SURFACE TENSION 2SPS Lectures January 2006

Wayne Lawton

Department of Mathematics National University of Singapore

http://math.nus.edu.sg/~matwml

matwml@nus.edu.sg

ABSTRACT

The Journal of Chemical Physics -- September 1, 2000 -- Volume 113, Issue 9, pp. 3882-3893

Spatial and energetic-entropic decomposition of surface tension in lipid

bilayers from molecular dynamics simulationsErik Lindahl and Olle Edholm

Theoretical Physics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden

We explain molecular cause of surface tension using thermodynamic concepts that explain the role of both energy and entropy – cutting edge concepts in biochemistry and life sciences.

ENTROPYThermodynamics

Statistical Mechanics

PdVqE 1st Law

2st Law

ofenergy Eby absorbedheat q

todone work PdV

TdSq of ure temperatT

ofentropy S

WkS logBoltzmann’s

tombstone

constant B s'k

smicrostate # W

EQUIPARTITIONEntropy of a System With Two Subsystems

2121211212 loglogloglog SSWkWkWWkWkS

21 EE )()( 2211 ESES therefore

EP Theorem Microstates have equal probability

12'211

'122

'211

'1 )()()()(0 EESEESEESEES

Hence 2nd Law 122

'21

'1

11 )()( TESEST

Corollary Two subsystems in thermal equilibrium with constant total energy will maximize

EP Theorem (Boltzmann) Each translational or rotational component of the random thermal motion of a molecule has an average kinetic energy 2/kT

A = HELMHOLTZ FREE ENERGYWe consider a constant volume system whose entropy S = S(E) that is in thermal equilibrium with an infinite reservoir that has temperature T

Theorem Energy will flow into / out of the system so as to minimize A(E) = E – TS(E)

Proof At thermal equilibrium the total entropy is

))()(()()( ESEESTEEAEEA therefore for every value of

is maximized 0)()( 1 ETESEESE

0)( 1 ETTERemark 0)(1))((1)( 1'' TTESTEA

We consider a system consisting of molecules that can be in states 1 or 2 having respective energies

ENDOTHERMIC REACTIONS

21,EETheorem The fraction p of moleculesin state 1 satisfies )/()(log 121 kTEEp

p

Proof For a system of N molecules the binomial theorem and Stirling approx

)]1log()1(log[)1(/)( 21 ppppkTEppENpA

and the result follows since 0)(' pA

Enthalpy

SURFACE TENSION THERMODYNAMICS

reaAPVEH Guggenheim-Hill [1] incorporate

Gibbs Free Energy TSHG Systems in therm. equil. minimize G

G0 reaATSPV

reaAVPSTE 2nd Law&surf. ten.into

hence into

TSSTAAVPPVE rearea

VEreaVSrea A

ST

A

E

,,

P

rea TAS

TUTORIAL PROBLEMS

3. Study the role of entropy in the chemical equilibrium formula in http://en.wikipedia.org/wiki/Chemical_equilibrium

1. Boltzmann’s formula uses the natural log and log W gives information in nats. How many bits of information = 1 nat ?

2. Derive A(p) endothermic in reactions

1. Carry out experiments described inhttp://www.iit.edu/~smile/ch8623.html

RESEARCH PROJECTS

2. Carry out experiments described in [3]

[1] Chemistry of Interfaces, M. J. Jaycock and G. D. Parfitt, Ellis Horwood, Chichester, 1986.

REFERENCES

[2] Dynamics of Surface Phenomena, P. Joos, Ridderprint, Utrecht, 1999.

[3] Science with Soap Films, D. Lovett, Institute of Physics Publishing, Bristol, 1994.