Entropy of Concentrated Systems

Story begins with a virus that killed tobacco plants Tobacco mosaic virus (TMV)

Structure of TMV

Organized rods led to plant death Conventional wisdom: van der Waals attraction between rods causes organization. But Lars Onsager thought about differently. Organized rods discovered by Stanley (Berkeley).

Phase behavior of non- interacting rods Some viruses are exact opposites of flexible polymer chains.

Disk of influence For an isotropic solution, rods must rotate freely. i.e. no other rods must lie within the "disk of influence". Disk vol.=Excluded vol.=V ex =  L 2 d/4=L 2 d, ignoring prefactors L d

Disk of influence When neighboring rods enter disk of influence, entropy is reduced. allowed configurations disallowed configurations

Critical concentration Solute volume fraction in disk = d 2 L/L 2 d=d/L. If overall volume fraction  >d/L then disks will be invaded. For TMV,  critical =18/300<1%. L d

Aligned state entropy At a high enough concentration, aligned state entropy exceeds unaligned state entropy, and spontaneous alignment is obtained.

Numerical values of critical concentration  >d/L simple analysis  >4d/L Onsager analysis  >10d/L Flory analysis Bottom line: Rods spontaneously form a nematic phase when  >d/L upto a pre-factor.

Nematic liquid crystal n director

Nematic liquid crystal n director Only orientational order (crystal), no positional order (liquid) center of mass locations

Alignment responds to quickly to electric fields (ms) V onV off not birefringentbirefringent

LCD displays

Simulations  >d/L simple analysis  >4d/L Onsager analysis (1949)  >10d/L Flory analysis Bottom line: Rods spontaneously form a nematic phase when  >d/L upto a pre-factor. Much or our knowledge about hard spheres, rods, disks etc. is based on simulations.

Probability of inserting an additional sphere Computer simulations showed that the probability of inserting a sphere vanishes as the sphere volume fraction approaches However, closed packed sphere volume fraction (HCP) is In 0.67<  <0.74 range, hard spheres must be crystalline!

Imagine evaporating solvent in a hard sphere suspension

Sphere can lie anywhere in circle with equal probability-no springs holding the crystal at the mean position

 =1/1.49=0.67  =1/1.37=0.73 At constant P and N there would be a jump in V (or  ).

Hard sphere colloids

Non-spherical, patchy colloids dimertrimertetramer 1  m Dave Pine, NYU Colloidal silicon!

Photonic structure Norris, U. Minnesota

Photonic properties Comparison of theoretical and experimental band structure. (a) transmittance at normal incidence (111 plane). (b) band diagram along 111 direction (c,d,e) band diagrams along 110 direction showing band gaps. Can we control the flow of light using photonic crystals and thus use photons for logic circuits instead of electrons? A diamond structure exhibits the largest photonic bandgap.

Entropy of Concentrated Systems Fundamental questions: Can ordered states be found in a system with E=0 ? Can an ordered state have higher entropy than a disordered state at the same temperature?