a The mathematics of packing spheres

Transcription

1 1 Of oranges and mayonnaise: a The mathematics of packing spheres Rob Farr & Rob Groot September 2011

2 π / ?

3 3 π /

4 In 2 dimensions it s clear: 4 Any packing of equal circles can be split up into triangles The triangle with the highest packing is this one, with 91% coverage: Magically this triangle can Magically, this triangle can be used to tile the plane:

5 In 3 dimensions there is a fly in the ointment: 5 The best packed polyhedron (at 78%) can t be fitted together to tile all of space. So the globally best packed state is not the locally best packed state

6 Different kinds of packing: 1. Crystalline packing of monodisperse spheres 6 History: 1611 Kepler conjecture for monodisperse sphere packing: π/18 1/ from face centred cubic Gauss proved this for regular packings 1997 Thomas Hales provided a full proof... Face Centred Cubic

7 Crystalline packing of monodisperse spheres 7 Seen in colloidal crystals with enough time to equilibrate: Temper rature Gas Coexiste ence of gas and fcc FCC crystal Not po ossible Volume fraction Note: Because interaction is hard sphere, no T dependence

8 Random close packing of monodisperse spheres 8 At high volume fractions, it takes a long time to form crystals Nature 413, 694 (2001) Dil ute Concen ntrated Liq. Coexistence f.c.c?? The system can get trapped in a disordered glassy state, even at a volume fraction where it should crystallize.

9 Random close packing of monodisperse spheres 9 If the system is compressed further in the glassy state, there is a maximum packing fraction which is achievable for this random, disordered state, which is φ max 0.64 First studied in detail by Bernal in 1956, & appears to be a y, pp well defined state (still controversial see Torquato et al.).

10 Why do we care? 10

11 The emulsion clock 11 Most foods are emulsions. Both water and oil phases are usually structured. Low Fat Margarine HIGH OIL Margarine Mayonnaise Dressing HIGH OIL Sauce W/O O/W Soup HIGH WATER Liquid id Margarine VLF Dairy Spread Fresh Cheese HIGH WATER Non-Dairy Cream Ice Cream

12 The importance of random close packing Viscosity of (hard) particle suspensions 12 In fluid systems of hard particles, viscosity is related to how far below RCP you are (e.g. Krieger-Dougherty equation) η r = 1 φ φmax [ η]φ [ ] max Rheology of hard spheres J. Chem Phys. 83(9), 4717 (1985) At low shear rate, the particles will be randomly arranged At high shear rate may get aligned into string phases (higher φ max ) Vis scosity Volume fraction

13 The importance of random close packing Yield stress of emulsions & elastic particle gels You can go above RCP if the spheres are deformable, and then you get a solid. The elastic and yield properties of emulsions (e.g. full fat mayo.) are related to how far above RCP they are. 13 ld stress Yiel CSLM of dense emulsion J. Coll. Int. Sci. 79 (1996) 439

14 The importance of random close packing Specific surface area and permeability D Arcy flow through packed beds (oil through sandstone, coffee through grounds) is related to average pore sizes, specific surface area and so indirectly to packing fraction 14

15 Packing of different size distributions 15 What if you mix two powders and tap them down to random close packing? How much space will they occupy? Or two different emulsions what will the viscosity be? You can run computer simulations of packing, but it is not easy, and each simulation takes hours or days even with the latest code [Rob Groot]. The only remaining option is to think harder...

16 What does polydispersity do to R.C.P.? Think of packing spheres with two different sizes: 16 (a) (b) In general, this increases the close packing vol. fraction: You can sometimes think of packing small spheres in the gaps between big ones, Or grouping small spheres to form large ones which Or grouping small spheres to form large ones, which frees up space.

17 Random Close Packing of bidisperse systems The bidisperse case (mixture of two different sizes) has been long studied but even this simplest case is not trivial 17 t Mass of these particles presen Two parameters: Ratio R of sphere sizes, and Relative amounts of the two types (w) Sphere diameter

18 Random Close Packing of bidisperse systems 18 mass fraction in larger spheres (Data from R. Groot)

19 Predicting the volume fraction of polydisperse hard sphere packings 19 Difficulty 1: Small spheres may rattle around between large spheres. However, if they are similar il size, they might just interfere with the packing of the large spheres. Difficulty 2: It is not just one question: we want to know the maximum packing fraction of any distribution (mixtures of 2, 3, ) different spheres. What are we are trying to approximate?

20 The mathematical zoo 20 3 Numbers Function Numbers 9 g(x)=12j 0 (x) Functions Operator Functions g(x)=cosh(x) Functions Numbers Functional? (Exercise for the reader) Numbers Functions There are other animals, like functionalals and operatorators, but not many have been successfully domesticated.

21 What we are looking for 21 We want to approximate the functional F, where F : P ( D ) a φ 3 D max P 3D (D) is the size distribution of the spheres (could be anything) P 3D size D This is much more difficult than approximating a function How can we simplify the problem?

22 Mapping the 3D problem onto 1D Suppose we start t with our sphere distribution, ib ti and create a collection of rods in the following way: 22 This gives: ( L L P 1D 3 D L P ) ( D ) dd D If the 3D arrangement of spheres were true RCP, then the length fraction of the rods would be exactly φ RCP

23 Mapping the 3D problem onto 1D 23 How do we pack this collection of rods in 1D, in such a way that it represents a section through a 3D RCP state? There must exist a many-body potential V ( x, x,..., x 1 2 N ) which is minimized for the correct positions of the rods. V will be very complex, reflecting topological properties of 3D space (interactions of next nearest neighbours etc.). However, the potential V has some basic properties:

24 Basic properties of the potential V (1) It should lead to a maximum packing fraction that is unchanged if all the rods (or spheres) are magnified by an equal amount. 24 (2) The potential should be "hard : it is either 0 or. (3) The interaction between large rods should reach through small rods, so very small rods can rattle around in the gaps between the large rods, while the large rods carry the load. (4) Interaction range for very unequal rods should be determined by size of the smallest rod, so small rods can form a random packed system in between the large rods.

25 What is the simplest 2-body potential with these properties? Suppose we have rods on a line: and we introduce a strange hard particle interaction: ti The rods bump into each other before they actually touch: L 1 L 2 minimum separation is f L 1, L ) min( 2 Free to move cannot get closer This gives our approximation to V

26 Results for bidisperse spheres 26 But, how accurate is this approximation to the potential V?

27 Results for log-normal distribution 27 P3 D ( D) 1 [ln( D / D exp Dσσ 2 π 2 σ 2 0 )] 2 σ=0.66

28 Results for log-normal distribution 28 P3 D ( D) 1 [ln( D / D exp Dσσ 2 π 2 σ 2 0 )] 2

29 Results for tridisperse spheres 29 1 : 3 : 9

30 30 Conclusions It is a long story... but: We now have a fast and jolly accurate method for predicting RCP for any given sphere size distribution This could be useful for design of systems which are This could be useful for design of systems which are approximately hard sphere.

31 31 Quick advertisement An free and open-source program called spherepack1d will soon be available on the Source Forge website: This will make predictions for arbitrary mixtures of sphere sizes, including mixtures of lognormal distributions.