Extrinsic Point Defects: Impurities

Transcription

1 Extrinsic Point Defects: Impurities Substitutional and interstitial impurities Sol solutions, solubility limit Entropy of ing, eal solution model Enthalpy of ing, quasi-chemical model Ideal and regular solutions Solubility from Gibbs free energy Interaction between impurities and vacancies References: Porter and Easterling, Ch..3 Ragone, Thermodynamics of Materials, Ch. 3.3 University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

2 Impurities Impurities (extrinsic point defects - atoms which are different from the host. Even the most pure materials contain some impurities and the concentration of the impurities is, in most real materials, comparable to or exceeds the concentration of the equilibrium intrinsic point defects. Example: Very pure metals % - one impurity per 6 atoms; equilibrium vacancy concentration at T m - one vacancy per ~ 4 atoms May be intentional or unintentional Examples: carbon added in small amounts to iron makes steel, which is stronger than pure iron boron added to silicon change its electrical properties lloys - tures of components Example: sterling silver is 9.5% silver 7.5% copper alloy. Stronger than pure silver. Material is described in terms of "impurity concentration rather than alloy composition if the amount of component in matrix is small types: substitutional impurities substitute the host atoms in the lattice interstitial impurities located in interstitial lattice sites University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

3 Sol solutions Sol solutions are made of a host (the solvent or matrix which dissolves the minor component (solute. The ability to dissolve is called solubility. Solvent: in an alloy, the element or compound present in greater amount Solute: in an alloy, the element or compound present in lesser amount Sol Solution: homogeneous maintains crystal structure contains randomly dispersed impurities (substitutional or interstitial Second Phase: as solute atoms are added, new compounds / structures are formed, or solute forms local precipitates Solubility Limit of a component in a phase is the maximum amount of the component that can be dissolved in it Example: Cu and i are mutually soluble in any amount (unlimited sol solubility, while C has a limited solubility in Fe. Whether the addition of impurities results in formation of sol solution or second phase depends the nature of the impurities, their concentration, temperature, pressure University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

4 Substitutional and interstitial sol solutions Substitutional sol solutions: Max solute concentration 5 at% e.g. Cu-i (unlimited sol solubility Factors for high solubility: tomic size factor - atoms need to fit solute and solvent atomic radii should be within ~ 5% Crystal structures of solute and solvent should be the same Electronegativities of solute and solvent should be comparable (otherwise new intermediate phases are encouraged Generally, more solute goes into solution when it has higher valency than solvent Interstitial sol solutions: ormally, max. solute concentration % e.g.. at% of C in α-fe (CC. Factors for high solubility: For fcc, bcc, hcp structures the vos (or interstices between the host atoms are relatively small atomic radius of solute should be significantly less than solvent University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

5 Equilibrium concentration of impurities? In contrast with intrinsic point defects, the concentration of impurities is often fixed and cannot attain the value that corresponds to the equilibrium at given T and P The equilibrium in a phase with impurities is usually described in terms of their solubility Instead of the formation energies and entropies used for the intrinsic defects, the solubility limit is defined by enthalpies and entropies of ing Gibbs free energy of a binary solution: Let s conser a binary solution of and atoms that have the same crystal structures in their pure states and can be ed in any proportions - has unlimited sol solubility, e.g., Cu i mol of homogeneous sol solution contains X mol of and X mol of. X and X are the mole fractions of and in the alloy. X X. ring together X mol of & X mol of G G Two steps of ing: G step G. Mix & to make a homogeneous solution G ΔG X G G XG XG X G ΔG is the change of the Gibbs free energy X caused by the ing University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei G ΔG ΔH TΔ S

6 Gibbs free energy of ing ΔG ΔH TΔ S ΔH H H - heat of ing of the components (heat of formation of a solution Δ S S S - difference in entropy between ed and uned states (entropy of formation of a solution Model systems: eal solution interactions between atoms -, - and - are entical, and ΔH. The free energy change upon ing is only due to the change in configurational entropy: ΔG TΔ S statistical or quasi-chemical model heat of formation (ΔH is evaluated by counting bonds between atoms of different type. The assumption is that the interatomic distances and bond energies are independent of composition. regular solution random arrangement of atoms in a solution is assumed (no clustering or compositional ordering University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

7 Δ S Configurational entropy of ing S S S k ln Therefore Δ S S - there is only one way the atoms can be arranged before ing if we have tot objects and spec of them are special, the number of ways the objects can be arranged (number of microstates is tot! Ω!(! Remember for vacancies we had tot number of lattice sites, spec n number of vacancies For ing of particles of type with atoms of type : Ω ΔS ΔS k lnω k ln (!!! (! Using Stirling formula: ln! ln [ ln(!-ln!-ln!] k ( ln( -(!! spec tot [ - ln - ln ] k [ X lnx X lnx ] k ln ln R spec X /( X /( for mol, a X a, X a, a k R University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

8 for eal solution ΔH and G G G ΔG Gibbs free energy of eal solution [ lnx X lnx ] ΔG TΔ S RT X G XG XG [ lnx X lnx ] G XG XG RT X G G T ΔG T G T G T T X G T ΔG How is the position of the minimum of the Gibbs free energy curve related to the solubility limit? How does it change when T increases? X University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

9 Enthalpy of ing: quasi-chemical model ΔG ΔH TΔ S ΔH > ΔH < - ing is endothermic (heat absorbed - ing is exothermic (heat released quasi-chemical model: ΔH is only related to the bond energies between adjacent atoms if there are P, P, P bonds of each type, the internal energy of the solution is E P E P E P E 3 types of bonds: bond (energy E bond (energy E bond (energy E if z is the coordination number, z P P z P P z P P P z P E z P E z P E P E z E z E energy of uned components P E E E energy of ing, ΔE University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

10 Enthalpy of ing of a regular solution E E Enthalpy of ing (heat of formation: ΔH H H ΔE P E E E If E - the solution is eal: ΔH If E E E > - ΔH > atoms tend to be surrounded by atoms of the same type If E E E < - ΔH < atoms tend to be surrounded by atoms of different type for small differences between E and (E E / (and for high T we can still conser a random arrangement of atoms in a solution (regular solution model. Then E E P z tot X X and ΔH ΩX X where Ω ztot E ΔH Ω > Ω X University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

11 Ω < ΔH -TΔ S Gibbs free energy of a regular solution ( lnx X lnx ΔG ΔH TΔS ΩX X RT X Ω <, ΔH < exothermic ing, favorable at all T For high Ω and low T, P max - an ordered alloy could be formed the assumption of random ing is not val, solution is not regular, ΔH ΩX X ΔG X Ω >, high T Ω >, ΔH lowt ΔH For Ω >, ΔH > ing (formation of - pairs is avoed at low T. ΔG ΔG t high T entropy helps to. t low T clustering may occur solution is not regular. -TΔ S X -TΔ S X University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

12 The closer is the minimum of the Gibbs free energy curve G α (X to the axes X, the smaller is the maximum concentration of in phase α. Solubility from Gibbs free energy G β liqu T α X T T α αl α β liqu βl β X What is the temperature dependence of the solubility of in? Have to find the temperature dependence of the minimum of G α (X [ lnx X lnx ] reg G XG XG ΩXX RT X dg dx - minimum of G(X University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

13 G reg Solubility from Gibbs free energy XG XG ΩXX RT[ XlnX XlnX] ( - X G X G Ω( - X X RT[ ( - X ln( - X X lnx ] dg dx -G G Ω Ω X RT -ln - X ( ( - X ( - X lnx X X -G [ ( lnx ] G Ω Ω X RT -ln -X X G -G Ω -X ( X RTln G -G Ω RTln( X if X is small (X G(X min X G exp G RT Ω - sol solubility of in α increases exponentially with T (similar to vacancy concentration University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

14 Solubility from Gibbs free energy X X vs. T T T vs. X X G exp G Ω RT T e T supersaturation with phase separation α and β precipitation of -rich intermediate phase or compound X α αl liqu βl β T α β X University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei

15 Interaction between impurities and vacancies vacancy-impurity complexes Impurity atoms may interact with extrinsic point defects (vacancies, divacancies, self-interstitials, etc. and form complexes if there is an attractive interactions between the defects. We can calculate the equilibrium concentration of impurity-vacancy complexes in a way similar to our analysis of concentrations of vacancies and divacancies. n c eq v tot n v eq v eq n imp n n n n n v tot v eq c z hb exp kt c eq c nimp z hb exp kt n c eq n v eq n imp equilibriu m number of complexes number of impurities z coordinati on number h c b equilibriu m number of indivitual vacancies binding enthalpy of an impurity - vacancy complex Thus, for positive binding energies, impurities can substantially increase the total vacancy concentration, especially at low T Example: at T 7 K, in an fcc metal with impurity fraction of.5% (n imp / 5-3 and a binding enthalpy of. ev, v ntot 3.eV 5 exp.35 v 4 n.8673 ev/k 7 K eq University of Virginia, MSE 6: Defects and Microstructure in Materials, Leon Zhigilei