Surveying Free Energy Landscapes: Applications to Continuum Soft Matter Systems

Transcription

1 Surveying Free Energy Landscapes: Applications to Continuum Soft Matter Systems Halim Kusumaatmaja Department of Physics, University of Durham

2 Acknowledgement David Wales (Cambridge) Discussions on various numerical techniques Apala Majumdar (Bath) Collaboration on the LC eample

3 On the Menu Today The Basic Idea Motivation - Typical questions Methods - Eploring the free energy landscapes Energy Minimization Nudged Elastic Band Method Eigenvector Following Results - Specific Systems Wetting on Chemically Striped Surfaces Liquid Crystals Lipid Vesicles

4 The Basic Idea Two-dimensional energy contour plot: F (, y) The global minimum All relevant minima

5 The Basic Idea Two-dimensional energy contour plot: F (, y) The global minimum All relevant minima Saddle point

6 The Basic Idea Two-dimensional energy contour plot: F (, y) The global minimum All relevant minima Saddle point Minimum Energy Pathway Energy barriers Connectivity Global view of the free energy landscape

7 The Basic Idea Our Goal: Study the evolution of the free energy landscapes as function of the system parameters. The skeleton represents perhaps the most interesting points on the landscape. The global minimum All relevant minima Saddle point Minimum Energy Pathway Energy barriers Connectivity Global view of the free energy landscape Degrees of freedom: All possible configurations

8 Competing Pathways More than one possible minimum energy pathways Systematic algorithm to enumerate these pathways? saddle point 1 min 1 min 2 saddle point 2

9 Motivation

10 Wetting States Superhydrophobic Surfaces Suspended - Good Collapsed - bad Reentrant Post Geometry: Tuteja et al.

11 Membranes D Surface G Surface P Surface Cubic Membranes Seddon et al. Multicomponent Membranes Encapsulation of Nanoparticles Hu et al. Dasgupta et al.

12 Multistable Liquid Crystal Devices Zenithal Bistable Display Spencer et al. Defect State - bright Continuous State - dark Nematic Liquid Crystal Wells Liquid Crystal Colloids Diagonal State Rotated State Tkalec et al.

13 Methods H. Kusumaatmaja, J. Chem. Phys. 142, (2015)

14 Wetting on Chemically Striped Surfaces Wetting on Striped Surfaces Contact Angle cosθ = γ SV γ SL γ LV hydrophilic hydrophobic Free Energy Function Surface Energies Landau Free Energy Double well potential Ψ = γ LV A LV + γ SV A SV + γ SL A SL ' Ψ = ψ b + ε 2 φ 2 * ), dv + ψ ( + A da V ψ b = 1 ( 4ε φ 2 1) 2 A Surface Energy ψ A = hφ A

15 Continuum vs Discrete Continuum ' Ψ = ψ b + ε 2 φ 2 * ), dv + ψ ( + A da V A Discrete 1' ψ b dv = ε 1 2 φ 2 ijk φ * ) 4 ijk, ΔΔyΔz ( + V ijk ε 2 φ 2 dv = ε 2 φ 2 ΔΔyΔz ijk V ijk hφ A da = hφ ijk ΔΔy A ijk A Minimum/Saddle Points 1st Derivative δψ δφ! ( ) = 0 1 Minimum/Saddle Points 1st Derivative Ψ φ ijk = 0 2nd Derivative δ 2 Ψ δφ (! )δφ! 1 ( ) 2 2nd Derivative 2 Ψ φ ijk φ i' j'k' All positive eigenvalues for a minimum. One negative eigenvalue, the others are positive for a transition state. Typically 10 6 degrees of freedom.

16 Finding Minimum Free Energy States The global minimum All relevant minima

17 Basin-Hopping Three sub-steps: 1. Random perturbations Option 1: φ ijk φ ijk + d ξ Option 2: Deform the contour of the shape 2. Minimizations use LBFGS algorithm (or others) 3. Acceptance criterion Option 1: use Metropolis algorithm Option 2: accept all minima (random search)

18 Finding barriers and pathways Generally two classes of method: 1. Single ended methods start from a minimum 2. Double ended methods start with two minima of interests The global minimum All relevant minima Saddle point Minimum Energy Pathway

19 (Doubly-) Nudged Elastic Band Basic Algorithm: 1. Make an initial guess for a set of images between the two minima

20 (Doubly-) Nudged Elastic Band Basic Algorithm: 1. Make an initial guess for a set of images between the two minima Linear Interpolations - Inefficient Need to Morph the shape

21 (Doubly-) Nudged Elastic Band Basic Algorithm: 1. Make an initial guess for a set of images between the two minima 2. Connect these images by springs Spring potential energy: α V spring (( ) 2 ( s α,+ ) 2 ) = k 2 sα,

22 (Doubly-) Nudged Elastic Band Basic Algorithm: 1. Make an initial guess for a set of images between the two minima 2. Connect these images by springs 3. Rela using projections from the true gradient and from the spring force component

23 (Doubly-) Nudged Elastic Band Basic Algorithm: 1. Make an initial guess for a set of images between the two minima 2. Connect these images by springs 3. Rela using projections from the true gradient and from the spring force component Perpendicular component of the true gradient: g α = g α ( g α ˆ τ α ) ˆ τ α Avoid all images collapsing to minima

24 (Doubly-) Nudged Elastic Band Basic Algorithm: 1. Make an initial guess for a set of images between the two minima 2. Connect these images by springs 3. Rela using projections from the true gradient and from the spring force component Parallel component of the spring force: g α = k( s α,+ s α, ) ˆ τ α Maintain equal distance between images Avoid corner-cutting

25 (Doubly-) Nudged Elastic Band Basic Algorithm: 1. Make an initial guess for a set of images between the two minima 2. Connect these images by springs 3. Rela using projections from the true gradient and from the spring force component Parallel component of the spring force: g α = k( s α,+ s α, ) ˆ τ α Maintain equal distance between images Avoid corner-cutting

26 Hybrid Eigenvector-Following Further optimise using hybrid eigenvector-following: 1. First, remember that we want to find saddle point of inde 1 2. Uphill step in one eigendirection 3. Minimization in the tangent space Candidate for a transition state Optimize using eigenvector-following Obtain a minimum energy path Keep iterating until all minima and transition states are found!

27 Hybrid Eigenvector-Following Further optimise using hybrid eigenvector-following: 1. First, remember that we want to find saddle point of inde 1 2. Uphill step in one eigendirection 3. Minimization in the tangent space 4. Remove zero eigenvectors Zero Eigenvectors Arise when periodic boundary conditions are used y z In this eample, for the y-direction: ˆ e = 1 N $ & % y φ ijk y φ ' NNyNz ) y ( φ 111

28 Many Ways to Rome With Joao Louis Carabetta

29 Results

30 Wetting on Chemically Striped Surfaces Surface Energies Ψ = γ LV A LV + γ SV A SV + γ SL A SL Contact Angle cosθ = γ SV γ SL γ LV Clear reaction pathway

31 Wetting on Chemically Striped Surfaces Clear reaction pathway Minimum Energy Path

32 Vesicle Shapes I Helfrich Free Energy Ψ = A κ $ ' & ) 2 % R 1 R 2 ( 2 da + PV + γa Curvature Energy Volume Conservation Area Conservation Energy scales for bending: e.g. for a vesicle of size 1 micron.! 1 F ~ κ b # " R 2 $ &A ~ κ b k B T % Energy scales for shearing: ~0. This is a consequence of membrane fluidity. For eample, when where there is a spectrin (cytoskeleton) network (e.g. red blood cells), this contribution is non-zero and can be important. Energy scales for stretching: e.g. for a vesicle of size 1 micron to be stretched by 1%. " F ~ κ s A$ ΔA # A % ' & 2 ( ) 2 $ 1 60 k B T nm nm " % ' # 100 & k B T For this reason, the membrane area is also assumed to be constant, when considering vesicle shapes that arise due to bending energy.

33 Vesicle Shapes I Helfrich Free Energy Ψ = A κ $ ' & ) 2 % R 1 R 2 ( 2 da + PV + γa Curvature Energy Volume Conservation Area Conservation Landau Free Energy [Du et al. (2004)] Ψ = κε 2 Δφ 1 2 ( + * ( φ 2 1)φ - dv + 1 ) ε 2, 2 k V V V 0 V ( ) k A ( A A 0 ) 2 Reduced volume ( ) ( ) = / 2 ν = V 0 / 4π /3 A 0 /4π Reaction pathway unclear

34 Vesicle Shapes I Helfrich Free Energy Ψ = A κ $ ' & ) 2 % R 1 R 2 ( 2 da + PV + γa Curvature Energy Volume Conservation Area Conservation Landau Free Energy Ψ = κε 2 Δφ 1 2 ( + * ( φ 2 1)φ - dv + 1 ) ε 2, 2 k V V V 0 V ( ) k A ( A A 0 ) 2 Reduced volume ( ) ( ) = / 2 ν = V 0 / 4π /3 A 0 /4π

35 ν = Vesicle Shapes II

36 Vesicle Shapes II ν =

37 Liquid Crystals Eperimental Setup Tsakonas et al., APL 90, surface anchoring

38 Liquid Crystals H. Kusumaatmaja and A. Majumdar, Soft Matter 11, 4809 (2015) Landau-de Gennes Free Energy Ψ = κ el 2 Q 2 da + ( αtrq 2 b2 3 TrQ3 + c 2 * ) 4 TrQ2 A A Bending Energy Bulk Energy # Where the Q-tensor is defined as Q = Q 11 Q 12 & % ( = s 2n n I $ Q 12 Q 11 ' ( ) ( ) da, in two dimensions. Dimensionless form Ψ = [ ] κ el Q 2 + Q 2 da A As for a scalar order parameter Ψ = f ({ Q 11,Q 12 } ijk ) ( Q 2 + Q 2 1 ) da A + surface anchoring A W ( Q Q wall ) 2 dl Anchoring Strength W

39 Rotated and Diagonal States Rotated States Beyond the minimizers: 1. Pathways and barriers 2. Competing pathways? 3. Connectivity 4. Stability of Minima Diagonal States

40 One Possible Pathway

41 Strong Anchoring Pathways - ½ defect + ½ defect Pair of defects

42 Order Parameter s (a) (b) (c) - ½ defect (d) (e) (f) Remember that: Q = s( 2n n I) + ½ defect (g) (h) (i) Pair of defects

43 Strong Anchoring Pathways + ½ defect Pair of defects

44 Energy Profiles Strong Anchoring Varying Anchoring Strengths

45 Pathways Depend on Anchoring Strengths Medium Anchoring Strength Two Degenerate Pathways Weak Anchoring Strength

46 Medium Anchoring Two Degenerate Pathways

47 Weak Anchoring

48 Energy Profiles Strong Anchoring Varying Anchoring Strengths

49 Catastrophe Events Fold Catastrophe Cusp Catastrophe

50 Only Diagonal States Are Stable Reminiscent of Rotated State Minimum Transition State Minimum

51 Only Diagonal States Are Stable Four equivalent pathways

52 Take Home Messages 1. Tools to survey the free energy landscapes of Landau models in details, including how they vary with system parameters, boundary conditions, and eternal perturbations. Most, if not all, relevant minima Most, if not all, transition states and competing minimum energy paths Connectivity of the landscapes Catastrophe events/phase transitions 2. Cannot do dynamics. However, the descriptions and discretizations of the functionals are fully compatible with other methods. 3. Can take advantage of schemes already developed for handling comple boundaries (corrugations, curved boundaries, etc) from other methods. 4. Scalar, vector, and tensor fields can be treated on equal footing. Landau free energies are very popular in physics, chemistry, and materials science. We are interested in testing the limits of the methods. 5. Relatively inepensive (especially compared to atomistic calculations). Typical minimization runs take O(minutes). Run in parallel Typical pathway runs take O(1 hour).

53 Thank you for listening!