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Module 3: Defects, Diffusion and Conduction in Ceramics Introduction

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Module 3: Defects, Diffusion and Conduction in Ceramics Introduction Objectives_template Module 3: Defects, Diffusion and Conduction in Ceramics Introduction In this module, we will discuss about migration of the defects which happens via an atomistic process called as diffusion. Diffusivity of species in the materials is also related to their physical properties such as electrical conductivity and mobility via Nernst-Einestein relation which we shall derive. As we shall also see, the conductivity in ceramics is a sum of ionic and electronic conductivity and the ratio of two determines the applicability of ceramic materials for applications. Subsequently framework will be established for understanding the temperature dependence of conductivity via a simple atomistic model leading to the same conclusions as predicted by the diffusivity model. Subsequently we will look into the conduction in glasses and look at some examples of fast ion conductors, material of importance for a variety of applications. Presence of charged defects in ceramics also means the existence of electrical potential gradients, in addition to the chemical gradient, which results in a unified equation for electrochemical potential. Finally, we will look at a few important applications for conducting ceramic materials. The Module contains: Diffusion Diffusion Kinetics Examples of Diffusion in Ceramics Mobility and Diffusivity Analogue to the Electrical Properties Conduction in Ceramics vis-à-vis metallic conductors: General Information Ionic Conduction: Basic Facts Ionic and Electronic Conductivity Characteristics of Ionic Conduction Theory of Ionic Conduction Conduction in Glasses Fast Ion Conductors Examples of Ionic Conduction Electrochemical Potential Nernst Equation and Application of Ionic Conductors Examples of Ionic Conductors in Engineering Applications Summary Suggested Reading: file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_1.htm[5/25/2012 3:03:41 PM] Objectives_template Physical Ceramics: Principles for Ceramic Science and Engineering, Y.-M. Chiang, D. P. Birnie, and W. D. Kingery, Wiley-VCH Principles of Electronic Ceramics, by L. L. Hench and J. K. West, Wiley Electroceramics: Materials, Properties, Applications, by A. J. Moulson and J. M. Herbert, Wiley file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_1.htm[5/25/2012 3:03:41 PM] Objectives_template Module 3: Defects, Diffusion and Conduction in Ceramics Diffusion 3.1 Diffusion Diffusion causes changes in the microstructures to take place in processes such as sintering, creep deformation, grain growth etc. Diffusion is also related to transport of defects or electronic charge carriers giving rise to electrical conduction in ceramics. Ionic conductors are used in variety of applications such as chemical and gas sensors, solid electrolytes and fuel cell. For example, an oxygen sensor made of ceramic ZrO2 is used in automobiles to optimize the fuel/air ratio in the engines. Atomic diffusion rates and electrical conductivity are largely governed by defect types and their concentration where concentration is a function of temperature, partial pressure of oxygen or pO2, and composition. file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_2.htm[5/25/2012 3:03:41 PM] Objectives_template Module 3: Defects, Diffusion and Diffusion in Ceramics Diffusion Kinetics 3.2 Diffusion Kinetics Typically diffusion is explained on the basis of compositional gradients in an alloy which act as driving force for diffusion. Thermodynamically speaking, this amounts to gradient in the chemical potential which drives the migration of species from regions of higher chemical potential to lower chemical potential so that system reaches a chemical equilibrium. The atomic flux as a result of driving force is expressed in terms of chemical composition gradient, also called as Fick’s law(s). These laws are briefly explained below. For detailed discussion on diffusion, readers are referred to standard text books on diffusion.1,2 3.2.1 Fick's First Law of Diffusion It states that atomic flux, under steady-state conditions, is proportional to the concentration gradient. It can be stated as (3.1) where J is the diffusion flux with units moles/cm2-s, and basically means the amount of material passing through a unit area per unit time; D is the proportionality constant, called as diffusion coefficient or diffusivity in cm2/s; x is the position in cm; and c is the concentration in cm3 . The negative sign on the R.H.S. indicates that diffusion takes place from regions of higher concentration to lower concentration i.e. down the concentration gradient. Diffusivity is a temperature dependent parameter and is expressed as D = D0 exp (-Q/kT) where Q is the activation energy, k is 2 Boltzmann's constant and D0 is the pre-exponential factor in cm /s. 3.2.2 Fick's Second Law of Diffusion : Strictly speaking it is not a law, but rather a derivation of the first law itself. It predicts how the concentration changes as a function of time under non-steady state conditions . It can be derived from Fick's first law easily. For further discussions, one can refer to “Diffusion in Solids”, a classic book written by Paul Shewmon and published by Wiley or other NPTEL courses related to Phase Transformations and Diffusion. (3.2) file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_3.htm[5/25/2012 3:03:41 PM] Objectives_template where t is the time in seconds. Other terms are defined above. 1Diffusion in Solids, Paul Shewmon, Wiley 2Diffusion in Solids, Martin Glicksman, Wiley- Interscience file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_3.htm[5/25/2012 3:03:41 PM] Objectives_template Module 3: Defects and Diffusion in Ceramics Diffusion Kinetics 3.2.3 Diffusivity: A Simple Model Figure 3. 1 Schematic of the planes of atoms with arrows showing the cross- movement of species As shown in Figure (3.1), a schematic diagram shows atomic planes, illustrating 1-D diffusion of species across the planes . Flux from position (1) to (2) is written as (3.3) where n1 is no. of atoms at position (1) and G is the jump frequency i.e number of atoms jumping per second (atoms/s) Similarly, Flux from plane (2) to (1) is expressed as (3.4) -1 where n2 is the number of atoms at (2) and G is the jump frequency in s . in both the above expressions, factor ½ is there because of equal probability of jump in +x and -x directions. Now, the net flux, J, can be calculated as (3.5) Concentration is defined as file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_4.htm[5/25/2012 3:03:41 PM] Objectives_template (3.6) if area is considered as unit area (=1) and λ is the distance between two atomic planes. Figure 3. 2 Schematic diagram showing concentration gradient between two planes of atoms Concentration gradient can be written as (note the minus sign) (3.7) Hence, flux can now be expressed as where D = ½ λ2 t with unit cm2/s in 1-D and can easily show to become D = 1/6 λ2 t in a 3-D cubic co-ordination scenario. In general, diffusivity can be expressed as (3.9) where γ is governed by the possible number of jumps at an instant and λ is the jump distance and is governed by the atomic configuration and crystal structure. file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_4.htm[5/25/2012 3:03:41 PM] Objectives_template Module 3: Defects, Diffusion and Conduction in Ceramics Diffusion Kinetics 3.2.4 Temperature Dependence of Diffusivity Now, equation (3.9) can further be modified by replacing the jump frequency, G , which, by Boltzman statistics, is defined as (3.10) where ν is the vibration frequency in s-1, ΔG* is the activation energy of migration in J and k is Boltzmann Constant (J/K). Further, ΔG* can be written as (3.11) where ΔH* is the enthalpy of migration and ΔS* is the associated entropy change. Now, substitution of equation (3.10) in equation (3.11) leads to OR or where pre-exponential factor . Equation (3.12) explains the thermally activated nature of diffusivity showing an exponential temperature dependence resulting in significant increase in the diffusivity upon increasing the temperature. file:///C|/Documents%20and%20Settings/iitkrana1/Desktop/new_electroceramics_14may,2012/lecture13/13_5.html[5/25/2012 3:03:41 PM] Objectives_template Module 3: Defects, Diffusion and Conduction in Ceramics Examples of Diffusion in Ceramics 3.3 Examples of Diffusion in Ceramics 3.3.1 Diffusion in lightly doped NaCl NaCl is often used a conducting electrolyte and it a good case for proving an example. Consider the ease of NaCl containing small amounts of CdCl lIn such scenario, for each Cd ion, Cd occupying Na 2 site with an extra positive charge and a sodium vacancy with one negative charge is created according to the following defect reaction: • x CdCl → Cd + 2Cl + V ' (3.13) 2 Na Cl Na In addition, NaCl will also have certain intrinsic sodium and chlorine vacancy concentration (VNa' and • VCl ) due to Schottky dissociation, depending on the temperature. In such a scenario, the diffusivity of sodium ions is governed by vacancy diffusion and can be worked out as (3.14) where ΔGNa* is the migration free energy for sodium vacancies and VNa' is the sodium vacancy concentration. This diffusivity of vacancy is dependent on vacancy’s concentration which is related to the dopant concentration. However, the diffusivity dependence
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