University of Minnesota, Minneapolis

Transcription

1 Coarse Graining of Electric Field Interactions with Materials Prashant Kumar Jha Ph.D., CMU Adviser Dr. Kaushik Dayal Funded by Army Research Office Research Talk University of Minnesota, *Figure :

2 Goal Introduction 1 1. Electrostatics in nanostructures 3. Multiscale method for ionic solids at finite temperature 2. Electrostatics in random media Nanotube Image -

3 Motivation Electrostatics interaction Storage devices Ferroelectric RAM Piezoelectric sensors Introduction 2 Finite temperature Thermal fluctuations of atoms Coupling of deformation, electric field with temperature (a) Hard drive (c) Piezoelectric sensor (b) Ferroelectric RAM (a) (b) (c)

4 Long range interactions Introduction 3 Field at X due to charge/dipole at Y Charge/dipole at Y Expansion of kernel G for charge distribution

5 Introduction Long range interactions 4 Field at X due to charge/dipole at Y Charge/dipole at Y Charge distribution Quadrupole distribution Dipole distribution Figure: Marhsall and Dayal 2013

6 Introduction Long range interactions 5 Linear Elasticity Electrostatics Energy density depends on polarization field over whole material domain

7 Introduction Long range interactions 6 Continuum limit of electrostatic energy : polarization field in a material

8 Length scales 7 Continuum Length scale : L Size of material point : Atomic spacing : Macroscopic field vary at the scale Interested in limit Continuum mechanics Fields vary at fine scale compared to size of material Continuum limit approximations Atomic spacing is fine compared to scale at which fields vary Figure: Marhsall and Dayal 2013

9 Continuum limit 8 Average energy of atoms in Sphere B r (0) Two equivalent approach Scaled potential

10 Continuum limit 9 Energy of domain Accuracy increases as increases

11 Electrostatics energy: Periodic media 10 Figure: Marhsall and Dayal 2013

12 Scaling on charge density field: Periodic media 11 Energy of one unit cell due to charge distribution in material point x

13 Continuum limit and two scale homogenization 12 The general theory of homogenization by Tartar Homogenization and two-scale convergence by Allaire Modeling materials: Continuum, atomistic, and multiscale techniques by Tadmor and Miller On the Cauchy-Born rule by Ericksen The elastic dielectric by Toupin Internal variables and fine-scale oscillations in micromagnetics by James and Muller Micromagnetics of very thin films by Gioia and James From molecular models to continuum mechanics by Blanc, Le Bris, and Lions 2002.

14 Homogenization with random fields 13 In Blanc, Le Bris, and Lions 2007, they consider the homogenization of short range atomic forces for random media. In Blanc, Lions, Legoll, and Patz 2010, homogenization in one-d is considered. The thermal fluctuations are modeled as random field. Other related works are: Blanc, Le Bris, and Lions Chapter 7 of Jikov, Kozlov, and Oleinik 1994 gives brief introduction to stationarity and ergodicity and considers the homogenization of Poissons equation with random coefficient. In chapter 3 of Bensoussan, Lions, and Papanicolaou, stochastic homogenization of Poissons equation and diffusion equation is considered. A book Random heterogeneous materials by Salvatore 2002 is another reference on materials with randomness.

15 Stationary and Ergodic random field 14 Stationarity Similar to periodic media average over unit cell is independent of unit cell Ergodicity Spatial average

16 Dynamical system 15 We follow Jikov, Kozlov, and Oleinik 1994 as a reference for probability theory.

17 Examples Periodic field 16 Quasiperiodic field

18 Birkhoff Ergodic theorem 17 (1) Chapter 7, Theorem 7.2 Jikov, Kozlov, and Oleinik (2) Linear Operators by Dunford and Schwartz.

19 Random media: Charge density field 18

20 Random media: Electrostatic energy 19

21 Random media: Local energy 20

22 Random media: Local energy 21

23 Random media: Local energy 22

24 Random media: Local energy 23

25 Random media: Non-local energy 24 Taylor s series expansion Go to infinity, unless term in bracket is zero By Ergodic theorem

26 Random media: Non-local energy 25

27 Random media: Non-local energy 26

28 Random media: Discussion 27

29 Nanostructures Cross-section is of few atomic thickness Long in axial direction Translational, and/or rotational symmetry 28 Nanostructure and macroscopically thick structures in a continuum limit Nanotube Image:

30 Objective nanorod 29

31 Nanostructures: Charge density field 19

32 Nanostructures: Result 20 Interaction energy due to charges within material point. If net charge in unit cell is zero. No long range interaction

33 Nanostructures: Discussion 30

34 Nanostructures/thin films behave differently 31 Estimate of dipole energy for 1-D, 2-D and 3-D materials At distance r net dipole is 1 Along the circumference of circle of r, net dipole is 2*pi*r At the surface of sphere of radius r, net dipole is 4*pi*r*r Dipole field kernel decays fast for 1-D and 2-D materials

35 Thank you!