The Nature of the Glassy State: Structure and Glass Transitions

Journal of Materials Education Vol. 34 (3-4): 69 - 94 (2012)

THE NATURE OF THE GLASSY STATE: STRUCTURE AND GLASS TRANSITIONS

Ioannis M. Kalogeras 1,2 and Haley E. Hagg Lobland 2

1 Section of Solid State Physics, Department of Physics, University of Athens, Panepistimiopolis, Zografos 157 84, Greece; [email protected], [email protected]. 2 Laboratory of Advanced Polymers & Optimized Materials, (LAPOM), Department of Materials Science & Engineering and Department of Physics, University of North Texas, 1155 Union Circle #305310, Denton, TX 76203-5017, USA; [email protected].

Key words: crystals, glasses, polymer composites, amorphous structure, glass transition, free volume, Voronoi tessellation, binary radial distribution function

ABSTRACT

Materials in the glassy state have become an increasing focus of research and development and are found in a variety of commercial products and applications. While non-crystalline materials are not new, their often unpredictable properties and behavior continue to elude neat systems of classification. For purposes of teaching, there is presently a need for further explication of glasses, especially given their high importance and the wide extent of glassy materials in existence. This work is thus intended to address that need. Voronoi polyhedra have been used to represent amorphous (glassy) structures with considerable success; however, the system and procedure are not well taught and understood as, for example, are Miller Indices for describing crystals. The present article provides a practical update on results extracted from Voronoi polyhedra analyses of simulated and real physical systems. There is an alternative approach to characterization of amorphous and liquid structures, namely the binary radial distribution function, which is explained. Above all this article discusses the nature of the glassy state.

INTRODUCTION majority of solid materials already in use or under evaluation for novel applications are in a In the earliest approaches – dating back to the non-crystalline state1,2. In spite of that, Bronze and Iron Age – the atomic structures of instruction in Materials Science and Engineer- naturally occurring crystalline materials were ing (M.S.E.) and in related disciplines remains amended by introducing lattice defects, e.g. by largely focused on crystals. An attempt to forging. Nowadays, in a century of tremendous remedy this situation was made in this Journal industrial and technological growth, the just more than a decade ago, in an article 70 Kalogeras and Hagg Lobland

explaining the representation of amorphous erature Tm) or become super cooled; this is structures by Voronoi polyhedra and under- shown by the temperature dependence of the lining their effectiveness in isolating even specific volume, entropy or enthalpy, under subtle differences among the possible physical constant pressure, as illustrated in Fig. 1a. The states of a system3. Changes in several metric particles (atoms, molecules or ions) forming properties of a Voronoi polyhedron –which is crystalline materials are arranged in orderly defined as the convex region of space closer to repeating patterns, with elementary building its central particle (ion, atom, molecule, blocks (unit cells) extending to all three spatial monomer, chain segment, etc.) than to any dimensions. The structures of crystalline solids other– convey information on microstructural depend (predictably) on the chemistry of the fluctuations of various causes. There are in fact material and the conditions of solidification two methods of characterization of amorphous (starting temperature and cooling rate, ambient materials; the second one is based on binary pressure, etc.), and can be described easily in radial distribution functions. Both approaches detail by combining crystallographic notions are discussed in the remainder of this article. with diffraction/scattering data4-6. Super cooled To begin, we address the fundamental liquids, on the other hand, demonstrate a rather distinctions between glasses and crystals and intriguing behavior. Upon further cooling between the glassy and liquid states. below the Tm, their particles progressively lose translational mobility, so that around the so- called glass transition temperature (Tg) GLASSES vs. CRYSTALS rearrangement to “regular” lattice sites is practically unfeasible; this behavior is As a liquid is cooled from a high temperature, it distinctive for the amorphous structures may either crystallize (at the melting temp- described as glasses or vitreous solids.

Figure 1. (a) Typical temperature dependence of the specific volume (v), entropy (S), or enthalpy (H) of glasses and crystals. Path (1) is not possible in, for example, atactic polymers lacking a crystalline ordering state. (b) Temperature dependence of the isobaric expansivity (also called coefficient of thermal expansion) –1 [α = V (ϑV/ϑT)P], heat capacity [Cp = (ϑH/ϑT)P], and log10 of viscosity (η), in the region of Tg.

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The term liquid-glass transition –frequently abbreviated to glass transition– signifies the reversible transition in amorphous materials (including amorphous regions within semi crystalline materials) from a molten or rubber- like state into a hard and relatively brittle state. The question of what phase transition underlies the glass transition is a matter of continuing research. Evidently, this process bears no connection with the first-order phase transitions in Paul Ehrenfest’s (1933) classification scheme7. Those exhibit a discontinuity in the first derivative of the Gibbs energy (G) with respect to some thermodynamic variable: crystallization proffers a characteristic example, as discontinuities emerge in both density and specific volume (v = (ϑG/ϑP)T, P = pressure) versus temperature (T) plots. On the other hand, it is observed in glasses that intensive thermodynamic variables such as the thermal expansivity and heat capacity (second-order derivatives) exhibit a smooth step (formally a discontinuity at Tg) upon cooling (or heating) through the glass transition range (Fig. 1b). Figure 2. A kinetic feature of the glass transition This fact supports connections of the glass phenomenon demonstrated in the temperature transition phenomenon with a second-order variation of the isobaric volume of polyvinyl acetate. 8 phase transition . Nevertheless, the glass Black dots represent equilibrium volumes, while the transition is not a transition between states of half-moons correspond to the volumes observed thermodynamic equilibrium: it is widely 0.02 h and 100 h after sample quenching (adapted believed that the true equilibrium state is from Ref. 10). always crystalline. Only by annealing or ageing system’s viscosity reaches the threshold of η = (when time, t, constitutes the prime exper- 13 12 imental parameter) is structural relaxation 10 Poise (10 Pa⋅s), an unfounded supposi- facilitated and the structure enabled to explore tion. In dilatometric studies the glass transition lower energy minima – inaccessible during the temperature is located at the intersection cooling process – allowing a progressive shift between the cooling V – T curve for the glassy to a more stable (lower energy and entropy) state and the super cooled liquid (Fig. 1a), state. Given the fact that the glass transition is which typically gives a value of the Tg dependent on the history of the material (e.g., approximately equal to 2Tm/3; a feature 11 12 see Figure 2)8 and on the rate of temperature recognized as early as 1952 and applauded change, it seems reasonable to consider it as as “a good empirical rule” with the addition that merely a dynamic phenomenon, extending over “symmetrical molecules such as poly(vinyl- a range of temperatures and defined by one of idene chloride) tend to have ratios about 0.06 several conventions9. smaller than unsymmetrical molecules such as 12 polypropylene”. In dynamical experiments, Interestingly, different operational definitions such as in isothermal dielectric spectroscopy of the glass transition temperature are in use, studies (where a frequency- and time-dependent and several of them are endorsed as scientific electric field E(ω, t), the stimulus, interacts with standards. For example, in rheological studies the electric dipoles in a many-particle system), one considers Tg to be the temperature at which Tg is defined as the temperature where the

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“structural” relaxation time (τ) reaches the amorphous solid are not yet well understood value 100 s. In such experiments, τ pertains to and are therefore still a subject of much the time-scale of system’s shift to equilibrium research. after, for example, a thermal, mechanical or electrical perturbation9; structural relaxation (or Consider again, what happens during the glass intermolecular rearrangement) proceeds by transition. Refer to Fig. 1a: upon cooling below collective motions of structural elements (ions Tm, molecular motion slows, and below Tg the in metallic glasses, and chain segments in rate of change of volume (or enthalpy) amorphous polymers, etc). Even in the case of decreases to a value similar to that of the differential scanning calorimetry (DSC), which crystalline solid. Since slower cooling rates is a routine thermal analysis method9, one will allow a longer time for particles to sample find several definitions of Tg. All these arbitrary different configurations (maintaining the liquid- definitions generate largely dissimilar estim- state equilibrium longer), the value of the Tg ates; at best, values of Tg for a given substance necessarily decreases with a slower rate of agree within a few Kelvin. Multiple glass cooling. Additionally, viscosity (as well as the transitions may appear in multiphase (comp- structural relaxation time) is very sensitive to osite) systems, providing information on the temperature near the Tg: some liquids exhibit a state of mixing and the strength of interaction significant viscous slow-down near the glass between the components. transition, presumably owing to relaxation processes. Although it has been reported that As the liquid passes through its Tg during the Vogel-Tammann-Fulcher-Hesse (VTFH) cooling, its viscosity increases by as much as 17 equation14,15 orders of magnitude (Fig. 1b), but static structural parameters (e.g., the static structure (1a) factor13, S(q)) change almost imperceptibly. The absence of long-range order is a distinctive (where C is a system-specific constant) – but not the only – difference between glasses represents this behavior reasonably well16, it and crystals. The glass exhibits a long-range was later stated17 that “there is no compelling structure close to that observed in the super evidence for the VTFH prediction that the cooled fluid phase, while displaying solid-like relaxation time diverges at a finite temp- mechanical properties on the timescale of erature”. In a separate attempt to model the practical observation. Both in a glass and in a behavior of super cooled liquids, Angell used crystal it is only the vibration degrees of fragility plots (Angell plots), where the freedom and some rotational motions that logarithm of a dynamical quantity (commonly, remain active whereas translational motion is η or τ) are plotted versus Tg/T (e.g., see Fig. 3), arrested. to classify liquids on a scale from strong to fragile18-20. In Angell’s classification scheme the word “fragility” is used to determine the GLASSES vs. LIQUIDS tendency of materials to form glasses, in contrast to its colloquial meaning, which more Having discussed the features distinguishing closely relates to the brittleness of a solid glasses from crystals, we now address the material. Formally, fragility reflects to what distinctions between the glass and liquid states. degree the temperature dependence of the We are familiar generally with the differences viscosity, relaxation time, or the resistivity of between a liquid and a glass. However, for the glass former, deviates from the Arrhenius certain purposes the existing analytical behavior; strong glass-formers nearly exhibit an techniques do not provide sufficient structural Arrhenius-type dependence details to quantitatively distinguish glass from a liquid phase. The molecular processes (1b) governing the translation of a liquid into a rigid

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Figure 3. Fragility (or Angell) plots: logarithm of viscosity (log η) vs. reduced temperature (Tg/T) plot for glass forming liquids. Inset: heat capacity change at the glass transition for selected systems.20 which implies a simple thermally-activated between translational diffusion and viscosity behavior. Moreover, according to this and also between rotational and translational classification, liquids are also distinguished by diffusion that occurs below approximately 16 their structures: strong liquids such as SiO2 and 1.2Tg. Proportionality between the said GeO2 (network oxides) have tetrahedrally properties is evident at higher temperatures but coordinated structures while non-directional no longer holds as the glass transition is neared. dispersive forces and complex coordination are Isothermal dielectric relaxation studies of some characteristic of molecules in fragile liquids materials indicate that at sufficiently high (such as o-terphenyl). The classification of temperatures the liquid exhibits a single liquids as strong versus fragile glass-formers is relaxation mechanism, while at moderate super one that continues to be used and continues as a cooling the liquid may exhibit two different subject of investigation. Apart from providing a relaxation mechanisms16. Such observations relationship to viscosity behavior, the classi- highlight the importance of Arrhenius plots (i.e. fication system provides for some correlation to τ(T) vs. T–1 plots) in studies of system dynamics structural features, although we shall see later and their potential effectiveness for distin- there are alternative approaches to such guishing the glassy and liquid states; and we relationships. shall see later how a geometrical approach may be used to that end. Other changes observed in liquids at temperatures near the glass transition We cannot leave such a discussion without temperature seem to provide additional clues mention of thermodynamics. Important in the for distinguishing the glassy state from the present analysis is entropy, especially as it is liquid state. There appears a decoupling possible to calculate the entropy difference

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between the (super cooled) liquid and sampling distinct potential energy minima, with crystalline states. At the melting temperature, further distinction of vibrations within a entropy of the liquid state is higher than that of minimum. A formal separation of configure- the corresponding crystal. The difference tional and vibrational contributions to a liquid’s between the two decreases with the temperature properties is therefore possible16. Another owing to higher heat capacity of the liquid state. interesting idea coming out of the landscape (Importance of the heat capacity function is picture is the notion that super cooled fragile addressed elsewhere by Angell21.) It would liquids are dynamically heterogeneous: possibly seem from this trend that the entropy difference consisting, in instantaneous moments, of mostly would soon vanish, however the kinetics of the non-diffusing molecules with just a few ‘hot glass transition intervene16. It has therefore spots’ of mobile molecules, a theory evidently been proposed that the glass transition is a supported by experimental and computational “kinetically-controlled manifestation of an work16. Also noteworthy of the landscape view- underlying thermodynamic transition to an point is that it gives a conceivable interpretation ideal glass with a unique configuration”.16 A for the decoupling of the primary (α) and formula of Adam and Gibbs promotes this secondary (β) dynamic mechanical and connection22: dielectric relaxation modes, originating, respectively, from cooperative (long-range) and (1c) weakly or non-interacting (localized) motions of structural units. where C is again a system-specific constant, and Sc is the configurational entropy. The The glass transition phenomenon is often Adam-Gibbs theory provides a picture of the treated also as a purely dynamic transition; behavior leading to the glass transition. there is no singularity, except that with According to this theory, a decrease in the decreasing temperature the dynamics become number of available configurations that the so slow that the system behaves as a solid. The system can sample leads to a viscous slow- most famous approach of this kind is the mode- down (mentioned earlier) close to the transition coupling theory (MCT), which describes a non- temperature16. Adam and Gibbs invoke the idea linear feedback mechanism that links shear- of cooperatively rearranging regions (CRRs) in stress relaxation, diffusion, and viscosity. The their derivation of Eq. (1a) but do not indicate result of this would be structural arrest the size of such regions nor provide a means to occurring as a dynamic singularity16. Although distinguish CRRs from one another. In spite of some features of relaxation dynamics of liquids the aforementioned weakness in the Adam- are described well by MCT, it does not give us Gibbs theory, Eq. (1a) describes the behavior of a theory of the glass transition and therefore a super cooled liquids quite well. Furthermore the particular means of predicting the transition outworking of the theory suggests an important from liquid to glass. connection between dynamics and thermodynamics and also between kinetic fragilities In summary, we have not yet arrived at a and thermodynamic fragilities16. coherent theory of super cooled liquids and glasses. The behavior of many liquids near the Super cooled liquids and the glass transition can glass transition has been described, but not all also be considered from the viewpoint of the that behavior is thoroughly explained. In their potential energy landscape. The so-called review of the topic Debenedetti and Stillinger16 energy landscape picture provides a convenient posit that the energy landscape perspective can framework within which to interpret existing explain qualitatively much of the behavior. data and purported concepts. Further details of What remains then is to establish theoretical the system are discussed later in this review. In perspectives with quantitative support. Related brief, the landscape picture establishes attempts should be –at least in part– built on the molecular motion at low temperatures as dissimilarities existing between liquid and

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glassy structures, which can be described by positions of large enough (micron-sized) means of the mathematical concepts of the particles from microscopy techniques (e.g., radial distribution function and the Voronoi tranditional or confocal microscopy29). In polyhedra. addition, given a pair potential energy function u(r), the RDF can be found either via computer simulation methods or through approximate THE RADIAL DISTRIBUTION functions30. Relations involving both g(r) and FUNCTION FOR SPATIAL u(r) can be used to calculate important DESCRIPTION OF NONCRYSTALLINE equilibrium thermodynamic quantities, such as, SYSTEMS the potential energy

Having spent some considerable time discuss- (3a) ing the nature of the liquid, glass, and crystalline states with regard to their physical the macroscopic pressure, properties and behaviors, we now advance to discussion on spatial descriptions of amorphous (3b) systems. Our picture of glassy structures has considerably improved – and given shape (kβ being Boltzmann’s constant), and the iso- –1 beyond the instructive balls and sticks model– thermal compressibility (κT = –V (ϑV/ϑP)T,N) by detailed images of Monte Carlo (MC) and 27,30,31. Such relations are of particular value for molecular dynamics (MD) computer simul- non-crystalline systems like liquids, solutions, 23 ations of amorphous cells . The mathematical dense gases and amorphous solids, for which concept of the radial distribution function alternative methods are limited. Note, however, (RDF) g(r), also known as the pair correlation that the results will not be as accurate as 3,24,25 function , has provided a means for directly calculating these properties because of distinguishing subtle differences among the the averaging involved in the calculation of 26 amorphous solid and liquid states of matter , g(r). The usability of RDFs in determining especially with regard to structure. structural parameters, like the structural coordination number z and its changes with The binary radial distribution function is a physical transformations of the system, has 2/N measure of the probability, ρ (0, r), of finding been extensively demonstrated24,32. a particle within an arbitrary reference frame and located in a spherical shell of an infinitesi- For crystals the RDF approach provides a series mal thickness at some distance r from a of delta functions along any crystallographic reference center13. That is, the RDF is defined direction. The radial distribution function of a as liquid is intermediate between the gas and the solid (Fig. 4), with a small number of peaks at (2) short distances, superimposed on a steady decay to a constant value at longer distances (dilute where ρ = N/V is the average number density of gas pattern, see Fig. 4a). The presence of a peak N-particle system (e.g., a fluid) in a container of indicates a particularly favored reparation 2 volume V. ρg(r)4πr dr is the number of distance for the neighbors of a given particle, particles at a distance between r and r + dr. The thus providing valuable structural information. calculation of g(r) involves averaging the In the solid state, maxima and minima appear number of particles at a distance r from any with positions explained in terms of particle in the system and dividing that number coordination shells of particles packed around 2 by the element of volume 4πr dr. It is thus the central reference particle (Fig. 4c). The possible to measure g(r) experimentally with presence of an atom at the origin of coordinates 27 neutron scattering or x-ray scattering excludes other particles from all distances 28 diffraction data, or by simply extracting the smaller than the radius of the first coordination

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Figure 4. Example g(r) plots of MD simulated (a) 500 particle Mie (Lennard-Jones) gas (number density = 0.01 and temperature = 1.0), (b) 500 particle Mie liquid (0.5 and 1.0, respectively), and (c) a 2000 particle Mie crystal (1.0 and 0.5, respectively). Adapted from: http://matdl.org/matdlwiki/index.php/softmatter:Radial_Distribution_Function. shell, where g(r) has a maximum. It is worth with its dual the Delaunay tessellation– noting that structural alterations –as measured essential for our understanding of the structure by traditional static binary correlation functions of amorphous materials. Before examining in – appear insignificant in the tempera-ture range detail the method of applying Voronoi poly- of the glass transition phenomenon. Never- hedra to amorphous systems, it is instructive to theless, for reasons just stated, the RDF is still describe the scope of this approach. The useful for describing some points of structure as partition of space into Voronoi polyhedra and well as for calculating certain properties of analysis of their topological features and metric materials in the liquid and glassy states. properties has numerous applications in local structure characterizations of disordered systems. The kinds of systems studied with the VORONOI POLYHEDRA FOR SPATIAL aid of Voronoi polyhedra include: hard-35,36 or DESCRIPTION OF NONCRYSTALLINE soft-sphere glasses23,37; liquid metals25,26; SYSTEMS molten38 and hydrated39 salts; nanoporous inorganic membranes40; water at ambient In 1908 and 1909 a Ukrainian mathematician, conditions41, in super cooled42,43 and stretched Hrihory Voronoi, published his two papers33,34 states42, as well as at the vicinity of the critical defining a mathematical construct –together point43; and other hydrogen bonding liquids44.

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Polycrystalline microstructures in metallic Selected applications will be discussed, with alloys are commonly represented using Voronoi emphasis on structural descriptions and local tessellations45,46. However, the use of the particle-dynamics studies of macromolecular Voronoi method is not limited to the field of substances and composites (so-called polymer- condensed matter physics but finds applications based materials or PBMs). Our focus on this from biophysics47, biology, and physiology to particular class of amorphous materials is astrophysics48 (e.g. fragmentation of celestial dictated by their expanding usage as metal part bodies, for instance, to describe quantitatively substitutes in, e.g., automotive and aeronautics results of a collision of two meteorites), industries and other engineering applic- bioinformatics49 and mathematics50. Since ations66,67. Voronoi cells can be defined by measuring distances to areas as well as to points, the approach is being used in image segmentation, THE VORONOI-DELAUNAY APPROACH city planning, optical character recognition and other applications51. If one were to investigate zirconia ceramics, one could not adequately do so without a Now more to the subject at hand, the glassy working knowledge of Miller indices since that state: macromolecular systems are often found knowledge makes it easy to understand and in the glassy state, since irregular chain explain the structures and behaviors of the architecture may inhibit crystallization under crystalline material. As new tools to describe typical polymerization conditions, under amorphous materials are developed, it likewise compression of as-received amorphous spec- behooves one who works with non-crystalline imens, or even under very slow cooling of their materials to acquire knowledge of those melts. Consequently, applications of Voronoi- techniques. Therefore we shall explain in some Delaunay structural analysis to polymer science detail how to use Voronoi polyhedra and their have appeared at an increasing rate, with mathematical dual, Delaunay simplices, to important results extracted for simple linear- better define and describe glassy materials. chain polymers52-59, as well as for grafted polymers60, polymeric foams61, dense colloidal A Voronoi polyhedron is defined as the suspensions62 and biopolymers (proteins, lipid (usually) convex region of space closer to its membranes, etc.)63,64. central particle than to any other. It is constructed – according to a unique The starting point for the description of mathematical procedure – for individual polymer structure is the concept of densely physical entities, hereafter called particles (e.g., packed, entangled, random Gaussian coils65, a ions, atoms, radicals, molecules or polymer notion derived from studies on polymer chain segments)3,52,68,69. In this procedure, each solutions and melts. Like inorganic glasses and particle is principally characterized by the glass-forming liquids, the structure of an location of its geometrical center and by the amorphous polymer exists in a metastable state size and shape of the surrounding polyhedron. with respect to its crystalline form (although in For a set of points on a flat surface (2D space), certain circumstances, for example in atactic with each point representing a particle, links are polymers, there is no crystalline analogue of the drawn between neighboring points (the so- amorphous phase). Therefore, experimental and called Delaunay triangulation process). Then, theoretical approaches dealing with the short- for each link a line is drawn perpendicular to it range dynamics of microstructural elements and and passing through a point equidistant from with the spatial description of polymers are the terminal points. By removing the Delaunay crucial in our attempt to provide interpretations triangulation, the bisectors produce polygons of these phenomena on the microscopic level. around the particles. For each particle, the In the next section, a description of the smallest polygon so constructed is the Voronoi Voronoi-Delaunay method is provided. polygon; the sum of all polygons constitutes the

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Voronoi diagram (illustrated in Figure 5). only shifted diagonally and equal to unit cells. This application of the Voronoi-Delaunay method to a square crystal lattice provides a simple illustration of the duality between Voronoi and Delaunay tessellations.

The procedure for constructing Voronoi polyhedra and Delaunay simplices in 3D is illustrated in Figure 6. The topological difference between these objects is that the Voronoi polyhedron represents the environment of individual particles whereas the Delaunay simplex represents the ensemble of neighboring particles (for conceptualization of this, consider again comparison to the crystal lattice). Furthermore, whereas the Voronoi polyhedra may differ topologically (i.e., they may have different numbers of faces and edges), the Figure 5. Delaunay diagram (thick lines) and Delaunay simplices are always topologically Voronoi diagram (thin lines) for a given group of equivalent. Ιn three-dimensional space, two particles –represented by dots– in 2D space. prevailing configuration types have been found based on several models of solids and Extending the above construction to 3D, with monoatomic liquids: Delaunay simplices close bisectors as planes instead of lines, is in form to a regular tetrahedron (“good” straightforward. A quadruplet of geometrical tetrahedra) or to a quarter of a regular neighbors (i.e. particles whose Voronoi octahedron (quatroctahedra)70. The partitioning polyhedra meet at a common vertex) forms of space attained in the way just outlined another basic topological object called a constitutes the Voronoi tessellation process. Delaunay simplex; the four particles are called When periodicity in local arrangement reaches a simplicial configuration. One used to working long-range order the above process becomes with crystalline materials could easily apply the identical to the Wigner-Seitz unit cell method same approach to a simple 2D square lattice. for crystalline solids4,71 and the construction of The crystals are represented by Voronoi the first Brillouin zone (although the fcc polyhedra, in this case squares (cubes in 3D). Brillouin zone leads to a bcc Voronoi The Delaunay simplices are identical squares polyhedron, and vice-versa).

(a) Irregular point set (b) Delaunay tessellation (c) Voronoi tessellation

Figure 6. Partitioning of 3D space (a cube in the present case) containing a randomly placed set of particles (dots) into Voronoi polyhedra.

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Since the shape of the Voronoi polyhedra and largest void that can be inscribed inside the the arrangement of the Delaunay tetrahedra tetrahedron without overlapping the provide a measure of the local environment and particles3,23,25,35. of particle packing in amorphous systems, several topological characteristics and metric By analyzing metric properties – such as the properties are in use. For each constructed number of faces, polyhedron shape and volume, polyhedron, Euler’s formula72 and related distributions – of an assembly of Voronoi polyhedra it becomes possible to Nυ – Ne + Nf = 2 (4a) appraise important phenomena and properties connects the number of vertices (Nυ), edges (Ne) of non-crystalline materials. Notably these and faces (Nf). In addition, since each vertex is include the ability to monitor changes in the the intersection of exactly three faces, and local structure and the free volume distribution hence that of exactly three edges, whereas each as the collection of particles passes – with edge connects exactly two vertices, it follows decreasing temperature – from the liquid or that rubbery to the glassy state (glass transition 2Ne = 3Nυ. (4b) phenomenon), and the possibility to describe percolative problems (e.g., phase transitions, Nf provides information on the number of geometric neighbors of the central particle, and thermal or electrical conductivity while the area of a face is related to the distance percolation thresholds in polymer nano- of the corresponding neighbor (i.e., closer composites). The 3D Voronoi tessellation neighbors generally share larger polyhedral method is now a well-established tool for faces). Voronoi polyhedra23,37,54 are commonly geometrical description of the structure of amorphous polymers from sub-nano to macro- measured by their volume (Vp), total surface area (A), shape factor (η'), scale levels. Construction of Voronoi diagrams is a non-trivial problem for which a few (5a) algorithms have been proposed with variable success23,36,69,73,74; a related routine has been offered with MatLab 6.5TM (by The Mathworks and curvature (C) Inc.), as well as with Materials Studio75 and (5b) other software packages (e.g., the VORONOI program, provided by CAPCPO and running TM where li is the length of edge i of the within AutoCAD ). polyhedron and θi is the angle between the normals of the intersecting faces37,54. The reciprocal of the Voronoi polyhedron volume FREE VOLUME CONSIDERATIONS IN characterizes the local density around the AMORPHOUS MATERIALS particle. By definition η' is 1 for a sphere and increases with increasing deviation from a We have made already a brief reference to free spherical shape (e.g., it equals 1.33, 1.35 and volume; and no discussion of structure as we 1.91 for a truncated octahedron, a rhombic are attempting at present can be complete dodecahedron and a cube, respectively). For the without further explication of this important Delaunay simplex, descriptions include the aspect. Within the macroscopic volume of a f tetrahedricity (Γ), material, the so-called “free volume” υ constitutes an equilibrium property of the (5c) system at temperatures exceeding Tg. The free volume is of paramount importance in relation (where li is the length of the i-th simplex edge to thermomechanical characteristics and and lAV is the average edge value for the engineering applications of most glassy simplex), along with octahedricity (O), materials, especially of polymer-based perfectness (S), and void size (υT), i.e., the materials. Molecular motions in the bulk state

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of polymeric materials are considered to depend unoccupied volume is no longer treated as a on the presence of structural voids, also known “hole” or “free” volume, but rather it is as “vacancies” or “holes” of molecular size associated with the immediate environment of (typical hole volumes of 0.02-0.07 nm3), or individual groups (e.g. side-chains) and the imperfections in the packing order of topology of the chains. Therefore it will change molecules. These holes are collectively as the polymer is heated, deformed78, or mixed described as free volume, a term also used to with another material (e.g., in the form of describe the excess volume that can be miscible binary polymer or oligomeric organic redistributed freely without energy change. At + polymer blends80-82). It should be noted, sufficiently fast cooling rates from the rubbery however, that the Voronoi polyhedron volume state to T < Tg, the spatial position of chains, only gives an indication of the volume available with their folds and constraints, remains fixed in the periphery of a particle; quantitative free- and unrelaxed to a large extent. Part of the volume estimates necessitate subtraction from unoccupied free volume spontaneously perishes the polyhedron volume of the hard-core volume with a concomitant reduction of both the bulk of the particle enclosed (i.e., the incompressible volume and the equilibrium end-to-end distance volume occupied by the particle at absolute of chains. The residual υf strongly depends on zero temperature and an infinitely high the starting temperature of the quenching pressure). process, the cooling rate, and the conformational characteristics of the polymer chain. Several reports highlight the importance of the notion of the distribution of free volume, rather Quantitative determination of free volume is than that of the total free volume of a material. performed infrequently, and that despite the fact Free volume distribution contributes signif- that several well-established methods exist. For icantly to several thermal, mechanical and example, in glass-forming liquids, molecular rheological properties of a polymer system. Its simulations are used to examine changes of evolution in the course of different time- local structure and free volume in the dependent thermomechanical and physico- temperature region approaching the glass chemical treatments (drawing or stretching transition. For probing pore sizes, pore size deformation, physical aging, curing, etc.) is a distributions and pore connectivities of poly- matter of interest for material scientists and meric systems, techniques including positron engineers. Details of the fine structure of annihilation lifetime spectroscopy (PALS), amorphous systems can be derived by computer modeling, and numerical analyses of considering individual “atomic” polyhedra. amorphous cells have been extensively When such systems are analyzed in terms of implemented54,76. Moreover, computer simul- Voronoi polyhedra they show wide variations ations provide equilibrated polymer con- in packing density on the atomic/monomer figurations which permit Voronoi tessellations scale, with a characteristic skewed distribution; of structural groups (atoms, compounds, in fact, the width of the monomer Voronoi monomers, chains, etc.) to be constructed and volume distribution is regarded as a measure of also examined along the macromolecular chain; amorphicity. density fluctuations, ranging from very narrow54 to very wide52, and anisotropic The combined molecular dynamics and distribution of free volume53,56 have become Voronoi tessellation analysis of amorphous evident in that way. Additionally, through poly(trimethyl terephtalate) (PTT)55, a model computer simulations collections of particles linear polymer resembling polyethylene53, and can be classified into liquid- or solid-like of other types of unentangled linear chains57,58, categories on the basis of free volume and on has successfully addressed the applicability of the basis of shape and distortion of the Voronoi the Voronoi approach in free volume polyhedra and Delaunay simplices23,37,77. When distribution determinations. It is worth noticing described in such a geometric way, the that atomistic simulation studies of “simple”

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linear polymer chains are rather common, since The free volume distribution in polymer such systems encompass the essential attributes systems is expected to be influenced by tacticity of connectivity, constrained flexibility and van and side group substitution. By comparing pore der Waals interactions. Jang and Jo55, for distributions obtained from PALS measure- example, analyzed chain conformation ments with the static free volume distribution characteristics, such as bond orientation, the obtained from amorphous cells simulated using dihedral angle, and Voronoi volume tessell- the Voronoi tessellation of space, Dammert, et ation, and showed that the PTT chain under al.83 have explored these features using poly(p- extension undergoes conformational changes methyl styrene) and polystyrene as model different from those under compression. systems. The results indicated broader hole Voronoi volume distribution was found to distributions for the syndiotactic specimens broaden with strain under extension and shrivel compared to the more atactic samples. On the under compression, with a concomitant other hand, modeling revealed that the methyl decrease in free volume. Stachurski52,82 substituent broadens the distribution of free provided another illustrative example of this volumes considerably; a behavior documented behavior through a computer simulation study in the positron annihilation results as longer of a cell of poly(methyl methacrylate) lifetimes and larger volumes of the holes. The (PMMA), comprising 9000 atoms. The analysis maxima in the free volume hole size of such virtual amorphous cells revealed large distribution were of smaller values for the periodic density fluctuations, which in polystyrenes than for the poly(p-methyl uncrosslinked polymers led to – and provided styrene)s. Interestingly, calculations also evidence for – the concept of constriction revealed the presence of a large number of points; at locations along the chain where the undersized holes (due to the spacing among local Voronoi volume is minimal, the functional groups in the polymers), with surrounding atoms act as constrictions on the dimensions outside the PALS measuring molecular chain within52,78. sensitivity.

Figure 7. Average volume of the Voronoi polyhedron around the particle located at the n position along the chain (symmetric positions with respect to the centre of the chain are averaged over n, with n ≤ M/2, M = chain length). Note the larger volume of the polyhedra surrounding the chain ends (n = 1). Inset: chain length dependence of the Voronoi volume of the inner polyhedra (from Ref. 57).

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The chain length dependence of the glass (–CH2–), and junction (>CH–) groups exhibit transition temperature in polymers is often different Voronoi polyhedra shapes with explained by the higher free volume volumes decreasing by group in the same order surrounding chain ends57,76,84. The Voronoi as they are here mentioned (Fig. 8). Chain-end polyhedron volume associated with a chain end volumes are the most sensitive to temperature, has been reported to exceed the average indicating higher mobility for these units. Voronoi volume57,59,76 (Figure 7). Similar Moreover, chain ends dominantly localize at the arguments apply to both non-deformed material’s surface. This striking result involved structures and model systems experiencing the observation that while the ratio of surface extensional strain53. For polymers with either groups was only 24% of all atoms, the ratio of flexible chains or strong intermolecular ends at the surface was 91% out of all ends. interactions, the nucleation of cavities upon This “preference” has direct consequences in deformation occurs preferentially near the chain the interpretation of accelerated relaxation ends58. Simulations results of the effect of dynamics, namely that as the surface is stretch and the resulting molecular orientation approached, we observe a gradual change along on the free volume distribution in a poly- with increasing relaxation frequencies. ethylene-type polymer confirmed the exper- Furthermore there is a concomitant depression imental observation that chain alignment (due of the global glass transition temperature to increasing extensional strains) causes a reported for polymers in the form of free- decrease both in the total number of voids and standing or supported (on to repulsive surfaces) the number-average void size. At the same time ultrathin films87. Therefore we infer a there is an increase in the number of larger, significant relationship between chain topology, more elliptical empty cavities (voids) in the free volume, and the glass transition polymer85 due to the extensional strain. The temperature. Such connections are just effective density increases as result of a beginning to be fully appreciated and utilized in decrease in the total free volume but it is not research pertaining to the glassy state. distributed evenly; free volume associated with atoms located away from the ends (i.e., atoms generating the so-called “inner” polyhedra) decreases, while the free volume associated with atoms located at the molecular ends increases with stretch. In a recent simulation study of the nucleation and growth of voids (cavitation process) that precedes craze formation, and the early crazing itself, occurring in rod-containing polymer nanocomposites, it was found that the Voronoi volume can anticipate void formation and that it can also be used as a predictor of failure, particularly in composite materials86.

The free volume is important to understanding the glassy state especially as it applies to relaxation dynamics – and we may recall from Figure 8. Histogram of the frequency distribution of section 3 the significance of relaxation times in the inverse polyhedron volume 1/Vp at 300 K for our characterization of glasses. Here we expand various groups. The exceptional peak at 1/Vp = 0 the discussion to include relationships with corresponds to open polyhedra. I indicates a small f chain structure and υ . The Voronoi space shoulder near 1/Vp = 0.03 (end groups), the division of topologically different groups has Gaussian-type peak II at 1/Vp = 0.05 corresponds to 60 been explored through MD simulations of a internal groups, and peak III near 1/Vp = 0.08 500-mer polyethylene model chain linked by 50 corresponds to junction groups. The Voronoi hexyl groups as side-chains (i.e., a grafted polyhedra of the three types of centers are also 60 polymer having 52 ends). End (CH3–), internal shown.

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LOCAL STRUCTURE, GLASS already examples of how Voronoi-type cells DYNAMICS AND THE GLASS can be used for modeling thermal88 and TRANSITION electrical89-91 conductivities of various polymer nanocomposites. Gerhardt and coworkers90,92, We are now equipped to examine in further for example, describe for compression molded detail the local structure — as described by polymer-matrix composites the formation of a Voronoi polyhedra — of amorphous PBMs and separated network microstructure in which the the relationship to glass dynamics and the glass originally circular-shaped matrix particles reach transition. A promising use of Voronoi polyhedral shapes upon compression. networks involves its application to percolation Specifically, for polymethyl methacrylate analysis of amorphous structures. Using (PMMA)/carbon black (CB)89, acrylonitrile Voronoi networks, one can study diffusion and butadiene styrene (ABS)/CB91, and percolation properties of complex systems. PMMA/indium tin oxide (ITO) Percolation theory is commonly used to nanocomposites90, the polymer phases were describe natural phenomena that feature a roughly equivalent to ordered Voronoi cells continuous phase transition. For a given (i.e., truncated tetrahedra, with 8 faces and 18 network, finding the critical point pc (i.e., a edges) in the microstructure with the filler probability, mass or volume fraction) at which nanoparticles aggregating along the edges of the percolation transition occurs is a problem of the polyhedra, thereby forming a structure particular interest. The glass-to-rubber or glass- comparable to nanowires (illustrated in Figure to-liquid transitions in amorphous organic or 9). Through this approximation it was made inorganic systems as well as the conductor-to- possible to interrelate the radii of the initially insulator transition in insulating organic spherical matrix particle and filler (rp and rf, matrices with conductive inclusions constitute respectively) with the edge lengths of the important percolative problems. There exist deformed matrix particle and filler (ap and af,

Figure 9. (a) Transmission electron microscopy (TEM) image of ITO nanoparticles. Images in (b), (c) and (d) are illustrations of ITO-coated polymer-matrix particles where the filler is depicted as small particles and the matrix is depicted as large particles: (b) before compression molding, (c) during compression molding, and (d) after compression molding in the final composite with an ITO content near the percolation threshold.90

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respectively), and finally obtain theoretical In spite of the several differences reported, predictions for the critical volume fraction of there is generally a lack of signatures of the the filler90. Self-assembled structures such as glass transition in the particle positions on the these are the driving force for extremely low molecular level. To overcome this lack of percolation thresholds, phenomena observed markers, a number of studies have focused on through electrical impedance analysis of relationships between local structures, samples with varied filler content. Thus we characterized by Voronoi polyhedron volume, have multiple instances where the measured and local dynamics, characterized by particle properties of amorphous PBMs are described displacements62. In an interesting study back in through Voronoi networks. 1997, Jund and coworkers94 executed an atomistic study –for a simulated 1000-particle The information presented so far provides system– of the mechanism of the glass unambiguous evidence for the physical transition, analyzing the volume and surface significance of some simplicial particle distributions of Voronoi polyhedra above and configurations. For instance, Voronoi poly- below the system’s Tg. A saturation of the hedra with large circumradii denote low density density fluctuations was verified as a signature configurations, and vice-versa. Along these of the transition in the system cooled from the liquid state. Moreover, evaluating deviations of lines, the Voronoi-Delaunay technique has been the cell shapes from a regular dodecahedron, utilized for studying subtle differences in the the authors observed that the fraction of non- structure of liquid and solid (quenched) 70 pentagonal cell faces increases with phases . Structural signatures in the form of temperature in the glassy phase, at the expense percolation thresholds of Delaunay net- 26,70 of 5-edged faces, while at the glass transition works and increase in icosahedral ordering the trend is reversed and a majority of 93 near Tg have been observed in cases of pentagons is recovered in the liquid state. simulated amorphous solids and simple liquids. Another research group95,96 approached the Structural information has been extracted by glass transition phenomenon by distinguishing studying the percolation thresholds of networks between “liquid-like” and “solid-like” defects: of Delaunay simplices of different “coloring”, the concentration, , of liquid-like defects – where each color denotes Delaunay sites of defined by them as small particles enclosed in identical form (i.e., with identical metric heptagons or octagons, and large particles properties). For example, in a case study of the enclosed in pentagons or even squares– was molecular dynamics configurations of liquid, shown to nearly vanish in the glass state. A super cooled and quenched rubidium, typical scale parameter was defined as 26 95 Medvedev and coworkers indicated that the ξ ≡ , and found to diverge at Tg , in a way Delaunay simplices develop macroscopic analogous to the divergence of the relaxation aggregates in the form of percolative clusters. time of a viscous fluid as temperature In the liquid state, clusters result from low- approaches Tg. density atomic configurations. These macroscopic structural organizations in the liquid Gil Montoro and Abascal23 performed state permit extensive motions, like those in microcanonical MD simulations of a model shear flow. In contrast to the low-density glass-sphere system featuring Mie (Lennard- atomic configurations of the liquid state, nearly Jones) interactions. This study revealed that tetrahedral high-density configurations con- both the width and asymmetry of the Voronoi tribute to cluster formation in the solid state. In volume distribution increases when going from the liquid state, the low-density cluster goes the solid to the liquid and finally to the gas (i.e., across the whole material; in the amorphous a simulated non-interacting fluid) state (Fig. 10a). Furthermore, the nonsphericity α solid state, the high-density cluster percolates distribution (α = RA/3V , R being the average across the whole glass. p

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Figure 10. (a) Voronoi polyhedron volume distribution functions about the mean (V* = Vp/Vp(mean)), and (b) nonsphericity (α) distribution functions, for simulated solid (S), liquid (G) and gas (G) systems. Replotted graphs from Ref. 23. curvature radius of the convex body) in a cooperative manner. Numerous applic- demonstrated the highest versatility in ations of the PEL approach have been reported, describing structural differentiations among the including the establishment of links between its three states (Fig. 10b). topography and the dynamics for binary Lennard-Jones glasses97, and the identification of significant confinement-induced differences Establishing connections between local among model bulk and free-standing polymer structure characteristics and vibrational proper- films99. Jain and de Pablo100 performed detailed ties in glassy systems remains an open issue. A computer simulations of a model super cooled valuable theoretical framework in this pursuit is polymer, near its apparent Tg, exploring the role the potential energy landscape (PEL, mentioned of the local “inherent” structure of particles earlier), where energy is partitioned into basins (i.e., of their Voronoi polyhedron) on motions connected by saddle points—a system, which on the PEL101,102. Their results indicated that the represents the complicated dependence of time of vibration of a particle in a metabasin 97 energy on configuration . When studying correlates with the structure of its Voronoi dynamics in the glassy state one assumes a polyhedron and with the number of its separation of time scales as the system neighbors; the largest metabasins corresponded approaches the glass transition temperature; to particles whose average Voronoi volume was short-time motions are considered to occur via close to the value expected on the basis of the intrabasin vibrations about a particular structure density, and whose approximate number of (a local potential energy minimum), while long- neighbors approached 12 (icosahedral order- time motions take place via occasional ing). The local distortion around a particle, activated jumps over saddle points into measured in terms of the tetrahedricity of the neighboring basins. In an amplification of this Delaunay simplices, also revealed that particles concept, the picture of “metabasins” has been 98 with a higher degree of local distortion are introduced . Each metabasin consists of several likely to transition faster to a neighboring meta- local minima separated by low energy barriers; basin. The above arguments stress the the α-relaxation (the signature of the glass- significance of the identification of structural transition process in dynamic experiments) motifs in understanding the influence of occurs via jumps between neighboring chemical structure on the dynamics in glass metabasins, with molecular motions proceeding formers.

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On a totally different –yet still related– direction, Luchnikov and coworkers37 studied atomic configurations in a simple soft-sphere model glass and demonstrated the usability of the Voronoi-Delaunay structural approach in deciphering correlations between normal-mode vibrations with parameters characterizing local- structure perfectness (i.e., O, S and Γ, defined above). In combination with a conventional harmonic vibrations analysis the conclusion drawn was that in real space the lowest- a frequency quasi-localized vibrations can be envisaged as being caused by instabilities in local geometry.

VORONOI TOPOLOGIES OF MICROSCOPY IMAGES

A majority of theoretical studies based on statistical physics that incorporate the Voronoi- Delaunay approach analyze objects existing within virtual space created by a computer. Nevertheless, in the discipline of applied materials science geometric analyses of b Voronoi topologies involve –on a more and more frequent basis– real structural information acquired, among other techniques, from Figure 11. (a) SEM image of a polymeric foam scanning electron (SEM), atomic force (AFM) 61 (×100), and (b) its Voronoi diagram. and scanning tunneling (STM) microscopy 61,90,103-105 images . Nearly two decades ago, 61 105 Jacobs and coworkers recently discussed a Stange, et al. , using AFM and STM, followed quantitative method for the comparative the evolution of spin-coated polystyrene films Voronoi analysis of various polymeric foam on silicon surfaces from individual isolated morphologies. At the foundation of their molecules to a continuous film. At a critical analytical technique are parameters related to polymer concentration in toluene (the the average Voronoi cell area, cell area polymer’s solvent), they observed the formation distribution, and foam homogeneity. As a first of 2D Voronoi tessellation-like networks of step in this approach, cell walls are visualized polystyrene molecule aggregates, which they with SEM analysis of a cross section of the subsequently discussed in terms of a specific foam (Fig. 11a). Using only the perimeter of the failure mechanism leading to film rupture in 106 cells as markers, the cell structure appearing in spin-coating processes. Recently, Song et al. such images is subsequently converted into a used Voronoi diagrams and bond-orientation grid (using MatLabTM) after which the Voronoi correlation functions to analyze microscope diagram is constructed (Fig. 11b). The images of “structured” microporous polymer homogeneity of the foams could thus be films (i.e., films with closely packed hexagonal described in terms of cell area, perimeter and arrays of pores). A direct measurement of the number of faces. Using poly(styrene-co-methyl open pore sizes and their distribution was made methacrylate) foamed with supercritical carbon possible. dioxide as model system, Jacobs et al.61

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demonstrated the ability of the above method to suspension, in excellent agreement with the reveal even slight changes in a foam’s homo- simulation results. geneity. The accuracy of the constructed Voronoi diagrams is clearly dependent on the perspicuity of the SEM pictures; use of GLASS TRANSITIONS IN BINARY sufficient contrast between the voids and the SYSTEMS continuous phase is mandatory. Microstructure is a critical determinant of Similar descriptors are applied to morph- various thermophysical and mechanical ological characterizations of fiber reinforced properties of inorganic + organic composites composites in terms of Voronoi polygons103,104. and of miscible organic blends. Moreover For example, the transverse spatial distribution microstructure is a key feature for materials in of glass fibers in continuous fiber reinforced selected applications such as drug delivery epoxy microstructures has been described using systems and composite solid state dye laser Voronoi polygon areas and nearest neighbor matrices, among others. While the method of distances as spatial descriptors103. In such an material processing makes some contribution to approach, Voronoi tessellations with fibers the resultant microstructure, that structure for located at cell centers were generated from organic polymer blends is also inherently SEM micrographs. Convergence towards global dependent on the balancing among inter- and distributions was observed (for both metric intra-molecular interactions and the ensuing properties) at increasing sample sizes for most local density fluctuations. Molecular packing is of the systems studied. Statistical tests provided clearly reliant on chemical composition, estimates of the size of the volume element polymer chains conformation and configur- (i.e., a sampling area dimension), which is ation, and the degree and relative strength of representative of the global microstructure and enthalpic and entropic factors. Given the inhomogeneity in each material. At the scale of complexity of the underlying mechanisms in this “representative” volume element any such polymer systems, a usually inhomo- sample taken of the actual composite structure geneously dispersed population of structural is deemed equivalent, and representative, of the voids appears, with dimensions extending even entire microstructure. up to ~10 nm. Connected to all these attributes of amorphous binary systems is the glass Concentrated colloidal suspensions are also of transition – and with it relationships to interest as they may exhibit a distinct glass microstructure. transition whose value is a function of particle concentration or density. Using dense colloidal Microstructural peculiarities are evident in suspensions as a model system, Conrad and miscibility studies of binary polymer79,80 and coworkers62 searched for correlations among drug + polymer mixtures81,107. Notably, the the Voronoi volume and particles’ displace- glass transition temperature of these systems ment. Particle positions were imaged by manifest anomalous dependencies on com- confocal microscopy using fluorescently position (φ, as a mass fraction) (Figs 12, 13). labeled polymer spheres. Results indicated that Even in cases of athermal binary mixtures, the scaled (by the standard deviation) there is ample experimental evidence in binary distribution of Voronoi volumes around the mixtures that the thermodynamically predicted average Voronoi volume is universal for smooth and monotonic Tg(φ) variation, dispersions with a hard interparticle potential confined within the transition temperature (i.e., super cooled liquids containing “hard region of the constituents, is often violated. spheres” with a Lennard-Jones type potential). This is the case irrespective of the problems The scaled distributions were shown to fall onto encountered in defining or locating the Tg of a a universal curve over a wide range of volume given system – an issue also partly related to fractions of the polymer spheres in the colloidal kinetic attributes of the glass transition.

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Figure 13. Tg vs. composition dependence of poly(styrene-co-N,N-dimethylacrylamide) [with 17 mol % of N,N-dimethylacrylamide] (SAD17) + poly(styrene-co-acrylic acid) [with 18, 27 or 32 mol % acrylic acid]80. Thick lines are fits to the BCKV equation (6).

with φ1 denoting the mass fraction of the low-Tg component. Eq. (6) is more recently called the BCKV equation81. The quadratic polynomial on its right side, centered around 2φ1 – 1 = 0, is defined to represent deviations from linearity. The type and level of deviation is primarily described by parameter a0, which mainly reflects differences between the strength of Figure 12. (a) Tg vs. composition dependence of Intraconazole (ITZ) + PLS-630 blends, and (b) hetero- (intercomponent) and homo- compositional variation of density ρ and excess (intracomponent) interactions. The magnitude E mixing volume V (per gram of mass) for the same and sign of the higher-order parameters a1 and mixture. Compiled from data appearing in Ref. 107. a2 is dictated by composition-dependent energetic contributions from hetero-contacts, entropic effects and structural heterogeneities Quantitative description of the deviations from (e.g., nanocrystalline phases). Along these ideality has been attempted through various lines, the number and magnitude of the theoretical and semi-empirical equations81, with parameters required to represent an the most system-inclusive approach only very experimental Tg(φ) pattern provide quantitative recently proposed (by Brostow, Chiu, measures of a system’s complexity. Evidently, Kalogeras and Vassilikou-Dova)108 in the form irregularly positive or negative –or even of the function sigmoidal– deviations from the linear mixing rule are strongly linked to the compositional Tg = φ1Tg,1 + (1 – φ1)Tg,2 + dependence of the total free volume and the 2 + φ1(1 – φ1)[a0 + a1(2φ1 – 1) + a2(2φ1 – 1) ] free volume distribution around pertinent chain segments. All these “asymmetric” entropic or (6) enthalpic contributions are expected to reflect

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in the volumes and numbers of faces of the Considering the increased depth and image composite polyhedra of structural particles resolution of contemporary microscopy (e.g., monomers for polymers and molecular apparatuses discussed in the previous section, polyhedra for oligomeric organics). one may attempt performing V-D analysis of electron miscoscopy images of miscible blends Voronoi analysis of binary systems is in as spin casted or solvent evaporated thin films, principle possible, e.g., by using MD for example. However, development of simulations to obtain equilibrated sufficient suitable codes remains an unsolved structures109,110 and developing appropriate problem. Likewise, selection criteria are needed analysis codes. For example, there was recently for distinguishing among the two different a report109 of MD simulation of the populations of "composite" polyhedra in a compatibility of chitosan (CS) + poly(vinyl binary mixture; for example, the ones generated pyrrolidone) (PVP) biomedical mixtures. by drug molecules and those encompassing Although the Voronoi tessellation analysis was preselected polymer segments (i.e., the not conducted, the binary RDFs could be elementary unit of a monomer). calculated from amorphous cells obtained for selected blend compositions. The RDF behavior (Figure 14) confirmed the supposition that the SUMMARY components’ miscibility arose from hydrogen Considerable literature exists regarding the bond formation among the –C=O group of PVP representation of structures of amorphous and the –CH OH group of CS. A Voronoi- 2 materials. Of the available approaches, the Delaunay analysis of that particular mixture Voronoi-Delaunay tesselation technique and the would be of high interest to our present radial distribution function are usable for any discussion, especially given the strongly material structure – crystals included. The sigmoidal shape of its T vs. composition g experimental information and theoretical pattern111 and its classification as a “high- analyses discussed in this review exemplify complexity” binary system81. their efficacy for quantitative local-geometry 112 investigation of various types of non-

crystalline solids. In view of that, both approaches warrant inclusion in Materials Science Education (MSE) courses and should be taught along with the venerated crystallographic methodology.

A brief scheme of an introductory MSE module, or a pertinent educational presentation that makes use of the information previously presented, is given in Table 1. Several computational-geometry routines and applets are available113-119 for instructors interested in an attention-grabbing interactive demonstration of the construction of Voronoi diagrams and Figure 14. RDF calculation for a miscible 80/20 (molar fraction) CS + PVP blend, using the Delaunay triangulations in any random set of hydrogen atom of the hydroxyl methyl group of particles. Representative tools proffer the TM 114 chitosan (solid line) and the hydrogen atom of the – Java applets of Paul Chew (Cornell 115 NH2 group of chitosan (dotted line) relative to the University), Andreas Pollak or Christian location of the oxygen atom of the carbonyl group of Raskob116 (University of Hagen), which were 95 PVP . created as part of their diploma dissertations or

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related activities. The VoroGlide applet117 Diagrams in Biology” describes a program that (University of Hagen) is another interesting constructs weighted Euclidean and power program for interactive Voronoi diagrams, diagrams for computer simulation and analysis offering a single step animation mode for the of a system of growing plants120. It also incremental Delaunay construction and a demonstrates application of power diagrams to recorder for teaching purposes. There are also a problems from the fields of biology and number of websites providing educational ecology. Sites such as this, along with the programs that can serve as sources of interest mentioned applets, provide an opportunity for for the novice in the field. For example, a practical exercise of the concepts outlined in webpage under the title “Weighted Voronoi Table I and discussed in this review.

Table I. Proposed plan of an introductory MSE module, related to the Voronoi-Delaunay representation and analysis of non-crystalline structures and other applications.

Steps Visual tools Training resources (suggestions) (JavaTM applets, etc.)

1. Introducing basic notions a) States of matter: Gas, Liquid, Solid. b) The solid state: Glasses vs. Crystals. Fig. 1. c) Glasses vs. Liquids: The glass transition. Fig. 1; Fig. 2; Fig. 3.

2. Structure descriptors a) RDFs Fig. 4 (and comparison with δ-function RDFs of typical crystals).

b) The Voronoi-Delaunay (V-D) approach: Fig. 5; Fig. 6 (and Real-time practicing Construction and metric properties. comparison with with V-D graphs113-118 crystallographic notions).

3. V-D analysis: case studies a) Free-volume distribution in polymers. Fig. 7; Fig. 8. b) Local structure and transitions. Fig. 9; Fig. 10. c) Polymeric foam morphologies. Fig. 11.

4. Ideas for future applications a) Modeling glass transitions in binary Fig. 12; Fig. 13; Fig. 14. systems. b) and more, from instructor’s research Voronoi & biology120 field… Voronoi & fractals121,122

5. Time for relaxation: The Voronoi game! 123 A two-player game based on a simple geometric model for the “competitive facility location”. Competitive facility location studies the placement of sites by competing market players. The geometric concepts are combined with game theory arguments to study if there exists any winning strategy.

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