Disordered Structures. Part 2

Transcription

1 Disordered Structures Part 2

2 Composites and mixtures Consider inhomogeneities on length scales > Å Phase separation two (or multi-) phase mixtures Mixtures of particles of different kinds - solids, powders Voids porous materials Columnar structure in thin films Composites mixtures of two or more materials on length scales larger than 10 Å Distinct from random alloys, which are mixtures on atomic scale

3 Composite structures Volume fraction, f Dilute dispersion of particles, f small Separate grain strucutre: Particles in a matrix material Aggregate structure: Random mixture of particles (or crystallites) of two materials

4 Clustering of particles Smaller or larger clusters of particles in a matrix material Connected clusters of particles Percolation threshold at critical volume fraction Source: da.nieltiggemann.de Source: wikimedia.org

5 Separated grain structure Pair distribution function Low f: Hard spheres, radius R. r R g ( 0 < r) = { 2 1 r > R Higher f: Percus-Yevick approximation Higher-order distribution functions become more important as f increases Aggregate structure: Describe in general by partial pair distribution functions (and higher ones) Binary composite: g 2,AA (r), g 2,AB (r), g 2,BB (r)

6 Ex: Metal-insulator composite Co particles in Al 2 O 3 Electron-beam evaporation Analysis of electron micrograph

7 Local geometry distributions Problems with distribution functions High order g n contain enormous amounts of data Functions with n>3 are impractical to determine, and even the three-point function in most cases We need a better approach to describe the composite structures Local porosity theory was developed within geophysics (R. Hilfer). Local density functions, which depend on length scale Connectivity taken into account by local percolation probability functions Can be used as input to theories for a range of physical properties

8 Local density function Divide an object into M equal measurement cells One-cell local density function µ ( i= 1 ( L)) Depends on the length, L, of the cell Contains information on fluctuations of volume fraction,f M 1 f, L) = M δ ( f f i µ(f,l) Depends on L L small: two δ-functions L large: δ-function at average f, f ave f

9 Local geometry entropy At small and large length scales, µ is determined by the average volume fraction, f ave. Optimal information on the structure can be found at an intermediate length scale Minimum of Information measure or entropy I(L) 1 I ( L) = µ ( f, L)log( µ ( f, L)) df 0 L

10 Local percolation probability Probability that two points on opposite surfaces of a measurement cell of length L are connected by a path of the same component Fraction of percolating local geometries 1 p ( L) = λ( f, L) µ ( f, L) df 0 λ(f,l) The local percolation probability λ(f,l) depends also on the length of the cell f

11 Example: Local porosity analysis Sintered glass powder (Haslund et al. JAP 1994) Local porosity distributions, various L SEM picture of pore space (250 µm beads) Peak at medium porosity φ=1-f

12 Continued Entropy function: Minimum at L=40 Percolation probability of bulk samples Needs 3-dim geometry characterization - tomographic methods. May be different in x- y- and z-directions Percolation in all three principal directions described by λ 3 (f,l) (Haslund et al. J. Appl. Phys. 1994) Local percolation probability cannot be measured on a 2-dim image. Probability of blocking in all three directions: λ 0 (f,l)

13 Three-dimensional case Measurement on sandstones by X-ray microtomography: (Biswal et al, Physica A, 1998)

14 Percolation theory A lattice with sites randomly occupied with probability p Adjacent occupied sites are connected and belong to the same cluster Finite clusters Infinite cluster connects opposite sides of a sample and appears when p>p c Source: A. Aharony and D. Stauffer, Percolation theory

15 Percolation theory Continuum percolation: Particles randomly placed in a volume Percolation threshold, p c, or critical volume fraction f c. Two-dim: p c =0.593 (sq) f c ~0.45 Three-dim: p c =0.312 (sc) f c ~0.16 Percolation clusters can be described by scaling theory Many geometrical properties follow power laws Average number of sites of finite clusters ~ lp-p c l -γ Probability that a site belongs to the infinite β cluster P ( p) ~ ( p pc ) Others

16 Infinite cluster The infinite percolation cluster (red) also exhibits scaling Correlation length ξ ( = ξ p pc p) 0 ν Number of sites within distance r at p=p c and for R<ξ at other p s N ( r) ~ r d f Source: terrytao.wordpress.com

17 Fractals Dilation symmetry structure invariant upon a change of scale Self-similar structures look similar at different magnifications (scaling) Self-affine structures different scaling behaviour in different directions Multifractals many scaling behaviours Described by a fractal (Hausdorff) dimension D f Dtop D f E D top is the topological dimension E is the dimension of the embedding space

18 Dimensionality Line, length L Length in units of L and L/r are related by L/r Λ( L / r) = Surface: Area relation A ( L / r) = r r 1 2 Λ( L) A( L) L/r Solid volume V ( L / r) = r 3 V ( L) Fractal: Measure M M D f ( L / r) = r M ( L)

19 Example: von Koch curve A deterministic fractal is constructed iteratively In each iteration a line of length L is replaced by N (N=4 here) segments of length L/r (here r=3) Λ( L / 3) = 3 D f = ln N D f / ln r Λ( L) = ln 4 / ln The mathematical fractal is obtained after infinitely many iterations In physics, we must of course have lower and upper cutoffs to the fractal structure Source: nd.edu

20 Other examples L=rR L=r 2 R Source: mathaware.org Sierpinski gasket N=3 r=2 D f = ln 3 / ln 2 = Particle cluster: N=5 r=3 D f = ln 5 / ln 3 = Three-dim cluster D f = ln 7 / ln 3 = 1.771

21 Box dimension The volume of a fractal can be determined by covering it with boxes of size L Image size L max Number of covering boxes N(L) the measure r=l max /L, put N(L max )=1 N( L) L L = max D f Source: fast.u-psud.fr

22 Example: Co particles Co particle clusters deposited by evaporation in 10 Torr of argon. Particle radius ~10 nm Box counting (ImageJ) D f =1.68

23 Mass dimension Convenient for clusters of particles Minimum length scale: Particle size R Log M M D f ( L) = M ( rr) = r M ( R) Log r Density depends on length scale r Log ρ ρ( r) ~ r D f E Log r

24 Infinite percolation cluster p>p c, correlation length ξ Mass within a box of size ξ M ( ξ ) 3 ~ ξ P ( p) ~ ξ 3 β / ν In general M(r) ~ r E-β/ν when r<ξ. The fractal dimension has been found to be D f =91/48 for E=2 and D f ~2.48 for E=3 Fractal for length scales below the correlation length Not fractal above the correlation length

25 Distribution functions Fractal structures can be described by their pair distribution function M(r) is proportional to the integrated RDF The PDF is proportional to the density function ρ(r). Inner cutoff: Radius of the building blocks. Atoms, molecules or particles Outer cutoff: Radius of gyration of aggregate, R g, or correlation length Fractal structure R<r<ξ Homogeneous (density equal to mean density) for r>ξ.

26 Pair distribution function A fractal aggregate of particles g 2 ( r) = A r R g D f 3 f co ( ) r R g f co ( x) ~ exp( Cx α ) Numerical simulations indicate that α~d f. For connected percolating aggregates replace R g by ξ and f co might be different. D f 3 r r In this case g 2 (r)~1 for r>>ξ. g2( r) 1 = B fco( ) ξ ξ

27 Aggregating metal particles Al particles evaporated in 3 Torr He (+O2) Fractal structure: g 2 (r) ~ r D f-2 D f ~1.75 Shadowing (projection) effects important D f =1.9

28 Fractal surfaces The surface of a volume fractal has a fractal structure with the same fractal dimension An interface separating two non-fractal materials may also be fractal. Such rough surfaces are usually self-affine. They scale differently in the plane and in the z- direction. Scaling relation for the height (or rms roughness), δ, of the surface.

29 One-dimensional surface Scaling relation δ(bx) ~ b H δ(x) Hölder exponent H Ex: δ(4x) ~ 2 δ(x) hence H=1/2 Fractal dimension D f = 2-H In general D f =E-H

30 Box dimensions Cover the curve with boxes of length L=L max /N In each length L we need L H /L boxes N(L) ~ NL H-1 ~ L H-2 We assumed that L<<h max. Local dimension D f =2-H If L>h max, then we need just one box in each interval and N(L) ~ L -1 Global dimension D f =1 A self-affine structure has two fractal dimensions!

31 Rough surfaces Real surfaces of for example thin films often exhibit a more complex scaling Rms roughness Log δ ~<h> β δ = ( h h ) 2 1/ 2 ~L H δ ( L, h ) ~ Thin film deposition: <h> ~ deposition time L H f ( h / L z ) Log L Exponent β=h/z The two regions are separated by correlation length L c.

32 Particle-cluster aggregation model Particle-cluster aggregation Particles are launched one by one The particles perform a random walk until they collide and stick to a growing cluster Common structure in physics Basis for models of e.g.: Dielectric breakdown, lightning, snowflakes

33 Particle-cluster aggregation

34 Box counting In twodimensional case (E=2) the DLA model has a fractal dimension D f ~1.65 In threedimensional case: D f ~ 2.4

35 Example: Viscous fingering Pattern formed by displacing liquid epoxy from a 2-dimensional porous material consisting of a single layer of glass spheres between two plastic sheets Similar patterns: Dielectric breakdown

36 Cluster-cluster aggregation Cluster-cluster aggregation A number of particles make random walks When they collide clusters are formed which are also allowed to perform random walks Particles and clusters collide and form larger and larger clusters D f ~ 1.75 (E=3) Source: fisica.ufc.br

37 Exp: Clusters of Co particles Fractal dimension ~

38 Simulation link