation

Start: Start: y(1) = yisolated surface(1) y0 = 2yisolated surface(n)

Determine y and y0 y0 = y0 + a*(y(n)-y0) y(n) = ycalculation(n)

dy via Poisson- y(1) = y(1) +Δy Boltzmann equation

dy via boundary condition

R > threshold dy check dif,2

Determination of all y

Rdif,3 > threshold y0 check

Simulation finished

Figure 4.1: A schematic representation of the determination of the dimensionless potential in a double surface calculation. The dimensionless potential at the surface is y(1). y(n) is the value at the middle of the gap between the surfaces. 38 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES the interpretation of the results. In this paragraph, the initial settings of the simulations for non- isolated surface simulations are deﬁned. The pK used in this simulations for aluminium hydroxide is 10 and for silica is 7.5. The density of the sites, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2 [6, 9, 11]. In all the simulations the distance between the surfaces is set to 10 nm.

4.3.1 The step size The first initial setting, which will be determined, is the step size. If the step size is decreased, the simulation result will become more accurate, but the simulation time will also be increased. Therefore, an optimal step size should be determined. The results for the different step sizes in an acidic environment are shown in figure 4.2, while the results for an basic environment are shown in appendix D. Figure 4.2.a to 4.2.d show that the step size only has a small effect of the dimensionless potential on the simulations using an acidic surface (silica), if the step size is less than 1 nm. Figure 4.2.d shows a small influence of the step size when it is decreased from 0.01 nm to 0.001 nm. Figure 4.2.e to 4.2.h show that there is no influence of the step size for amphoteric surfaces (aluminium hydroxide) when the step size is smaller than 0.1 nm. The results in the appendix show that the step size has no effect on the accuracy of the simulation in a basic environment. However, the step size should be smaller than 1 nm to prevent imaginary results. Concluding, a step size of 0.1 nm gives accurate results. To ensure reproducibility the step size used in the simulations is 0.01 nm. Furthermore, the step size is decreased by a factor ten (the step size will become 0.001 nm) when the distance between the surfaces is less then 2 nm. This ensures at least 20 step sizes between the surfaces.

4.3.2 The relative difference of the surface potential, Rdif,2 Besides the step size, the relative difference of the surface potential is also an important initial setting. The effect of Rdif,2 is determined for both acidic (silica) and amphoteric (aluminium hydroxide) surfaces, at different pH and concentration. The results for an acidic environment are shown in figure 4.3, while the results of the simulations in a basic environment are given in appendix D. Figure 4.3.a to 4.3.d show the effect of Rdif,2 for acidic surfaces. Only at low concentration the effect of Rdif,2 is noticeable. These results show that a Rdif,2 of 0.01 % gives reproducible results. Figure 4.3.e to 4.3.h show that Rdif,2 has no effect on the dimensionless potential. In a basic environment, the Rdif,2 has no effect on the dimensionless potential for both acidic and amphoteric surfaces. Therefore, it is concluded that a Rdif,2 of 0.01 % is sufficient to obtain reproducible results.

4.3.3 The relative diﬀerence of y0, Rdif,3

The relative difference of y0, Rdif,3, is also an initial setting, which should be determined. The effect of Rdif,3 is determined in a basic environment in appendix D, while the effect in an acidic environment is shown in figure 4.4. The effect is determined for two concentrations and both acidic and amphoteric surfaces. Figure 4.4.a and 4.4.b show that there is almost no influence of Rdif,3. Rdif,3 has a more pronounced influence at lower concentration, as shown in figure 4.4.c and 4.4.d. From these results, no conclusion can be drawn with respect to Rdif,3 needed to obtain reproducible results, for amphoteric surfaces. For amphoteric surfaces, there is almost no influence of Rdif,3 on the dimensionless potential, as shown in figure 4.4.e and 4.4.f. At lower concentrations, figure 4.4.g and 4.4.h, the Rdif,3 should be at least 0.01 % to ensure reproducible results. To determine, which value of Rdif,3, should be used to ensure reproducible results, the effect of Felectrostatic Felectrostatic Rdif,3 on the R . The effect of Rdif,3 is determined by determining the relative R 4.3. THE INITIAL SETTINGS 39

(a) Silica, 0.1 M (b) Silica, 0.1 M, close up

(e) Aluminium hydroxide, 0.1 M (f) Aluminium hydroxide, 0.1 M, close up

(g) Aluminium hydroxide, 1 mM (h) Aluminium hydroxide, 1 mM, close up

Figure 4.2: The effect of the step size on the accuracy of the non-isolated surface simulations in a acidic environment, pH of 4. The lengths given in the legend are the step sizes. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity of the solvent is 78. The used relative difference for the surface potential, Rdif,2, and the relative difference for y0, Rdif,3, are 0.01 % for the silica simulations and 0.001 % for the aluminium hydroxide simulations. The ions of water are taken into account. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. 40 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES

(a) Silica, 0.1 M (b) Silica, 0.1 M, close up

(e) Aluminium hydroxide, 0.1 M (f) Aluminium hydroxide, 0.1 M, close up

(g) Aluminium hydroxide, 1 mM (h) Aluminium hydroxide, 1 mM, close up

Figure 4.3: The effect of the relative difference of the surface potential, Rdif,2, on the accuracy of the non-isolated surface simulations in a acidic environment, pH of 4. The percentages given in the legend are the relative differences. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity of the solvent is 78. The used step size is 0.01 nm. The used relative difference for y0, Rdif,3, is 0.01 % for the silica simulations and 0.001 % for the aluminium hydroxide simulations. The ions of water are taken into account. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. 4.3. THE INITIAL SETTINGS 41

(a) Silica, 0.1 M (b) Silica, 0.1 M, close up

(e) Aluminium hydroxide, 0.1 M (f) Aluminium hydroxide, 0.1 M, close up

(g) Aluminium hydroxide, 1 mM (h) Aluminium hydroxide, 1 mM, close up

Figure 4.4: The effect of the relative difference of y0, Rdif,3, on the accuracy of the non-isolated surface simulations in a acidic environment, pH of 4. The percentages given in the legend are the relative differences. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity of the solvent is 78. The used step size is 0.01 nm. The used relative difference for the surface potential, Rdif,2, is 0.01 % for the silica simulations and 0.001 % for the aluminium hydroxide simulations. The ions of water are taken into account. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. 42 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES

Felectrostatic Table 4.1: The eﬀect of the Rdif,3 on the calculated R . All the values are determined using the relative diﬀerence between a certain simulation and the most accurate simulation, as shown in equation 4.26.

Rdif,3 Silica 0.1 M Silica 1 mM Aluminium hydroxide 0.1 M Aluminium hydroxide 1 mM 10 % 17.0 % 25.3 % 20.7 % 10.0 % 1 % 2.87 % 5.47 % 3.55 % 1.70 % 0.1 % 0.31 % 0.71 % 0.39 % 0.20 % 0.01 % 0.04 % 0.08 % 0.04 % 0.02 % 0.001 % - 0.01 % - 0.00 % 0.0001 % - - - -

(a) Silica, 0.1 M (b) Silica, 1 mM

Figure 4.5: The effect of the ions in the water on the non-isolated surface simulations in a acidic environment, pH of 4. The blue lines are the simulation without water ions and the red lines are the simulations with water ions. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity is 78. The step sizes are 0.01 nm and the relative differences, Rdif,2 and Rdif,3, are 0.01 % for silica and 0.001 % for aluminium hydroxide. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density at the silica surface, N, is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. 4.4. THE VALIDATION 43

for each simulation, with respect to the lowest Rdif,3 in which the simulation has no imaginary numbers. The equation used for determination is equation 4.26.

Felectrostatic Felectrostatic R − R |lowest Rdif,3 Rdif,4 = (4.26) Felectrostatic R |lowest Rdif,3

The results of the calculations are shown in table 4.1. These show that all the simulations have less than 0.1 % deviation when Rdif,3 is less than 0.1 %. The simulations in the basic environment, in appendix D, show that the dimensionless potential Felectrostatic does not differ when Rdif,3 is less than 0.1 %. When the effect of Rdif,3 on the R is determined for these simulations, there is a relative difference of maximally 0.04% when the Rdif,3 is less than 0.1 %. Concluding, Rdif,3 is 0.01 % for all simulations. When the distance between the surfaces is less than 2 nm, Rdif,3 is decreased to 0.0001 % to increase the reproducibility.

4.3.4 The ions in the water In all the simulations performed for the initial settings, the ions in water are taken into account. However, it should be determined if it is important to take these into account. The results of the simulations in an acidic environment are presented in figure 4.5, while the results in a basic environment are presented in appendix D. Figure 4.5.a and 4.5.c show that the effect of the ions of water is negligible at relative high concentrations of electrolyte for both kinds of surfaces. However, when the concentration of the electrolyte is lowered to concentrations comparable to the concentration of the ions in water, figure 4.5.b and 4.5.d, the effect of the ions in water is more pronounced. The effect for amphoteric surfaces is smaller, because the maximal surface charge is already reached at a pH of 4, as shown in figure 3.5. For the simulations in a basic environment, in appendix D, similar results is obtained. The- refore, the ions of water have a large influence on the dimensionless potential in simulations with similar concentration of electrolyte and concentration of ions of water. The ions of water should thus be taken into account.

4.4 The validation

The Poisson-Boltzmann model for two non-isolated surfaces is defined by the governing equations as discussed above. However, to ensure the validity of the simulations the results of the model will to compared to theory and simplified models. The numerical model and the charge regulation model used in the model are already validated, using the isolated surface model. In this paragraph, the determination of the pressure between the plates is validated. Then, the determination of the free energy via the indirect route is validated using a simplified model, the Linear Superposition Approximation (LSA). Following, the direct route is validated using the results of the direct route. Finally, the implementation of the Stern layer is checked.

4.4.1 The pressure The pressure between the surfaces should be constant over the distance between the plates [29]. Therefore, the pressure between the plates is determined at all distances from the surfaces. The results of these simulations in an acidic environment are shown in figure 4.6, while the results of the simulations in a basic environment and at different distances between the surfaces are shown in appendix D. Figure 4.6 and the results in appendix D show that the pressure is constant over the distance, for different pH, concentration and distance between the surfaces. Therefore, the determination of the pressure is validated. Besides the determination of the pressure, the determination of the dimensionless potential at halfway the distance is also validated, because the pressure depends on the dimensionless potential and the gradient of the dimensionless potential. 44 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES

(a) Silica, 0.1 M (b) Silica, 1 mM

Figure 4.6: The pressure at diﬀerent distances from the surface, which are 6 nm apart, in an acidic environment, pH of 4. The temperature is 298 K and the relative permittivity is 78. The kind of surface and the concentration of the 1:1-electrolyte are stated in the name of the sub ﬁgure. The used pK in this simulations for aluminium hydroxide is 10 and for silica is 7.5. The density of the sites, N, at the silica surface is 8 nm−2 and 10 nm−2 at the aluminium hydroxide surface [6, 9, 11].

(a) Simulation results (b) Results P.M. Biesheuvel 2004 [7]

Figure 4.7: The validation of the implementation of the indirect for the determination of the electrostatic free energy. The electrostatic free energy is compared to the electrostatic free energy determined via the Linear Superposition Approximation (LSA). The concentration of the 1:1- electrolyte solution used is 0.1 M. The density of sites at the surface is set to 1 nm−2. The radius of the particle is set to 100 nm. The solid lines represent the simulated results, while the dotted lines represent the results obtained via the LSA. 4.4. THE VALIDATION 45

4.4.2 The electrostatic free energy via the indirect route The determination of the electrostatic free energy via the indirect route can be validated using a simplified model, the Linear Superposition Approximation (LSA). This approximation determines the potential between two surfaces by adding the potentials of two isolated surfaces [28]. The force over radius determined using LSA, is equal to equation 4.27, for a surface in a 1:1 electrolyte solution. In this equation, the first R is the radius of the particle and the second R the gas constant. F c y electrostatic,LSA (x) = πRT64 ∞ tanh2 s,∞ e−κH (4.27) R κ 4 The results of the LSA are valid when the region between the plates can be separated in three parts. In the middle part, the potential should be sufficiently low that it obeys the linearised Poisson-Boltzmann equation. In the other two parts, the potential should only depend on the nearest wall [28]. These conditions can only be met at large separation distances. So the electrostatic free energy determined via the indirect route should be similar to the results obtained at large distances, while it might differ at smaller distances. The results of the LSA and the simulations performed with the model are shown in figure 4.7.a. The difference between the simulation results and the LSA results are small at large distances and larger at small distances, as expected. Furthermore, the results presented in figure 4.7.a are similar to the results found by P.M. Biesheuvel (2004), as shown in figure 4.7.b [7]. Therefore, the determination of the electrostatic free energy via the indirect route is validated.

4.4.3 The electrostatic free energy via the direct route To validate the electrostatic free energy determined via the direct route, the results can be compared with the LSA. However, the results could also be validated using the already validated electrostatic free energy via the indirect route. Therefore, the results of the electrostatic free energy of the direct route are compared to the results of the indirect route in parity plots. The results for the acidic (silica) and amphoteric (aluminium hydroxide) surfaces are shown in figure 4.8. Figure 4.8 shows a parity plot of the electrostatic forces over the radius determined via the indirect and the direct route. It is notable that the results are similarly to y = x. However, at F small distances ( R is then large), the results via the direct route are slightly lower. The lower value of the direct route is probably caused by the small amount of integration regions in these simulations (they contain only 20 step sizes). The direct route is thus validated for acidic surfaces for all n:n-electrolytes and n:m-electrolytes. For the amphoteric surfaces only the simulations with n:n-electrolytes and 1:n-electrolytes (n:1-electrolytes) for pH of 4 (12) are validated. The results the n:1-electrolytes (1:n-electrolytes) for a pH of 4 (12) are shown in figure 4.9. As shown in the figure, the electrostatic free energy determined via the direct route is partly negative. When the results are compared to similar 1:n-electrolytes (n:1-electrolytes), the contribution of the diffuse layer is about twice as large, which is also found in acidic surfaces. However, the surface contribution of the free energy is similar, while the surface contribution is also increased when acidic surfaces are used. This leads to the conclusion that the surface contribution would be the problem. The effect of the n:1-electrolytes (1:n-electrolytes) on the positive (negative) surfaces should be studied in more detail to explain and correct the problems.

4.4.4 The Stern layer Finally, the implementation of the Stern layer should be validated. First of all, the effect on the dimensionless potential and the gradient of the dimensionless potential are determined. As stated in paragraph 4.1.4, the gradient in the dimensionless potential should be linear within the Stern layer, while ideal behaviour is expected in the diffuse layer. Furthermore, the dimensionless potential should be a smooth function. In figure 4.10.a and 4.10.c, the effect of Stern layer on the dimensionless potential is shown for an acidic environment. Both curves are smooth. Furthermore, the gradient of the dimensionless 46 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES

(a) Silica, pH 4, n:n (b) Silica, pH 4, n:m

(e) Aluminium hydroxide, pH 4, n:n (f) Aluminium hydroxide, pH 4, n:m

(g) Aluminium hydroxide, pH 12, n:n (h) Aluminium hydroxide, pH 12, n:m

Figure 4.8: Parity plots of the electrostatic free energy determined via the indirect route and via the direct route. The temperature is 298 K, the concentration of the electrolyte is 0.1 M and the relative permittivity is 78. The kind of electrolyte solution used is stated in the legend of the ﬁgures. The pH of the simulation and the kind of surface are stated in the name of the ﬁgure. The used pK in the simulations for silica is 7.5 and 10 for the aluminium hydroxide simulations [9, 11]. The density of the sites at the surface is 1 nm−2. 4.4. THE VALIDATION 47

(a) pH 4, 2:1 (b) pH 12, 1:2

Figure 4.9: Parity plots of the electrostatic free energy determined via the indirect route and via the direct route for aluminium hydroxide. The temperature is 298 K, the concentration of the electrolyte is 0.1 M and the relative permittivity is 78. The kind of electrolyte solution used is stated in the legend of the ﬁgures. The pH of the simulation is stated in the name of the ﬁgure. The used pK in this simulations is 10 [9, 11]. The density of the sites at the surface is 1 nm−2.

(a) Dimensionless potential, Silica (b) ∇y, Silica

Figure 4.10: The eﬀect of the introduction of the Stern layer on the dimensionless potential and the gradient of the dimensionless potential, ∇y, in an acidic environment, pH of 4. The red doted line represents the Stern layer, while the blue line represents the dimensionless potential or ∇y. The temperature is 298 K, the concentration of the 1:1-electrolyte is 0.1 M and the relative permittivity is 78. The kind of surface used is stated in the name of the ﬁgure. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2 [6, 9, 11].The capacity of the Stern layer is 1 F m−2. 48 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES

(a) Silica (b) Aluminium hydroxide

Figure 4.11: The eﬀect of the introduction of the Stern layer on the pressure in an acidic environment, pH of 4. The red doted line represents the Stern layer, while the blue line represents the pressure. The temperature is 298 K, the concentration of the 1:1-electrolyte is 0.1 M and the relative permittivity is 78. The kind of surface used is stated in the name of the ﬁgure. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2 [6, 9, 11]. The capacity of the Stern layer is 1 F m−2.

(a) n:n (b) n:m

Figure 4.12: Parity plots of the electrostatic free energy determined via the indirect route and via the direct route, for silica surfaces in an acidic environment, pH of 4. The temperature is 298 K, the concentration of the electrolyte is 0.1 M and the relative permittivity is 78. The kind of electrolyte solution used is stated in the legend of the ﬁgures. The used pK in the simulations for silica is 7.5 [9, 11]. The density of the sites at the surface is 1 nm−2. The capacity of the Stern layer is 1 F m−2. 4.4. THE VALIDATION 49

(a) pH-pK = 6, n:n (b) pH-pK = 1, n:n

(e) pH-pK = 6, n:m (f) pH-pK = -6, n:m

Figure 4.13: Parity plots of the electrostatic free energy determined via the indirect route and via the direct route for amphoteric surfaces. The temperature is 298 K, the concentration of the electrolyte is 0.1 M and the relative permittivity is 78. The kind of electrolyte solution used is stated in the legend of the figures. The used pK in the simulations for aluminium hydroxide is 10 [9, 11]. The density of the sites at the surface is 1 nm−2. The capacity of the Stern layer is 1 F m−2. 50 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES potential is ideal in the diffuse layer, while the gradient of the dimensionless potential is linear in the Stern layer, as shown in figure 4.10.b and 4.10.d. Similar results are obtained for the simulations in a basic environment, shown in appendix D show similar results. The implementation of the Stern layer in the dimensionless potential is thus validated. The introduction of the Stern layer, also influences the determined pressure. The pressure will not be constant in the Stern layer, because equation 4.6 is based on ideal behaviour. The effect of the Stern layer on the pressure is determined in figure 4.11 and appendix D. Both the simulations in an acidic environment (figure 4.11) and in a basic environment (appendix B) show a steep increase in pressure within the Stern layer, while the pressure remains constant in the diffuse layer, which is the expected effect on the pressure. Besides the influences on the pressure, the introduction of the Stern layer does not influence the determination of the electrostatic free energy via the indirect route. If the pressure used in the calculations is the pressure of the diffuse layer, the electrostatic free energy is already validated in paragraph 4.4.2. The effect of the Stern layer has a larger effect on the calculation of the electrostatic free energy via the direct route. The new equations of the direct route can thus be validated using the results of the indirect route. Figure 4.12 shows the parity plots for the direct and the indirect route in an acidic environment for an acidic surface. The figure shows that the results via the indirect route and the for the Stern layer adjusted direct route are similar. Similar results are obtained for acidic surfaces in a basic environment. Therefore, the implementation of the Stern layer for the electrostatic free energy via the direct route is validated for silica surfaces. The direct route is also validated for amphoteric surfaces. The results of these simulations are shown in figure 4.13. The figure shows that the implementation of the equations for the direct route in paragraph 4.1.4 give similar results to the indirect route when |pH − pK| ≥ 6. The results for these simulations are validated for both n:n-electrolytes and 1:n-electrolytes (n:1-electrolytes) for positively (negatively) charged surfaces. However, when |pH − pK| < 6, the results obtained via the direct route are lower than the results obtained via the indirect route, as shown in figure 4.13.b and 4.13.c. The effect must be caused by the implementation of the surface contribution, because the effect is only obtained in the simulations with a Stern layer. In these simulations, there is only a modification of the surface contribution of the free energy. The modifications of the direct route due to the Stern layer for amphoteric surfaces should be studied in more detail, to explain and correct the differences.

4.5 Discussion

The results for the determination of the electrostatic free energy via the direct route for amphoteric surfaces showed problems for the n:1-electrolytes (1:n-electrolytes) for positively (negatively) charged surfaces. The electrostatic free energy determined via the direct route is partly negative. When the results are compared to similar 1:n-electrolytes (n:1-electrolytes), the contribution of the diffuse layer is about twice as large, which is also obtained for acidic surfaces. However, the surface contribution is similar, while an increase in the surface contribution is obtained for silica surfaces. This leads to the conclusion that the surface contribution would be the problem. The effect of the n:1-electrolytes (1:n-electrolytes) on the positively (negatively) surfaces should be studied in more detail to explain and correct the problems. The effect of the Stern layer on the dimensionless potential and the gradient of the dimensionless potential is only validated using the governing equations of the model. To validate this part of the model, another procedure for the calculation of the Stern layer should be determined. When the Stern layer is introduced, the determination of the electrostatic free energy via the direct route is only similar to the indirect route when |pH − pK| ≥ 6. The implementation of the Stern layer has no effect on the diffuse contribution of the electrostatic free energy. Therefore, the problems are caused by the surface contribution of the free energy. The modifications of the direct route due to the Stern layer for amphoteric surfaces should be studied in more detail, to explain and correct the differences. 4.6. CONCLUSION 51

Table 4.2: Initial settings for the Poisson-Boltzmann model for two non-isolated surface. The second column is the default for simulations with a distance of 2 nm or less.

Default Default (2 nm) Step size 1 × 10−11 m 1 × 10−12 m The relative difference of the surface potential, Rdif,2 0.01% 0.01% The relative difference of y0, Rdif,3 of silica 0.01% 0.0001% The relative difference of y0, Rdif,3 of aluminium hydroxide 0.001% 0.0001% The ions in water taken into account taken into account

4.6 Conclusion

In this chapter, the Poisson-Boltzmann model for non-isolated surfaces is described. The model uses the modiﬁed DLVO-theory. The theory is modiﬁed using the Stern layer and the charge regulation boundary condition. The model is able to determine the electrostatic free energy via either the direct or the indirect route. This electrostatic free energy is then combined with the van der Waals forces, to determine the total interaction force. For the model, some initial settings are determined, as shown in table 4.2. Besides, the determination of the pressure is validated. Using the Linear Superposition Ap- proximation, the indirect route for the determination of the electrostatic free energy is validated. The direct route is validated for acidic and amphoteric surfaces. However, the direct route cannot be used for simulations using n:1-electrolytes (1:n-electrolytes) for positively (negatively) charged surfaces. Following, the implementation of the Stern layer is validated for the dimensionless potential, the gradient of the dimensionless potential and the pressure. The implementation of the Stern layer in the direct route for acidic surfaces is also validated, while the implementation of the Stern layer in the direct route for amphoteric surfaces is only validated for |pH − pK| ≥ 6. 52 CHAPTER 4. THE POISSON-BOLTZMANN MODEL FOR NON-ISOLATED SURFACES Chapter 5

Simulation results for isolated surfaces

To validate the model, the numerical results of the non-isolated surfaces will be compared to experimental results. To enable verification, all the parameters should be known. However, the charge density of the surface, the capacity of the Stern layer, the thickness of the Stern layer, the relative permittivity of the solvent and the temperature are not known in the experiments. Therefore, it should be determined if several parameters have an effect on the dimensionless potential. To determine these influences, the Poisson-Boltzmann model for isolated surfaces, as described in detail in chapter 3, is used to determine the effect of the electrolyte solution, the relative permittivity, the pH, the charge density of the surface and the temperature. The results of these simulations are presented in this chapter and appendix E.

5.1 The acidic surface

For the acidic surface, 104 different simulations are performed. All the simulations are compared to the base case. The base case for the simulations in an acidic environment is given in table 5.1. In the simulations, the electrolyte concentration, the kind of electrolyte, the relative permittivity of the solvent, the pH (and therefore the pH-pK), the charge density at the surface and temperature are varied in the ranges shown in table 5.1. The simulations, which use the base case for a basic environment, are given in appendix E. In all the simulations, the Stern layer is presented in the figure. The relative permittivity of the Stern layer is assumed similar to the relative permittivity of the diffuse layer. Due to the decrease in the relative permittivity of the solvent in the Stern

Table 5.1: The base case of the simulations for isolated surfaces of the acidic surface in an acidic environment. The ranges in which the parameters are varied during the simulations are also given.

Base case Ranges 2 −1 5 The concentration of electrolyte (c∞) in mM 10 10 −10 The valency of the positive ions (zp) 1 1 - 5 The valency of the negative ions (zm) 1 1 - 5 The pH-pK -3.5 -8 - 9 The pK 7.5 - The charge density at the surface (N) in nm−2 8 1 - 12 The temperature (T) in K 298 278 - 313 The relative permittivity of the solvent (εr) 78 65 - 85 The thickness Stern layer in nm 0.69 - The capacity of the Stern layer in F m−2 1.00 -

53 54 CHAPTER 5. SIMULATION RESULTS FOR ISOLATED SURFACES

(a) c∞ (b) n:n

Figure 5.1: The effect of electrolyte solution on the dimensionless potential for an acidic surface in an acidic environment, pH of 4. In figure 5.1.a, the effect of the concentration of electrolyte, c∞, is shown. Figure 5.1.b to 5.1.d show the effect of the kind of electrolyte. For these simulations, the base case of table 5.1 is used. c∞ (in figure 5.1.a) or the kind of electrolyte (in figure 5.1.b to 5.1.d) is adapted to c∞ or the kind of electrolyte presented in the legend. layer, the thickness of the Stern layer will in reality be smaller than the 0.69 nm [8].

5.1.1 The effect of the electrolyte solution The electrolyte solution influences the potential and thus the thickness of the diffuse layer via both the concentration of the solution and the kind of electrolyte used. This can already be concluded from the governing equations of the Poisson-Boltzmann equation for isolated surfaces, which are given in chapter 3. The results for the effect of the electrolyte solution in an acidic environment are given in figure 5.1. Figure 5.1.a shows that the concentration of the electrolyte solution has a large influence on the thickness of the diffuse layer. The thickness decreases when the concentration is increased. This influence is to be expected from the governing equations in chapter 3. The effect is explained physically by the enhancement of the screening of the surface charge, due to the increase of the concentration of the ions in the diffuse layer. Furthermore, the concentration of the electrolyte also influences the potential at the surface. Increasing the concentration increases the gradient of the potential. This leads to a decrease in the surface potential at which the gradient of the potential is equal to the surface charge. The effect of the electrolyte concentration in a basic environment is similar to the results obtained in an acidic environment, as shown in appendix E. Figure 5.1.b to 5.1.d show the effect of the electrolyte on the dimensionless potential. All the figures show a decrease in the thickness of the diffuse layer with an increase of n of the n:n-electrolyte, 1:n-electrolyte or n:1-electrolyte. This increase can be explained by the higher effective charge of the ions, either the charge is increased or the concentration of the positive ions 5.1. THE ACIDIC SURFACE 55

Table 5.2: The dimensionless potential at the surface for diﬀerent electrolytes (n:n, 1:n and n:1). The numerical results are obtained using the base case of table 5.1 and E.1 and the ranges for the electrolytes in the same tables

pH 4 pH 10 n n:n 1:n n:1 n:n 1:n n:1 1 -0.0363 -0.0363 -0.0363 -8.0889 -8.0889 -8.0889 2 -0.0260 -0.0276 -0.0277 -7.8013 -7.8021 -7.9919 3 -0.0226 -0.0241 -0.0241 -7.6995 -7.7003 -7.9319 is increased. This leads to a better screening of the charge of the particles. This behaviour is also expected from the Poisson-Boltzmann equation. If the dimensionless potential of the n:n-electrolyte, the 1:n-electrolyte and the n:1-electrolyte are compared, the dimensionless potential decreases from the 1:n-electrolyte via the n:1-electrolyte to the n:n-electrolyte in both figure 5.1.b to 5.1.d and the results in appendix E. The surface potentials are given in table 5.2. This is to be understandable, because there are two effects due to the change in the kind of the electrolyte. The increase of the charge of the positive ions will increase the screening directly. However, when the charge of the negative electrolyte is increased, the dimensionless potential at the surface also decreases. This can be explained by the increase in the concentration of the electrolyte. This indirect effect is less pronounced as the direct effect. Table 5.2 shows that the effect of the 1:n-electrolyte compared to the n:1-electrolyte for the results in an acidic environment are comparable, while a clear difference is shown in a basic environment. This is explained by the Poisson-Boltzmann equation. The correlation between the gradient of the potential and the valency of the positive ions is exponential, while the effect of an increased relative concentration (vp) is only linear as shown in equation 5.1.

c 2 −zpy −zmy H,∞ −y σ = 2c∞ε0εrRT vpe + vme + e + ··· (5.1) c∞ cH,∞ y cH,∞ cH,∞ ··· + e −vp − vm − − c∞ c∞ c∞

Therefore, the effect of the n:1-electrolyte is considerably larger (because it is exponential correlation) then the effect of the 1:n-electrolyte, when the potential is high. So, the n:1-electrolyte and the n:n-electrolyte will give similar results, while the effect of the 1:n-electrolyte will be considerably smaller. However, when the potential is lower, the exponential effect can be approximated linearly. Therefore, the difference between the two effects will decrease at lower potentials. This leads to almost similar results for the n:1-electrolyte and the 1:n-electrolyte at low potentials, while the effect of the n:n-electrolyte is larger.

5.1.2 The eﬀect of relative permittivity of the solvent

The electrolyte solution also has an influence on the dimensionless potential via the relative permittivity of the solvent. Therefore, the effect of the relative permittivity of the solvent is determined in figure 5.2.a and appendix E. The results show a different result for each permittivity, the maximal difference is 16 %. Especially, the difference between the simulations is large close to the surface. The results in appendix E show almost no difference between the simulations, maximally 3 %. Figure 5.2.a and the results in appendix E show that an increase in the permittivity causes a decrease of the dimensionless potential of the wall. However, the diffuse layer is not decreased. This is can be explained by the combination of equation 5.1 and 5.2. 56 CHAPTER 5. SIMULATION RESULTS FOR ISOLATED SURFACES

(a) εr (b) pH-pK

Figure 5.2: The effect of the relative permittivity, εr, and pH-pK on the dimensionless potential for an acidic surface in acidic environment, pH of 4. In figure 5.2.a, the effect of εr is shown. Figure 5.2.b shows the effect of pH-pK. For these simulations, the base case of table 5.1 is used. εr (in figure 5.2.a) or pH-pK (in figure 5.2.b) is adapted to the εr or pH-pK presented in the legend.

s 2 2F c∞ −z y −z y cH,∞ −y ∇y = −sign (σs) vpe p + vme m + e + ··· (5.2) RTε0εr c∞ cH,∞ y cH,∞ cH,∞ ··· + e − vp − vm − − c∞ c∞ c∞

When the relative permittivity, εr, is increased, the surface charge increases as shown by equation 5.1. Equation 5.2 shows a decrease of the gradient of the potential.

5.1.3 The effect of the pH Besides the effect of the electrolyte solution via the concentration, the kind of electrolyte and the relative permittivity, the electrolyte solution will also influence the diffuse layer via the pH. In paragraph 3.4, the effect of the pH on the dimensionless potential at the surface is already shown. When the pH is lowered with respect to the pK, the surface potential is decreased. The same result is obtained in figure 5.2.b. Due to the lower surface potential, the thickness of the diffuse layer decreases. The decrease in the surface potential is caused by the lower surface charge. The effects of a lower surface charge on the ions in the electrolyte solution will be smaller. The diffuse layer will thus decrease with decreasing pH. Furthermore, the pH of the point of zero charge might be determined from figure 5.2.b. Ac- cording to literature, the point of zero charge is reached asymptotically. This is shown in the figure. Although the blue line of pK-pH is -4 is not shown in the figure, the dimensionless surface potential of this pH-pK value is not zero, but -0.012. Therefore, no pH can be assigned the point of zero charge.

5.1.4 The effect of the charge density at the surface As shown in the previous paragraph, the surface potential of the surface has a large influence on the dimensionless potential. Besides the pH and the pK, the charge density of the surface also changes the surface potential. Therefore, the effect of the charge density on the dimensionless potential is determined in both an acidic environment and a basic environment, as shown in figure 5.3.a and appendix E, respectively. Figure 5.3.a shows that the charge density has a large effect on the dimensionless potential of the surface. This is also expected. A decrease in the charge density decreases the effective charge 5.2. THE AMPHOTERIC SURFACE 57

(a) N (b) T

Figure 5.3: The effect of the charge density, N, at the surface and the temperature, T, on the dimensionless potential for an acidic surface in a acidic environment, pH of 4. In figure 5.3.a, the effect of N is shown. Figure 5.3.b shows the effect of T in K. For these simulations, the base case of table 5.1 is used. N (in figure 5.3.a) or T (in figure 5.3.b) is adapted to N or T presented in the legend. of the surface and thus increases the potential. Furthermore, the lower surface charge will lead to a decrease of the thickness of the diffuse layer. In a basic environment, similar results are obtained.

5.1.5 The eﬀect of the temperature

Finally, also the temperature might influence the double layer, because the Poisson-Boltzmann equation includes the entropy effect, which will increase when the temperature is increased. To determine the influence of the temperature, the temperature is changed. However, all the indirect effects, such as a change in the pK-value or a change in the relative permittivity of the solvent, are not taken into account. The results of these simulations are given in figure 5.3.b and appendix E. Figure 5.3.b shows that the effect of the temperature is minimal. A temperature change of 30 K does not influence the dimensionless potential at the surface (a maximal difference of 7 % is obtained). This might be caused by the relative small range with respect to the value (30 K with respect to 278 K). A similar result is obtained for the surface in a basic environment. The difference between the different temperatures in a basic environment is 1 %. The effect of the temperature will thus become smaller at higher dimensionless potentials. It should be taken into account that the potential changes more rapidly with the temperature, due to the definition of the dimensionless potential.

5.2 The amphoteric surface

For the amphoteric surface, 97 different simulations are performed. All the simulations are va- riations on the base case, which is shown in 5.3 for an acidic environment. In the simulations, the electrolyte concentration, the kind of electrolyte, the relative permittivity of the solution, the pH (and therefore the pH-pK), the charge density at the surface and temperature are varied in the ranges shown in table 5.3. The simulations, which use the base case for a basic environment, are given in appendix E. In all the simulations, the Stern layer is presented in the figure as a layer of 0.69 nm. The relative permittivity of the Stern layer is assumed similar to the relative permittivity of the diffuse layer. Due to the decrease in the relative permittivity in the Stern layer, the thickness of the Stern layer will in reality be smaller than the 0.69 nm [8]. 58 CHAPTER 5. SIMULATION RESULTS FOR ISOLATED SURFACES

Table 5.3: The base case of the simulations for isolated surfaces of the amphoteric surface in an acidic environment. The ranges in which the parameters are varied during the simulations are also given.

Base case Ranges 2 −1 5 The concentration of electrolyte (c∞) in mM 10 10 −10 The valency of the positive ions (zp) 1 1 - 5 The valency of the negative ions (zm) 1 1 - 5 The pH-pK -6 -9 - 4 The pK 10 - The charge density at the surface (N) in nm−2 10 1 - 12 The temperature (T) in K 298 278 - 313 The relative permittivity of the solvent (εr) 78 65 - 85 The thickness Stern layer in nm 0.69 - The capacity of the Stern layer in F m−1 1.00 -

(a) c∞ (b) n:n

Figure 5.4: The effect of electrolyte solution on the dimensionless potential for an amphoteric surface in an acidic environment, pH of 4. In figure 5.4.a, the effect of the concentration of electrolyte, c∞, is shown. Figure 5.4.b to 5.4.d show the effect of the kind of electrolyte. For these simulations, the base case of table 5.3 is used. c∞ (in figure 5.4.a) or the kind of electrolyte (in figure 5.4.b to 5.4.d) is adapted to c∞ or the kind of electrolyte presented in the legend. 5.2. THE AMPHOTERIC SURFACE 59

(a) εr (b) pH-pK

Figure 5.5: The effect of the relative permittivity, εr, and the pH-pK on the dimensionless potential for an amphoteric surface in an acidic environment, pH of 4. In figure 5.5.a, the effect of εr is shown. Figure 5.5.b shows the effect of pH-pK. For these simulations, the base case of table 5.3 is used. εr (in figure 5.5.a) or pH-pK (in figure 5.5.b) is adapted to the εr or pH-pK presented in the legend.

5.2.1 The eﬀect of the electrolyte

For amphoteric surfaces, the electrolyte solution influences the diffuse layer via two parameters, the kind of electrolyte and the concentration of the electrolyte. The effect of the concentration of the electrolyte is shown in figure 5.4.a. This figure shows that the thickness of the diffuse layer decreases with an increasing concentration, as is also shown for acidic surfaces. This effect is also obtained in a basic environment, as shown in appendix E. The effect is explained by the enhancement of the screening of the surface charge, due to the increase of the concentration of the ions in the diffuse layer. The higher concentration of the ions is caused by the enhanced concentration of the ions in the bulk, which leads to less attraction to the bulk of the solution. The concentration of the electrolyte has only a small influence on the dimensionless potential of the surface, as shown in figure 5.3.a and appendix E. The surface potential is maximal when a pH of 4 (the pH-pK is -6) is used, while the potential is almost minimal at a pH of 12 (the pH-pK is 2), as shown in figure 3.7.b. This asymptotic behaviour guarantees a similar value of the dimensionless potential. However, when the pH is similar to pK a similar effect of the concentration on the surface potential as shown for acidic surfaces is obtained. Figure 5.4.b to 5.4.d show the influence of the kind of electrolyte on the diffuse double layer. If the charge of the ions, n, is increased, all the figures show a decrease of the thickness of the diffuse layer. These results are also obtained in a basic environment, as shown in appendix E. This is explained by the more effective screening of the surface charge by ions with a higher effective charge. The effect by the increase of the charge of the negative ions is explained by the relative higher concentration of the positive ions. Figure 5.4.b to 5.4.d show that the n:1-electrolyte are comparable to the n:n-electrolyte, while the 1:n-electrolyte have less influence of the dimensionless potential. In a basic environment, the 1:n-electrolyte is comparable to the n:n-electrolyte. This difference between the acidic and basic results can be explained by the difference in the sign of the charge of the surface. The increased screening for a 1:n-electrolyte (n:1-electrolyte) for a positive (negative) surface charge is caused by the higher charge of the counter-ions. The effect of the n:1-electrolyte (1:n-electrolyte) for a positive (negative) surface charge is caused by the higher relative concentration of the counter- ions. However, increasing the relative concentration is a less effective manner of decreasing the dimensionless potential. 60 CHAPTER 5. SIMULATION RESULTS FOR ISOLATED SURFACES

(a) N (b) T

Figure 5.6: The effect of the charge density, N, at the surface and the temperature, T, on the dimensionless potential for an amphoteric surface in an acidic environment, pH of 4. In figure 5.6.a, the effect of N is shown. Figure 5.6.b shows the effect of T. For these simulations, the base case of table 5.3 is used. N (in figure 5.6.a) or T (in figure 5.6.b) is adapted to the N or T presented in the legend.

5.2.2 The effect of relative permittivity of the solvent The electrolyte solution also has an influence on the dimensionless potential via the relative permittivity of the solvent. Therefore, the effect of the relative permittivity is shown in figure 5.5.a and appendix E. The results show almost no difference when using the maximal or the minimal relative permittivity (maximally 2 %) in acidic environment. The difference between the results in a basic environment are slightly larger 4 %. The results in figure 5.5.a and appendix E both show an increase (decrease) in the potential at the wall when the relative permittivity is decreased for a positive (negative) surface. However, the gradient of the potential decreases. This can be explained by the governing equations for the surface charge and the potential gradient, as shown in paragraph 5.1.2.

5.2.3 The effect of the pH The electrolyte concentration, the kind of electrolyte and the relative permittivity are not the only parameters through which the solution influences the thickness of the diffuse layer. The pH also influences the thickness of the diffuse layer via the surface potential. A increase (decrease) of the dimensionless potential of the surface with a positive (negative) surface potential causes a increase in the thickness of the diffuse layer. This effect is shown in figure 5.5.b. Figure 5.5.b shows no diffuse double layer when the pH is equal to the pK. Because the surface potential is zero at these surfaces, there is no charge at the surface. Therefore, the concentration of the ions will be equal at all distances from the surface and the potential in the surface will be zero. When the pH is higher than the pK, the amount of protons near the surface decreases (due to the decrease in pH). This leads to a more negative charge, according to the following reaction.

− 1 + + 1 MOH 2 + H MOH 2 While a lower pH, induces an increase of the number of protons near the surface. This leads to a more positive charge at the surface. This explains the positive potentials at pH-pK lower (higher) than zero and the increase (decrease) in the potential for a lower (higher) pH-pK.

5.2.4 The effect of the charge density at the surface The dimensionless potential at the surface is thus an important parameter for the thickness of the diffuse layer. The dimensionless potential at the surface depends on the charge density of the surface. The effect of the charge density is shown in figure 5.6.a and appendix E. 5.3. CONCLUSION 61

Figure 5.6.a shows that the charge density has a large effect on the dimensionless potential of the surface. This is also expected. A decrease in the charge density decreases the effective charge of the surface and thus the potential. Furthermore, the lower surface charge will lead to a decrease in the diffuse layer. In a basic environment, similar results are obtained.

5.2.5 The effect of the temperature Besides the effects of the surface and the effects of the electrolyte solution, the temperature might also influence the dimensionless potential. The effect of the temperature is determined in figure 5.6.a and appendix E. In these simulations, the direct influence of the temperature on the dimensionless potential is determined. All secondary effect, like changes in the pK-value or the relative permittivity of the surface, are not taken into account. Both results show there is little to no effect of the temperature. A temperature difference of 30 K results in a 0.3 % difference in an acidic solution and 0.2 % in a basic solution. This small difference is explained by the small temperature change with respect to the absolute value of the temperature ( 30 K with respect to 278 K). The effect of the temperature on the dimensionless potential will however increase due to the secondary effects of the temperature. Furthermore, there is also a larger effect of the temperature on the potential, because of the definition of the dimensionless potential.

5.3 Conclusion

In this chapter, the effect of different parameters on the dimensionless potential are determined. The electrolyte concentration, the kind of electrolyte, the pH and the charge density at the surface show a large influence on the dimensionless potential for both acidic and amphoteric surfaces. However, the effect of a n:1-electrolyte versus a 1:n-electrolyte is less pronounced at low potentials (as shown for the acidic surfaces), which is caused by the comparable contribution of both the enhanced concentration in a n:1-electrolyte and the enhanced charge of the ions in a 1:n-electrolyte (for a negatively charged surface, for a positive surface the contribution will be the other way around). The relative permittivity has little to no effect on the dimensionless potential (2 - 4 %). Ho- wever, at low surface potentials the effect will become important; the effect increases to 16 %. Furthermore, the effect of the temperature is very small (0.2 - 7 %). However, in these simulations the secondary effects of the temperature change, like changes in the pK-value and the relative permittivity of the solvent, are not taken into account. Introducing these effects will cause the effect of the temperature to be much larger. 62 CHAPTER 5. SIMULATION RESULTS FOR ISOLATED SURFACES Chapter 6

Simulation results for non-isolated surfaces

The Poisson-Boltzmann model for non-isolated surfaces, as described in chapter 4, is used for the simulation of several acidic and amphoteric surfaces. To ensure the model gives realistic results, the simulation results are compared to the experimental results presented in chapter 2. The F experimental R of the silica and aluminium hydroxide colloidal can be compared directly with the simulation results. However, the distances between the titania nanosheets cannot be determined F directly. These distances can be compared with the secondary minima of the R -graphs.

6.1 The experiments with silica

First of all, the AFM results of the silica particles will be simulated. The used constants for these simulations are shown in table 6.1. The used pK is within 10% of the value obtained in literature. Furthermore, the density of the sites on the surface is 9 nm−2. This is higher than the value found in literature. The Hamaker constant is within the range found in literature, 8.0 · 10−21-1.6 · 10−21 J [6, 32, 33]. Figure 6.1 shows the numerical results and the experimental results for all the cases using the parameters in table 6.1. Figure 6.1.a shows that the numerical results and the experimental results are in very good comparison when the distance between the particles is more than 1 nm. The difference between the experimental results and the numerical results are probably caused by specific interaction of the ions with the surface. Due to complexation of the ions with the surface, the effective surface charge is lowered. This leads to a lower dimensionless potential and thus a decrease in the interaction force between the particles. To compensate the lower surface charge of the experiments in the numerical simulations, the capacity of the Stern layer is increased and the temperature is decreased. It is recommended, to implement the specific adsorption of ions into the model. This will increase the accuracy of the numerical result. Besides, the effect of temperature and field strength on the relative permittivity are not taken into account. Figure 6.1.b shows that the numerical results are not similar to the experimental results. This is probably also caused by the specific adsorption of ions. Due to the specific adsorption, the surface charge and thus the dimensionless potential are lowered this might cause a secondary

Table 6.1: The used parameters for the simulation of the experimental results for silica

−2 −2 c (mM) pH shift (nm) pK N (nm ) T (K) Cstern (F m ) dvdW (nm) A (J) 1 10 4.1 0.8 7.1 9 290 0.77 0.34 4 · 10−21 2 10 10.3 0.8 7.1 9 290 0.77 0.34 4 · 10−21 3 1000 5.3 0.8 7.1 9 290 0.77 0.34 4 · 10−21

63 64 CHAPTER 6. SIMULATION RESULTS FOR NON-ISOLATED SURFACES

(a) 1 (b) 2

Figure 6.1: The comparison of the AFM results for the repulsion between aluminium surfaces in a CaCl2 solution and the numerical simulations. The numbers in subscript of the ﬁgure represent the cases in table 6.1. The distances in the experimental results are shifted by 0.8 nm.

(a) 1 (b) 2

Figure 6.2: The comparison of the experimental SFA results for the repulsion between aluminium surfaces in a NaBr solution with a pH of 3 and the numerical simulations. The numbers in the subscript of the ﬁgure are the cases in table 6.2. 6.2. THE EXPERIMENTS WITH ALUMINIUM 65

Table 6.2: The used parameters for the simulation of the experimental results for aluminium

−2 −2 c (mM) pH pK N (nm ) T (K) Cstern (F m ) dvdW (nm) A(J) 1 110 2.5 8 5 290 0.31 0.25 8 · 10−20 2 10 2.5 8 5 290 0.31 0.25 8 · 10−20 3 1 2.5 8 5 290 0.31 0.25 8 · 10−20 minimum in the interaction forces. Furthermore, the relative permittivity is independent on the temperature and the field strength. Figure 6.1.c has a very good comparison with the experimental results. In these experiments, there might also be influenced of the specific adsorption of ions. The difference between simulations with a high concentration and a low concentration is the concentration at the surface, when the surfaces are at a certain distance, with respect to the concentration at the surface for isolated surfaces. The concentration is increased relatively more in the case of a low concentration. The adsorption of the ions will thus be relatively more when the concentration is low. Therefore, the F effect of specific adsorption might be more pronounced. The effect on the R will then be larger at low concentrations.

6.2 The experiments with aluminium

The SFA experiments using two aluminium surface are simulates using the constants stated in table 6.2. The first difference between the results and the experiments is the concentration of the first case. The concentration is 10 % higher than the concentration of the experiments. However, the results could not be simulated using the concentration of the experiment. Furthermore, the pH is decreased to 2.5, which is within the error margin of the pH of the experiment. Furthermore, the density of sites is relatively low, while the Hamaker constant is relatively high [6, 32, 33]. The results of the simulations are given in figure 6.2. The figure clearly shows that the numerical results are similar to the experimental results when high concentrations are used, figure 6.2.c. However, when the concentration is decreased the numerical results do not describe the experimental results, figure 6.2.a and 6.2.b. This is probably caused by the fact that the specific adsorption of the ions is not taken into account in the simulations. The specific adsorption will decrease the dimensionless potential of a non-isolated surface more than the dimensionless potential of an isolated surface, because isolated surfaces are F only influenced on one surface. Therefore, the R will decrease when the specific absorption is taken into account. The effect of the introduction of the specific adsorption will be larger when the concentration is lower. The difference between simulations with a high concentration and a low concentration is the concentration at the surface, when the surfaces are at a certain distance, with respect to the concentration at the surface for isolated surfaces. The concentration is increased relatively more in the case of a low concentration. The adsorption of the ions will thus be relatively more when the concentration is low. Therefore, the effect of specific adsorption might be more F pronounced. The effect on the R will then be larger at low concentrations.

6.3 Colloidal solution of titania nanosheets

The swelling of the titania precursor is measured experimentally, as shown in ﬁgure 2. The distances between the layers of titania is more than 10 Å, therefore the swelling is osmotic swelling. This type of swelling can be described by the DLVO theory. The used parameters of the simulation of the swelling of the parameters is shown in table 6.3. To simplify the numerical simulation the pH is set to 12.4. It should be determined if the pH has inﬂuence on the numerical results. Furthermore, the pK-value used in the simulations is high. This might be caused by the 2D- morphology of the nanosheets. Besides, the density of sites is relatively high, but short numerical 66 CHAPTER 6. SIMULATION RESULTS FOR NON-ISOLATED SURFACES

Table 6.3: The used parameters for the simulation of the experimental results for titania

−2 −2 c (mM) pH pK N (nm ) T (K) Cstern (F m ) dvdW (nm) A (J) 1 26.0 12.4 7 10 290 0.49 0.50 6 · 10−20 2 30.0 12.4 7 10 290 0.49 0.50 6 · 10−20 3 34.5 12.4 7 10 290 0.49 0.50 6 · 10−20 4 38.5 12.4 7 10 290 0.49 0.50 6 · 10−20 5 43.0 12.4 7 10 290 0.49 0.50 6 · 10−20

Table 6.4: The comparison between the experimental and numerical results for titania nanosheets. The numbers in the ﬁrst column represent the cases in table 6.3. The numbers in the second and third column are the distances measured experimentally and determined numerically in nm, respectively.

Experimental Numerical 1 12.2 11.0 2 10.7 10.5 3 9.6 10.1 4 8.7 9.7 5 8.0 9.1

F simulations show that the effect of the density of the sites on the R -graph is small. The used Hamaker constant is within the range obtained in literature [32, 33]. The numerical results and the experimental results of the swelling are shown in table 6.4. The F distances in the table for the numerical results are the minima of the R -graphs. The determination of the distance from the numerical results has an error due to read out of the results of about 0.3 nm. The table shows that at medium distances (case 2 and 3) the experimental and numerical results are in good comparison. However at smaller distances or larger distances, the numerical results are different. At larger distances (case 1) , the numerical results are under estimated, while at smaller distance (case 4 and 5) the results are over estimated. The differences can be caused by F the specific adsorption of ions. This will change the shape of the R -graphs and thus the position of the minima of the graphs. Besides the effect of the specific adsorption, the swelling might still be in its first stage in case 4 and 5. The swelling can be called osmotic swelling for clays when the distance between the layers is more than 10 Å, which is caused by the successive build up of water layers. However, due to the stronger interaction in the titania precursor, a bulky ion should be exchanged by the interlayer. This causes a initial swelling of the precursor. Following, the layers have to be hydrated. This causes an extra swelling of the precursor. Therefore, the swelling of the cases with a small distance might not be in the range of osmotic swelling.

6.4 Discussion

The numerical results for silica and aluminium particles do not match the experimental results. The difference between the experimental and the numerical results might be caused by the indepen- dency of the relative permittivity on the temperature and the field strength. Besides, the specific adsorption of ions is not taken into account in the simulations. The absorption will decrease the dimensionless potential of the non-isolated surfaces more than the dimensionless potential of the isolated surfaces, because it is affect by both surfaces. The difference between simulations with a high concentration and a low concentration is the concentration at the surface, when the surfaces are at a certain distance, with respect to the concentration at the surface for isolated surfaces. The concentration is increased relatively more in the case of a low concentration. The adsorption of the ions will thus be relatively more when the concentration is low. Therefore, the effect of 6.5. CONCLUSION 67

F specific adsorption might be more pronounced. The effect on the R will then be larger at low concentrations. The numerical results of the titania nanosheet do not match the experimental results at small distances and large distances between the layers. This might be caused by the specific adsorption of ions at the surface as is the case for aluminium and silica particles. The differences at small distances might also be caused by the first stage of swelling. For clays, the first stage of swelling stops at 10 Å, which is caused by the build up of successive layers of water molecules. However, due to the stronger interaction in the titania precursor, a bulky ion should be exchanged by the interlayer. This causes a initial swelling of the precursor. Following, the layers have to be hydrated. This causes an extra swelling of the precursor. Therefore, the swelling of the cases with a small distance might not be in the range of osmotic swelling.

6.5 Conclusion

The numerical results for the silica and aluminium particles do not match the experimental results. This might be caused by the influence of the temperature and the field strength on the relative permittivity and the specific adsorption of ions on the surface. The numerical results for the swelling of the titanium precursor can be simulated very well at medium distances. The differences at small and large distances might be explained by the specific adsorption of ions. Furthermore, the difference between the numerical and experimental results at small distances between the layers might be caused due to the fact that the simulations are not yet in the range of the osmotic swelling. 68 CHAPTER 6. SIMULATION RESULTS FOR NON-ISOLATED SURFACES Chapter 7

Conclusions

The aim of this report is to develop a numerical model which is able to determine the eﬀect of the electrolyte solution, the permittivity of the solvent, the pH and the temperature on the colloidal stability of oxide nanosheets (especially the titania nanosheets made by R. Besselink (2010) [5]). The numerical model is based on the DLVO-theory. The DLVO-theory is adjusted due to the unrealistic elements. The main conclusions will be discussed in the following two paragraphs.

The modiﬁed Poisson-Boltzmann models

To enable the determination of the effect of several parameters on the colloidal stability, two models are developed. Both models are based on the DLVO-theory. Due to some unrealistic elements, the DLVO-theory is adjusted. First of all, in the DLVO-theory the ions are assumed to be point charges. The ions are thus infinitesimal small particles. However, the ions have a finite size. To take this into account the Stern layer is introduced in the model. Furthermore, the DLVO-theory assumes a constant charge at the surface. However, the charge of an oxidic surface depends on the degree of protonation of the oxidic surface. Therefore, the constant charge model is changed to a charge regulation model: the 1-pK model. First of all, the Poisson-Boltzmann model for isolated surfaces is developed. The model determines the dimensionless potential at a certain distance from the oxidic surface. For this model, some initial settings are determined, to ensure the reproducibility of the results. Using the analytical solution of the DLVO-theory for n:n-electrolytes, 1:2-electrolytes and 2:1-electrolytes, the used numerical model is validated. Furthermore, the implementation of the charge regulation model is validated using the degree of ionisation of the surface and the analytical solution of the DLVO-theory for n:n-electrolytes. The implementation of the Stern layer in the model is validated using the definitions of both the Stern layer and the diffuse layer. Secondly, the Poisson-Boltzmann model for non-isolated surfaces is developed. This model determines the dimensionless potential between two similar oxidic surfaces. The electrostatic free energy is determined via the indirect route (the integration of the pressure) and the direct route (integration of the dimensionless potential). The electrostatic free energy is combined with the Van der Waals forces to obtain the interaction forces. For this model, there are some initial settings determined to ensure the reproducibility of the simulation results. The numerical method and the charge regulation model of the Poisson-Boltzmann model for non-isolated surfaces are similar to the Poisson-Boltzmann model for isolated surfaces. Therefore, the numerical model and the charge regulation model are already validated. The new parts in the numerical model and the determination of the pressure are validated, using the properties of the pressure. The indirect and the direct method for acidic and amphoteric surfaces is validated using the Linear Superposition Approximation. However, the results of the direct route for the n:1-electrolytes (1:n-electrolytes) for positively (negatively) charged amphoteric surfaces could not be validated.

69 70 CHAPTER 7. CONCLUSIONS

The implementation of the Stern layer is validated via several routes. First of all, the changes in the dimensionless potential and the gradient of the dimensionless potential are validated using the definitions of the diffuse layer and the Stern layer. Furthermore, the calculation of the free energy via the indirect route did not change, when the pressure of the diffuse layer is used in the calculations. Therefore, the changes in the direct route due to the implementation of the Stern layer are validated using the indirect route. The changes in the direct route are validated for acidic surfaces. However, the changes could only be validated for amphoteric surface with a |pH − pK| ≥ 6.

The simulation results

Using the models described above several simulations are performed. First of all, the effects of the different parameters on the isolated surface are determined. The simulations of isolated surfaces show that the concentration of the electrolyte, the kind of electrolyte, the pH and the charge density have an influence on the dimensionless potential for isolated surfaces. The effect of the temperature is small (0.2 - 7 %). However, the secondary effect of the temperature, like changes in the pK value and the relative permittivity of the solvent, are not taken into account. The effect of the relative permittivity is small at high surface potentials (2 - 4 %). However, at low surface potentials the chosen relative permittivity has a large effect (maximally 16 %). The numerical results for the silica and aluminium particles do not match the experimental results. This might be caused by the influence of the temperature and the field strength on the relative permittivity and the specific adsorption of ions on the surface. The numerical results for the swelling of the titanium precursor can be simulated very well at medium distances. The differences at small and large distances might be explained by the specific adsorption of ions. Furthermore, the difference between the numerical and experimental results at small distances between the layers might be caused due to the fact that the simulations are not yet in the range of the osmotic swelling. Chapter 8

Recommendations

There are also some recommendations for further research. These points will be discussed in this chapter.

The modiﬁed Poisson-Boltzmann models The protonation model The Poisson-Boltzmann model used in this report, determines the surface charge via the 1-pK model. This model takes into account only 1 type of surface groups. However, spectroscopic measurements show that there are multiple types of oxygen containing groups on a metal (hydr)oxide surface. In order to take into account all these surface groups the 1-pK model should be replaced by the MUSIC model or multi site complexation. This will ensure a more realistic description of the surface groups.

The validation of the Stern layer In this report, the implementation of the Stern layer with respect to the dimensionless potential and the gradient of the dimensionless potential is only validated using the governing equations of the model. So the implementation should be validated in all cases. To really validate this part of the model, another procedure for the calculation of the dimensionless potential and the gradient of the dimensionless potential, when the Stern layer is implemented, should be determined.

The direct route for the electrostatic free energy The implementation of the direct route for the determination of the electrostatic free energy gave some problems for amphoteric surfaces. First of all, the electrostatic free energy determined via the direct route is not equal to the electrostatic free energy determined via the indirect route for n:1-electrolytes (1:n-electrolytes) for positively (negatively) charged surfaces. The results obtained via the direct route are partly negative. The diffuse contribution is doubled when a n:1-electrolyte (1:n-electrolytes) for positively (negatively) charged surfaces is used, which is also found for acidic surfaces. However, the surface contribution for 1:n-electrolytes and n:1-electrolytes are similar, while for acidic surfaces an increase in the surface charge is obtained. Therefore, the problems are probably caused by the surface contribution of the calculation of the electrostatic free energy. The effect of the n:1- electrolytes (1:n-electrolytes) on the positively (negatively) charged amphoteric surfaces should be studied in more detail to explain and correct the problems. Secondly, when the Stern layer is introduced, the determination of the electrostatic free energy via the direct route is only similar to the indirect route when |pH − pK| ≥ 6. The implementation of the Stern layer has no effect on the diffuse contribution of the electrostatic free energy. Therefore,

71 72 CHAPTER 8. RECOMMENDATIONS the problems are caused by the surface contribution of the free energy. The modiﬁcations of the direct route due to the Stern layer for amphoteric surfaces should be studied in more detail, to explain and correct the diﬀerences.

The simulation results The comparison between the experiments and the numerical results The numerical results for silica and aluminium particles do not match the experimental results. The difference between the experimental and the numerical results might be caused by the indepen- dency of the relative permittivity on the temperature and the field strength. Besides, the specific adsorption of ions is not taken into account in the simulations. The absorption will decrease the dimensionless potential of the non-isolated surfaces more than the dimensionless potential of the isolated surfaces, because it is affect by both surfaces. The difference between simulations with a high concentration and a low concentration is the concentration of the ions at a certain distance from the surface with respect to the concentration for isolated surfaces. The concentration is increased relatively more in the case of a low concentration. The adsorption of the ions will thus be relatively more when the concentration is low. The effect of adsorption on the difference between the dimensionless potential of isolated and non-isolated surfaces will thus be more pronounced. F Therefore, the effect on the R will be larger at low concentrations. Therefore, it is recommended to introduce the specific adsorption into the model. The numerical results of the titania nanosheet do not match the experimental results at small distances and large distances between the layers. This might be caused by the specific adsorption of ions at the surface as is the case for aluminium and silica particles. Therefore, the specific adsorption of the ions should be implemented in the model. The differences at small distances might also be caused by the first stage of swelling. For clays, the first stage of swelling stops at 10 Å, which is caused by the build up of successive layers of water molecules. However, due to the stronger interaction in the titania precursor, a bulky ion should be exchanged by the interlayer. This causes a initial swelling of the precursor. Following, the layers have to be hydrated. This causes an extra swelling of the precursor. Therefore, the swelling of the cases with a small distance might not be in the range of osmotic swelling. To check if the swelling is already osmotic, it is recommended to introduce the hydration energy in the model.

The effect of the parameters The effect of the different parameters, like the temperature, the kind of electrolyte and the electrolyte concentration, on the colloidal stability are not determined in this report. It is recommended to perform these simulations. This will give ranges in which the nanosheets form a stable colloidal solution. 73 74 CHAPTER 9. LIST OF SYMBOLS

Chapter 9

List of symbols

Symbols A The Hamaker constant J c The concentration mol m−3 d The thickness of the Stern layer m E The electric ﬁeld V m−1 F The Faraday constant C mol−1 F The force N −1 [H]s The concentration of protons at the surface mol L I The ionic strength mol m−3 K The equilibrium constant of the surface reaction m3mol−1 N The number of sites mol m−2 P The pressure Pa pH The logarithm of the concentration of protons ions with a base 10 - pK The logarithm of the equilibrium constant with a base 10 - R The gas constant J mol−1K−1 r The radius m T The temperature K V The free energy N mol1m−1J−1 v The proportionality constant of the ions - x The direction of calculation m y The dimensionless potential z The charge -

α The degree of ionisation of the surface - ∆ The diﬀerence between two values - −1 ε0 The permittivity of vacuum F m εr The relative permittivity of the solvent - κ The Debye length m ψ The electrochemical potential V σ The charge density C m−1 θ The relative number of sites - 75

Subscripts BC Boundary condition d At the outer Helmholtz plane electrostatic Electrostatic h Protonated H+ protons ions i Ion i LSA Linear Superposition Approximation m Negative ions OH− Hydroxide ions p Positive ions PB Poisson-Boltzmann equation s At or of the surface sw Of an isolated surface vdW Of the Van der Waals forces 0 At the mid plane ∞ At inﬁnite distance from the surface

Abbreviations AFM Atomic Force Microscope DLVO Derjaguin-Landau-Verwey-Overbeek LSA Linear Superposition Approximation SFA Surface Force Apparatus TBA+ tetrabutylammonium cations 76 CHAPTER 9. LIST OF SYMBOLS Bibliography

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Appendix A

The Matlab scripts for the isolated surface simulations

The simulations using the Poisson-Boltzmann model for isolated surfaces are performed using the Matlab scripts in table A.1. The table shows if the ions of water and/or the Stern layer are taken into account in the script. All the models are based on the governing equations shown in chapter 3. First, the input parameters of the isolated surface models will be discussed. Following, the diﬀerent parts of the script will be described.

A.1 The input parameters

The diﬀerent input parameters for the isolated surface simulations are given in table A.2. The input parameters have to be inserted in the order of table A.2. The input parameter ds can be omitted, when a model, which does not take into account the Stern layer, is used for the simulation of an isolated surface. Furthermore, if the parameter ds is used, the thickness of the Stern layer should be inserted using the amount of stepsizes. The input parameter nstep should also be inserted using the amount of stepsizes. The input parameter surface can be either 1 or 2. If 1 is used, the oxide surface is simulated as an acidic surface. If 2 is inserted, an amphoteric surface is simulated. The input parameter numcase can be either 1, 2 or 3. When 1 is inserted, the Euler method is used as the discretisation method, while changing to 2 or 3 the midpoint method or the third-order Runga Kutta method are used, respectively.

A.2 The outline of the script

The script starts with the selection of the default values for nstep, stepsize and numcase if these parameters are not given as input. Following, the Faraday constant, the vacuum permittivity

Table A.1: The diﬀerent Matlab scripts for the isolated surface simulations. The second column shows if the script takes into account the ions of water and the third column shows if the script takes into account the Stern layer.

The ions of water The Stern layer PB_single_wall_exp_newBC_E.m yes no PB_single_wall_exp_newBC_E_noHOH.m no no PB_single_wall_exp_newBC_E_stern.m yes yes PB_single_wall_exp_newBC_E_stern_noHOH.m no yes

83 84APPENDIX A. THE MATLAB SCRIPTS FOR THE ISOLATED SURFACE SIMULATIONS

Table A.2: Input parameters for the Matlab script for an isolated surface simulation, in order.

Parameter Unit Default value ds The thickness of the Stern layer stepsize - c The concentration of the electrolyte mM - zp The zp -- zm The zm -- vp The vp -- vm The vm -- pH The pH of the electrolyte - - N The number of charged sites on the oxide surface mol m−2 - K The equilibrium constant of the protonation reaction mol L−1 - epsr The relative permittivity of the solvent - - T The temperature K - surface The kind of oxide surface used - - nstep The number of steps for determination of y stepsize 104 stepsize The step size used m 10−11 numcase The numerical model used - 3 and the gas constant are defined in the script. Furthermore, the concentration of protons and hydroxide ions at infinite distance from the surface are determined. Using these concentrations, the relative concentration at infinite distance from the surface of the water ions can be determined, when these are taken into account. Now all constants are determined, the dimensionless potential at the wall is determined. This is done via the scheme presented in figure A.1. The dimensionless potential is set to zero. The gradient of the potential is then determined using the Poisson-Boltzmann equation (when the Stern layer is taken into account this calculation is at the outer Helmholtz plane) and using the 1-pK model. The results of the two methods are then compared. If the relative difference is more than a set Rdif , a small value ∆y is added to the previous dimensionless potential. This is repeated until the relative difference between the gradient determined via the Poisson-Boltzmann equation and via the 1-pK model is less than Rdif . The dimensionless potential used in the calculation of the gradient via the 1-pK model is the dimensionless potential at the wall. Now that the dimensionless potential and the gradient of the dimensionless potential are known at the surface, the dimensionless potential and the gradient can be determined at all the distances from the surfaces until nstep is reached. The dimensionless potential in the Stern layer is determined using the linear decrease of the dimensionless potential. The dimensionless potential in the diffuse layer is determined via the numcase chosen in the input parameters. For the determination of the free energy between two non-isolated surfaces, the diffuse contribution of a surface at infinite distance of another surface is determined at the end of the script. Finally, an output struct is created. In this struct, the input parameters are stored in input. Furthermore, the determined gradient of the potential, the potential and the diffuse contribution of the free energy are stored in values. The necessary values (the number of iterations, the step size, the number of steps, the Rdif and the ∆y) are stored in nesval. A.2. THE OUTLINE OF THE SCRIPT 85

Start: y(1) = 0

dy via Poisson-Boltzmann y(1) = y(1) +Δy equation

dy via boundary condition

Rdif > threshold dy check

Determination of all y

Simulation finished

Figure A.1: A schematic representation of the numerical method for the determination of the dimensionless potential. The dimensionless potential at the surface is y(1). 86APPENDIX A. THE MATLAB SCRIPTS FOR THE ISOLATED SURFACE SIMULATIONS Appendix B

The Poisson-Boltzmann model for isolated surfaces

The results of the simulations for the determination of the initial settings and the validation of the Poisson-Boltzmann model for isolated surfaces could not all be presented in chapter 3. Therefore the results are partly presented in this appendix.

B.1 The initial settings

B.1.1 The step size

The results presented in figure B.1.a to B.1.d show that using larger then 0.1 nm as the step size for the isolated surface simulations of acidic surfaces (silica) might lead to large differences between the real solution and the obtained results. When smaller time steps are used virtually no difference can be found between the results. Figure B.1.e to B.1.h show that a step size of 0.01 nm is also necessary in amphoteric surfaces (aluminium hydroxide) to obtain results which do not differ from the results with smaller step sizes.

B.1.2 The relative diﬀerence, Rdif

All the results presented in figure B.2 show that increasing the accuracy, represented by the relative difference, Rdif , does not really influence the results. In figure B.2.b, B.2.d and B.2.h show that a relative difference of 10% gives slightly different results. Therefore an accuracy of at least 1% should be chosen for both acidic and amphoteric surfaces.

B.1.3 The ions in the water

The results in figure B.3 show the effect of the ions of water on the dimensionless potential. Figure B.3.a and B.3.c do not show a difference between the two simulations. So at high concentrations of electrolyte, the effect of the ions in the water is minimal in an acidic environment, only a small influence is obtained for amphoteric surfaces (aluminium hydroxide). At low concentrations, the ions in the water will induce a different potential as shown in figure B.3.b and B.3.d. So the ions of water should be taken into account at low concentrations for both surfaces and high concentrations when amphoteric surfaces are used in basic environments.

87 88 APPENDIX B. THE POISSON-BOLTZMANN MODEL FOR ISOLATED SURFACES

(a) Silica, 0.1 M (b) Silica, 0.1 M, close up

(e) Aluminium hydroxide, 0.1 M (f) Aluminium hydroxide, 0.1 M, close up

(g) Aluminium hydroxide, 1 mM (h) Aluminium hydroxide, 1 mM, close up

Figure B.1: The effect of the step size on the accuracy of the isolated surface simulations in a basic environment, pH of 10 for silica and pH of 12 for aluminium hydroxide. The lengths given in the legend are the step sizes. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity is 78. The used relative difference, Rdif , is 0.01% and the ions of water are taken into account. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. B.1. THE INITIAL SETTINGS 89

(a) Silica, 0.1 M (b) Silica, 0.1 M, close up

(e) Aluminium hydroxide, 0.1 M (f) Aluminium hydroxide, 0.1 M, close up

(g) Aluminium hydroxide, 1 mM (h) Aluminium hydroxide, 1 mM, close up

Figure B.2: The effect of the relative difference of the determination of the dimensionless potential, Rdif , at the surface on the accuracy of the isolated surface simulations in a basic environment, pH of 10 for silica and pH of 12 for aluminium hydroxide. The percentages given in the legend are the different Rdif used. The temperature is 298 K and the relative permittivity is 78. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The step size used is 0.01 nm and the water ions are taken into account. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. 90 APPENDIX B. THE POISSON-BOLTZMANN MODEL FOR ISOLATED SURFACES

(a) Silica, 0.1 M (b) Silica, 1 mM

Figure B.3: The effect of the ions in the water of the isolated surface simulations in a acidic environment, pH of 10 for silica and pH of 12 for aluminium hydroxide. The blue lines are the simulation without water ions and the red lines are the simulations with water ions. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity is 78. The step sizes are 0.01 nm and the relative difference, Rdif , is 0.01%. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density at the silica surface, N, is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. B.2. THE VALIDATION 91

(a) Dimensionless potential, Silica (b) ∇y, Silica

Figure B.4: The eﬀect of the introduction of the Stern layer on the dimensionless potential and the gradient of the dimensionless potential, ∇y, in an basic environment, pH of 10 for silica and pH of 12 for aluminium hydroxide. The red doted line represents the outer Helmholtz plane, while the blue line represents the dimension less potential or ∇y. The temperature is 298 K, the concentration of the 1:1 electrolyte is 0.1 M and the relative permittivity is 78. The kind of surface used is stated in the name of the ﬁgure. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. The capacity of the Stern layer is 1 F m−2.

B.2 The validation B.2.1 The Stern layer The results in figure B.4 show the effect of the Stern layer on the dimensionless potential and the gradient in the dimensionless potential in a basic environment. Figure B.4.a and B.4.c show a fluent course of the dimensionless potential. Figure B.4.b and B.4.d show that there is a difference in the slope between the Stern layer and the diffuse layer. As expected, the slope of the gradient of the dimensionless potential is zero in the Stern layer, while it is exponential in the diffuse layer. Therefore, the implementation of the Stern layer resulted in the expected dimensionless potential and gradient in the dimensionless potential. 92 APPENDIX B. THE POISSON-BOLTZMANN MODEL FOR ISOLATED SURFACES Appendix C

The Matlab scripts for non-isolated surface simulations

In this appendix, the Matlab script for the simulation of non-isolated surfaces will be discussed in detail. There are two kinds of non-isolated surface simulations: simulations with a single distance and simulations with multiple distances.

C.1 Matlab script for a single distance between the surfaces

The simulations using the Poisson-Boltzmann model for non-isolated surfaces with a certain distance between the surfaces are performed using the Matlab scripts in table C.1. The table shows if the ions of water and/or the Stern layer are taken into account in the script. All the models are based on the governing equations shown in chapter 4. First, the input parameters of the isolated surface models will be discussed. Following, the diﬀerent parts of the script will be described.

C.1.1 The input parameters The diﬀerent input parameters for the non-isolated surface simulations with a single distance are given in table C.2. The input parameters have to be inserted in the order of table C.2. The input parameter ds can be omitted, when a model, which does not take into account the Stern layer, is used for the simulation of an isolated surface. Furthermore, if the parameter ds is used, the thickness of the Stern layer should be inserted using the amount of stepsizes. The input parameters nstep and dis should also be inserted using the amount of stepsizes. The input parameter surface can be either 1 or 2. If 1 is used, the oxide surface is simulated as an acidic surface. If 2 is inserted, an amphoteric surface is simulated. The input parameter numcase can be either 1, 2 or 3. When 1 is inserted, the Euler method is used as the discretisation method, while changing to 2 or 3 the midpoint method or the third-order Runga Kutta method are used, respectively.

Table C.1: The diﬀerent Matlab scripts for the non-isolated surface simulations. The second column shows if the script takes into account the ions of water and the third column shows if the script takes into account the Stern layer.

The ions of water The Stern layer PB_double_wall_samesurface_2.m yes no PB_double_wall_samesurface_2_noHOH.m no no PB_double_wall_samesurface_2_stern.m yes yes PB_double_wall_samesurface_2_stern_noHOH.m no yes

93 94APPENDIX C. THE MATLAB SCRIPTS FOR NON-ISOLATED SURFACE SIMULATIONS

Table C.2: Input parameters for the Matlab script for a non-isolated surface simulation, in order.

Parameter Unit Default value ds The thickness of the Stern layer stepsize - dis The half of the distance between the surfaces stepsize - c The concentration of the electrolyte mM - zp The zp -- zm The zm -- vp The vp -- vm The vm -- pH The pH of the electrolyte - - N The number of charged sites on the oxide surface mol m−2 - K The equilibrium constant of the protonation reaction mol L−1 - epsr The relative permittivity of the solvent - - T The temperature K - surface The kind of oxide surface used - - nstep The number of steps for determination of y stepsize 104 stepsize The step size used m 10−11 numcase The numerical model used - 3

C.1.2 The outline of the script

The script starts with the selection of the default values for nstep, stepsize and numcase if these parameters are not given as input. Then the Rdif,3 of figure C.1 is selected, based on the distance and the solution parameters. Also the number of steps used is multiplied by ten, when the distance between the surfaces is less than 101. The obtained set of parameters is used for the isolated surface simulation. This is the first step of figure C.1. The resulting dimensionless potential and the diffuse contribution of the free energy are stored as variables. Following, the Faraday constant, the vacuum permittivity and the gas constant are defined in the script. Furthermore, the concentration of protons and the hydroxide ions at infinite distance from the surface are determined. Using these concentrations, the relative concentration at infinite distance from the surface of the ions of water can be determined, when these are taken into account. Then, the dimensionless potential can be determined. First, the a of figure C.1 is set depending on the difference between the y(dis) and the y0. This value will be used to determine the next y0. Using this y0, the dimensionless potential at the wall is determined via the method shown in figure C.1. The starting dimensionless potential at the wall is the dimensionless potential at the wall for an isolated surface. The gradient of the potential is then determined using the Poisson-Boltzmann equation (when the Stern layer is taken into account this calculation is at the outer Helmholtz plane) and using the 1-pK model. The results of the two methods are then compared. If the relative difference is more than a set Rdif,2, a small value ∆y is added to the previous dimensionless potential. This is repeated until the relative difference between the gradient determined via the Poisson-Boltzmann equation and via the 1-pK model is less than Rdif,2. The dimensionless potential used in this calculation of the gradient via the 1-pK model is the dimensionless potential at the wall. Now that the dimensionless potential and the gradient of the dimensionless potential are known at the surface, the dimensionless potential and the gradient can be determined at all the distances from the surfaces until dis is reached. The dimensionless potential in the Stern layer is determined using the linear decrease of the dimensionless potential. The dimensionless potential in the diffuse layer is determined via the numcase chosen in the input parameters. This leads to a new value of the dimensionless potential at dis stepsizes from the surface, y(dis). This value is compared to y0. If the relative difference between the two values is more than the Rdif,3, the previous calculations will be repeated until the relative difference is less than Rdif,3. C.1. MATLAB SCRIPT FOR A SINGLE DISTANCE BETWEEN THE SURFACES 95

Figure C.1: A schematic representation of the numerical method for the determination of the dimensionless potential. The dimensionless potential at the surface is y(1). 96APPENDIX C. THE MATLAB SCRIPTS FOR NON-ISOLATED SURFACE SIMULATIONS

Table C.3: The diﬀerent Matlab scripts for multiple non-isolated surface simulations. The second column shows if the script takes into account the ions of water and the third column shows if the script takes into account the Stern layer.

The ions of water The Stern layer PB_multiple_same_wall_calculations_2.m yes no PB_multiple_same_wall_calculations_2_noHOH.m no no PB_multiple_same_wall_calculations_2_stern.m yes yes PB_multiple_same_wall_calculations_2_stern_noHOH.m no yes

The dimensionless potentials used in the last calculation will be the dimensionless potentials used in further calculations. The pressure between the plates can then be determined using the Maxwell stress and the osmotic pressure. Furthermore, the free energy can be determined via the direct route as described in chapter 4. Finally, an output struct is created. In this struct, the input parameters are stored in input. Furthermore, the determined gradient of the potential, the potential, the pressure, the degree of ionisation and the free energy and all the contributions are stored in values. The necessary values (the number of iterations, the step size, the number of steps, the Rdif,3, the Rdif,2 and the ∆y) are stored in nesval.

C.2 Matlab script for multiple distances between the surface

The simulations using the Poisson-Boltzmann model for non-isolated surfaces at multiple distances are performed using the Matlab scripts in table C.3. The table shows if the ions of water and/or the Stern layer is taken into account in the script. All the models are based on the governing equations shown in chapter 4. First, the input parameters of the isolated surface models will be discussed. Following, the diﬀerent parts of the script will be described.

C.2.1 The input parameters The diﬀerent input parameters for the non-isolated surface simulations at multiple distances are given in table C.4. The input parameters have to be inserted in the order of table C.4. The input parameters ds and dvdw can be omitted, when a model, which does not take into account the Stern layer, is used for the simulation of an isolated surface. Furthermore, if the parameter ds is used, the thickness of the Stern layer should be inserted using the amount of stepsizes. The input parameter nstep should also be inserted using the amount of stepsizes. The input parameter surface can be either 1 or 2. If 1 is used, the oxide surface is simulated as an acidic surface. If 2 is inserted, an amphoteric surface is simulated. The input parameter numcase can be either 1, 2 or 3. When 1 is inserted, the Euler method is used as the discretisation method, while changing to 2 or 3 the midpoint method or the third-order Runga Kutta method are used, respectively.

C.2.2 The outline of the script The script starts with the selection of the default values for nstep, stepsize and numcase if these parameters are not given as input. Following, a for-loop is entered. In this loop, the distance between the surfaces is varied. For each distance, the non-isolated surface simulation for a single distance is performed. The results of this simulation are stored. Furthermore, the pressure between the plates is integrated as depicted in chapter 4 using the midpoint method. The Van der Waals forces are also determined. The output struct of this C.2. MATLAB SCRIPT FOR MULTIPLE DISTANCES BETWEEN THE SURFACE 97

Table C.4: Input parameters for the Matlab script for multiple non-isolated surface simulations, in order.

Parameter Unit Default value ds The thickness of the Stern layer stepsize - dvdw The thickness of the Van der Waals Stern layer stepsize - c The concentration of the electrolyte mM - zp The zp -- zm The zm -- vp The vp -- vm The vm -- pH The pH of the electrolyte - - N The number of charged sites on the oxide surface mol m−2 - K The equilibrium constant of the protonation reaction mol L−1 - epsr The relative permittivity of the solvent - - T The temperature K - surface The kind of oxide surface used - - nstep The number of steps for determination of y stepsize 104 stepsize The step size used m 10−11 numcase The numerical model used - 3 script contains a group results, which contains the simulation results of the non-isolated surface simulations with the diﬀerent distances. Furthermore, the resulting values for the Van der Waals forces, the electrostatic force and the distance between the surface is stored in the part called values. 98APPENDIX C. THE MATLAB SCRIPTS FOR NON-ISOLATED SURFACE SIMULATIONS Appendix D

The Poisson-Boltzmann model for two non-isolated surfaces

The results of the simulations for the determination of the initial settings and the validation of the Poisson-Boltzmann model for non-isolated surface simulations could not all be presented in chapter 4. Therefore the results are partly presented in this appendix.

D.1 The initial Settings

D.1.1 The step size Figure D.1 shows the effect of the step size on the accuracy of the non-isolated surface simulations in a basic environment. All the figures show almost no difference between the results using different step sizes. Figure D.1.b, D.1.d, D.1.f and D.1.h show that there is a small difference between the simulations with a step size of 1 nm and the other simulations. All the simulations with a step size of 1 nm result in imaginary dimensionless potentials. Therefore, the step size should be smaller than 1 nm to obtain reproducible results.

D.1.2 The relative diﬀerence of the surface potential, Rdif,2

The effect of the relative difference of the surface potential, Rdif,2, on the accuracy of the determination of the dimensionless potential in a basic environment, is shown in figure D.2. All the results in the figure show that the relative difference has no effect on the dimensionless potential in a basic environment. At low concentration, there are some small differences, as shown in figure D.2.d and D.2.h. However, the effect is minimal.

D.1.3 The relative diﬀerence of y0, Rdif,3

Figure D.3 shows the effect of the relative difference of y0, Rdif,3, in a basic environment. The effect of Rdif,3 on the dimensionless potential is very small at high concentrations as shown in figure D.3.a, D.3.b, D.3.e and D.3.f. The effect of Rdif,3 becomes clear at lower concentrations. In figure D.3.d and D.3.h it is shown that only the simulations with a relative difference smaller than 0.1 % are similar. According to these results a relative difference of 0.01 % is sufficient to obtain reproducible results. It should be noted that the results with a Rdif,3 of 0.0001 % are imaginary for all cases with a high concentration. Felectrostatic The large effect of Rdif,3, suggests that Rdif,3 will also influence the R . The effect of Felectrostatic Rdif,3 is determined by determining the relative R for each simulation, with respect to the lowest Rdif,3 in which the simulation has no imaginary numbers. The results of these calculations

99 100APPENDIX D. THE POISSON-BOLTZMANN MODEL FOR TWO NON-ISOLATED SURFACES

(a) Silica, 0.1 M (b) Silica, 0.1 M, close up

(e) Aluminium hydroxide, 0.1 M (f) Aluminium hydroxide, 0.1 M, close up

(g) Aluminium hydroxide, 1 mM (h) Aluminium hydroxide, 1 mM, close up

Figure D.1: The effect of the step size on the accuracy of the non-isolated surface simulations in a basic environment, pH of 10 for silica and pH of 12 for aluminium hydroxide. The lengths given in the legend are the step sizes. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity of the solvent is 78. The used relative difference for the surface potential, Rdif,2, and the relative difference for y0, Rdif,3, are 0.01 % for the silica simulations and 0.001 % for the aluminium hydroxide simulations. The ions of water are taken into account. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. D.1. THE INITIAL SETTINGS 101

(a) Silica, 0.1 M (b) Silica, 0.1 M, close up

(e) Aluminium hydroxide, 0.1 M (f) Aluminium hydroxide, 0.1 M, close up

(g) Aluminium hydroxide, 1 mM (h) Aluminium hydroxide, 1 mM, close up

Figure D.2: The effect of the relative difference of the surface potential, Rdif,2, on the accuracy of the non-isolated surface simulations in a basic environment, pH of 10 for silica and pH of 12 for aluminium hydroxide. The percentages given in the legend are the relative differences. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity of the solvent is 78. The used step size is 0.01 nm. The used relative difference for y0, Rdif,3, is 0.01 % for the silica simulations and 0.001 % for the aluminium hydroxide simulations. The ions of water are taken into account. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. 102APPENDIX D. THE POISSON-BOLTZMANN MODEL FOR TWO NON-ISOLATED SURFACES

(a) Silica, 0.1 M (b) Silica, 0.1 M, close up

(e) Aluminium hydroxide, 0.1 M (f) Aluminium hydroxide, 0.1 M, close up

(g) Aluminium hydroxide, 1 mM (h) Aluminium hydroxide, 1 mM, close up

Figure D.3: The effect of the relative difference of y0, Rdif,3, on the accuracy of the non-isolated surface simulations in a basic environment, pH of 10 for silica and pH of 12 for aluminium hydroxide. The percentages given in the legend are the relative differences. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity of the solvent is 78. The used step size is 0.01 nm. The used relative difference for the surface potential, Rdif,2, is 0.01 % for the silica simulations and 0.001 % for the aluminium hydroxide simulations. The ions of water are taken into account. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density, N, at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. D.2. THE VALIDATION 103

Felectrostatic Table D.1: The eﬀect of Rdif,3 on the calculated R . The eﬀect of Rdif,3 is determined by Felectrostatic determining the relative R for each simulation, with respect to the lowest Rdif,3 in which the simulation has no imaginary numbers.

Rdif,3 Silica 0.1 M Silica 1 mM Aluminium hydroxide 0.1 M Aluminium hydroxide 1 mM 10 % 17.1 % 3.19 % 16.9 % 14.0 % 1 % 2.97 % 1.69 % 2.85 % 2.32 % 0.1 % 0.40 % 0.20 % 0.31 % 0.25 % 0.01 % 0.03 % 0.02 % 0.03 % 0.02 % 0.001 % - 0.00 % - 0.00 % 0.0001 % - - - -

are shown in table D.1. The table shows that the eﬀect of Rdif,3 is less than 0.04 % when Rdif,3 is smaller than 0.1 %. Thus this shows that Rdif,3 should be smaller than 0.1 %.

D.1.4 The ions in the water The effect of the ions of water on the dimensionless potential in a basic environment is shown in figure D.4. Figure D.4.a shows that there is no effect of the ions of water for a silica surface when the concentration of the electrolyte is higher than the concentration of the ions of water. When the concentration of the electrolyte is decreased, figure D.4.b, the ions of water cause a decrease in the dimensionless potential. For aluminium hydroxide, the ions of water already influence the dimensionless potential at higher concentrations. However, the pH of these simulations is higher than in the silica simulations. The concentration of the ions in water is comparable to the concentration of the electrolyte. This explains the influence at high concentrations.

D.2 The validation D.2.1 The pressure The results in figure D.5 show the pressure between the plates in a basic environment. All the figures show a constant pressure between the surfaces. Figure D.6 shows that the pressure is also constant at different distances.

D.2.2 The Stern layer Figure D.7 shows the effect of the introduction of the Stern layer on the dimensionless potential and the gradient of the dimensionless potential in a basic environment. Figure D.7.a and D.7.c show a smooth function for the dimensionless potential of acidic and amphoteric surfaces in a basic environment. Furthermore, the gradient of the dimensionless potential is linear in the Stern layer, as shown in figure D.7.b and D.7.d. In the diffuse layer, ideal behaviour is obtained. Figure D.8 shows the effect of the Stern layer on the pressure in a basic environment. The pressure is increased drastically within the Stern layer. In the diffuse layer the pressure is still constant. The introduction of the Stern layer does not influence the calculation of the electrostatic free energy via the indirect route. Therefore, the validation of the changes in the direct route can be done comparing them with the indirect route. The parity plots obtained for the simulations in an basic environment are shown in figure D.9. Both graphs show that the results of the direct and the indirect route are similar. 104APPENDIX D. THE POISSON-BOLTZMANN MODEL FOR TWO NON-ISOLATED SURFACES

(a) Silica, 0.1 M (b) Silica, 1 mM

Figure D.4: The effect of the ions in the water on the non-isolated surface simulations in a basic environment, pH of 10 for silica and pH of 12 for aluminium hydroxide. The blue lines are the simulation without water ions and the red lines are the simulations with water ions. The concentration of the used 1:1-electrolyte solution is stated in the name of the sub figure. The temperature is 298 K and the relative permittivity is 78. The step sizes are 0.01 nm and the relative differences, Rdif,2 and Rdif,3, are 0.01 % for silica and 0.001 % for aluminium hydroxide. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density at the silica surface, N, is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2. D.2. THE VALIDATION 105

(a) Silica, 0.1 M (b) Silica, 1 mM

Figure D.5: The pressure at diﬀerent distances from the surface, which are 6 nm apart, in a basic environment, pH of 10 for the silica surfaces and pH of 12. The temperature is 298 K and the relative permittivity is 78. The kind of surface and the concentration of the 1:1-electrolyte are stated in the name of the ﬁgure. The used pK in this simulations for aluminium hydroxide is 10 and for silica is 7.5 [9, 11]. The density of the sites, N, at the silica surface is 8 nm−2 and 10 nm−2 at the aluminium hydroxide surface. 106APPENDIX D. THE POISSON-BOLTZMANN MODEL FOR TWO NON-ISOLATED SURFACES

(a) Silica, 0.2 nm (b) Silica, 10 nm

Figure D.6: The pressure at varying distances from the surface, in an acidic environment, pH of 4. The temperature is 298 K and the relative permittivity is 78. The kind of surface and the distance between the surfaces are stated in the name of the ﬁgure. The concentration of the 1:1-electrolyte is 0.1 M. The used pK in this simulations for aluminium hydroxide is 10 and for silica is 7.5 [9, 11]. The density of the sites, N, at the silica surface is 8 nm−2 and 10 nm−2 at the aluminium hydroxide surface D.2. THE VALIDATION 107

(a) Dimensionless potential, Silica (b) ∇y, Silica

Figure D.7: The eﬀect of the introduction of the Stern layer on the dimensionless potential and the gradient of the dimensionless potential, ∇y, in a basic environment, pH of 10 for silica and pH of 12 for aluminium hydroxide. The red doted line represents the Stern layer, while the blue line represents the dimensionless potential or ∇y. The temperature is 298 K, the concentration of the 1:1-electrolyte is 0.1 M and the relative permittivity is 78. The kind of surface used is stated in the name of the ﬁgure. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2 [6, 9, 11]. The capacity of the Stern layer is 1 F m−2.

(a) Silica (b) Aluminium hydroxide

Figure D.8: The eﬀect of the introduction of the Stern layer on the pressure in an acidic environment, pH of 10 for silica and pH of 12 for aluminium hydroxide. The red doted line represents the Stern layer, while the blue line represents the pressure. The temperature is 298 K, the concentration of the 1:1-electrolyte is 0.1 M and the relative permittivity is 78. The kind of surface used is stated in the name of the ﬁgure. The pK is 7.5 and 10 for silica and aluminium hydroxide, respectively. The charge density at the silica surface is 8 nm−2 and for the aluminium hydroxide surface 10 nm−2 [6, 9, 11]. The capacity of the Stern layer is 1 F m−2. 108APPENDIX D. THE POISSON-BOLTZMANN MODEL FOR TWO NON-ISOLATED SURFACES

(a) n:n (b) n:m

Figure D.9: Parity plots of the electrostatic free energy determined via the indirect route and via the direct route, for silica surfaces in a basic environment, pH of 10. The temperature is 298 K, the concentration of the electrolyte is 0.1 M and the relative permittivity is 78. The use kind of electrolyte solution is stated in the legend of the ﬁgures. The used pK in the simulations for silica is 7.5 [9, 11]. The density of the sites at the surface is 1 nm−2. The capacity of the Stern layer is 1 F m−2. Appendix E

Simulation results for isolated surfaces

The numerical results for the eﬀect of the concentration of the electrolyte, the kind of electrolyte, the charge density at the surface and the temperature could not all be presented in chapter E. Therefore, the results of the numerical simulations in a basic environment are presented in this appendix.

E.1 The acidic surface

The base case for the simulations in an basic environment is given in table E.1. In the simulations, the electrolyte concentration, the kind of electrolyte, the charge density at the surface and temperature are varied in the ranges shown in table E.1.

E.1.1 The effect of the electrolyte solution In figure E.1.a, the influence of the concentration of the electrolyte solution is shown. The figure shows that the concentration of the electrolyte has a large influence on the potential. When the concentration of the electrolyte is increased, the thickness of the diffuse layer is decreased. Further- more, the figure shows that the dimensionless potential at the surface decreases with increasing concentration. Figure E.1.b to E.1.d shows the effect of the kind of electrolyte used. All the figures show a decrease in the thickness of the diffuse layer with an increase in the n of the n:n-electrolyte, 1:n

Table E.1: The base case of the simulations for isolated surfaces of the acidic surface in a basic environment. The ranges in which the parameters are varied during the simulations are also given.

Base case Ranges 2 −1 5 The concentration of electrolyte (c∞) in mM 10 10 −10 The valency of the positive ions (zp) 1 1 - 5 The valency of the negative ions (zm) 1 1 - 5 The pH-pK 2.5 - The pK 7.5 - The charge density at the surface (N) in nm−2 8 1 - 12 The temperature (T) in K 298 278 - 313 The relative permittivity of the solvent (εr) 78 65 - 85 The thickness Stern layer in nm 0.69 - The capacity of the Stern layer in F m−1 1.00 -

109 110 APPENDIX E. SIMULATION RESULTS FOR ISOLATED SURFACES

(a) c∞ (b) n:n-electrolyte

Figure E.1: The effect of electrolyte solution on the dimensionless potential for an acidic surface in a basic environment, pH of 10. In figure E.1.a, the effect of the concentration of electrolyte, c∞, is shown. Figure E.1.b to E.1.d show the effect of the kind of electrolyte. For these simulations, the base case of table E.1 is used. c∞ (in figure E.1.a) or the kind of electrolyte (in figure E.1.b to E.1.d) is adapted to c∞ or the kind of electrolyte presented in the legend.

Figure E.2: The eﬀect of the relative permittivity, εr, on the dimensionless potential for an acidic surface, in a basic environment, pH of 10. For these simulations, the base case of table E.1 is used. εr is adapted to the εr presented in the legend. E.2. THE AMPHOTERIC SURFACE 111

(a) N (b) T

Figure E.3: The effect of the charge density, N, at the surface and the temperature, T, on the dimensionless potential for an acidic surface in a basic environment, pH of 10. In figure E.3.a, the effect of N is shown. Figure E.3.b shows the effect of T. For these simulations, the base case of table E.1 is used. N (in figure E.3.a) or T (in figure E.3.b) is adapted to N or T presented in the legend. electrolyte or n:1-electrolyte. The results for the n:n-electrolyte and the n:1-electrolyte are similar. The results for 1:n-electrolyte show a slightly smaller effect on the thickness of the diffuse layer. Furthermore, these results do not show a real difference in the potential at the surface.

E.1.2 The effect of relative permittivity of the solvent Figure E.2 shows the effect of the relative permittivity of the solvent on the dimensionless potential. There is a minimal effect on the dimensionless potential of maximal 3%. An increase in the relative permittivity causes an increase of the potential at the wall, while the gradient of the dimensionless potential decreases. This causes a crossing of the results with different relative permittivities.

E.1.3 The effect of the charge density at the surface Figure E.3.a shows the effect of the charge density of the surface in a basic environment. The figure shows that the surface potential decreases when decreasing the surface charge density. Due to a decrease in the charge density, the effect of the surface charge on the solution will be reduced. This results in a smaller diffuse layer, which is shown in figure E.3.a

E.1.4 The effect of the temperature Figure E.3.b shows the effect of the temperature on the diffuse layer. The temperature has no direct effect on the dimensionless potential. The differences between the dimensionless potential of 278 K and 308 K is only 1 %.

E.2 The amphoteric surface

The base case for the simulations in a basic environment for amphoteric surfaces is given in table E.2. In the simulations, the electrolyte concentration, the kind of electrolyte, the charge density at the surface and temperature are varied in the ranges shown in table E.2.

E.2.1 The eﬀect of the electrolyte Figure E.4.a shows the eﬀect of concentration on the dimensionless potential. An increase in the concentration causes the gradient of the dimensionless potential to increase substantially. However, the increase of the concentration only slightly decreases the dimensionless potential at the surface. 112 APPENDIX E. SIMULATION RESULTS FOR ISOLATED SURFACES

Table E.2: The base case of the simulations for isolated surfaces of the amphoteric surface in a basic environment. The ranges in which the parameters are varied during the simulations are also given.

Base case Ranges 2 −1 5 The concentration of electrolyte (c∞) in mM 10 10 −10 The valency of the positive ions (zp) 1 1 - 5 The valency of the negative ions (zm) 1 1 - 5 The pH-pK 2 - The pK 10 - The charge density at the surface (N) in nm−2 10 1 - 12 The temperature (T) in K 298 278 - 313 The relative permittivity of the solvent (εr) 78 65 - 85 The thickness Stern layer in nm 0.69 - The capacity of the Stern layer in F m−1 1.00 -

(a) c∞ (b) n:n

Figure E.4: The effect of the electrolyte solution on the dimensionless potential for an amphoteric surface in a basic environment, pH of 12. In figure E.4.a, the effect of the concentration of electrolyte, c∞, is shown. Figure E.4.b to E.4.d show the effect of the kind of electrolyte. For these simulations, the base case of table E.2 is used. c∞ (in figure E.4.a) or the kind of electrolyte (in figure E.4.b to E.4.d) is adapted to c∞ or the kind of electrolyte presented in the legend. E.2. THE AMPHOTERIC SURFACE 113

Figure E.5: The eﬀect of the relative permittivity, εr, on the dimensionless potential for an amphoteric surface in a basic environment, pH of 12. For these simulations, the base case of table E.2 is used. εr is adapted to the εr presented in the legend.

(a) N (b) T

Figure E.6: The effect of the charge density, N, at the surface and the temperature, T, on the dimensionless potential for an amphoteric surface in a basic environment, pH of 12. In figure E.6.a, the effect of N is shown. Figure E.6.b shows the effect of T. For these simulations, the base case of table E.2 is used. N (in figure E.6.a) or T (in figure E.6.b) is adapted to N or T presented in the legend. 114 APPENDIX E. SIMULATION RESULTS FOR ISOLATED SURFACES

In figure E.4.b to E.4.d, the effect of the kind of electrolyte is shown. The increase of the charge of the ions, the n, decreases the thickness of the surface charge. The surface potential is again only slightly influenced by the increase of the charge. Furthermore, the results of the n:n-electrolyte and the n:1-electrolyte are similar, while the screening effect of the 1:n-electrolyte is less.

E.2.2 The effect of relative permittivity of the solvent Figure E.5 shows the effect of the relative permittivity of the solvent on the dimensionless potential in a basic environment. The difference between the results using the highest and the lowest relative permittivity is maximally 4 %. The results are thus comparable. Furthermore, upon close examination, the dimensionless potential at the wall is decreased with increasing relative permittivity. However, the gradient of the dimensionless potential is decreased when the relative permittivity is increased.

E.2.3 The effect of the charge density at the surface In figure E.6.a, the effect of the charge density on the dimensionless potential in a basic environment is shown. The figure shows a large effect of the charge density on both the diffuse layer and the dimensionless potential. Due to a decrease in the charge density of the surface, the surface potential will be lower. This leads to a decrease in the surface charge.

E.2.4 The effect of the temperature Figure E.6.b shows the effect of the temperature on the dimensionless potential. The increase of 30 K does not really influence the dimensionless potential. The difference between the results is only 0.2 %.