Structure of liquids and glasses
The structure of
The study of liquid and glass structure aims to gain insight into their behavior and physical properties, so that they can be understood, predicted and tailored for specific applications. Since the structure and resulting behavior of liquids and glasses is a complex
Pair distribution functions and structure factors
The pair distribution function (or pair correlation function) of a material describes the probability of finding an atom at a separation r from another atom.
A typical plot of g versus r of a liquid or glass shows a number of key features:
- At short separations (small r), g(r) = 0. This indicates the effective width of the atoms, which limits their distance of approach.
- A number of obvious peaks and troughs are present. These peaks indicate that the atoms pack around each other in 'shells' of nearest neighbors. Typically the 1st peak in g(r) is the strongest feature. This is due to the relatively strong chemical bonding and repulsion effects felt between neighboring atoms in the 1st shell.
- The attenuation of the peaks at increasing radial distances from the center indicates the decreasing degree of order from the center particle. This illustrates vividly the absence of "long-range order" in liquids and glasses.
- At long ranges, g(r) approaches a limiting value of 1, which corresponds to the macroscopic density of the material.
The
-
(1)
where q is the magnitude of the momentum transfer vector, and ρ is the number density of the material. Like g(r), the S(q) patterns of liquids and glasses have a number of key features:
- For small angle scattering, due to large scale structure or voids in the material.
- The sharpest peaks (or troughs) in S(q) typically occur in the q=1–3 ångström range. These normally indicate the presence of some medium range order corresponding to structure in the 2nd and higher coordination shells in g(r).
- At high-q the structure is typically a decaying sinusoidal oscillation, with a 2π/r1 wavelength where r1 is the 1st shell peak position in g(r).
- At very high-q the S(q) tends to 1, consistent with its definition.
Diffraction
The absence of
-
(2)
The g(r), which describes the probability of finding an atom at a separation r from another atom, provides a more intuitive description of the atomic structure. The g(r) pattern obtained from a diffraction measurement represents a spatial, and thermal average of all the pair correlations in the material, weighted by their coherent cross-sections with the incident beam.
Atomistic simulation
By definition, g(r) is related to the average number of particles found within a given volume of shell located at a distance r from the center. The average density of atoms at a given radial distance from another atom is given by the formula:
-
(3)
where n(r) is the mean number of atoms in a shell of width Δr at distance r.[1] The g(r) of a simulation box can be calculated easily by histograming the particle separations using the following equation
-
(4)
where Na is the number of a particles, |rij| is the magnitude of the separation of the pair of particles i,j. Atomistic simulations can also be used in conjunction with interatomic pair potential functions in order to calculate macroscopic thermodynamic parameters such as the internal energy, Gibbs free energy, entropy and enthalpy of the system.
Theories of glass formation and criterion
Structural theory of glass formation, Zachariasen
While studying glass, Zachariasen began to notice repeating properties in glasses. He postulated rules and patterns that, when atoms followed these rules, they were likely to form glasses. The following rules make up Zachariasen's theory, applying only to oxide glasses.[2]
- Each oxygen atom in a glass can be bonded to no more than two glass-forming cations
- The coordination number of the glass forming cation is 3 or 4
- The oxygen coordination polyhedra only share corners, not edges or faces
- At least 3 corners of every polyhedra must be shared, creating a continuous random network.
All of these rules provide the correct amount of flexibility to form a glass and not a crystal.
While these rules only apply to oxide glasses, they were the first rules to establish the idea of a continuous random network for glass structure. He was also the first to classify structural roles for various oxides, some being main glass formers (SiO2, GeO2 , P2O5), and some being glass modifiers (Na2O, CaO).
Energy criterion of K.H. Sun
This criterion established a connection between the chemical bond strength and its glass forming tendency. When a material is quenched to form glass, the stronger the bonds, the easier the glass formation.[3]
- If a bond strength is higher than 80 kcal per bond (high bond strength), it will be glass network forming, meaning it is likely to form a glass.
- If a bond strength is less than 60 kcal per bond (low bond strength), it will be glass network modifying, since it would only form weak bonds, it would disrupt glass forming networks.
- If a bond strength is between 60 and 80 kcal per bond (intermediate bond strength, it will be an intermediate. This means it will not form a glass on its own, but it partially can while combined with other network forming atoms.
Dietzel's field strength criterion
Dietzel looked at direct Coulombic interactions between atoms. He categorized cations using field strength where FS=zc/(rc+ra)2, where zc is the charge of the cation, and rc and ra are the radii of the cation and anion respectively. High field strength cations would have a high cation-oxygen bond energy.[4]
- If FS was greater than 1.3 (small cation with high charge[5]), it would be a glass network former.
- If FS was less than 0.4 (large cation with small charge[5]), it would be a glass network modifier.
- If FS was between 0.4 and 1.3 (medium-sized cation with medium charge[5]) it would be an intermediate.
These three criterion help establish three different ways to determine whether or not certain oxides molecules will form glasses, and the likeliness of it.
Other techniques
Other experimental techniques often employed to study the structure of glasses include nuclear magnetic resonance, X-ray absorption fine structure and other spectroscopy methods including Raman spectroscopy. Experimental measurements can be combined with computer simulation methods, such as reverse Monte Carlo or molecular dynamics simulations, to obtain more complete and detailed description of the atomic structure.
Network glasses
Early theories relating to the structure of glass included the crystallite theory whereby glass is an aggregate of
The structure of glasses differs from the structure of liquids just above the glass transition temperature Tg which is revealed by the XRD analysis[10] and high-precision measurements of third- and fifth-order non-linear dielectric susceptibilities.[11] Glasses are generally characterised by a higher degree of connectivity compared liquids.[12]
Alternative views of the structure of liquids and glasses include the interstitialcy model[13] and the model of string-like correlated motion.[14] Molecular dynamics computer simulations indicate these two models are closely connected [15]
Oxide glass components can be classified as network formers, intermediates, or network modifiers.[16] Traditional network formers (e.g. silicon, boron, germanium) form a highly cross-linked network of chemical bonds. Intermediates (e.g. titanium, aluminium, zirconium, beryllium, magnesium, zinc) can behave both as a network former or a network modifier, depending on the glass composition.[17] The modifiers (calcium, lead, lithium, sodium, potassium) alter the network structure; they are usually present as ions, compensated by nearby non-bridging oxygen atoms, bound by one covalent bond to the glass network and holding one negative charge to compensate for the positive ion nearby.[18] Some elements can play multiple roles; e.g. lead can act both as a network former (Pb4+ replacing Si4+), or as a modifier.[19] The presence of non-bridging oxygens lowers the relative number of strong bonds in the material and disrupts the network, decreasing the viscosity of the melt and lowering the melting temperature.[17]
The alkali metal ions are small and mobile; their presence in a glass allows a degree of
Crystalline SiO2
Glassy SiO2
In amorphous silica (fused quartz), the SiO4 tetrahedra form a network that does not exhibit any long-range order. However, the tetrahedra themselves represent a high degree of local ordering, i.e. every silicon atom is coordinated by 4 oxygen atoms and the nearest neighbour Si-O bond length exhibits only a narrow distribution throughout the structure.[6] If one considers the atomic network of silica as a mechanical truss, this structure is isostatic, in the sense that the number of constraints acting between the atoms equals the number of degrees of freedom of the latter. According to the rigidity theory, this allows this material to show a great forming ability.[25] Despite the lack of ordering on extended length scales, the tetrahedra also form a network of ring-like structures which lead to ordering on intermediate length scales (up to approximately 10 angstroms or so).[6] Under the application of high pressure (approximately 40 GPa) silica glass undergoes a continuous polyamorphic phase transition into an octahedral form, i.e. the Si atoms are surrounded by 6 oxygen atoms instead of four in the ambient pressure tetrahedral glass.[26]
See also
References
- ^ McQuarrie, D.A., Statistical Mechanics (Harper Collins, 1976)
- S2CID 240817562, retrieved 2021-11-11
- ^ Brow, R. K, Structural Theories of Glass Formation (PDF), pp. 5–6
- ^ Brow, R. K, Structural Theories of Glass Formation (PDF), pp. 5–6
- ^ a b c Brow, R. K, Advanced Vitreous State: The Structure of Glass (PDF), pp. 19–22
- ^ .
- .
- ^ W.H. Zachariasen (1932). "The Atomic Arrangement in Glass". J. Am. Chem. Soc. 54 (10): 3841. .
- S2CID 39062607.
- ^ Ojovan, M.I.; Louzguine-Luzgin, D. Revealing Structural Changes at Glass Transition via Radial Distribution Functions. J. Phys. Chem. B, 124, 3186–3194. (2020) https://doi.org/10.1021/acs.jpcb.0c00214
- ^ Albert, S.; Bauer, T.; Michl, M.; Biroli, G.; Bouchaud, J.-P.; Loidl, A.; Lunkenheimer, P.; Tourbot, R.; Wiertel-Gasquet, C.; Ladieu, F. Fifth-order susceptibility unveils growth of thermodynamic amorphous order in glass-formers. Science, 352, 1308–1311 (2016) https://www.science.org/doi/10.1126/science.aaf3182
- ^ M.I. Ojovan, W.E. Lee. Connectivity and glass transition in disordered oxide systems J. Non-Cryst. Solids, 356, 2534-2540 (2010)
- PMID 10046046.
- S2CID 16322630.)
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: CS1 maint: multiple names: authors list (link - S2CID 122849676.)
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: CS1 maint: multiple names: authors list (link - ISBN 978-0-323-39312-6.
- ^ ISBN 978-3-527-65026-2.
- ISBN 978-3-86644-907-7.
- ^ Zhu, Yuntian. "MSE200 Lecture 19 (CH. 11.6, 11.8) Ceramics" (PDF). Retrieved October 15, 2017.
- ISBN 978-3-527-31549-9.[page needed]
- ISBN 978-0-470-64089-0.
- ISBN 978-1-4831-6522-6.
- ISBN 978-1-315-66798-0.
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- .
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Further reading
- Egelstaff, P.A. (1994). An Introduction to the Liquid State. Oxford University Press. ISBN 978-0198517504.
- A. Zaccone (2023). Theory of Disordered Solids. Lecture Notes in Physics. Vol. 1015. Springer. ISBN 978-3-031-24705-7.
- Allen, M.P. & Tildersley, D.J. (1989). Computer Simulation of Liquids. Oxford University Press. ISBN 978-0198556459.
- Fischer, H.E., Barnes, A.C., and Salmon, P.S. (2006). "Neutron and x-ray diffraction studies of liquids and glasses". Rep. Prog. Phys. 69 (1): 233–99. S2CID 121691082.)
{{cite journal}}
: CS1 maint: multiple names: authors list (link - Kawazoe,Y. and Waseda, Y. (2010). Structure and Properties of Aperiodic Materials. Springer. ISBN 978-3642056727.)
{{cite book}}
: CS1 maint: multiple names: authors list (link - Santen, L. & Krauth W. (2000). "Absence of a Thermodynamic Phase Transition in a Model Glass Former". Nature. 405 (6786): 550–1. S2CID 4411550.