Configuron: Difference between revisions
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==[http://en.wikipedia.org/wiki/Hausdorff_dimension| Hausdorff dimension]== | ==[http://en.wikipedia.org/wiki/Hausdorff_dimension| Hausdorff dimension]== | ||
The Hausdorff dimension (d) generalizes the notion of the dimension of a real [[vector space]]. In particular, the Hausdorff dimension of a single point is zero, the Hausdorff dimension of a line is one, the Hausdorff dimension of the plane is two, of a solid is three, etc. The Hausdorff dimension can be thought of as the power of a set of space filling balls formally expressed by | The Hausdorff dimension (d) generalizes the notion of the dimension of a real [[Vector (spatial)|vector space]]. In particular, the Hausdorff dimension of a single point is zero, the Hausdorff dimension of a line is one, the Hausdorff dimension of the plane is two, of a solid is three, etc. The Hausdorff dimension can be thought of as the power of a set of space filling balls formally expressed by | ||
:<math>C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i>0\Bigr\}.</math> | :<math>C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i>0\Bigr\}.</math> | ||
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where C is the space (S)-filling Content of a countable number (the index number - i) of balls whose radii are dimensioned to produce the space-filling balls. | where C is the space (S)-filling Content of a countable number (the index number - i) of balls whose radii are dimensioned to produce the space-filling balls. | ||
In three dimensions, the balls can be spheres of many different radii and the volume of each ball is proportional to its r<sup>3</sup>. Hence the Hausdorff dimension, d = 3. In four dimensions, the balls can be hyperspheres of many different radii and the volume of each ball is proportional to its r<sup>4</sup>. Consider a sphere that changes its radius with time. At each time the sphere has a finite radius r<sub>t</sub> that differs from each t before and after. The volume calculated is proportional to r<sub>space-filling</sub><sup>4</sup> that equals the space occupied for all time. | In three dimensions, the balls can be spheres of many different radii and the volume of each ball is proportional to its r<sup>3</sup>. Hence the Hausdorff dimension, d = 3. In four dimensions, the balls can be [[Multivariate normal distribution#Geometric interpretation|hyperspheres]] of many different radii and the volume of each ball is proportional to its r<sup>4</sup>. Consider a sphere that changes its radius with time. At each time the sphere has a finite radius r<sub>t</sub> that differs from each t before and after. The volume calculated is proportional to r<sub>space-filling</sub><sup>4</sup> that equals the space occupied for all time. | ||
[ | [http://en.wikipedia.org/wiki/Fractals| Fractals] often are spaces whose Hausdorff dimension strictly exceeds the [[Dimension#Lebesgue covering dimension|topological dimension]]. A 2-dimensional fractal has a Hausdorff dimension, d - 2<d<3. | ||
There is a symmetry change expressed by step-wise variation in the Hausdorff dimension (d) for bonds at the solid-liquid transition.<ref name=Ojovan/> In the solid state d=3 but for the liquid state d=d<sub>f</sub> (the fractal d) = 2.55 ± 0.05.<ref name=Ojovan1>{{ cite journal |author=Ojovan MI, Lee WE |title=Topologically disordered systems at the glass transition |journal=J Phys: Condens Matter. |year=2006 |volume=18 |issue=50 |month=Nov |pages=11507-20 |doi=10.1088/0953-8984/18/50/007 }}</ref> d<sub>f</sub> occurs at each broken bond. | There is a symmetry change expressed by step-wise variation in the Hausdorff dimension (d) for bonds at the solid-liquid transition.<ref name=Ojovan/> In the solid state d=3 but for the liquid state d=d<sub>f</sub> (the fractal d) = 2.55 ± 0.05.<ref name=Ojovan1>{{ cite journal |author=Ojovan MI, Lee WE |title=Topologically disordered systems at the glass transition |journal=J Phys: Condens Matter. |year=2006 |volume=18 |issue=50 |month=Nov |pages=11507-20 |doi=10.1088/0953-8984/18/50/007 }}</ref> d<sub>f</sub> occurs at each broken bond. | ||
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An amorphous substance is any in which there is no long-range order over the positions of its constituent particles; i.e., no translational periodicity. Some of the kinetic energy of these substances can be in the form of interparticle bonds. A broken interparticle chemical bond and associated strain-releasing local adjustment in centers of vibration form a configuron, an elementary configurational excitation in an amorphous material.[1]
Amorphous substances
The particles in an amorphous substance can be subatoms, atoms, ions, molecules, dust, crystallites, or grains, stones, boulders, or larger debris.
Amorphous substances can fall into the usual categories of solid, liquid, gas, or plasma. But some substances which are amorphous, such as sand are fluids.
Water as a liquid has much of the available kinetic energy expressed through additional degrees of freedom than water vapor. Some of this energy is in the form of intermolecular bonds. These bonds are a resistance to flow. Water has a resistance to flow that is considered relatively "thin", having a lower viscosity than other liquids such as vegetable oil. At 25°C, water has a nominal viscosity of 1.0 × 10-3 Pa∙s and motor oil has a nominal apparent viscosity of 250 × 10-3 Pa∙s.[2]
Viscous flow in amorphous materials such as water is a thermally activated process:[3]
- <math>{\mu} = A \cdot e^{Q_L/RT},</math>
where QL is the activation energy in the liquid state, T is temperature, R is the molar gas constant and A is approximately a constant.
With
- <math>Q_L = H_m\,</math>
where Hm is the enthalpy of motion of the broken hydrogen bonds.
Solid-liquid transition in amorphous substances
In principle, given a sufficiently high cooling rate, any liquid can be made into an amorphous solid. Cooling reduces molecular mobility. If the cooling rate is faster than the rate at which molecules can organize into a more thermodynamically favorable crystalline state, then an amorphous solid will be formed. Because of entropy considerations, many polymers can be made into amorphous solids by cooling even at slow rates. In contrast, if molecules have sufficient time to organize into a structure with two- or three-dimensional order, then a crystalline (or semi-crystalline) solid is formed. Water is one example. Because of its small molecular size and ability to quickly rearrange, it cannot be made amorphous without resorting to specialized hyperquenching techniques. These produce amorphous ice, which has a quenching rate in the range of metallic glasses.[4]
The higher the temperature of an amorphous material the higher the configuron concentration. The higher the configuron concentration the lower the viscosity. As configurons form percolating clusters, an amorphous solid can transition to a liquid. This clustering facilitates viscous flow. Thermodynamic parameters of configurons can be found from viscosity-temperature relationships.[4]
Short-range order
Like a liquid an amorphous solid has a topologically disordered distribution of particles but elastic properties of an isotropic solid. The symmetry similarity of both liquid and solid phases cannot explain the qualitative differences in their behavior.
Due to chemical bonding characteristics amorphous solids such as glasses do possess a high degree of short-range order with respect to local atomic polyhedra.[5] The amorphous structure of glassy silica has no long range order but shows local ordering with respect to the tetrahedral arrangement of oxygen atoms around silicon atoms.
Bond structure
One useful approach is to consider the bond system instead of considering the set of particles that form the substance.[4] For each state of matter we can define the set of bonds by a bond lattice model.[4] The congruent bond lattice for amorphous materials is a disordered structure. Moreover the bond lattices of amorphous solids and liquids may have different symmetries in contrast to the symmetry similarity of particles in a liquid or fluid and solid phases.
For an amorphous material a given unit can be delimited by its nearest neighbors so that its structure may be characterized by a distribution of Voronoi polyhedra filling the space of the disordered material. Molecular dynamics simulations have revealed that the difference between a liquid and glass of an amorphous material results from the formation of percolation clusters of broken bonds in the Voronoi network.[6]
Hausdorff dimension
The Hausdorff dimension (d) generalizes the notion of the dimension of a real vector space. In particular, the Hausdorff dimension of a single point is zero, the Hausdorff dimension of a line is one, the Hausdorff dimension of the plane is two, of a solid is three, etc. The Hausdorff dimension can be thought of as the power of a set of space filling balls formally expressed by
- <math>C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i>0\Bigr\}.</math>
where C is the space (S)-filling Content of a countable number (the index number - i) of balls whose radii are dimensioned to produce the space-filling balls.
In three dimensions, the balls can be spheres of many different radii and the volume of each ball is proportional to its r3. Hence the Hausdorff dimension, d = 3. In four dimensions, the balls can be hyperspheres of many different radii and the volume of each ball is proportional to its r4. Consider a sphere that changes its radius with time. At each time the sphere has a finite radius rt that differs from each t before and after. The volume calculated is proportional to rspace-filling4 that equals the space occupied for all time.
Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension. A 2-dimensional fractal has a Hausdorff dimension, d - 2<d<3.
There is a symmetry change expressed by step-wise variation in the Hausdorff dimension (d) for bonds at the solid-liquid transition.[4] In the solid state d=3 but for the liquid state d=df (the fractal d) = 2.55 ± 0.05.[7] df occurs at each broken bond.
Glass transition temperature of water
The glass transition temperature for water is about 136 K or -137°C. Factors in the formation of amorphous ice include ingredients that form a heterogenous mixture with water (such as is used in the production of ice cream), pressure (which may convert one form into another), and cryoprotectants that lower its freezing point and increase viscosity. Melting low-density amorphous ice (LDA) between 140 and 210 K through its transition temperature shows that it is more viscous than normal water.[8] LDA has a density of 0.94 g/cm³, less dense than the densest water (1.00 g/cm³ at 277 K), but denser than ordinary ice.
Amorphous ice is used in some scientific experiments, especially in electron cryomicroscopy of biomolecules.[9] The individual molecules can be preserved for imaging in a state close to what they are in liquid water.
Acknowledgements
The content on this page was first contributed by: Henry A. Hoff.
Initial content for this page in some instances came from Wikipedia.
References
- ↑ Angell CA, Rao KJ (1972). "Configurational excitations in condensed matter, and "bond lattice" model for the liquid-glass transition". J Chem Physics. 57 (1): 470–81. doi:10.1063/1.1677987.
- ↑ Raymond A. Serway (1996). Physics for Scientists & Engineers (4th Edition ed.). Saunders College Publishing. ISBN 0-03-005932-1.
- ↑ Ojovan MI, Lee WE (2004). "Viscosity of network liquids within Doremus approach". J Appl Phys. 95 (7): 3803–10. doi:10.1063/1.1647260. Text "month" ignored (help)
- ↑ 4.0 4.1 4.2 4.3 4.4 Ojovan MI (2008). "Configurons: thermodynamic parameters and symmetry changes at glass transition". Entropy. 10: 334–64. doi:10.3390/e10030334. Text "http://www.mdpi.org/entropy/papers/e10030334.pdf " ignored (help); Unknown parameter
|month=ignored (help) - ↑ Salmon PS (2002 pages=87-8). "Amorphous materials: Order within disorder". Nature Materials. 1 (2). doi:10.1038/nmat737. Check date values in:
|year=(help) - ↑ Medvedev NN, Geider A, Brostow W (1990). "Distinguishing liquids from amorphous solids: Percolation analysis on the Voronoi network". J Chem Phys. 93: 8337–42.
- ↑ Ojovan MI, Lee WE (2006). "Topologically disordered systems at the glass transition". J Phys: Condens Matter. 18 (50): 11507–20. doi:10.1088/0953-8984/18/50/007. Unknown parameter
|month=ignored (help) - ↑ "Liquid water in the domain of cubic crystalline ice Ic".
- ↑ Dubochet J, Adrian M, Chang JJ, Homo JC, Lepault J, McDowell AW, Schultz P (1988). "Cryo-electron microscopy of vitrified specimens". Q Rev Biophys. 21: 129–228.
See also
External links
- Discussion of amorphous ice at LSBU's website.
- Glassy Water from Science, on phase diagrams of water (requires registration)
- Structure of amorphous ice