Non-crystalline Materials

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Sheila Byrd
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1 Noncrystalline (or amorphous) solids by definition means atoms are stacked in irregular, random patterns. The general term for non-crystallline solids with compositions comparable to those of crystalline ceramics is glass, eg SiO 2. (building block SiO 4 4- ) Another material is glass-ceramics. Certain glass compositions (e.g., lithium aluminosilicates) can be devitrified (transformed from the vitreous, or glassy, state to the crystalline state) by an appropriate heat treatment. Tend to be fine grained with no porosity giving higher mechanical strength superior to many traditional ceramics. Also exhibit low C.T.E., making them resistant to fracture due to rapid DT, important in cookware. Non-crystalline Materials 2-D schematics of atomic scale structure of hypothetical: Crystalline oxide (LRO) Noncrystalline oxide (SRO) The open circles represent a nonmetallic atom and the solid circles represent a metal atom having same chemical composition. The noncrystalline material retains short-range order, SRO (the triangularly AO 3-3 coordinated building block), but loses long-range order, LRO (crystallinity) as in glass. The noncrystalline structure is referred to as Zachariasen model (random network theory) of glass structure which is the analog of the point 1 lattice we discussed with crystalline structures.

2 DL/L o or Processing of Non-crystalline Materials Noncrystallline solids can be made by several routes, e.g. rapidly cooling a liquid (supercooled liquid) silicate or allowing a silicate vapor to condense on a cool substrate effectively freezes in the random stacking of silicate building blocks, eg SiO 4 4- tetrahedra in SiO 2 glass. The supercooled liquid is the material cooled just below the melting point, where it still behaves like a liquid (deforming by viscous flow). The glass is the same material cooled to a sufficiently low temp. so that it s rigid solid (deforming by elastic mechanism). The atomic mobility of the material at these low temperatures is insufficient for the theoretically more stable crystalline structure to form. Thus two types of noncrystalline solids differing in mobility of their constituents: -low h liquid state and high h glassy state. The material is fluid when thermal energy dominates over interatomic binding energies, and vice versa for glassy solid. Slow cooling by shutting off furnace DL/L o (CTE) has 2 different slopes above and below T g True glass is a rigid solid with CTE similar to crystal Moderate quenching near T g gradual solidification occurs Rapid splatquenching Condensation from gas phase CTE below T g like a solid and above like a liquid. Thus T g is referred to as glass transition temperature. Specific volume plot is similar to a CTE plot. A comparison can be made by the addition of data for crystalline material (same composition as glass/liquid). Upon heating crystal undergoes modest CTE up to its metal point, T m, at which a sharp increase in specific volume occurs. Upon further heating, liquid undergoes greater CTE. Slow cooling of liquid would allow crystallization abruptly at T m and retracing of melting plot. Rapid cooling of liquid can 2 suppress crystallization producing a supercooled liquid.

3 Applications of Non-crystalline Materials Those semiconductors with structures similar to some ceramics can be made in amorphous forms also, amorphous semiconductors. More economic than producing high quality single crystals. Amorphous metals (metallic glass) are another noncrystalline class of materials. Liquid metals can be cooled very rapidly (1 C/microsecond) to prevent crystallization. Expensive process but worthwhile due to targeted properties such as corrosion resistance and mechanical strength. Melt spinning of metallic glass. The solid ribbon of amorphous metal is spun off at speeds that can exceed 1km/min. 3

Bernal Model The Bernal model is used to visualize an amorphous metal structure, produced by drawing lines between the centers of adjacent atoms. Bond angle and length vary.

4 Bernal Model The Bernal model is used to visualize an amorphous metal structure, produced by drawing lines between the centers of adjacent atoms. Bond angle and length vary. The resulting polyhedra are comparable to those illustrating grain boundary structure shown below (S5 boundary for FCC metal). High angle (36.9 ) tilt boundary Note the number of atoms along the boundary are common to each adjacent lattice (correspondence is known as coincidence site lattice, CSL). By extending the lattice grid on left grain, 1 (open circle) in 5 (solid circles) of the atoms on right grain is coincident with that lattice, the boundary is said to have S -1 =1/5, or S=5. The geometry of this overlap can also give = 2tan -1 (1/3) = Here polyhedra, are also irregular in shape, but lack any repetitive stacking arrangement. Coordination # is not fixed. Here polyhedra, formed by drawing straight lines between adjacent atoms in the grain boundary region, are irregular in shape due to misorientation angle, but reappear at regular intervals within each grain. Grain boundary dislocations occur within these boundary planes (for these high s.) Figure. S5 boundary for FCC metal in which [100] directions of two adjacent grains are orientated at 36.9 to each other. This is a 3-D projection with atoms on adjacent parallel planes to this page (solid vs. open) 4

Bernal Model Randomization of atomic packing in amorphous metals (shown below) generally causes no more than a 1% drop in density compared with the crystalline structure of the same composition.

5 Bernal Model Randomization of atomic packing in amorphous metals (shown below) generally causes no more than a 1% drop in density compared with the crystalline structure of the same composition. Crystalline solids can be made into glasses via radiation damgage, such as ion implantation and high energy electron beams. The energy imparted by the penetration of a crystal by a high energy particle can cause amorphization of a small volume through a process of multiple collisions (known as cascade). On right shows formation of region in GaAs formed as a result of high energy (temperature) Ne+ implantation (square shaped particle). These amorphous zones are several nanometers in diameter. The fast rate of heat removal from energetic region by surrounding crystalline matrix prevents rearrangement back to crystalline array. 5

6 Medium-Range Order in Non-crystalline Materials Imperfections such as chemical impurities can be defined relative to the uniformly noncrystalline structure. For example, addition of Na + ions to silicate glass substantially increases formability of the material in the supercooled state (i.e., viscosity is reduced). Na is known as glass modifier, breaks up the random network and leaves non-bridging oxygen ions. Thus it s no longer a Zachariasen structure which was uniformly and perfectly random. Another example, shows nonrandom arrangement of atoms of Ca 2+ modifier ions in a CaO-SiO 2 glass. The adjacent CaO 6 octahedra identified by neutron diffraction experiments (each Ca 2+ ion is coordinated by six O 2- ions in a perfect octahedral pattern) are arranged in a regular, edge-sharing (octahedra) fashion, in contrast to the random distribution of Na + ions above. This is an example of medium-range ordering (MRO), i.e. structural ordering occurs in range of a few nanometers between the well- known short-range order of SiO 4 tetrahedra and longrange randomness of irregular linkage of those tetrahedra. Thus, Zacharisen (CRN) model is an inadequate description of vitreous silica. 6

Zachariasen s Rules In SiO 2 glass, the Si atoms are four-functional and O atoms are two-functional. Each Si bonds to 4 oxygens with the O-Si-O bond angle (f) fixed at 109.28º.

7 Zachariasen s Rules In SiO 2 glass, the Si atoms are four-functional and O atoms are two-functional. Each Si bonds to 4 oxygens with the O-Si-O bond angle (f) fixed at º. No dangling bonds, thus T g high (1430K). Linking in this continuous random network (CRN) structure occurs through corner-to-corner connections of SiO 4 tetrahedra to share oxygen ions, so as to form Si-O-Si- linkages with variable Si-O-Si bond angles (b) between adjacent SiO 4 tetrahedra. 130º b 160º The mutual orientation of adjacent tetrahedra is defined by the angle y. It varies from 0º y 180º such that orientations of adjacent tetrahedra are uncorrelated. The bond lengths are quite comparable to crystalline SiO 2, thus the density of SiO 2 glass is similar to that of crystalline SiO 2. The principal constraints for the formation of a CRN structure in oxide glasses were first described by Zachariasen. He emphasized the importance of the polyhedra defined by connecting the oxygen anions arranged around a central cation. Rule 1: Each oxygen anion should be linked to not more than two cations. Rule 2: The network cation formers (Si 4+, P 5+, B 3+ ) have ability to form branching networks. Small, highly charged, and have a low coordination (3 or 4) thus bond strengths are high strong network. Rule 3: No shared edges or faces are allowed between the network formers; corner sharing (oxygen) is favorable because it promotes flexibility. Also, the highly charged cations are further apart which lowers the repulsive Coulombic forces while leaving flexibility in strong bonded network. Conversely, molecular groups consisting of edge & face shared units tend to have rigid geometries. Rule 4: At least three corners of each polyhedron should be shared to promote 3-D cross-linking. Rule 5: Modifier ions (Na+), modifies temp. and viscosity to make class more workable, are large with a small charge and generally lower Tg and Tm. Si O Si-O bond distance=1.6å O-O bond distance=2.6å 7

8 Radial Distribution Function For liquids and glasses we described a noncrystalline structural descriptor last time, T g. There is another equally important descriptor, the dimensionless pair (or radial) distribution function. Complete disorder is gas state where molecules are significantly separated from each other to have an identical probability of occupying all space available to them. Description becomes more complex when the elements of the gas liquid/glass disordered structure are close enough to each other to influence each others positions, as in liquids and glasses. First way: a portion of the disordered structure is considered as a giant molecule and the position M i (x i y i z i ) and nature of all its elements are defined, it s only possible to do this by computer simulations and comparing to experimental values. crystalline solid Second way: characterize the structure by a statistical descriptor through the distribution of certain distances to deduce certain results accessible to experiment. This consists of probability of finding a pair of atoms M i and M j defined by the vector: r ij r j r i where r i and r jthe position vectors of the atoms M i and M j : In an isotropic structure, as in liquid or glass, if all orientations of r ij are assumed equivalent, a description based on the distance r and scalar distribution function is sufficient. ij r ij For structures containing one kind of atom, a function r(r) is defined by the expression: 4 r 2 r( r) dr which represents the 8 average number of atomic centers situated in a spherical shell 8 or radius r and of thickness dr centered on any atom of the structure

9 Radial Distribution Function (continued) The function r(r) is obtained by successively taking each atomic center of the structure as the origin and calculating the resulting average value. The term radial density is applied to r(r), which represents the average number of atoms per unit volume situated at some distance from any atom in the structure taken as the origin. The form of the function is: r( r) 0 The successive maxima correspond to the most frequent average distances in the structure, the minima to the distances least frequently encountered. for values of r<d where D is the minimum distance of approach (contact distance) for values of r D the function presents a maximum corresponding to the presence of nearest neighbors. The regular arrangement of the neighbors close to the atom at the origin indicates the existence of short range order in the structure. For r r(r) tends toward the value r o =N/V which is the average density of N centers distributed in volume V. which shows that for larger The function: r(r)/r o = g(r) which is the For r : pair distribution function and represents g( r) 1 distances, the structural effect G( r) 0 disappears and all positions for the probability of the presence of a pair of the atoms are equally probable. atoms at distance r. The quantity G(r) = g(r) 1 is called the pair correlation function 9

10 Radial Distribution Function (continued) The function 4 r 2 r( r) is called the radial distribution function (RDF): It shows that the maxima and minima oscillating around the parabola 4 r 2 r o corresponding to the average distribution of centers in the structure and tending toward it at r. The area under the curve representing the RDF permits the calculation of the average number of centers situated between the two given distances, r 1 and r 2. It is instructive to apply the preceding method of description to the case of an ideal crystal structure, such as SC, FCC, BCC, HCP. The atoms are grouped on a series of spherical surfaces called coordination spheres centered on the atom chosen as the origin and following a regular numerical progression resulting from the geometry of the crystal lattice. The number N r of neighbors = CN; N r in first coordination sphere is the principal CN. Long range order Notice that the different coordination spheres come closer together as r increases the differences between the radii of the successive coordination spheres become less and less which is an effect inherent to the radial calculation 10 method used.

11 Radial Distribution Function (continued) Assume now that the atoms undergo displacements relative to their positions in the ideal crystal, e.g. by thermal vibrations around the rest position, thus the atoms are no longer situated on the ideal spheres considered previously, but are distributed in the spherical shell situated between r and r+dr from the ideal rest position. Let the number of atoms contained in such a shell be denoted by h(r)dr. The form of the function h(r): Effect of variations In atomic positions Disordered structure The ideal vertical lines have been replaced by peaked curves, the width of which increases with the amplitude of the displacements. The area under each peak corresponds to the number of atoms contained by the ideal sphere for the unperturbed structure. The structure still contains SRO and LRO, and its possible to distinguish the different coordination spheres. The RDF gives way to a continuous distribution where it s no longer possible to distinguish the successive coordination spheres with precision; it s not possible to define the exact CN, while it is possible to speak of average coordination shells to which can be associated average coordination numbers. The area under a RDF can be sub-divided into portions assigned to different maxima which is the same as attributing approximate average numbers to coordination shells. This deconvolution is somewhat arbitrary, because the areas present some successive overlap. If the deconvolution has a meaning for the nearest neighbors, it loses all significance for the high order coordination shells due to disappearance of LRO as structure blends statistically into a continuum. 11

12 Radial Distribution Function (continued) For structure containing several kinds of atoms, it becomes necessary to introduce RDF s for different atomic pairs, i.e. to define the partial pair distributions. In case of structure with two species, A and B, there would be three independent functions, g A-A, g A-B, g B-B, for the pairs A-A, A-B, and A-C, respectively. For three pairs A,B, and C: g A-A, g B-B, g C-C, g A-B, g A-C, g B-C, In general case of n-species, it is necessary to have n(n+1)/2 pair functions to define the structure. RDF can be obtained by radiation scattering techniques such as X-ray diffraction. To summarize to this point: 12

The Science and Engineering of Materials, 4 th ed Donald R. Askeland Pradeep P. Phulé. Chapter 3 Atomic and Ionic Arrangements

The Science and Engineering of Materials, 4 th ed Donald R. Askeland Pradeep P. Phulé Chapter 3 Atomic and Ionic Arrangements 1 Objectives of Chapter 3 To learn classification of materials based on atomic/ionic