Crystalline solid and liquid states

....The fundamental differences between the crystalline solid and liquid states is that there is long range order in the crystalline solid but there can only be local order in the liquid. Amorphous solids (e.g. ordinary glass) also have no long range order and there is little difference between the structures of liquids and amorphous solids. They can, however, be distinguished by the time scale of the motion. In a liquid a molecule explores the whole space occupied by the liquid in a relatively short time whereas the movement between adjacent positions in amorphous solids is more or less infinitely slow. For the purposes of assessing the role of molecular interactions on the nature of solids, liquids and gases, it is important to have a common means of describing their structures.





The Radial Distribution Function

The rapid molecular motion in a liquid and its lack of long range order make it difficult to describe. The function that is used to describe the average structure is the radial distribution function, usually written g(r). To construct g(r) we start by taking an instantaneous snapshot of liquid, as shown in a 2-D representation in the diagram alongside. With reference to the purple coloured molecule there are two groups of molecules, one whose centres are at a distance of about a molecular diameter from that of the reference molecule (green) and one whose centres are at about two molecular diameters (pink). The range of distances between the reference molecule and the molecules in the first coordination shell is much narrower than for the second coordination shell.

Taking the centre of the purple reference molecule as being at r = 0 we construct a histogram of the molecules at a given distance r from the reference molecule. A typical result would be as shown in the second diagram (distances correspond approximately to liquid argon). The first coordination shell gives rise to a noticeable cluster whereas the second coordination shell is more diffuse, and it becomes difficult to discern a pattern for subsequent coordination shells. We now repeat this process for every other molecule in the liquid, taking as reference molecules A, B, C, D, etc. for the same instantaneous snapshot (see figure below). Each gives a slightly different histogram. Finally we average the results of all the histograms. The resulting plot, shown below right for liquid argon at its triple point, gives the probability of finding a molecule at a given distance from any chosen central reference molecule.

David Wolff; Dept of Physical Chemistry;
http://physchem.ox.ac.uk/~rkt/teaching.html

Demonstration by java applets





We have investigated the size and shape distributions of Pd nanocrystals on SrTiO3(001) supports. Our results show that depending on reconstruction and temperature during deposition three nanocrystal shapes can be created. We have studied the growth shapes of these nanocrystals: huts grow preferentially along their (001) end facets, hexagons grow preferentially in width rather than height, and truncated pyramids can become elongated if their density is sufficiently high. Due to its diversity in supported nanocrystal shape, Pd on SrTiO3(001) should be regarded as a model system for the correlation of nanoparticle shape with optical and catalytic
properties.

...As long as nanocrystal nucleation is not defect mediated, the nucleation density is predicted to drop as the substrate temperature is increased, and this is indeed
what we find. However, the effect of lower nanocrystal density is that the nanocrystals are on average larger. During the postanneal the nanocrystals undergo atomic reorganization to reach their equilibrium shape. At a constant temperature, the time it takes for a particle to reach its equilibrium shape is proportional to the fourth power of its radius.


Silly F, 2005, Growth shapes of supported Pd nanocrystals on SrTiO3(001), Dept of Materials, Oxford

ora.ouls.ox.ac.uk


THE GLASSY STATE


The solid state: crystals and glasses

Organic matter often crystallizes only in part, or not at all. In particular, polymers often solidify in the form ofa glass, but there are a number of low molecular weight organic materials that also form glasses, and there are inorganic glass formers as well; eg. Silicon oxide used for windows. A glass displays the mechanical properties of a solid – similar to crystalline solids, but the structure is disordered as in a liquid. In thermodynamic terms, the glassy state is a non equilibrium state. However, particularly for polymeric glasses, the relaxation time to reach equilibrium can diverge to infinity at the Vogel-Fulcher temperature, well above absolute zero; thus the non-equilibrium state is no longer transient. Thermodynamic theories have been tailored to accommodate the permanent non-equilibrium nature of the glassy state, and to describe the glass transition above which the material regains its fluidity and re-approaches thermodynamic equilibrium. These are powerful theories, but currently no comprehensive molecular theory of the glass transition based on first principles is available. ….the most important fact about the glassy staqte in thecontext of organic semiconductors is that he glass is structurally disordered like a liquid. Consequently, unlike crystalline materials there is no translation symmetry and Bloch’s theorem does not apply.

Kelsall, Nanoscale, Wiley




…what Shakespeare wrote in The Winter's Tale Act I, Scene I namely: ".....physics the subject, makes old hearts fresh"
Cited in: Creation of the Nuclear Physics Laboratory at Oxford by Sir Denyse Wilkin-son
http://www.physics.ox.ac.uk/pp/dwb//wilkinson-talk.htm











sample size 16



sample size 32


CENTRAL LIMIT THEOREM



The distribution of an average tends to be Normal, even when the distribution from which the average is computed is decidedly non-Normal.

Thus, the Central Limit theorem is the foundation for many statistical procedures, including Quality Control Charts, because the distribution of the phenomenon under study does not have to be Normal because its average will be.

source:http://www.statisticalengineering.com/central_limit_theorem.htm
access through Oxford Dept of Continuing Education; Stats for Health Researchers Course; http://openmoodle.conted.ox.ac.uk/mod/forum/view.php?f=51








Entropy and probability

The transition from a detailed dynamical description to a statistical description entails discarding information. State variables that are regarded as having definite values in the dynamical description are regarded as having probability distributions in the statistical descrip-tion. In quantum statistical mechanics, distinct states can be enumer-ated, and the entropy, which can be stated

L = - k ∑ P(state) ln[P(state)]

Is a weighted measure of the number of possible states. This expres-sion yields the familiar result (the third law of thermodynamics) that a perfect crystal at absolute zero has zero entropy: the structure of the crystal is known and it is in the vibrational ground state with unit probability: 1 x ln(1) = 0 hence L = 0.
This result is equally true for any completely specified structure at absolute zero.



Information can increase free energy; probability, however, depends on the information one possesses. This may suggest that the entropy of an irregular solid at absolute zero is positive if these irregularities are unknown, but zero if they are completely described by some algo-rithm or external record (the complete specification just alluded to). If so then entropy is not a local property of a physical system.

Drexler E, 1992, Nanosystems; Wiley, p 83


Statistical thermodynamics or molecular thermodynamics is most commonly applied to relate macroscopic thermodynamic properties to statistical descrioptions of the behaviour of large numbers of molecules. In describing properties of bulk matter, statistica mechanics frequently describes probability distributions for underlying molecular variables, such as position and velocity. Statistical mechanics is commonly used to calculate quantitie such as pressure, entropy, and free energy based on averages taken over many molecules in thermal equilibrium. There is no fundamental difference between A)the statistical distribution of a dynamical quantity computed for many similar molecules at a particular instant, B)the statistical distributio computed for a single representative molecule over a long period of time, and C)the probability distribution for a single molecule at a single time. Accordingly, the concepts of pressure, entropy, free energy, andtheh like can be used to reason about the expect mean efficiency of a single nanomachine in a single operationsl cycle. The only caveat remains in the assumptions of equilibrium, where there is a subtle relationship between measurement and equilibrium.