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What Does Order and Disorder Mean? Order and Disorder designate the presence or absence of some symmetry or correlation in a many-particle system. In physics systems typically are ordered at low temperatures; upon heating, they undergo one or several phase transitions into less ordered states. Examples for such an order-disorder transition are: The melting of ice: solid-liquid transition, loss of crystalline order The demagnetization of iron by heating above the Curie temperature: ferromagnetic-paramagnetic transition, loss of magnetic order.
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The degree of freedom that is ordered or disordered can be translational (crystalline ordering), rotational (ferroelectric ordering), or a spin state (magnetic ordering). Depending on how the correlations decay with distance, one speaks of long-range order or short-range order. Long-range order (LRO): Short-range order (SRO): Differentiation „” of probalilities of Tendency for A-(B-) atoms to be preferencially surrounded by B- or A-atoms particular sublattice sites being occupied (correlation functions). by particular atoms. If a disordered state is not in thermodynamic equilibrium, it is quenched disorder. For instance, a glass is obtained by quenching (supercooling) a liquid. Some other quenched states are called spin glass, orientational glass and the opposite of quenched disorder is annealed disorder.
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Ordered Phases If the atoms in a substitutional solid solution, are randomly arranged each atom position is equivalent and the probability that any given site in the lattice will contain an A atom will be equal to the fraction of A atoms in the solution XA , similarly XB for the B atoms. In such solutions PAB, the number of A-B bonds, PAB = Na z XAXB (bonds mol-1) where Na Avogadros number, and z is the number of bonds per atom. Positionally ordered A B Probability of occupation: A 50% B 50% Probabilistically ordered PROBABILISTIC OCCUPATION
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is the difference between A-B bond energy and the average of A-A and B-B . = 0 , ∆H =0 - Ideal Solution. < 0 the atoms in the solution will prefer to be surrounded by atoms of mix the opposite type and this will increase P . Whereas > 0, P Will tend to be less than in a random solution. AB AB ∆Hmix= Ω Xa Xb Ω= Na z Ω <0 the no of A-B bonds is greater than this, the solution is said to contain short range order (SRO) s = PAB – PAB(random) PAB(max) –PAB(random) PAB(max)- max no. of bonds possible PAB(random)- no. of bonds in random solution.
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Cu-Au alloys Cu and Au – FCC structured and totally miscible.At Higher Temp: Cu or Au can occupy any site and the lattice can beconsidered as fcc with a “random” atom at each lattice( Fig 1.20 a).At Lower Temp: XCu=XAu =0.5 i.e 50/50 Cu/ Au mixture. (orderedstructure) (Fig 1.20 b) CuAu Superlattice. Cu3 Au Superlatice.
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The necessary condition for the formation of a superlattice, i.e. an ordered solid solution, isthat dissimilar atoms must attract each other more than similar atoms. In addition, the alloymust exist at or near a composition which can be expressed by a simple formula such as AB,A3B or AB3. The following are common structures: Cu3Au CuAu
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Disorder to Order Transformation An atomic disorder to order transformation is a change of phase. It entails a change in the crystallographic symmetry of the high temperature, disordered phase, usually to a less symmetric low temperature atomically ordered phase. This can be understood from a basic equation of phase equilibria in the solid state, namely the definition of the Gibbs Free Energy: G = H - TS where G is the Gibbs free energy ,H is the enthalpy, S is the entropy of the material Order Parameter Disorder OrderThermodynamic parameter, called the order parameter, ƞ which can be used as a measureof the extent of the transformation.This order parameter ƞ, is one which has an equilibrium value, so that we can alwayswrite: G 0 T , P since G, the Gibbs free energy is a minimum at equilibrium
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G = H - TS At constant T and P the system in equilibrium will be the one with the lowest Gibbs Free EnergyAt high temperatures the TS term dominates the phaseequilibria and the equilibrium phase is more “disordered”(higher entropy) than the low temperature equilibrium phase. ↑ T Disordered G H T S ↓ T OrderedExamples: Liquid to Solid Disorder to OrderIn both cases the high temperature equilibrium phase is more“disordered” than the low temperature “ordered” phase.
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A Phase Diagram Which Includes a Typical Disorder to Order Transformation
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Ordering Transformations: What Happens when Ordered Single Crystal( CuZn or Cu3Au) is Heated from low temperatures to above the disordering temperature. Here We quantify the degree of order in crystal by defining Long Range Order parameter L, rA X A L 1 X A rA → probability that A sublattice is occupied with the right atom XA → mole fraction of A in the alloy L → Long Range Order L=1 : Fully Ordered Alloy (all Atoms occupy their correct sites). L=0 : Completely Random Distribution
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At 0K : Min Free Energy by choosing Highly order Arrangement (L=1) : Lowest internal energy. Configurational Entropy =0; At higher temp: Min free energy state will contain some disorder. (diffusion) Entropy effects become increasingly important with rising temperature so L continuously decreases until above some critical ( Tc) L=0. Results for CuZn and Cu3Au Superlattices Second Order 1 Long Range Order ~ First Order Short Range Order CuZn Type 0 T TC No Sudden change in order at Tc : U and H Substantial change in order at Tc : will be continous (U ,H )disorder state> (U,H) Ordered StateAbove Tc it is impossible to distinguish separate sublattices extending over long distances and L=0;Solid solutions which have a negative enthalpy of mixing (Hmix < 0) prefer unlike nearestneighbours → show tendency for ordering phases at low temperature.
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Two Mechanism to create disordered superlattice ordered superlattice Continuous increase in SRO Mechanism Nucleation and Growth1. Continuous increase in SRO Local rearrangements occurring homogenously throughout the crystal which finally leads to LRO. 2. Nucleation and Growth Due to an energy barrier to the formation of ordered domain, transformation takes place with the help of nucleation and growth.
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The two are equivalent to spinodal decomposition and precipitation as mechanisms for formation of coherent zones in alloys with ∆Hmix >0 . 1st mechanism: Second order transformations or at very high supercooling below Tc. 2nd mechanism: more common.Atoms can order on either the A or B sublattice , the independently nucleated domains will be “out of phase”Domains Subsiquintly grow together a boundary will form (antiphase domain boundary (APB)) across whichatoms will have wrong neighbours.
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CuAu Cu Au CuAu CuAu a = 3.96Å, c = Lattice parameter(s) Wyck 3.67Å off Space Group P4/mmm (123) x y z positio Strukturbericht notation L10 n Pearson symbol tP4 Au1 1a 0 0 0 Other examples with this Au2 1c 0.5 0.5 0 TiAl structure Cu 2e 0 0.5 0.5
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Cu3Au Cu Au CuAu Cu3Au Lattice parameter(s) a = 3.75 Å Space Group Pm-3m (221) Strukturbericht notation L12 Pearson symbol cP4 Other examples with this structure Ni3Al, TiPt3