Equilibrium Defects Real crystals are never perfect, they always contain a considerable density of defects and imperfections that affect their physical, chemical, mechanical and electronic properties. The existence of defects also plays an important role in various technological processes. In this learning object, you will see what factor determines how many defects will be there in a material. Subject: MEMS Mentor: Prof. Gururajan Authors:

Learning Objectives After interacting with this Learning Object, the learner will be able to: Calculate the vacancy concentration for any metal given the values of ∆H v and “T” Appreciate the presence of defects that gives good properties to the material

Definitions: 1) Equilibrium is the state of minimum energy. 2) Gibbs free energy: The Gibbs free energy of a system is defined by the equation G = H – TS where H is the enthalpy, T the absolute temperature and S the entropy of the system. 3) Vacancy: A vacancy is a lattice position that is vacant because the atom is missing. 4) Configuration: The structural arrangement of atoms in a compound or molecule. The Configurational entropy of a system is given by S = k ln w where k is the Boltzmann’s constant and w is the number of different configurations of equal potential energy in which the system can exist. S = k ln where N = total number of sites, n v is the number of vacant sites ! ! !

Assumptions: 1)Stirling’s approximation: ln N! = N ln N – N 2) Vacancy concentration should be small enough that there is no interaction between two vacancies.

Derivation Let denote equilibrium concentration i.e. this concentration give minimum free Energy. Enthalpy of the solid is directly proportional to the number of vacancies added. Creating vacancies means breaking of bonds, this costs the system its internal energy Therefore, ∆H ≈ ∆H v X v where X v is the mole fraction of vacancies and ∆H v is the increase in enthalpy per mole of vacancies added and ∆H is the total increase. Each vacancy causes an increase of ∆H v /N a where N a is Avogadro’s number. There are 2 contributions to the increase in entropy - ∆S configurational which is due to the fact that there are many different ways in which the vacancies can be distributed in the lattice ∆S thermal due to the changes in vibrational frequencies of atoms around the vacancies. The largest contribution is by increase in configurational entropy and is given by ∆S configurational = ― R [X v ln X v + (1 - X v ) ln (1 - X v ) ] ( Refer to Assumption 1 ) ‏ ∆S thermal = X v ∆S v

Derivation (contd..) ‏ Thus, the total change in entropy is given by - ∆S = ∆S thermal + ∆S configurational ∆S = X v ∆S v ― R [X v ln X v + (1 - X v ) ln (1 - X v ) ] where ∆S v is the increase in the vibrational entropy of the crystal per mole of vacancies added. The molar free energy of the crystal containing X v mole of vacancies is therefore given by G = G A + ∆G G = G A + ∆H v X v – T ∆S v X v + R T [X v ln X v + (1 - X v ) ln (1 - X v ) ] …..Eqn. (i) ‏ With time, the number of vacancies will adjust so as to reduce G to a minimum. The equilibrium concentration of vacancies is given by the condition = 0 Differentiating Eqn (i) and making the approximation X v << 1 ( refer to assumption 2 ) gives ∆H v – T ∆S v + R T ln = 0

Derivation (contd..) ‏ Therefore the expression for is given by = exp. exp ………..eqn. (2) ‏ On R.H.S, The value of (∆S v /R) is between 1 and 2 hence its exponential i.e. the first term is a constant ≈ 3 and is independent of T ; ΔH v is close to 1 eV. Thus, the equilibrium vacancy concentration is finite and it increases exponentially with temperature. Substituting ∆ G v = ∆H v – T ∆S v in the above equation we get = exp

Examples of defects The electrical behavior of semiconductors, for example, is largely controlled by crystal imperfections. Copper or silver is added to gold to make gold harder and stronger, this is a defect added to gold. Addition of copper or silver creates a point defect by substituting gold atom by copper/ silver atom.

Graph: Variation of the free energy of a crystal with increase in the number of vacant lattice sites XvXv XvXv XvXv

Master Layout 1

Refer to master layout 1 Step No: 1 Instructions for the animatorAudio narration/ text to be displayed Display the master layoutThis is a 8 by 8 lattice in which blue colour spheres represent atoms and the white colour sphere represents vacant site. Is this the only possibility for a vacant site? Images copyrighted

Refer to master layout 1 Step No: 2 Instructions for the animator Audio narration/ text to be displayed Display the master layoutThe number of configurations of equal potential energy for the system is given by where N = total number of sites and n v is the number of vacant sites. What will be the number of configurations in this case? ! ! !

Refer to master layout 1 Step No: 3 Interact -ivity type Instructions for the user Instructions to the animator Audio narration Output/ Feedback Input box (multiple attempts) ‏ Refer to the formula and Type the answer in the box. Display the master layout and boxes as above. What will be the number of possible configurations in this case? The correct answer is 64. If user types in the wrong ans. display pop up – “Observe that N = 64 and n v = 1. Try again. ” Type your answer here ! ! ! Number of configurations =

Refer to master layout 1 Step No: 4 Instructions for the animatorAudio narration Display the master layout. Show the white circle moving to different places. When the white circle moves from one point to another, show a blue circle in the first point (which it leaves) ‏ There should be only one white circle throughout the lattice. Of these possible configurations we want to know for which configuration the system will have minimum free energy. Can this system have more than one defect in equilibrium?

Master Layout 2 Variation of the enthalpy of the system with increase in the number of vacant lattice sites Free energy XvXv

Refer to master layout 2 Step No: 5 Instructions for the animator Audio narration Display the master layout. Show the pink line appearing from bottom (i.e. zero) to up. The graph represents Gibbs free energy for the number of vacancy concentration. Observe that the enthalpy increases linearly with increase in number of vacancy concentrations. Creating vacancies requires breaking of bonds which costs the system its internal energy.

XvXv XvXv Master Layout 3 Variation of the free energy of a crystal with increase in the number of vacant lattice sites XvXv Free energy XvXv Equilibrium

Refer to master layout 3 animation Step No: 6 Instructions for the animator Audio narration Display the animation as done in previous slide. The vacancy concentration increases exponentially with temperature. The equilibrium point here corresponds to the the number of vacancies (i.e. vacancy concentration) for which the system has the minimum free energy. Refer to the derivation of vacancy concentration to find out how you can calculate vacancy concentration for any material.

Want to know more… (Further Reading) ‏ Graph (for reference) ‏ Animation Area Test your understanding (questionnaire) ‏ Lets Learn! Lets Sum up (summary) ‏ Instructions/ Working area Radio buttons (if any)/Drop down (if any) ‏ Interactivity options Sliders(IO1) ‏ / Input Boxes(IO2) ‏ /Drop down(IO3) ‏ (if any) ‏ Play/pauseRestart Output result of interactivity (if any) ‏ What will you learn Credits Definitions Examples of defects

Questionnaire MetalMelting point ( 0 C) ‏ H v (eV) ‏ Al Ag Cu Au Ni Pt Refer to the table given above and calculate the vacancy concentration of each metal at C its melting point 10 0 C below its melting point Answers to be provided Reference - Physical Metallurgy Principles by Reza Abbaschian, Lara Abbaschian, Robert E. Reed-Hill

Questionnaire NPTEL link will be provided by Prof. Gururajan

Links for further reading Books: 1) Raghavan, V., Materials Science and Engineering, a first course, Second edition, Prentice Hall of India, New Delhi 2) Jena, A. K., Chaturvdei, M.C., (1992), Phase transformation in materials, Englewoods Cliffs: Prentice Hall 3) Porter, D., Easterling, K., Phase Transformation in metals and alloys, 3 rd edition Weblinks: NPTEL link will be provided by Prof. Gururajan

Summary A vacancy is a lattice position that is vacant because the atom is missing. Equilibrium vacancy concentration is number of vacancies for which the system has minimum free energy. The internal energy increases linearly with vacancy concentration. There are two contributions to the increase in entropy due to creation of a vacancy site - – ∆S configurational which is due to the fact that there are many different ways in which the vacancies can be distributed in the lattice – ∆S thermal due to the changes in vibrational frequencies of atoms around the vacancies. The vacancy concentration increases exponentially with increase in temperature.