Charge mobility in dendrimers

This post was stimulated by a talk at last weeks COPE meeting and looking at this 2001 Phys. Rev. B paper from the groups of Samuel and Burn.


In the attached rough notes I try to provide a framework to answer questions such as:

What determines the charge mobility in an array of these systems? How can it be maximised?

What is the relative importance of the conjugated and non-conjugated components of the dendrimer?

I take it the non-conjugated surface groups are required to make these systems soluble.
It seems that the mobility falls off exponentially fast with the length of the non-conjugated molecules at the surface. (roughly an order of magnitude for every carbon in the chain?)

Understanding the notes may be made easier by ready some of my earlier organic electronics posts, especially this one on Hush-Marcus electron transfer theory.

Some key ideas for biophysics from basic physics and chemistry

Section 1.5 of Biological Physics by Nelson lists

1.5.1 Molecules are small

He describes how in 1773, Benjamin Franklin estimated the linear size of a single molecule from observing that one teaspoon of oil covered half an acre of pond!

1.5.2 Molecules are particular spatial arrangements of atoms

1.5.3 Molecules have well-defined internal energies

1.5.4 Low-density gases obey a universal law

Boltzmann's constant is universal, i.e., it does not depend on the chemical or structural details of the system being described.

At room temperature

k_B T = 4.1 pN nm (most important formula in this book!)

I tend to think k_B T = 25 meV since this is a useful scale in solid state physics.
But, Nelson deliberately writes it terms of a force scale (picoNewton) and length scale (nanometer) that is relevant to the physics of cells.

How to do better on exams (and discover new physical laws)

This is the title of section 1.4 in Phil Nelson's book, Biological Physics.
I really like the way he uses subsection titles which convey useful information. Here are some:

1.4.1 Most physical quantities carry dimensions

1.4.2 Dimensional analysis can help you catch errors and recall definitions

1.4.3 Dimensional analysis can also help you formulate hypotheses

I think something that we just have to keep "hammering" students on is keeping track of units at each step of a calculation and canceling them out.

The ten commandments of protein folding

It is always interesting to see what former students are up to. 15 years ago Bosco Ho was in an undergraduate Quantum Mechanics class I taught at UNSW.

While trying to find an answer to a student question on the PHYS3170 blog,
I found Bosco's Ten Commandments of Protein Folding.

Can we see visons?

The organic charge transfer salt -(ET)₂Cu₂(CN)₃ has attracted a lot of attention the past few years because there is significant experimental evidence that the ground state of the Mott insulating phase is a spin liquid.

I have been reading a very interesting theoretical paper by Qi, Xu, and Sachdev that presents a highly original (and exotic) explanation for thermal conductivity and nuclear magnetic resonance experiments on this material.

At low temperatures the NMR relaxation rate 1/T1 goes to zero as T^\eta with eta ~1.5 and the thermal conductivity (T)/T goes to zero in an activated fashion with a gap of about 0.5 K.

The authors propose the ground state is a Z2 spin liquid close to a quantum critical point with quasiparticles that are spin-1/2 bosons (spinons) and spinless bosons (visons).
It is shown that spinons dominate the nmr and visons the thermal conductivity.
The visons form a dilute Boltzmann gas with a bandwidth of about 8 K, which the authors claim corresponds to the peak observed in the heat capacity and thermal conductivity. Note that this bandwidth is only about 3 per cent of the exchange interaction J, which sets the energy scale for the spinons.

The visons correspond to low-energy singlet excitations and can be viewed as vortices in the Z2 gauge field associated with a liquid of resonating valence bonds.

These are bold hypotheses.

I worry how robust the thermal conductivity data is. Is there any chance that at these low temperatures the suppression is due to a decoupling of the magnetic excitations from the phonons, as was observed in cuprates and explained by Mike Smith. The experimentalists claim not.
The thermal conductivity data is inconsistent with the heat capacity data, i.e., (T)/T does not extrapolate to a non-zero value. So at least one of them must be wrong.

Static versus dynamic correlations in quantum chemistry

On Friday we had a great visit from Weitao Yang. One thing (among many others) I learnt from him was his definition of static versus dynamic correlations in quantum chemistry. This is something that people talk about but I have trouble getting a clear definition that I can understand.
Weitao illustrated this using the two-site Hubbard model and the hydrogen molecule.

Near the equilibrium geometry of the molecule, U/t is not large and a Hartree-Fock wavefunction is qualitatively but not quantitatively correct. The corrections to this are dynamical correlations.

At large separations, U/t is large, and the wavefunction approaches a Heitler-London state, and a Hartree-Fock wavefunction is qualitatively incorrect, e.g., it claims the H atoms will not bond to each other! This is static correlations.

Somehow I wonder if there is a way to quantify this distinction in terms of entanglement measures.

Why physicists and biologists need each other

I really enjoyed reading the rest of Chapter 1 of Biological Physics in preparation for thursday's class discussion. Section 1.3 has a nice discussion of the relationship between physical and biological sciences. Physicists seek the universal and simple in any system. In contrast, biologists when confronted with the complexities of the biosphere are more likely to emphasise "frozen accidents of history" and focus on details.

Figure 1.4 is nice but does not scan well and so I don't reproduce it here.

How does one synthesize these complementary approaches? First appreciate the value of each. Nelson suggests 3 steps for scientific advance:

"a) select a simplified but real model system for study
b) represent this system by a mathematical model with as few parameters and variables as possible.
c) deduce from the mathematical model some nonobvious, quantitative, and experimentally testable predictions."

He emphasises that a) and b) are inductive whereas c) is deductive.
a) and b) require a thorough knowledge of the biology.
Physicists need to be wary of proposing models that "lead to a large body of both theory and experiment culminating in irrelevant results."

Nelson points out that the best models may lead "to postulating entities whose very existence wasn't obvious from the observed phenomena."
Historical examples of this include:

Max Delbruck's deduction of the existence of a hereditary molecule (chapter 3)

the discovery of ions pumps and ion channels in cells (chapter 11,12)

George Gamow's proposal to find the genetic code