LECTURE 1: Disordered solids: Structure properties

Transcription

1 LECTURE 1: Disordered solids: Structure properties What is an disordered solid: examples amorphous solids or glasses inorganic compounds (e.g. SiO 2 /silicates, B 2 O 3 /borates, GeO 2 /germanates, P 2 O 5 /phosphates, V 2 O 5 /vanadates, As 2 O 5 /arsenate, SbO 5 /stibnates) organic compounds (e.g. polymers) elements (e.g. sulfur) metal alloys (e.g. Fe 80 B 20 ). Disordered from randomness: eg. Diffusion-limited aggregation Why are amorphous solids important? metallic glasses: transistors Silicate glasses: fiber optics How do we measure disorder? Pair correlation function, Structure function Energetics of disordered solids: Excess molar volume How are amorphous solids formed: glass transition temperature versus crystallization and melting temperature Suggested problems:

2 What are amorphous solids?

3 Nicolas et al, arxiv: (2017)

4 Crystals Amorphous solids Long-range order Regular atomic arrangement Short-range order Random atomic arrangement

5 B 2 O 3 (crystal) Borate glass Open Course MIT 3.017

6 Statistical measure of disorder: distribution functions Statistical description of atomic positions Describes important features of the structure Image analysis of optical or electron microscopy pictures Scattering experiments

7 Single particle distribution function N atoms in the volume V of the material Average density n " = $ % Single particle distribution function n & r = - δ(r r - ) Ensemble average over possible configurations (statistical description) n & r dr is the average number of atom centers in a volume element dr around r n & r dr can be interpreted as the probability to find an atom center in dr n & r dr

8 Two-particles distribution function N atoms in the volume V of the material n 2 r &, r 2 = δ r - r & δ(r 4 r 2 ) Probability to find atom centers simultaneously in a shell of thickness dr & around r & and a shell dr 2 around r 2 is n 2 r &, r 2 dr & dr 2 dr 2 n 2 r &, r 2 = N 1 n & (r & )

9 Atomic correlations Probability of having an atom at r 2 is correlated with the probability for an atom at r &, if these positions are not too far apart. This is expected due to inter-particle forces and constraints due to chemical bonds. Homogeneous material: n & (r & ) = ρ If r & r 2 1, atoms are uncorrelated and: n 2 (r &, r 2 ) ρ 2 Generally we can write n 2 (r &, r 2 ) = ρ 2 g(r &, r 2 )

10 Pair distribution function (PDF) General definition: g r &, r 2 = A B (r C,r B ) A C r C A C (r B ) Put r 2 = 0, then define r = r & r 2 Homogeneous material g r &, r 2 positions = g r = n 2 (r)/ρ 2 describes the correlations in the atom ρg r dr is interpreted as the probability to find an atom in dr, a vector distance r from another atom at the origin. ρ g r dr = N 1

11 Isotropic materials Isotropy: r is the distance between r & and r 2 ρg (r) can be integrated to give the average density of atoms in a spherical shell around the origin F G HIF dr ρg r = 4π 2 r 2 F G " g r " Δr = ρz r " Δr " Radial distribution function Z r Z r = 4π 2 g(r) Integrate Z r around the first peak -> Coordination number of the first shell

12 Pair Distribution Function (PDF) PDF show oscillations due to nearest neighbor, next nearest neighbor shells, etc. Oscillations damped out as r increases g r 1 for large r

13 PDF for an ideal crystal

14 PDF for ideal crystal vs glasses Open Course MIT 3.017

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16 Example of radial distribution function

17 Open Course MIT 3.017

18 Measure of disorder: Excess molar volume Energetics of bond density Ordered crystal has higher packing -> more bonds per unit volume Binding energy E OPQR < E TRQUU Binding energy <-> molar volume V V TRQUU > V OPQR Excess molar volume ΔV = V TRQUU V OPQR is a measure of disorder Open Course MIT 3.017

19 Glass formation from liquid Cooling curve Heating curve Open Course MIT 3.017

20 Glass formation from liquid Cooling curve Heating curve T m function of composition T m Open Course MIT 3.017

21 Glass formation from liquid Cooling curve Heating curve Open Course MIT 3.017

22 Glass formation from liquid Cooling curve Heating curve Open Course MIT 3.017

23 Glass formation from liquid T g depends on Cooling curve Heating curve (atom mobility) -1 X (complexity of crystal structure) X (cooling rate) X (composition) Open Course MIT 3.017

24 Glass formation from liquid Cooling curve Heating curve Excess Volume (disorder) Open Course MIT 3.017

25 LECTURE 2 Atomic diffusion: equilibrium and non-equilibrium structures Diffusion of atoms Fick s law Einstein s relation between mobility and diffusivity Equilibrium concentration Binary mixtures Diffusion limited aggregation, crystal growth Suggested problems:

26 Diffusion At any T different from 0K, all atoms irrespective of their aggregation (gas, liquid, solid, etc) are in constant motion Movement of atoms is associated with collisions -> single particle trajectory is a zigzag diffusive particle A collection of diffusive particles has an observable drift from high to lower concentrations

27 Source: wikipedia

28 Fick s law Flux = (conductivity) x (driving force) For atomic or molecular diffusion: conductivity = diffusivity constant D D reflects the mobility of the diffusing particles and depends on the environment: e.g. D is larger for particles in gases, than in liquids. Driving force = - concentration gradient J = D C J moles cm 2 = D dc s dx [D] = cm 2 /s moles cm gh cm

29 Steady-state concentration Equilibrium condition J = 0 D C = 0 C X

30 Fick s second law (dynamics) J O J OHyO = z{ zp dx Taylor series for dx 0: J OHyO = J O + z} ~ zo dx+.. J O C J OHyO z zo D z{ zo dx = z{ zp dx x x+dx z{ zp = DzB { zo B

31 Mixtures Mixtures of two or more different chemical species Alloys: mixtures of different metal elements Homogeneous mixtures: constituents are intermixed on an atomic scale to form a single phase solution Heterogeous mixtures: constituents do not mix and the mixtures contains two phases (oil and water)

32 Binary mixture phase diagram Composition X of a binary mixture Critical point for phase separation

33 Phase separation from a quench

34 Diffusion-limited aggregation: crystal vs amorphous structures Meakin, Jamtveit, PNAS 2009