Predicting ordered equilibrium structures for patchy particles

Predicting ordered equilibrium structures for patchy particles

Gerhard Kahl

Institut für Theoretische Physik and CMS, Technische Universität Wien, Austria

MECO 36, Lviv, April 5 ^th – 10 ^th 2011

Outline

1 Introduction

2 Model and theoretical methods

3 Results

4 Summary & Outlook

Patchy particles

Patchy particles are ...

(spherical) colloidal particles decorated on their surface by mutually attractive/repuslvie regions (patches)

patches can be realized via

◦ areas of different physical/chemical properties (magnetic particles grafted to a colloid, hydrophobic regions, ...)

◦ grafted polymers, double- and single-stranded DNAs, ...

◦ ...

consequences:

highly orientational effective interactions between patchy particles

selective bonding between particles

potential for complex self-assembly scenarios

ideal building blocks of larger entities

Patchy particles: example no. 1

from: O.D. Velevet al., Macromol.31, 190 (2010)

Patchy particles: example no. 2

from: D.J. Kraft, J. Groenewold, W.K. Kegel, Soft Matter5, 3823 (2009)

Patchy particles: example no. 3

triblock Janus particles

experiment

Q. Chen, S.C. Bae, and S. Granick, Nature496, 381 (2011)

theory

from: F. Romano and F. Sciortino, Nat. Mater.10, 171 (2011)

Patchy particles: theoretical models

spot-like patches

E. Bianchi,et al., Phys. Rev. Lett.97, 168301 (2006)

polyelectrolyte stars adsorbed on a charged colloid

Registered Charity Number 207890

www.chemistryworldjobs.com Upload your CV and profile today!

Discover something NEW

Your NEW recruitment site dedicated to chemistry and the chemical sciences Create a free account to get the most from chemistryworldjobs.com

Get headhunted Create a profile and publish your CV so potential employers can discover you Stay ahead of your competition Set up your job alerts and receive relevant jobs in your inbox as soon as they appear Discover your next career move Detailed searches by role, salary and location Save time and be the first to apply Online vacancy application Be efficient Bookmark jobs that interest you, so you can come back to them later

%HGLVFRYHUHG

ISSN 1463-9076 Physical Chemistry Chemical Physics

www.rsc.org/pccp Volume 13 | Number 14 | 14 April 2011 | Pages 6373–6712

COVER ARTICLE Bianchi et al.

Patchy colloids: state of the art and perspectives

HOT ARTICLE Goerigk and Grimme A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions Downloaded on 23 March 2011Published on 14 April 2011 on http://pubs.rsc.org | doi:10.1039/C1CP90037D

View Online

microscopic model

coarse grained model

see also:E. Bianchi, R. Blaak, and C.N. Likos, PCCP63, 1397 (2011);

E. Bianchi, G.K., and C.N. Likos, Soft Matter (submitted)

Outline

1 Introduction

2 Model and theoretical methods

3 Results

4 Summary & Outlook

Model

interparticle potential (2D)

two interacting patchy particles

V = V (r

, Θ

, Θ

)

J.P.K. Doye,et al., PCCP92197 (2007)

Theoretical methods – No. 1

goal:

predict ordered equilibrium configurations for the system i.e., configurations of minimum energy

ensemble:

NPT ensemble with (i) T = 0 and then (ii) T > 0 thus, minimizing Gibbs free energy leads to optimized

◦ lattice parameters

◦ equilibrium density (volume)

tools:

genetic algorithms

(Monte Carlo simulations)

Theoretical methods – No. 2

genetic algorithms – an efficient optimization tool

◦ very general optimisation strategies

◦ model natural evolutionary processes

(mutation, recombination, survival of the fittest)

◦ probe in an unbiased way the entire parameter space

◦ cope extremely well with rugged energy landscapes and high dimensional parameter spaces

idea:

1. treat any possible lattice structure as an individual 2. expose individual on the computer to an artifical evolution

condition for survival: minimize thermodynamic potential 3. mutation & recombination ⇒ large number of generations 4. ’in the end’ the ’fittest’ (i.e., energetically most favourable)

individual survies ⇒ ordered equilibrium structure

Theoretical methods – No. 3

Monte Carlo simulations

(a) NPT ensemble with variable cell geometry

◦ start from a random configuration

◦ vary temperature and/or pressure

◦ let particles find their energetically optimized arrangement (b) NPT ensemble & thermodynamic integration (Frenkel-Ladd)

⇒ thermodynamic properties at arbitrary T

Outline

1 Introduction

2 Model and theoretical methods

3 Results

4 Summary & Outlook

overview

1. self-assembly in two-dimensional systems

2. self-assembly in three-dimensional systems

3. formation of clusters

Results (2D) – No. 1

three-patch system

P = 0 . 5 P = 4 . 5

G. Doppelbauer, E. Bianchi, and G.K., JPCM22, 104105 (2009)

Results (2D) – No. 2

three-patch system

P = 0 . 5

P = 2 . 0

P = 4 . 0

P = 7 . 0

-2 -1 0 1 2 3 4 5 6

1 2 3 4 5 6

G⋆,U⋆,1/(ησ2)

P^⋆ U^⋆ 1/(ησ²) G^⋆=U^⋆+P^⋆/(ησ²)

thermodynamics

Results (2D) – No. 3

four-patch system

P = 0 . 5

P = 0 . 5

P = 6 . 0

P = 2 . 5

Results (2D) – No. 4

five-patch system with attractive and repulsive patches

P = 0 . 5

P = 1 . 5

P = 4 . 0

P = 8 . 0

Results (3D) – No. 1

four-patch system

90 120 150

g

lattice sum (energy)

(pressure vs. patch decoration )

double layers

150 120

090 2 4 6 8 10

g

P★

-2.00 -0.90

-1.45

open

bcc-like I bcc-like II

fcc-like II

fcc-like I bct-like Ibcc-like III hcp-like I hexagonal layershcp-like III

fcc III

fct I fct-like II -like II

U

hcp hcp-like VI hcp-like V

bct-like II

Results (3D) – No. 2

four-patch system

90 120 150

g

equilibrium volume

(pressure vs. patch decoration )

double layers

150 120

090 2 4 6 8 10

g

P★

0.72 1.84

1.28

open

bcc-like I bcc-like II

fcc-like II

fcc-like I bct-like Ibcc-like III hcp-like I hexagonal layershcp-like III

fcc III

fct I fct-like II -like II

V

hcp hcp-like VI hcp-like V

bct-like II

Results (3D) – No. 3

configurations (in two perpendicular views)

90 120 150

g

g ∼ 95

low-pressure phase

g ∼ 109

(tetrahedral patch arrangement)

low-pressure phase

Results (3D) – No. 4

configurations (in two perpendicular views)

90 120 150

g

g ∼ 124

low-pressure phase

high-pressure phase

g ∼ 150

low-pressure phase

high-pressure phase

Results (clusters) – No. 1

formation of clusters by patchy particles (tetrahedral symmetry) energies

Energy per particle

Energy

Number of particles

−1.6

−1.5

−1.4

−1.3

−1.2

−1.1

−1.0

−0.9

−0.8

−0.7

−0.6

0 5 10 15 20 25 30 35 40 45 50

cluster energies vs. cluster size

building entities

five-particle ring

six-particle ring

Results (clusters) – No. 2

disconnectivity graphs

−13 .0

−12.5

−12.0

−11.5

−11.0

−10.5

−10.0

−9.5

−9.0

−30.0

−29.0

−28.0

−27

.0

−26

.0

−25

.0

−24.0

n = 10 n = 20

D.J. Wales,Energy Landscapes(Cambridge, 2003)

Outline

1 Introduction

2 Model and theoretical methods

3 Results

4 Summary & Outlook

Summary & Outlook

System:

even simple models of patchy particles show a rich variety of self-assembly scenarios

structures can be understood in terms of competition of bond saturation (i.e., energy) and packing

Methods:

genetic algorithms are very reliable, powerful, and attractive optimization tools even for this delicate problem

Monte Carlo simulations perform not too good, in particular at high pressure values (particles get trapped due to their patches)

Outlook:

include finite temperature

thermodynamic integration with Monte Carlo simulations

within the harmonic approximation (including vibrations and

rotations)

Thank you!

Günther Doppelbauer TU Wien Emanuela Bianchi TU Wien

Eva Noya CSIC Madrid

David J. Wales University of Cambridge Dwaipayan Chakrabarti University of Cambridge

FWF for financial support