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 com- puting 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 dimen- sions, 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 nano- sheets. 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.