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Amorphous solid

An amorphous solid is a rigid material composed of atoms, molecules, or particles arranged in a disordered, non-repeating fashion, lacking the long-range translational symmetry and periodic lattice structure found in crystalline solids.^[1]^[2] Its atomic configuration resembles that of a liquid, with short-range order but no overall organization, resulting in a "frozen" supercooled state that maintains solidity at ambient temperatures.^[1]^[3] Amorphous solids form when a molten material is cooled rapidly enough to prevent the atoms from aligning into a crystal lattice, a process that minimizes molecular mobility and traps the disordered structure.^[1]^[4] Unlike crystalline solids, which exhibit anisotropic properties due to their ordered lattices, amorphous solids are isotropic, displaying uniform physical characteristics in all directions.^[2] They lack sharp melting points, instead softening gradually over a temperature range, and produce irregular fractures without defined cleavage planes when broken.^[2] X-ray diffraction patterns of amorphous solids are diffuse and poorly defined, reflecting their absence of long-range order, in contrast to the sharp peaks seen in crystals.^[2] These materials can span various classes, including ceramics, polymers, metals, and semiconductors, and often demonstrate enhanced solubility, reactivity, or mechanical resilience compared to their crystalline counterparts in specific contexts.^[1]^[5] Common examples of amorphous solids include silica glass (such as window glass or fused quartz, SiO₂), which forms the basis for optical fibers and laboratory ware due to its transparency and chemical inertness.^[1]^[2] Many polymers, like polystyrene, adopt amorphous structures and are widely used in packaging and insulation for their flexibility and lightweight nature.^[1] Other instances encompass natural volcanic glass like obsidian, a silica-rich (SiO₂) material, and synthetic amorphous metals, such as those in Liquidmetal® alloys (e.g., zirconium-beryllium-titanium-copper-nickel compositions), prized for their high hardness and elasticity.^[2]^[4]^[6] Even everyday items like cotton candy exemplify amorphous solids in food applications.^[1] Amorphous solids play a critical role in modern technology owing to their unique properties, enabling applications across diverse fields.^[7] In electronics, amorphous semiconductors like hydrogenated amorphous silicon (a-Si:H) are essential for thin-film solar cells and large-area displays due to their cost-effective deposition and optoelectronic performance.^[8]^[9] Amorphous metals find use in high-wear coatings for industrial equipment, such as refinery components, and in precision tools like medical scalpels, where their low friction and durability outperform traditional metals.^[4] In pharmaceuticals, amorphous forms enhance drug solubility and bioavailability, accelerating dissolution rates for better therapeutic efficacy.^[5] Additionally, their optical clarity and resistance to crystallization make them ideal for lenses, while in materials science, they exhibit intriguing phenomena like low-energy excitations absent in crystals, influencing research into elasticity and lattice dynamics.^[10]^[11]

Definition and Etymology

Definition

An amorphous solid is a non-crystalline solid characterized by a disordered atomic or molecular structure that lacks long-range order and translational symmetry, while maintaining short-range order where neighboring atoms adopt locally preferred coordination geometries similar to those in crystalline counterparts.^[12] This absence of periodicity means amorphous solids do not exhibit the sharp diffraction patterns typical of crystals when analyzed by X-ray scattering, instead producing broad, diffuse halos.^[13] In contrast to crystalline solids, which feature a periodic lattice arrangement leading to well-defined geometric shapes, sharp melting points, and anisotropic properties, amorphous solids display irregular external forms, gradual softening over a temperature range, and isotropic behavior due to their structural randomness.^[14] Unlike liquids, which flow under applied shear stress because their molecules can freely rearrange, amorphous solids resist such deformation and maintain a fixed shape, qualifying them as true solids despite their liquid-like atomic disorder.^[15] Representative examples of amorphous solids include silica glass used in windows, polymers like polystyrene in plastics, metallic glasses such as iron-boron alloys valued for their magnetic properties, and amorphous ice formed by rapid quenching of water vapor.^[13] The ideal glass represents a theoretical non-equilibrium state in which a supercooled liquid is kinetically arrested below its freezing point, preserving a frozen snapshot of the liquid's disordered configuration while exhibiting solid-like mechanical rigidity.^[16]

Etymology

The term "amorphous" derives from the Ancient Greek a- ("without") and morphḗ ("form" or "shape"), literally meaning "shapeless" or "without form," and entered scientific English via Modern Latin amorphus around 1731.^[17] This linguistic root reflected early perceptions of materials lacking the ordered, geometric structure of crystals, distinguishing them from well-formed minerals. Although the specific term "amorphous" emerged in the 18th century, Robert Hooke provided one of the earliest documented observations of non-crystalline materials in his 1665 work Micrographia, where he examined the microscopic texture of glass and noted its uniform, non-faceted appearance under magnification, contrasting it with crystalline substances like salts.^[18] In mineralogy, "amorphous" was applied to naturally occurring substances that lacked the external crystal faces typical of minerals, marking the term's initial adoption to categorize shapeless geological materials.^[19] By the 19th century, the concept was formalized in chemistry through studies of physical properties, with Hermann Kopp employing "amorphous" to describe non-crystalline forms of substances such as carbon, whose heat capacities deviated from those of their crystalline counterparts, thus broadening the term beyond mere visual description to include thermal and structural behaviors.^[20] In the 20th century, amid rapid advances in glass science and materials research, "amorphous solid" evolved to encompass a wider array of synthetic and processed materials, including polymers and metallic glasses, reflecting a shift from mineralogical origins to interdisciplinary applications in physics and engineering.^[21]

History

Early Observations

Ancient civilizations utilized amorphous solids, particularly glass, long before their structural nature was understood. In ancient Egypt, faience—a quartz-based material coated with a glassy, vitreous glaze—was produced as early as 4000 BCE, serving decorative and functional purposes such as beads and amulets. By around 1500 BCE, true glass objects, including vessels and ornaments, emerged in Egypt and Mesopotamia, crafted through melting silica with fluxes to form a non-crystalline material valued for its transparency and moldability, though artisans lacked knowledge of its atomic disorder.^[22]^[23] The development of microscopy in the 17th century enabled initial empirical observations of irregular structures in organic materials. Robert Hooke, in his seminal 1665 publication Micrographia, examined gums, resins, and similar tenacious substances, describing how they formed elongated, cohesive filaments without the regular facets or geometric patterns typical of crystals, attributing this to the "congruity of parts" in their fluid-like yet solid states.^[18] These findings highlighted the absence of ordered arrangements in such natural products, contrasting them with crystalline minerals observed contemporaneously.^[24] By the 19th century, chemists systematically identified substances that resisted crystallization, marking a shift toward recognizing amorphous solids as a distinct class. Jöns Jacob Berzelius isolated amorphous silicon in 1824 through the reaction of silicon tetrafluoride with potassium, noting its powdery, non-faceted form in contrast to crystalline silicon prepared later.^[25] Subsequently, Thomas Graham introduced the term "colloid" in 1861 to describe gelatinous materials like gels, rubber, and glue, which diffused slowly and failed to form crystals upon solidification, exhibiting jelly-like consistency due to their dispersed, non-crystalline organization.^[26] The term "amorphous," derived from Greek roots meaning "without form," was applied to such substances to denote their lack of crystalline structure.^[27] Early 20th-century experiments advanced understanding through controlled formation of amorphous states. Gustav Tammann conducted studies on undercooling liquids below their freezing points, demonstrating in works from the 1890s to 1900s that sufficient supercooling prevented nucleation and crystallization, yielding stable glassy solids from melts of metals, salts, and organics; for instance, he quantified crystallization rates as a function of undercooling depth, establishing kinetic barriers to crystal growth.^[28] These observations laid groundwork for viewing glasses as undercooled liquids frozen in disordered configurations.^[29]

Modern Developments

In the early 20th century, significant theoretical progress was made in modeling the atomic structure of amorphous solids, particularly silicate glasses. In 1932, William H. Zachariasen proposed the continuous random network (CRN) model, which describes the structure of glasses as a continuous, randomly connected network of polyhedra similar to those in their crystalline counterparts, but lacking long-range translational and orientational order.^[30] This model emphasized short-range order while allowing for distortions in bond angles and lengths, providing a foundational framework for understanding glass formation based on empirical rules such as limiting oxygen coordination to no more than two cations.^[30] Zachariasen's CRN paradigm was experimentally validated shortly thereafter through X-ray diffraction studies by B.E. Warren in 1934, confirming the absence of sharp Bragg peaks and the presence of diffuse scattering indicative of random atomic arrangements. A major experimental breakthrough occurred in the late 1950s with the discovery of metallic glasses, expanding the scope of amorphous solids beyond traditional oxide glasses. In 1960, Pol Duwez and colleagues at the California Institute of Technology successfully produced the first metallic glass by rapidly quenching a molten Au₇₅Si₂₅ alloy at cooling rates exceeding 10⁶ K/s using a splat-quenching technique, preventing crystallization and yielding a non-crystalline metallic structure.^[31] This innovation demonstrated that metallic systems could form amorphous phases under extreme non-equilibrium conditions, revealing unique properties such as high strength, elasticity, and corrosion resistance due to the absence of grain boundaries.^[31] The rapid quenching method paved the way for synthesizing a variety of amorphous metal alloys, influencing applications in magnetic materials and biomedical devices. During the 1970s and 1980s, experimental investigations uncovered universal vibrational and thermal anomalies in amorphous solids, distinguishing them from crystalline counterparts. The boson peak, a prominent excess in the vibrational density of states at frequencies around 1-3 THz, was first identified in low-frequency Raman scattering spectra of glasses, such as vitreous silica, highlighting deviations from Debye's phonon model due to disorder-induced quasi-localized modes. Concurrently, low-temperature studies (below 1 K) revealed universal properties including a linear specific heat term proportional to temperature (C_p ∝ T) and a quadratic dependence of thermal conductivity (κ ∝ T²), attributed to an ensemble of two-level tunneling systems (TLS) arising from atomic tunneling between metastable configurations in the disordered structure. These findings, consolidated in the soft-potential or tunneling model proposed in 1972, underscored the role of structural heterogeneity in governing low-energy excitations across diverse amorphous materials like polymers, oxides, and metals. From the 1970s onward, advances in amorphous semiconductors and computational modeling have deepened insights into their electronic and structural behaviors. Hydrogenated amorphous silicon (a-Si:H), first developed in the 1970s through glow discharge and plasma-enhanced chemical vapor deposition (PECVD), emerged as a key material for thin-film transistors and photovoltaics. Research in the 1990s focused on defect passivation and alloying (e.g., with germanium) to tune bandgaps and mitigate light-induced degradation via the Staebler-Wronski effect.^[32] Computational simulations, leveraging molecular dynamics with empirical potentials like the BKS model for silica (introduced in 1990), enabled large-scale modeling of glass formation and relaxation processes, revealing medium-range order and topological constraints absent in earlier models.^[33] In the 2020s, machine learning has revolutionized structure prediction, with generative deep learning frameworks automating the design of amorphous compositions and predicting properties like glass-forming ability from atomic datasets, achieving high accuracy in simulating complex oxides and alloys without exhaustive simulations.^[34] Recent 2025 studies have further advanced the field, exploring two-dimensional amorphous materials approaching the single-layer limit and using topology to reveal hidden structural rules, such as soft regions embedded in medium-range order.^[35]^[36] These AI-driven approaches, often integrated with ab initio calculations, facilitate rapid exploration of metastable phases, accelerating discoveries in energy storage and flexible electronics.^[37]

Structure and Formation

Atomic and Molecular Arrangement

Amorphous solids exhibit short-range order in their atomic and molecular arrangements, characterized by well-defined local bond lengths and angles that closely resemble those in their crystalline counterparts, but they lack the long-range translational periodicity that defines crystals. This local structural similarity arises from the tendency of atoms to adopt energetically favorable nearest-neighbor configurations during the solidification process, while the absence of a periodic lattice results from rapid cooling that traps the material in a metastable, disordered state. A foundational model for understanding this arrangement in oxide glasses is the continuous random network proposed by Zachariasen, which posits that the structure consists of corner-sharing polyhedra, such as tetrahedra in silicate glasses, forming an extended network without periodic repetition or energy wells that would favor crystallization. In this model, the oxygen atoms bridge the network-forming cations, ensuring that no oxygen is bonded to more than two cations and that the polyhedra share corners rather than edges or faces to maintain stability, while avoiding small rings (fewer than six members) that would impose undue strain. This framework explains the isotropic nature of glasses and their ability to form over a range of compositions without sharp melting points.^[30] The radial distribution function (RDF), which quantifies the probability of finding atoms at a given distance from a reference atom, captures this structural hierarchy in amorphous solids by displaying sharp peaks at short distances corresponding to first- and second-neighbor shells, followed by oscillations that gradually dampen without the persistent periodicity seen in crystals. These decaying oscillations reflect the preservation of local coordination environments beyond which the structural correlations fade, providing a quantitative measure of the disorder's extent. In metallic glasses, for instance, the RDF reveals subtle shifts in peak positions during relaxation processes that align with changes in compositional short-range ordering.^[38] Topological constraints further elucidate the mechanical implications of this atomic arrangement through the Phillips-Thorpe theory of rigidity percolation, which treats the network as a system of bonds and angles imposing constraints on atomic motion. In three dimensions, the network achieves optimal rigidity when the average coordination number reaches approximately 2.4, marking the percolation threshold where floppy modes disappear and the structure transitions from underconstrained to stressed-rigid, influencing properties like elasticity without relying on long-range order. This mean-field approach highlights how local connectivity determines global rigidity in covalent glasses such as chalcogenides. In amorphous polymers, the concept of voids and free volume accounts for the interstitial spaces arising from inefficient chain packing, where free volume represents the unoccupied regions that enable segmental motion and diffusion. These voids, distributed non-uniformly, contribute to the material's compliance and are quantified as the excess volume beyond a hypothetical close-packed state, playing a key role in phenomena like the glass transition by providing the space necessary for cooperative rearrangements. The free volume fraction typically ranges from 2.5% to 5% in polymers like polystyrene, decreasing with cooling and influencing transport properties.

Formation Processes

Amorphous solids, also known as glasses, form through processes that suppress crystallization by exploiting kinetic barriers or thermodynamic instabilities, preventing the material from achieving long-range atomic order during solidification or phase transitions. These methods rely on rapid structural arrest, where the system is driven into a metastable amorphous state far from equilibrium. The choice of process depends on the material's composition and desired morphology, with kinetic factors like cooling rates dominating in melt-based techniques and thermodynamic driving forces playing key roles in solid-state routes.^[39] One primary method is rapid quenching from the melt, which achieves extremely high cooling rates to bypass nucleation and growth of crystalline phases. In melt-spinning, a molten alloy is ejected onto a rotating chilled wheel, attaining cooling rates exceeding 10^6 K/s, as demonstrated in the production of metallic glasses like Fe-based alloys. This kinetic suppression of crystallization is thermodynamically favored in multi-component systems with deep eutectics, where the liquid viscosity increases rapidly near the glass transition temperature.^[40]^[41] Vapor deposition techniques, such as sputtering and thermal evaporation, produce amorphous thin films by condensing vaporized atoms or molecules onto a substrate under vacuum conditions that limit atomic mobility. Sputtering involves bombarding a target with ions to eject material, forming dense amorphous layers in materials like a-Si or chalcogenide films, while evaporation heats the source to generate a vapor flux for deposition. These processes enable control over film thickness and composition at rates from nm/s to μm/min, with amorphicity arising from the low substrate temperatures that hinder diffusive rearrangement.^[42]^[43] Solid-state amorphization transforms crystalline precursors into amorphous phases without melting, driven by accumulated defects or interdiffusion that destabilize the lattice. Ion implantation introduces high-energy ions into a solid, creating collision cascades that disrupt crystallinity, as seen in silicon where doses above 10^14 ions/cm² yield fully amorphous layers up to 100 nm thick. Mechanical alloying via ball milling repeatedly deforms and fractures powder particles, promoting amorphization in alloys like Zr-Al through shear-induced mixing and stored energy exceeding 10 kJ/mol. These methods highlight thermodynamic pathways where defect accumulation lowers the free energy barrier to the amorphous state.^[44]^[45]^[46] The sol-gel process synthesizes amorphous ceramics from solution precursors, involving hydrolysis and condensation of metal alkoxides to form a colloidal sol that gels into an oxide network, followed by drying and calcination. This wet-chemical route produces amorphous silica or titania gels at low temperatures (<100°C), with porosity tunable via pH and aging, offering advantages over melt methods for complex oxides. Pressure-induced amorphization, conversely, compresses crystalline solids beyond their stability limit, as in ice Ih transforming to high-density amorphous ice (HDA) at pressures above 1 GPa and temperatures below 150 K, where mechanical instability drives the polyamorphic transition.^[47]^[48] The ease of forming amorphous solids via these processes correlates with the fragility index of the parent liquid, a measure of how steeply the viscosity rises near the glass transition temperature (T_g). Strong liquids, like SiO_2 with fragility m ≈ 20, exhibit Arrhenius-like behavior and high glass-forming ability due to stable network structures resisting structural relaxation, requiring modest cooling rates (~1 K/s). Fragile liquids, such as o-terphenyl with m > 100, show non-Arrhenius divergence and poorer glass-forming ability, necessitating faster quenching (>10^4 K/s) to avoid crystallization, as fragility reflects the liquid's sensitivity to temperature changes in configurational entropy.^[49]^[50]

Fundamental Properties

Thermal Properties and Glass Transition

Amorphous solids exhibit distinct thermal behaviors that differ markedly from crystalline materials, particularly around the glass transition temperature T_g, where the material undergoes a kinetic arrest of structural relaxation. This transition marks the point at which the supercooled liquid's viscosity reaches approximately $10^{12} Pa·s, effectively freezing atomic or molecular motions on experimental timescales, typically occurring in the range of $10^2 to $10^3 K depending on the material composition, such as lower values for organic polymers and higher for inorganic glasses like silica. Unlike a true thermodynamic phase transition, the glass transition is rate-dependent, with T_g shifting to higher values under faster cooling rates due to incomplete relaxation.^[51] A hallmark of the glass transition is the discontinuous jump in specific heat capacity \Delta C_p at T_g, reflecting the onset of configurational contributions to heat capacity in the supercooled liquid state above T_g. This jump typically amounts to approximately 0.5 to 1 times the gas constant R per atom, providing a measure of the degrees of freedom unlocked during the transition, and is linked to the Prigogine-Defay relation, which assesses the consistency of the transition through ratios involving \Delta C_p, thermal expansion, and compressibility changes.^[51] Below T_g, the glass enters a non-equilibrium state, leading to enthalpy relaxation and physical aging, where stored excess enthalpy is gradually released over time as the structure evolves toward a more stable configuration, often resulting in increased density and reduced free volume.^[52] This aging process is thermally activated and can significantly alter properties like mechanical strength, with relaxation times following Vogel-Fulcher-Tammann behavior.^[53] The dynamics near T_g are further characterized by the fragility parameter m, defined as m = \left. \frac{d \log \tau}{d (T_g / T)} \right|_{T = T_g}, where \tau is the structural relaxation time; this dimensionless quantity classifies glass-formers as "strong" (low m \approx 20-40, Arrhenius-like behavior, e.g., SiO_2) or "fragile" (high m > 100, strongly non-Arrhenius, e.g., many organic liquids), influencing the sharpness of the transition and ease of glass formation.^[54] Underlying these kinetic aspects is the Kauzmann paradox, which arises from extrapolating the supercooled liquid's entropy below T_g to a hypothetical Kauzmann temperature T_K, where the liquid's configurational entropy would equal that of the crystal, potentially leading to an entropy crisis; this is averted in practice by the kinetic freezing at T_g > T_K, preventing the unphysical negative entropy excess. At low temperatures well below T_g, amorphous solids show deviations from the Debye model in specific heat, attributed to two-level systems.

Mechanical and Elastic Properties

Amorphous solids exhibit distinct elastic properties compared to their crystalline counterparts, primarily due to their disordered atomic structure. The Poisson's ratio, which measures the negative ratio of transverse to axial strain under uniaxial stress, typically ranges from 0.276 to 0.409 in metallic glasses, often higher than in many crystalline metals (around 0.25–0.35) because the lack of long-range order allows greater lateral expansion under compression. This elevated value arises from the isotropic nature of the disorder, enabling more uniform deformation responses.^[55] The shear modulus G and bulk modulus K characterize the resistance to shear and volumetric deformation, respectively, and are linked to the network's average coordination number \langle z \rangle through Maxwell's rigidity counting. In amorphous solids, rigidity emerges when \langle z \rangle exceeds the Maxwell threshold of $2d (where d is the dimensionality, typically 3 for bulk materials), with G vanishing at this isostatic point while K remains finite due to its dependence on radial constraints.^[56] For metallic glasses, G is approximately 30% lower than in corresponding crystals, reflecting softer shear modes from structural disorder, whereas K is only about 6% reduced, highlighting the relative stability of compressive responses.^[57] Plasticity in amorphous solids, particularly metallic glasses, occurs through localized shear transformation zones (STZs), which are cooperative clusters of 200–700 atoms that undergo irreversible shearing under applied stress. These STZs nucleate and propagate as the primary mechanism of plastic flow below the glass transition temperature, leading to shear banding and enabling high strength but limited ductility in many cases.^[58] The size and activation of STZs, typically 2.5–6.6 nm³, depend on strain rate sensitivity and correlate with overall deformability.^[58] The transition between brittleness and ductility in amorphous solids is governed by the role of free volume in the yielding process, where higher free volume facilitates STZ activation and distributed plasticity, promoting ductility, while lower free volume concentrates deformation into brittle shear bands. In metallic glasses, increased free volume—often tuned by processing conditions like cooling rate—enhances the critical strain for yielding by allowing atomic rearrangements without catastrophic fracture.^[59] This contrasts with brittle behavior in low-free-volume states, where shear bands propagate rapidly, leading to sudden failure.^[60] Under sustained or cyclic loading, amorphous solids display fatigue and creep, where creep involves time-dependent plastic deformation via gradual STZ accumulation under constant stress, resulting in sublinear strain growth.^[61] Fatigue, induced by cyclic stresses, promotes crack initiation along shear bands and surface modifications, often reducing endurance limits compared to crystalline alloys due to localized damage accumulation.^[62] Near the glass transition temperature, both shear and bulk moduli soften, influencing these long-term responses, though ambient-temperature classical mechanics dominate typical applications.^[57]

Low-Temperature Universal Behavior

At low temperatures, typically below 1 K, amorphous solids exhibit universal thermal and acoustic anomalies that distinguish them from crystalline counterparts, primarily attributed to the presence of tunneling two-level systems (TLS). These TLS arise from atoms or groups of atoms that can tunnel between two nearly degenerate equilibrium positions in the disordered structure, leading to weakly bound quasiparticle excitations with energies on the order of thermal energies at cryogenic conditions. This behavior is observed across a wide range of insulating glasses, such as silica and polymers, independent of their chemical composition, highlighting the intrinsic role of structural disorder. The specific heat of amorphous solids at these temperatures includes a linear term C = \gamma T, where \gamma is the coefficient arising from TLS excitations, typically in the range of $10^{-4} to $10^{-3} J/mol K². This contrasts with the cubic T^3 dependence expected from phonon contributions in Debye theory for crystals, as the TLS provide a constant density of states at low energies. Experimental measurements on materials like vitreous silica confirm this linear contribution dominates below 0.5 K, with \gamma values around 0.5 mJ/mol K² for many glasses.^[63] The tunneling model for TLS posits a uniform distribution of these systems in energy splitting \varepsilon and tunneling parameter \Gamma, given by P(\varepsilon, \Gamma) = P_0, where P_0 is a material-independent constant on the order of 1 per eV per cm³. Here, \varepsilon = \sqrt{\Delta^2 + \Gamma^2}, with \Delta as the asymmetry between the two wells. This assumption of uniformity in the parameter space explains the saturation of low-energy excitations and their weak coupling to phonons via deformation potentials, enabling the linear specific heat and other universals. The model, originally proposed for ionic tunneling, has been validated through fits to thermal data across diverse amorphous materials.^[63] Thermal conductivity \kappa in amorphous solids follows a T^2 dependence at very low temperatures (below ~1 K), resulting from phonon scattering by TLS. Long-wavelength phonons resonant with TLS energy splittings cause resonant scattering, while one-phonon relaxation processes contribute, leading to this quadratic regime before a plateau emerges around 5-10 K. Measurements on fused silica and other glasses show \kappa / T^2 values spanning nearly three orders of magnitude but universally scaling as T^2 in the quantum regime, underscoring the TLS dominance.^[63] At slightly higher temperatures (~10-50 K), the vibrational density of states in amorphous solids reveals a boson peak, an excess over the Debye \omega^2 prediction, centered at frequencies \omega_b \approx 1-3 THz. This feature, observed via Raman or neutron scattering in materials like silica glass, corresponds to quasi-localized modes arising from disorder-induced anharmonicities, contributing to enhanced low-frequency excitations. The peak height and position vary modestly with glass type but remain a hallmark of amorphous vibrational spectra.^[64]^[65] Acoustic properties further illustrate these universals, with attenuation showing a linear T dependence at millikelvin temperatures due to TLS-phonon interactions, transitioning to a broad peak in internal friction around 5-20 K. This peak, prominent in ultrasonic measurements on glasses like boron oxide, reflects thermal relaxation of TLS resonant with phonon frequencies, with the friction Q^{-1} reaching ~10^{-3} near 10 K before decreasing. Such behavior is nearly universal, though peak positions shift slightly with frequency and material.^[63]^[66]

Characterization Techniques

Diffraction and Scattering Methods

Diffraction and scattering methods are essential for characterizing the atomic structure of amorphous solids, which lack long-range translational order and thus do not produce sharp Bragg peaks characteristic of crystalline materials. Instead, these techniques reveal diffuse scattering patterns that provide information on short- and medium-range order through the analysis of broad halos and diffuse features. By employing X-rays, neutrons, or electrons as probes, researchers can extract structural metrics such as the radial distribution function (RDF) or pair distribution function (PDF), which describe the probability of finding atoms at specific interatomic distances. These methods average over relatively large volumes, typically on the order of nanometers cubed, offering ensemble-averaged insights into the disordered atomic arrangements.^[67] X-ray diffraction (XRD) is a primary technique for studying amorphous solids, where the diffraction patterns exhibit broad, symmetric halos rather than discrete peaks, reflecting the absence of periodic lattice planes. The position and width of these halos correspond to the average interatomic distances and structural disorder, respectively. To quantify local structure, the total scattering data from XRD can be Fourier-transformed to obtain the pair distribution function (PDF), which provides a real-space representation of atomic pair correlations up to several nanometers. This approach has been instrumental in analyzing materials like amorphous silica and metallic glasses, enabling the identification of coordination numbers and bonding environments. Synchrotron-based XRD enhances resolution due to higher flux and energy, allowing for finer details in the PDF.^[68]^[69]^[67] Neutron diffraction complements XRD by offering sensitivity to light elements and isotopic variations, which is crucial for amorphous solids containing hydrogen, oxygen, or other low-atomic-number atoms where X-ray scattering lengths are similar. The technique's isotope sensitivity arises from the distinct neutron scattering lengths of isotopes like hydrogen and deuterium, allowing selective probing of specific atomic species in multicomponent systems. Total scattering neutron diffraction captures both Bragg-like (though diffuse) and diffuse components, enabling the extraction of the RDF through Fourier analysis of the structure factor over a wide Q-range. This has been applied to study amorphous alloys and polymers, revealing chemical short-range order that is obscured in X-ray data. Facilities like spallation sources provide high-intensity neutrons for such measurements.^[70]^[71]^[72]^[73] Electron diffraction, particularly in transmission electron microscopy (TEM), allows for the investigation of local order in amorphous solids at the nanoscale by selecting small areas, typically 10-100 nm in diameter, through apertures. Selected area electron diffraction (SAED) patterns from amorphous regions show ring-like diffuse scattering, indicative of isotropic short-range order without crystalline domains. The high spatial resolution of electron beams enables mapping of structural heterogeneity in thin samples, such as amorphous thin films or nanoparticles. Analysis of these patterns via radial integration yields intensity profiles similar to XRD, from which reduced PDFs can be derived to assess local atomic arrangements. This method is particularly useful for confirming amorphicity in regions too small for bulk diffraction techniques.^[74]^[75]^[76] Wide-angle X-ray scattering (WAXS) extends the utility of X-ray methods by focusing on higher scattering angles (typically 5° to 50° 2θ), which probe medium-range order (up to 1-2 nm) in amorphous solids beyond the short-range correlations captured by small-angle scattering. WAXS patterns feature broad peaks that reflect correlated atomic arrangements over multiple coordination shells, such as in chalcogenide glasses or metallic amorphous alloys. By combining WAXS with PDF analysis, researchers can distinguish polyamorphic phases with differing medium-range packing densities. This technique is often performed in situ during processing, like rapid quenching, to track structural evolution.^[77]^[78]^[68] Despite their power, diffraction and scattering methods have inherent limitations when applied to amorphous solids. The absence of sharp Bragg peaks precludes direct indexing of lattice parameters, requiring indirect modeling to interpret diffuse data. Moreover, these techniques inherently average the structure over the probed volume, which for laboratory XRD or neutron setups can span micrometers, potentially masking nanoscale heterogeneities. High-resolution variants like aberration-corrected TEM electron diffraction mitigate this to some extent but are limited to thin specimens. Complementary spectroscopic methods can provide additional site-specific information to overcome these averaging effects.^[79]^[80]^[67]

Spectroscopic and Absorption Techniques

Spectroscopic and absorption techniques are essential for probing the local electronic, vibrational, and chemical environments in amorphous solids, where long-range order is absent. These methods provide insights into short-range atomic arrangements and bonding without relying on periodic structures, complementing global structural analyses like diffraction in one sentence. Key techniques include X-ray absorption fine structure (XAFS), Raman and infrared (IR) spectroscopy, nuclear magnetic resonance (NMR), and X-ray photoelectron spectroscopy (XPS), each targeting specific aspects of disorder and local heterogeneity. XAFS encompasses extended X-ray absorption fine structure (EXAFS) and X-ray absorption near-edge structure (XANES), offering element-specific information on the local coordination environment around absorbing atoms in amorphous materials such as oxide glasses. EXAFS analyzes the oscillatory fine structure beyond the absorption edge, yielding precise interatomic distances with resolutions down to approximately 0.01 Å and coordination numbers within ±10%, as demonstrated in studies of silicate and borate glasses containing transition metals like Fe and Ti. For instance, in amorphous Fe-doped silicates, EXAFS has measured Fe-O bond lengths of 1.87–1.92 Å, revealing variations due to local distortions absent in crystalline counterparts. XANES, focusing on the near-edge region, elucidates coordination geometry and oxidation states through features like pre-edge peaks and edge shifts; in Ti-containing glasses, pre-edge intensities indicate average coordination numbers of 5.4–5.8, reflecting tetrahedral-to-octahedral transitions under pressure or compositional changes. Raman and IR spectroscopy interrogate vibrational modes, which are broadened by structural disorder in amorphous solids, providing signatures of short-range bonding and phonon density of states. In crystalline materials, selection rules limit active modes to specific symmetries, producing sharp peaks, whereas amorphous solids exhibit contributions from the full phonon spectrum, resulting in broad bands with widths of several hundred cm⁻¹ due to distributions in bond lengths and angles. For example, in amorphous silicon, Raman spectra show a transverse optical mode broadened to below 550 cm⁻¹, contrasting the narrow 520 cm⁻¹ peak in crystalline silicon, while IR complements by detecting dipole-active modes influenced by local asymmetry. These techniques have been applied to inorganic glasses and epitaxial films, such as amorphous GeTe, where combined far-IR and Raman reveal shifts in low-frequency modes indicative of disorder-induced softening. NMR spectroscopy captures chemical shift distributions that highlight site-to-site heterogeneity in amorphous solids, arising from varied local magnetic environments. In solid-state NMR, broadened lineshapes—typically 2–6 ppm for ¹³C and 0.6–1.8 ppm for ¹H—reflect conformational and intermolecular variations, as seen in amorphous pharmaceuticals like AZD4625, where dynamic nuclear polarization enhances resolution to map hydrogen bonding networks. By comparing experimental shifts to predicted distributions from molecular simulations, NMR identifies dominant local motifs, such as clustered versus isolated molecular sites, enabling atomic-level structure determination in non-crystalline drugs. XPS determines surface composition and oxidation states in amorphous solids by measuring photoelectron binding energies from the top few nanometers. In amorphous alloys like Cu₅₀Ti₅₀, XPS reveals enriched Ti oxides at the surface with binding energies indicating Ti⁴⁺ states, differing from bulk compositions due to preferential oxidation during preparation. Similarly, for Fe₇₀Cr₁₀P₁₃C₇ amorphous ribbons, XPS quantifies passive oxide layers incorporating P, supporting models of corrosion resistance through semiquantitative elemental ratios and chemical shift analysis. These techniques excel at elucidating short-range order in amorphous solids, where their local probes—nanometer-scale coherence lengths and element selectivity—overcome limitations of methods requiring periodicity, as in Raman's sensitivity to bond-level disorder in electroceramics.

Microscopy and Imaging Methods

Microscopy and imaging methods play a crucial role in visualizing the nanoscale heterogeneity inherent to amorphous solids, where the lack of long-range order makes traditional crystallographic techniques insufficient. These approaches provide direct spatial information on atomic arrangements, surface features, and structural variations at resolutions down to the atomic scale, revealing medium-range order and defects that influence material properties. Unlike averaged signals from diffraction or spectroscopy, imaging techniques capture local variations, enabling the study of non-uniformity in materials such as glasses, polymers, and metallic alloys. Atomic electron tomography (AET) has emerged as a powerful technique for reconstructing three-dimensional atomic positions in thin samples of amorphous solids, overcoming the challenges posed by their disordered structures. By acquiring a series of two-dimensional projections from multiple tilt angles using aberration-corrected scanning transmission electron microscopy (STEM), AET employs iterative algorithms to determine the 3D coordinates of individual atoms, achieving localization precision better than 1 Å in materials like amorphous tantalum oxide. This method has been applied to determine the full atomic structure of an amorphous solid for the first time, demonstrating liquid-like packing with short-range order similar to the parent liquid. Recent advancements, such as ptychographic AET, further enhance resolution by incorporating phase information from overlapping probes, allowing visualization of sub-angstrom features in beam-sensitive amorphous samples. Fluctuation electron microscopy (FEM) quantifies medium-range order in amorphous solids by measuring the variance in the scattering intensity from nanometer-scale volumes, providing insights into structural correlations beyond short-range atomic packing. In this technique, dark-field electron images are acquired at various probe positions, and the normalized variance of the scattered intensity serves as a measure of paracrystalline-like order, with characteristic length scales typically 0.5–2 nm detectable in materials such as amorphous silicon and germanium. FEM has revealed consistent medium-range order in hydrogenated amorphous silicon films regardless of deposition method, correlating with enhanced stability against the Staebler-Wronski effect.^[81] The method's sensitivity to nanoscale heterogeneity makes it particularly useful for distinguishing subtle structural differences in non-crystalline semiconductors and oxides.^[82] Scanning tunneling microscopy (STM) offers atomic-resolution imaging of surface topography and defects in amorphous solids, leveraging quantum tunneling currents to map electronic density variations at the vacuum-solid interface. For instance, in hydrogenated amorphous silicon surfaces, STM reveals local protrusions and depressions associated with dangling bonds and voids, providing direct correlation between surface morphology and bulk defect formation during growth. This technique has been used to observe defect evolution in amorphous carbon films, where irradiation-induced interstitial migrations lead to observable lattice rearrangements and topographic changes.^[83] STM's ability to operate under ultra-high vacuum conditions minimizes contamination, enabling precise studies of surface relaxations in metallic glasses and chalcogenide alloys.^[84] Cryogenic transmission electron microscopy (cryo-TEM) is essential for imaging beam-sensitive amorphous solids like polymers and biomolecules, preserving their native hydrated or vitrified states through rapid freezing to form amorphous ice. In polymer blends, cryo-TEM combined with 4D-STEM visualizes phase separation and nanoscale domains in amorphous-crystalline mixtures, such as polystyrene-block-polybutadiene, with resolutions sufficient to track morphological evolution during processing. For biomolecules embedded in amorphous matrices, cryo-TEM captures structural details of protein aggregates or lipid vesicles without dehydration artifacts, as seen in studies of vitrified monoclonal antibody solutions revealing oligomeric heterogeneity.^[85] This approach extends to functional materials, where low-temperature operation reduces diffusion and maintains structural integrity during observation.^[86] Despite these advances, microscopy of amorphous solids faces significant challenges, particularly radiation damage and sample preparation. High-energy electron beams induce bond breaking and atomic displacement in non-crystalline materials, leading to structural alterations that obscure true atomic arrangements; for example, organic amorphous solids suffer rapid radiolysis, necessitating low-dose imaging strategies and cryogenic cooling to mitigate damage rates by factors of 10–100. Sample preparation is equally demanding, as amorphous solids often require ultrathin sections (below 50 nm) via focused ion beam milling or vitrification, which can introduce artifacts like contamination or bending in beam-sensitive specimens such as amorphous silicon. These hurdles demand specialized protocols, including liquid-nitrogen cooling and correlative imaging, to ensure reliable nanoscale visualization.^[87] Computational validation of reconstructed images can occasionally aid in assessing these artifacts.^[88]

Computational and Modeling Approaches

Computational modeling of amorphous solids relies on simulation techniques that capture the disordered atomic arrangements and dynamics without long-range order. Molecular dynamics (MD) simulations solve Newtonian equations of motion to evolve atomic positions over time, enabling the study of structural formation through rapid quenching from a high-temperature melt. In these quench simulations, systems are cooled at rates on the order of 10^9 to 10^14 K/s to mimic glass formation, often using empirical potentials like the Lennard-Jones (LJ) potential to describe pairwise interactions between atoms. A seminal example is the Kob-Andersen binary LJ mixture, which models a supercooled liquid prone to vitrification, allowing investigation of relaxation dynamics and medium-range order in metallic glasses.^[89] Monte Carlo (MC) methods complement MD by facilitating equilibrium sampling in the glassy regime below the glass transition temperature (Tg), where kinetic trapping hinders standard simulations. Techniques like swap Monte Carlo exchange particle sizes or configurations between replicas at different temperatures, enhancing ergodic exploration and accessing ultra-stable states deep in the supercooled phase. This approach has been applied to two-dimensional glass-forming systems, revealing thermodynamic properties such as vanishing configurational entropy at low temperatures.^[90]^[91] Density functional theory (DFT) provides accurate electronic structure calculations for small clusters or limited-size models of amorphous solids, treating quantum mechanical effects explicitly. In annealing simulations, DFT optimizes atomic configurations by minimizing energy on potential energy surfaces, yielding realistic structures for materials like amorphous silicon where classical potentials fall short. For instance, DFT-based relaxation of silicon supercells produces radial distribution functions consistent with experimental observations, highlighting local bonding motifs in disordered networks.^[92]^[93] Advancements in machine learning potentials, particularly neural network-based interatomic potentials developed since the 2010s, enable large-scale simulations of amorphous alloys by approximating quantum-accurate energies and forces from training data. These potentials, trained on DFT datasets, surpass traditional empirical models in transferability across phases, allowing MD-like simulations of systems with thousands of atoms, such as amorphous carbon or metallic glasses, to probe structural heterogeneity and dynamics efficiently.^[94]^[95] Validation of these models against experiments, such as through comparison of simulated radial distribution functions (RDFs) with diffraction data, ensures structural fidelity. For example, MD simulations incorporating bias potentials to refine RDFs have matched large-angle X-ray scattering results for amorphous polymers, confirming interatomic correlations and assembly motifs. Similarly, ab initio MD models of hydrogenated amorphous silicon demonstrate that while RDFs provide baseline agreement, complementary vibrational spectra offer stricter benchmarks for local order. Simulations are benchmarked against techniques like X-ray or neutron scattering to verify short- and medium-range structures without delving into specific property computations.^[96]^[97]

Applications and Phenomena

Thin Films and Nanotechnology

Amorphous solids play a pivotal role in thin film technologies and nanotechnology, where their lack of long-range order enables unique properties such as isotropy and tunable electronic characteristics at the nanoscale. In thin films, typically ranging from a few nanometers to hundreds of nanometers in thickness, amorphous materials are deposited to form uniform layers that serve as active components in devices, leveraging their ability to maintain structural homogeneity without grain boundaries. Nanotechnology applications extend this to nanostructured composites and films, where amorphous matrices host nanocrystals or other nanostructures, enhancing mechanical and functional performance.^[98] One prominent application is in amorphous semiconductors, particularly hydrogenated amorphous silicon (a-Si:H), which is widely used as the absorber layer in thin-film solar cells due to its low-cost deposition and suitable bandgap for photovoltaic conversion. a-Si:H solar cells achieve power conversion efficiencies of approximately 10-15% in laboratory settings, with optimized structures reaching up to 12.71% for ultrathin absorbers around 1 μm thick. This efficiency stems from the material's high absorption coefficient in the visible spectrum, allowing effective light harvesting in films as thin as 300 nm, though challenges like the Staebler-Wronski effect limit long-term stability.^[99]^[100] Metallic glass thin films, formed by rapid quenching techniques, offer exceptional mechanical properties for microelectromechanical systems (MEMS), including high strength-to-weight ratios and elastic limits exceeding 2% strain. These films, often composed of alloys like Zr-Cu-Ni, exhibit yield strengths over 1 GPa while maintaining densities comparable to crystalline metals, making them ideal for robust, lightweight MEMS components such as sensors and actuators. Their amorphous structure suppresses dislocation motion, providing superior wear resistance and fatigue life compared to polycrystalline counterparts.^[101]^[98] In nanotechnology, amorphous nano-composites enhance toughness by embedding nanocrystals within an amorphous matrix, creating interfaces that deflect cracks and promote shear banding for improved ductility. For instance, crystalline-amorphous nano-laminates or core-shell structures in metal-glass systems can achieve significantly enhanced fracture toughness compared to pure amorphous metals, by distributing stress and preventing catastrophic failure. This design exploits the nanocrystals' ability to pin shear bands in the matrix, balancing high strength (often >2 GPa) with enhanced plasticity.^[102]^[103] Key deposition techniques for achieving uniform amorphous thin films include pulsed laser deposition (PLD) and atomic layer deposition (ALD), which enable precise control over film thickness down to sub-10 nm layers. PLD uses high-energy laser pulses to ablate targets, producing stoichiometric amorphous films with high density and minimal substrate heating, suitable for complex oxides. ALD, in contrast, employs sequential self-limiting surface reactions for conformal, pinhole-free coatings on high-aspect-ratio nanostructures, ensuring uniformity even on 3D features. These methods are essential for nanotechnology, as they preserve the amorphous state during growth.^[104] A major challenge in amorphous thin films is stress-induced crystallization, where intrinsic or extrinsic stresses during deposition or operation promote nucleation of crystalline phases, degrading the desired isotropic properties. Compressive stresses exceeding 1 GPa can lower the energy barrier for crystallization in materials like a-Si:H or metallic glasses, leading to phase separation and reduced performance in devices; mitigation strategies include alloying or controlled annealing to manage stress without triggering unwanted ordering.^[105]^[106]

Superconductivity and Electrical Uses

In amorphous solids, electrical conduction often deviates from crystalline counterparts due to structural disorder, which localizes electron wavefunctions and impedes diffusive transport. This phenomenon, known as Anderson localization, arises when disorder strength exceeds a critical threshold, transforming extended states into localized ones and leading to insulating behavior even at zero temperature. In amorphous semiconductors, where charge carriers occupy localized states near the band edges, conduction proceeds via thermally activated hopping between these sites. At low temperatures, nearest-neighbor hopping dominates, but as temperature decreases further, carriers favor longer-range hops to minimize energy barriers, resulting in variable range hopping (VRH) conduction. The seminal Mott model for VRH in three dimensions describes the DC conductivity \sigma as

\sigma = \sigma_0 \exp\left[ -\left( \frac{T_0}{T} \right)^{1/4} \right],

where \sigma_0 is a prefactor related to the density of states and wavefunction overlap, T_0 characterizes the localization length and density of states at the Fermi level, and T is temperature; this T^{-1/4} dependence has been experimentally verified in materials like amorphous silicon and chalcogenide glasses. The metal-insulator transition (MIT) in amorphous solids marks the boundary between metallic and insulating regimes, often modeled by percolation theory, which treats conduction as the formation of a continuous network of interconnected metallic regions amid insulating ones. As the metal concentration or disorder varies, the system approaches a percolation threshold where the conductivity exhibits critical scaling behavior, with the correlation length diverging as \xi \sim |p - p_c|^{-\nu}, p being the fraction of metallic bonds and p_c the critical percolation probability. In amorphous alloys like Si-Au or Ge-Au, this transition aligns with the Mott criterion for localization, where the dimensionless parameter k_F l \approx 1 (with k_F the Fermi wavevector and l the mean free path) separates extended from localized states, enabling tunable resistivity over orders of magnitude. Amorphous superconductors, such as molybdenum-germanium alloys (a-MoGe), maintain superconductivity despite disorder, with critical temperatures T_c reaching up to 7 K in thin films, where atomic-scale randomness enhances electron-phonon coupling and pairing compared to crystalline analogs. This disorder-induced enhancement allows persistent superconductivity even near the superconductor-insulator transition, as probed by varying film thickness or composition. Such properties make amorphous solids ideal for superconducting electronics, particularly in Josephson junctions, where uniform thin films serve as electrodes or barriers, enabling high-critical-current densities and reduced quasiparticle tunneling for applications in quantum computing and sensitive detectors. Thin-film devices based on these materials, like SQUID magnetometers, benefit from the lack of grain boundaries, ensuring reproducible Josephson coupling and low noise.^[107]

Thermal and Protective Applications

Amorphous silica aerogels, characterized by their highly porous structure composed of interconnected amorphous silica nanoparticles, serve as exceptional thermal insulators in various applications, including building envelopes, cryogenic systems, and aerospace components. Their ultralow thermal conductivity, typically around 0.01 W/m·K at ambient conditions, arises from the minimized solid conduction paths and suppressed gas conduction within the nanoscale pores under vacuum or low pressure.^[108] This property makes them superior to traditional insulators like fiberglass, enabling efficient heat management in extreme environments while maintaining structural integrity due to the rigid, amorphous network.^[109] Amorphous carbon materials, including diamond-like carbon (DLC) films and soot-derived coatings, play a critical role in thermal barrier applications for high-temperature systems such as gas turbine engines. DLC coatings, with their amorphous sp³-rich structure, provide robust thermal protection by reflecting heat and reducing substrate temperatures, while exhibiting low thermal conductivity and high oxidation resistance up to 500°C.^[110] In turbine blades, these coatings mitigate thermal stresses, extending component lifespan under operating conditions exceeding 1000°C, and their tunable amorphous morphology allows for optimized adhesion and fracture toughness.^[111] The reusable thermal protection system of the Space Shuttle exemplifies the use of amorphous solids in extreme heat shielding, employing low-density tiles fabricated from 99.8% pure amorphous silica fibers. These tiles, with a porosity of approximately 90%, form a rigidized insulation that withstands peak re-entry temperatures up to 1600°C on the orbiter's underside, primarily through radiative heat dissipation and minimal conductive transfer.^[112] The amorphous fiber structure ensures low thermal conductivity (around 0.1 W/m·K) and rapid response to thermal gradients, preventing structural failure during hypersonic flight.^[113] Chalcogenide-based phase-change materials (PCMs), such as Ge-Sb-S-Se-Te alloys, utilize transitions between amorphous and crystalline phases for applications in non-volatile memory devices. These materials switch states at controlled temperatures (e.g., glass transition near 215°C), enabling high-density data storage through structural reconfiguration with enhanced cycling stability.^[114] This amorphous-to-crystalline switching provides reliable performance in electronics, with thermal stability indicated by crystallization temperatures around 170–200°C.^[115] Amorphous polymers, such as polystyrene, are integral to lightweight radiation shielding composites, where their non-crystalline matrix accommodates high-Z fillers like lead oxide nanoparticles to attenuate gamma rays effectively. In polystyrene-PbO nanocomposites, the amorphous polymer enables uniform dispersion of 52 nm PbO particles at loadings up to 35 wt%, achieving linear attenuation coefficients up to 3.5 times higher than pure polymer at 0.059 MeV, while reducing half-value layer thickness to 0.25 cm.^[116] These materials excel in space and medical shielding due to their flexibility, low density (1.03 g/cm³), and enhanced shielding efficiency (e.g., 70% less lead mass than metallic shields) across energies from 0.06 to 1.3 MeV.^[117]

Pharmaceutical and Biological Uses

Amorphous solids play a crucial role in pharmaceutical applications due to their enhanced solubility and bioavailability compared to crystalline counterparts, particularly for poorly water-soluble drugs. For instance, amorphous indomethacin exhibits significantly higher aqueous solubility—up to several orders of magnitude greater than its crystalline form—leading to improved dissolution rates and oral bioavailability in vivo.^[118]^[119] However, this thermodynamic instability often results in recrystallization, or devitrification, which can revert the material to a less soluble crystalline state, posing challenges for long-term storage and efficacy.^[120] To mitigate these stability issues, amorphous solid dispersions (ASDs) are commonly formulated by dispersing the drug within a polymer matrix, which inhibits devitrification through antiplasticization effects and molecular interactions that reduce molecular mobility. Polymers such as polyvinylpyrrolidone (PVP) or hydroxypropyl methylcellulose (HPMC) are frequently used as stabilizers, enhancing physical stability by increasing the glass transition temperature and preventing nucleation during storage or dissolution.^[121]^[122] Common preparation methods include spray-drying, where a drug-polymer solution is atomized and rapidly dried to form amorphous particles, and hot-melt extrusion, which involves melting and extruding the mixture to produce stable ASDs with controlled release profiles.^[123]^[124] These techniques have enabled the approval of numerous ASD-based drug products by the U.S. Food and Drug Administration, with 48 such formulations authorized between 2012 and 2023 as of 2023, demonstrating their growing clinical impact.^[125] In biological contexts, amorphous solids appear naturally as precursors in biomineralization processes, notably amorphous calcium phosphate (ACP), which serves as an intermediate phase in bone formation. ACP nanoparticles are transiently deposited in the extracellular matrix of developing bone, providing a flexible, hydrated precursor that transforms into crystalline hydroxyapatite under physiological conditions, facilitating rapid mineralization and structural adaptation.^[126]^[127] This amorphous phase's high solubility and reactivity enable efficient ion transport and integration into collagen scaffolds, underscoring its role in skeletal development and repair.^[128]

Occurrence in Nature

Amorphous solids occur widely in natural geological processes, particularly where rapid cooling prevents crystallization. Obsidian, a type of volcanic glass, forms when silica-rich lava from viscous eruptions cools extremely quickly, resulting in an amorphous structure with no crystalline minerals. This natural glass is composed primarily of silica (SiO₂) and minor oxides, exhibiting a brittle texture and conchoidal fracture ideal for sharp edges.^[129]^[130] Humans have exploited obsidian's properties since prehistoric times, crafting tools, weapons, and arrowheads from its razor-sharp flakes due to its homogeneous composition and ease of flaking.^[131] Impact glasses, such as tektites, represent another geological manifestation of amorphous solids, generated during hypervelocity meteorite collisions with Earth's surface. These events produce intense heat and pressure, melting target rocks and ejecting molten material that solidifies into glassy droplets or fragments upon atmospheric re-entry, forming non-crystalline structures rich in silica and alumina. Tektites are found in strewn fields across continents, like the Australasian field covering millions of square kilometers, and their aerodynamic shapes confirm high-speed formation.^[132]^[133] In biological systems, amorphous silica manifests as opal-A, a hydrated form (SiO₂·nH₂O) biosynthesized by organisms for structural support. Diatoms, unicellular algae, construct intricate frustules—cell walls—from this non-crystalline silica, enabling their role in marine productivity and the silicon cycle. Similarly, certain sponges, such as demosponges, incorporate opal-A into spicules and skeletal frameworks, contributing significantly to oceanic silica sequestration and export fluxes. These biogenic structures dissolve more readily than crystalline silica, influencing nutrient availability in aquatic environments.^[134]^[135] Soils host amorphous components essential for fertility and structure, including humic substances and allophane. Humic substances, derived from decomposed organic matter, are complex, amorphous macromolecules that bind soil particles, enhance water retention, and chelate nutrients like iron and phosphorus. These dark, polyelectrolyte-like materials form through microbial processes and persist in amorphous states, resisting crystallization. Allophane, a short-range ordered aluminosilicate (Al₂O₃·SiO₂·nH₂O), occurs in weathered volcanic soils and andisols, acting as a gel-like adsorbent for anions and cations due to its X-ray amorphous nature and high surface area.^[136]^[137]^[138] Atmospheric aerosols include amorphous sulfate phases that influence radiative forcing and cloud processes. Sulfate aerosols, often as ammonium sulfate or sulfuric acid particles, can adopt semi-solid or glassy states at low temperatures and humidities, particularly when mixed with organics, increasing viscosity and altering particle dynamics. These glassy sulfates in the troposphere and stratosphere promote heterogeneous ice nucleation, affecting cirrus cloud formation and potentially amplifying climate warming by altering precipitation efficiency. Volcanic injections of sulfur dioxide further enhance sulfate glass formation, leading to temporary cooling via enhanced aerosol scattering.^[139]^[140]

Phase Behavior

Structural Relaxation

Structural relaxation in amorphous solids refers to the time-dependent evolution of their atomic or molecular structure toward a more stable configuration below the glass transition temperature (Tg). This process occurs as the material, initially in a non-equilibrium state formed by rapid cooling, gradually minimizes its free energy through cooperative rearrangements. Unlike crystalline solids, amorphous materials lack long-range order, leading to heterogeneous dynamics where local regions relax at different rates, resulting in overall structural changes over extended periods. Physical aging, a key manifestation of structural relaxation, involves densification and a decrease in enthalpy as the amorphous solid approaches its supercooled liquid equilibrium state. During aging, the volume contracts due to the reconfiguration of atomic bonds, such as the reduction of less stable homopolar linkages in favor of more energetically favorable heteropolar ones, leading to increased packing density. Concurrently, the enthalpy decreases as the system sheds excess energy stored during quenching, with losses on the order of 0.5–1 eV per atom in some metallic glasses. These changes unfold over broad time scales, from seconds in accelerated conditions near Tg to years at ambient temperatures, reflecting the sluggish kinetics inherent to glassy dynamics.^[141] The relaxation kinetics during aging are often described by the stretched exponential function, also known as the Kohlrausch-Williams-Watts (KWW) form:

\phi(t) = \exp\left[-\left(\frac{t}{\tau}\right)^\beta\right]

where \phi(t) is the relaxation function, \tau is the characteristic relaxation time, and \beta (0 < \beta < 1) is the stretching exponent that captures the non-exponential, distributed nature of the process in disordered systems. This form, originally proposed by Kohlrausch in 1854 for charge decay in insulators, empirically fits structural relaxation in glasses, with typical \beta values around 0.3–0.6 near Tg, indicating broad heterogeneity in relaxation times. The stretched exponential arises from trapping models where diffusive motion in disordered landscapes leads to subdiffusive behavior, providing a conceptual framework for the cooperative yet spatially varying rearrangements in amorphous solids.^[142] Under applied stress, physical aging exhibits non-linear effects, where the relaxation rate accelerates or decelerates depending on the stress magnitude and direction relative to the material's aging history. In nonlinear viscoelastic models, stress perturbs the energy landscape, enhancing local mobility and causing faster structural evolution compared to isothermal aging without load, as seen in temperature-jump experiments on polymer glasses. This nonlinearity is quantified through material time scaling, where the effective relaxation time shortens under tensile stress but may lengthen under compression, leading to asymmetric responses that deviate from linear superposition principles. Such effects are critical in modeling the viscoelastic behavior of amorphous polymers under mechanical loading.^[143] The Johari-Goldstein (JG) β-relaxation serves as a secondary process and precursor to the primary α-relaxation in amorphous solids, initiating localized motions that facilitate the larger-scale structural changes of the α process. Observed below Tg, the JG relaxation involves short-range collective atomic rearrangements, such as bonding switches in metallic glasses that increase local density by favoring solute-solvent interactions over solvent-solvent pairs, with activation energies around 25–30 RTg. This β process, universal across glassy materials, acts as a dynamic heterogeneity driver, coupling to the α relaxation by providing the initial fluctuations necessary for cooperative diffusion in the viscous regime.^[144] These relaxation processes have significant implications for the mechanical stability of amorphous solids, as aging-induced densification and enthalpy reduction enhance rigidity but can also introduce brittleness. For instance, increased structural relaxation in polymer glasses correlates with higher Young's modulus and indentation hardness, improving resistance to deformation but reducing tensile strength in compacted forms due to diminished irreversible work absorption. In metallic glasses, JG-mediated relaxation modulates the transition from anelastic to plastic deformation, influencing shear stability and overall durability under load. Thus, controlling structural relaxation is essential for tailoring mechanical performance in applications like coatings and structural materials.^[145]

Polyamorphism and Transitions

Polyamorphism refers to the existence of multiple distinct amorphous phases in a single-component material, differing in density and structure, much like polymorphism in crystals. Recent studies have identified intermediate-density amorphous ices (MDA), suggesting a possible continuum of amorphous structures between low-density and high-density forms under certain pressure and temperature conditions.^[146]^[147] In the case of water, two prominent amorphous forms are low-density amorphous ice (LDA), with a density of approximately 0.94 g/cm³ at ambient pressure, and high-density amorphous ice (HDA), with a density of about 1.17 g/cm³. LDA features a tetrahedral hydrogen-bonded network similar to ice Ih, while HDA exhibits a more collapsed structure with increased coordination numbers around 5. These phases form under varying pressure conditions: LDA through vapor deposition or hyperquenching of liquid water, and HDA via compression of crystalline ice Ih at low temperatures (e.g., 77 K) to around 1 GPa.^[148]^[149] The transition between LDA and HDA is a first-order-like process, occurring discontinuously at pressures near 0.6 GPa and temperatures around 77 K, accompanied by a ~20% volume change and hysteresis upon reversal. This polyamorphic switch highlights how external pressure can induce structural reorganization in amorphous solids without crystallization. A related phenomenon is the very high-density amorphous ice (VHDA), formed by annealing HDA at pressures above 1 GPa and temperatures near 125 K, featuring even higher coordination (around 6) and density. These pressure-induced transformations underscore the sensitivity of amorphous phases to thermodynamic conditions, enabling distinct metastable states.^[150]^[151] The liquid-liquid transition (LLT) hypothesis posits that in the supercooled regime of water (below 235 K at ambient pressure), a first-order phase separation occurs between low-density liquid (LDL) and high-density liquid (HDL) phases, mirroring the LDA-HDA polyamorphism upon vitrification. Proposed in simulations showing a critical point at ~220 K and 0.1 GPa, with recent 2025 estimates refining it to approximately 200 K and 0.13 GPa, the LLT explains anomalies like density maxima and compressibility divergences in supercooled water.^[152]^[153] Experimental evidence emerged from ultrafast heating of HDA ice at 205 K under 2.5–3.5 kbar, where x-ray scattering revealed HDL formation followed by LDL domain growth during decompression, confirming a discontinuous structural shift distinct from crystallization. This supports the idea that amorphous polyamorphism reflects underlying liquid phase behavior in the deeply supercooled "no-man's land." In amorphous solids derived from undercooled melts, crystallization proceeds via nucleation and growth pathways, where nucleation—the formation of critical embryos—often decouples from subsequent crystal growth due to differing temperature dependencies. For glass-formers like metallic alloys (e.g., Zr-Ti-Cu-Ni-Be) or pharmaceuticals (e.g., ibuprofen), nucleation rates peak near or slightly above the glass transition temperature (T_g), driven by thermodynamic undercooling (ΔT = T_m - T, where T_m is the melting point), while growth maxima occur closer to T_m where mobility is higher. Heterogeneous nucleation dominates at surfaces or impurities, leading to polymorphic sequences where a metastable crystal form nucleates first, followed by transformation to the stable phase; for instance, in l-arabitol, nucleation maximizes at 5°C (T_g ≈ -14°C), while growth peaks between 60–95°C (T_m ≈ 101°C). These pathways determine the stability of amorphous materials, with deep undercooling favoring glass formation over rapid crystallization.^[154] Metastable phase diagrams for glass-formers map the undercooled liquid, amorphous, and crystalline regions, incorporating polyamorphic boundaries and avoiding true equilibrium crystallization lines. In water, such diagrams depict the supercooled liquid extending to ~235 K, with the LLT critical point connecting LDL/HDL to LDA/HDA upon cooling, and a "no-man's land" below 227 K where rapid ice nucleation hinders access. For general glass-formers, these diagrams highlight the glass-forming range bounded by T_g (isothermal vitrification) and T_m, with pressure axes revealing polyamorphic transitions; for example, in metallic systems, metastable extensions predict amorphous matrix compositions up to 28% solute before crystallization intervenes. These diagrams guide processing conditions to stabilize amorphous phases.^[155] High-pressure neutron scattering provides direct structural evidence for polyamorphism, revealing distinct pair correlation functions for LDA and HDA. Early studies at ~1 GPa showed HDA's first diffraction peak shifting to higher Q (indicating shorter O-O distances ~2.8 Å vs. 2.95 Å in LDA), confirming the collapsed network without crystalline order. More recent inelastic neutron scattering on HDA under pressure (e.g., 0.2–2 GPa) tracked the LDA-HDA transition, observing mode softening and density jumps consistent with a first-order change. These techniques validate the polyamorphic phases and their pressure-driven interconversions in amorphous solids.

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