Modelling across different time and length scales

Transcription

1 18/1/007 Course MP5 Lecture 1 18/1/007 Modelling across different time and length scales An introduction to multiscale modelling and mesoscale methods Dr James Elliott 0.1 Introduction 6 lectures by JAE, one computing class (31/1/07) Builds on courses MP and MP3 (assumed knowledge) Concerned with computational methods for addressing the large range of time and length scales of phenomena relating to materials properties In the first six lectures, we will cover: Introduction and basic concepts Levels of coarse-graining in polymers Lattice methods: Monte Carlo, lattice chain model Continuum methods, particle based simulations Free energy functional methods Examples of hierarchical modelling 1

2 18/1/ Recommended reading Introduction to Polymers, R.J. Young and P.A. Lovell, (AN6a.40) Introduction to Soft Matter, I.W. Hamley, (Pf55) Principles of condensed matter physics, P.M. Chaikin and T.C. Lubensky, CUP. (Cavendish 5 C 5) Bridging the gap between atomistic and coarse-grained models of polymers J. Baschnagel et al, Adv. in Polymer Science 15, (000) Lattice gases and cellular automata, Bruce M. Boghosian, Future Generation Computer Systems 16, (1999) Lattice-gas models of phase separation D. H. Rothman and S. Zaleski, Rev. Mod. Phys. 66, 1417 (1994) 0.3 Supplementary reading Scaling Concepts in Polymer Physics, P.-G. de Gennes, Cornell University, Ithaca, NY. Handbook of Stochastic Methods, C.W. Gardiner, Springer, Berlin, (Cavendish 39 G 10) The Physics of Polymers, nd Edition, G. Strobl (AN6c.14) Stochastic Processes in Physics and Chemistry, N.G. Van Kampen., North-Holland, Amsterdam.

3 18/1/ Reminder about online teaching resources You can find copies of the overheads and handouts for this lecture at: There are copies of overheads, handouts, example sheets and computing class handouts for this course (MP5) and most others in the MPhil Online communities sage.caret.cam.ac.uk 1.1 What is multiscale modelling? 3

4 18/1/ Where does mesoscale modelling fit in? Nanoscale structures play a crucial role in determining material properties Action of material depends on the precise form of the nanoscale units Structures are processing pathway dependent - need to control this Mesoscale modelling unifies materials design - links atomistic scale with FEM/CFD and QSPR It is the crucial link between our microscopic understanding of a system and the corresponding macroscopic description Why do mesoscale modelling? Typically, interactions on one scale lead to structures with effective interactions on the next scale up For example, force field descriptions of atomistic interactions (cf. MP3 lecture 6) are the result of effective electronic orbitals and valence force interactions If one had unlimited computing resources, one could hence derive everything from the electronic structure level, by solution of the Schrödinger equation However, this is neither feasible computationally, nor desirable from a scientific point of view because as a result of such simulations one would be overwhelmed by irrelevant data 4

5 18/1/ Why do mesoscale modelling? Moore s law describes exponential growth in CPU power Applications of mesoscale modelling Block-copolymers and surfactants Liquid crystalline polymers Dislocation structures in metals Nanotechnology Biomedical materials In general: any type of self-organising material 5

6 18/1/ Applications of mesoscale modelling Block copolymers for drug delivery At Temperature > CMT : Hydrophilic Hydrophobic Corona Core With hydrophobic drugs Hydrophobic Drug Applications of mesoscale modelling Block copolymers for drug delivery Intensity X-section (nm) 6

7 18/1/ Application of mesoscale modelling MesoDyn (lecture 5) allows prediction of morphology Applications of mesoscale modelling Defect textures in Liquid Crystalline Polymers (LCPs) 7

8 18/1/ Applications of mesoscale modelling 1.5 Definition of mesoscale Does mesoscale have a precise meaning? A mesoscale level can be defined if at that length (and associated time) scale one can assume that the degrees of freedom pertaining to a smaller scale will be in equilibrium when seen from that scale I.e. they have a relaxation time which is much shorter than the time scale of interest It can therefore be used as a generic definition of mesoscale, that it is any intermediate scale at which the phenomena at the next level down can be regarded as always having equilibrated, and at which new phenomena emerge with their own relaxation times 8

9 18/1/ Brownian motion Classic example of a mesoscale phenomenon Named after Robert Brown ( ), a naturalist who observed this motion within plant cells, and later also in liquids enclosed in minerals Importantly, he showed that this motion is not related to life as had previously been thought Independently, this type of motion was predicted by Einstein, as a test for kinetic theory Seemingly a ubiquitous physical phenomenon Brownian motion JAVA applet ( Relevance of BM to mesoscale concept It forms a link between the atomistic and the macroscale Kinetic theory is based on an atomistic picture of matter, given by the atomistic particle nature plus the kinetics of these particles (following Newton s laws) On the other hand, thermodynamics is a macroscopic formulation, with state potentials (cf. MP3 lectures 1-3) Thermodynamics, despite its name, does not actually include a detailed description of dynamics!! Nevertheless, changes in state can be described even for non-reversible systems, leading to fluxes such mass or heat diffusion 9

10 18/1/ Statistical thermodynamics of BM Mean kinetic energy of Brownian particle is given by the equipartition theorem (MP3 1.13) as: 1 m v = 3 k B T or, for just the x-component of the velocity m vx = 1 kbt vx kbt / m 1 = However, mesoscale observations of the particle velocity are on a much larger time scale τ : V x = 1 τ ( x( t + τ) x( t) ) Einstein relationships In 1905, during his annus mirabilis, Einstein considered the motion of the Brownian particle as a statistical random walk, and derived the following important relationships: l x = Nl = τ t where N is the number of steps of length l in the random walk, with mean time τ between collisions (cf. MP Einstein equation for diffusion coefficients) This equation displays the characteristic Brownian dependence of displacement on the square root of time 10

11 18/1/ Einstein relationships As the number of steps N becomes large, the Central Limit Theorem says that the probability density function of the particle displacement becomes Gaussian πl ( x) = τ t 1/ x exp l W τ t This is a solution of the diffusion equation: ρ = D t x ρ where D = x / t links the macroscopic diffusivity with the microscopic jumps in particle position ρ( x,0) = δ( x) The Langevin approach Paul Langevin ( ) reformulated the problem of Brownian motion into a more generalised framework He supposed that the particle moves in some dissipative medium (i.e. experiences frictional forces) dv = γv dt 1 γ = µ m Of course, this cannot be the whole story, as the solution to this equation for a particle with initial velocity v 0 is v = v 0 exp ( γt) which is clearly wrong 11

12 18/1/ The Langevin approach What is missing are the thermal kicking forces of the surrounding particles, which are represented by some average time-dependent force The result is known as the Langevin equation dv dt + γv = F( t) and F(t) is called the Langevin force Additional terms represent the neglected degrees of freedom (when γ = 0, recover molecular dynamics) In order for the equation to faithfully describe Brownian motion, the Langevin force must obey certain properties Properties of the Langevin force The first is due to the fact that the particle velocity should, on average, decay exponentially from any initial value v 0 in the same way as for the macroscopic law ( 0 0 ( ) v t) v = v exp γt This can only happen if the time averaged force is zero F( t) = 0 The second is due to the fact that the kicks experienced by the particle are independent of each other, and occur instantaneously 1

13 18/1/ Properties of the Langevin force If one observes the motion of the particle over typical intervals τ, where τ» t, then the velocity of the particle will appear to vary little even though there have been many kicks or fluctuations This means that F(t) and F(t +τ) should be uncorrelated in time, which can be expressed by: F( t ) F( t + τ) = 6µ kbtδ( t) Thirdly, and finally, it is generally assumed that higher order correlators of F(t) are zero. This then defines the underlying fluctuations as Gaussian Sometimes referred to as white noise 1.1 Solution of the Langevin equation With these three premises, the Langevin equation (1.10.) can be solved to yield a mean squared displacement: x = kbt mγ At short times γt 6kBT γt 1 3 ( 1 e ) + t + e e 3 γ t x = 3 mγ kbt t m γ γ x t Over long times kbt x = 6Dt D = mγ 0.5 x t 13

14 18/1/ The fluctuation-dissipation theorem Notice that the magnitude of the fluctuations of the particle displacements are related to their frictional mobility µ x = 6µ ktt B This is a manifestation of the fluctuation-dissipation theorem (cf. MP3.7.) There is a deep connection between the level of thermal noise and frictional damping which act together to create a well-defined macroscopic thermodynamic state 1.14 Other examples of the F-D theorem Johnson-Nyquist noise in electrical circuits caused by thermal fluctuations in motion of electrons Kramers-Kronig relations Linear response theory Onsager s regression hypothesis 14

15 18/1/ Mesoscale modelling The preceding discussion has hopefully sown in your minds the seed of an idea how to adapt computer modelling methods to model mesoscale processes The key is always to get rid of the uninteresting degrees of freedom via some F-D argument, and then observe the resultant behaviour of the system However, there are many different methods to do this We have also completely neglected the influence of hydrodynamic effects, where long range forces between particles are mediated by an intervening fluid Hence, in the forthcoming lectures, we will study a wide variety of different mesoscale modelling technique 1.16 Mesoscale modelling techniques continuum Brownian dynamics Rouse model DFT diffusive Monte Carlo L. chain L. director lattice SPH DPD Stokesian dynamics hydrodynamic L. gas L. Boltzmann 15

16 18/1/ Summary We began by introducing the concept of multiscale modelling, and indicating where mesoscale methods fit into the hierarchy of length and time scales We then looked at some examples of mesoscale modelling, and refined our definition of the term mesoscale The phenomenon of Brownian motion was studied in some detail, leading up the Langevin formulation and a discussion of the fluctuation-dissipation theorem Finally, the various types of mesoscale modelling techniques which will be discussed in forthcoming lectures were summarised and classified 16