1

Nanoscale Physics and Modeling

  Lectures: Tue and Thu, 5:00-6:20pm HSS 1138

  Instructor: Gaurav Arya, Assistant Professor of

Nanoengineering [garya@ucsd.edu]

  URL: http://maeresearch.ucsd.edu/arya/MAE207.html

  Office hours: 5pm Friday, #2304 Atkinson Hall

  Grading: 50% Homework assignments and 50%

Computer assignments

2

Course Syllabus

  Intra and intermolecular interactions: ionic, covalent, and metallic bonding; van der Waals, charge-charge, charge-dipole, and dipole-dipole, hydrogen bonding, pi-pi stacking, and hydrophobic interactions

  Molecular mechanics: mathematical description of molecules and their interactions (force fields)

  Energy minimization: Simplex, Newton Raphson, steepest descent, conjugate gradient, simulated annealing

  Statistical mechanics: ensembles, Boltzmann distribution, partition function, entropy and free energy, thermodynamic properties from statistical mechanics, molecular simulations

  Molecular dynamics: Newton’s equations of motion, periodic boundary conditions, thermostats, integrators, ensembles

3

Course Syllabus (cont…)

  Monte Carlo: Monte Carlo integration, importance sampling, Markov chains, detailed balance, biased sampling, ensembles

  Calculation of equilibrium/dynamic properties: radial distribution functions, autocorrelation functions, transport properties, free energy computation

  Stochastic simulations: Langevin and Brownian dynamics; Gillespie algorithm, hydrodynamics

  Interactions between surfaces: Hamaker constants, Poisson-Boltzmann equation, Debye-Huckel theory, DLVO theory

  Polymer physics: distributions and scaling properties, Flory-Huggins theory, lattice polymers, Rouse theory

  Self-assembly: critical micelle concentration, DNA hybridization, membranes, phase behavior versus micellization

4

Reference Materials   A. R. Leach, "Molecular Modeling: Principles and Applications", Addison

Wesley Longman, Essex, England (2001)

  D. Frenkel and B. Smit, "Understanding Molecular Simulations: From

Algorithms to Applications", Academic Press, San Diego, California (1996)

  M. P. Allen and D. J. Tildesley, "Computer Simulation of Liquids", Clarendon

Press, Oxford (1990)

  D. Chandler, “Introduction to Modern Statistical Mechanics”, Oxford

University Press, Oxford (1987).

  G. D. J. Phillies, “Elementary Lectures in Statistical Mechanics”, Springer-

Verlag, New York (2000)

  K. A. Dill and S. Bromberg, “Molecular Driving Forces”, Garland Science, New

York (2002)

  J. Israelachvili, “Interfacial and Surface Forces”, Academic Press, London (1991)

5

Lecture 1 Intra and intermolecular interactions

6

Types of forces   Intramolecular forces

  Ionic bond

  Covalent bond

  Metallic bond

  Intermolecular forces

  Charge-charge interactions

  charge-dipole, Dipole-dipole interactions

  Dispersion/London interactions

  Pi-pi stacking interactions

  Hydrogen bonding

 Entropic forces (not fundamental)

  Hydrophobic interactions

  Depletion interactions …

Compare to fundamental forces defined by physicists

•  Gravity •  Electromagnetism •  Strong interaction •  Weak interaction

7

Ionic bonding

 When two atoms of widely different electronegativities combine

  E.g. KBr (Br holds tighter to its outer pair of electrons than K)

 K -> K+ + e- I = +420 kJ/mol

 Br + e- -> Br- I = -325 kJ/mol

 Net effect is energy gain of 95 kJ/mol (unfavorable)

 But attraction between opposite ions K+ and Br- in a lattice reduces energy

 Energy of one ions pair = -589 kJ/mol => net effect now becomes favorable

  Infact, ions are rarely found in pairs but form an infinite lattice (KBr crystal) so net electrostatic attraction even larger = -672 kJ/mol

€

U =q1q24πεε0

1r r q1

q2

ε0: permittivity of vacuum; ε: dielectric constant

Coulomb law:

8

Covalent bonding

  Chlorine has seven valence electrons 3s23p5

  The two 3p orbitals in the two Cl atoms share the single unpaired electron by overlapping in between the two atomic centers (bonding orbital)

  That way nucleus-electron attraction maximized and nucleus-nucleus repulsion minimized (screening)

Two atoms of large and similar electronegativities prefer to share the electrons => that way both atoms attain full valence shells

9

Metallic bonding

  Consider Na: Single electron in 3s orbital, and

3p orbitals empty

  If the 3s orbitals share the two unpaired

electrons in the regions, the 3p orbitals still

remain empty (unfavorable)

  Also, if this sharing occurs between the atoms,

the (+)ve charge on nucleus on either side of

the pair will become very accessible to the

electrons from other Na atoms

  Other Na atoms join the chain, and so and so

forth, a lattice forms (Na lattice)

  Electrons shared between chain of Na atoms are

considered “delocalized”

Elements with similar but low electronegativities (metals) form metallic bonds

10

Charge-charge intermolecular interactions

 Two charges q1 and q2 interact through Coulomb law:

 But charge-charge interactions are much weaker in biological systems (WHY??)

  Dielectric constant of water is large

  Electrostatic screening by the salt

€

U =q1q24πεε0

1r

€

U ~ q1q2exp(−κr)

rinverse Debye

length

ε0: permittivity of vacuum; ε: dielectric constant

11

Electrostatic interactions at work

Compaction of genomic DNA through histone proteins

Properties of DNA

Salt bridges in proteins (Barnase)

between Asp and Arg

12

Charge-dipole interactions

 Can be repulsive and attractive depending upon the charge sign and position of the charge relative to the dipole

 In reality, dipoles are free to rotate and choose favorable positions, therefore charge-dipole interactions are always attractive (Boltzmann averaging)

+ +

–

r Electrostatic energy depends as 1/r2 if the charge interacts with a fixed dipole

Electrostatic energy depends as 1/r4 if the charge interacts with a free dipole (HOMEWORK)

Hydration of ions

θ

€

U =qµ4πεε0

cosθr 2

; µ = qll

(HOMEWORK)

13

Dipole-dipole interactions (Keesom forces)

 Can be repulsive and attractive depending upon the charge sign and position of the charge relative to the dipole

 Again, dipoles are usually free to rotate and choose favorable positions, and the average Boltzmann averaged energy is always negative

 We can also have induced-dipole/

dipole interactions (1/r6)

Electrostatic energy depends as 1/r3 if two dipoles interacts with each other

Electrostatic energy depends as 1/r6 if the dipoles free to rotate

+

–

r

14

Hydrogen-bonding: Special case of dipole-dipole interaction

 Special type of dipole-dipole bond that exists between an electronegative atom and a hydrogen bonded to another electronegative atom (O-H···N)

 Weaker than covalent and ionic bond

 Typical length of a hydrogen bond ~2 Angstroms

H-bonding in DNA H-bonding in α-helix

15

Instantaneous dipole-induced dipole (dispersion/London forces)

 One of the most basic interactions between atoms (even spherical ones)

 Weak at room temperatures (<kBT) but can strong enough to condense gases at lower temperatures

H2 molecule (average)

H2 molecule (instantaneous view)

H atom (instantaneous view)

What happens when another H2 molecule approaches?

OR

Also 1/r6 dependence?

Homework problem: Prove this

16

Van der Waals interactions (dispersion + Keesom forces)

 Large magnitude range for dispersion interactions (0.05-40 kJ/mol)

  Increases with the size of the atoms (no. of electrons)

CHCl3 61.2°C CCl4 76.8°C

Boiling Points

Example 2:

Example 1:

17

van der Waals interactions at work

 Attractive interactions between carbon nanotubes which makes their separation so difficult

 Geckos feet that allows to walk upside down

 Liquefaction of gases (nitrogen,methane, etc)

 Crystallization of many proteins

18

Pi-Pi Stacking Interactions

 Attractive interaction between stacked aromatic rings through mutual overlap of the p orbitals of Π-conjugated systems

 As strong a driving force as H-bonding for DNA double helix formation

Can also have other Pi interactions: cation-Pi; sigma-Pi; etc

19

Hydrophobic Interactions

 Effective attraction that arises between nonpolar molecules in water solvent - this minimizes the disruption of the favorable H-bonded network of water molecules

 Mostly an entropic effect

Hydrophobic collapse of protein folding Lipid bilayer formation