The Ontology of Chemistry_Jaap van Brakel

Substances


The Ontology of Chemistry


Jaap van Brakel

1. Terminological Preliminaries

This article aims to offer a preliminary introduction to the “rough ground” of the ontology of chemistry. Although chemistry has often been described as the study of the transformation of substances, there is no generally agreed upon definition of (chemical) substance. ¹ In this article we will address the variety and classification of a great variety of stuff. No sharp line will be drawn between the physical, chemical, and material sciences. The issue of the nature of the relations between different “levels” (molar, molecular, quantum mechanical, etc.) is not addressed here (see the article on ‘Reduction, Physicalism, and Supervenience’ in Part 5 of this Volume). In this article all such ‘levels’ are assumed to be equally real.¹The German normalisation institute DIN has declared substance (Stoff ) “nicht normfähig” (not capable of being defined).Materials such as paper, gold, steel, glass, or helium are the product of producing and processing practices (making gold, dye-stuffs, superconductors, glassy metals, buckminsterfullerene, etc.). Such processing practices include operations that separate and mix materials as well as operations that aim at chemical transformation. Chemical operations are typically followed by physical and/or mechanical “unit operations” to purify and concentrate (separate, isolate) the product (so called down-stream processing). ²²Separation methods form a neglected core of chemistry. The Dutch word for chemistry is “scheikunde” (German: Scheidekunst), literally meaning “knowledge and art of separation,” a method “utterly different from other modes of analysis known in the empirical sciences” [Klein, 2008,38]. For ‘unit operations’ see the article on ‘Chemical Engineering’ in Part 6 of this Volume.Such practices and prototypical examples of stuff such as water and gold provide quasi transcultural universal intuitions of “being more or less pure” (isolation and purification), “being the same stuff” (identification), and having dispositions or inclinations to behave in particular ways in particular circumstances. When Archimedes allegedly devised a method to ascertain whether the crown of Hiero was pure gold or gold alloyed with silver, he was appealing to a notion of pure substance. Presumably, pure substances possess unchanging characteristics that distinguish them from one another. Density (as an intensive property) is one of the most accessible properties “which makes a difference”. Various materials, some more pure, others less so, some of them (quite) homogeneous, others less so, become increasingly reliable and stable actors in experimental and technological interactions [Nordmann, 2006].Materials or stuffs are identified using properties such as taste, density, composition, melting point, reactivity, ³ which are specific properties independent of form, size, mass (within certain ranges) and may be identified by any kind of observation or measurement. Materials which only differ in size or shape are considered to be the same stuff. ⁴ Using separation and purification operations, a purer substance is obtained with characteristic reproducible behaviour (invariant properties) within reproducible contextual conditions (in particular temperature and pressure). One well-chosen property (density, melting or boiling point, solubility in different solvents) will often suffice to identify sameness of substance; a stuff has many essential properties. However, it always remains possible that some property has been overlooked which later requires further or alternative classifications, if only because new materials are constructed all the time and the range of contextual conditions is continually expanded. In general it is an open (empirical) question whether some particular property may serve well to distinguish two substances or two distinct forms of the same substance. ⁵³Among the many properties of stuffs Aristotle mentions, we find, passing over problems of translation: substances are liable to congeal, melt, being softened. They can be (more or less) flexible, breakable, comminuable, impressible, elastic, compressible, malleable, cutable, kneadable, combustible, fumigable, inflammable, etc. [Düring, 1944].⁴The same stuff (still to be defined) may have (quite) different properties, not only at extreme pressures or temperatures, but also in the nanometer range.⁵Colour may depend on the grain size of what is the same substance. Some properties of substances may depend on so called hysteresis phenomena; for example, the coefficient of friction of a metal depends on many aspects of its history of fabrication.If two arbitrarily chosen parts of some material display the same characteristics with respect to a number of suitably chosen properties, then this stuff can be said to be uniform or homogeneous. Of course it is always possible that what was thought to be homogeneous is not homogeneous at a later stage of inquiry. Homogeneity is relative to the accuracy and scale of observation. Although the transparency and uniform colour of glass objects suggest uniform material, many glasses are inhomogeneous at levels from a few nm up to a few µm.An important material property is that prototypical stuff such as water or gold may occur in more than one phase or state of aggregation, such as solid, liquid, or gas/vapour. Aristotle observed that water freezes in winter. ⁶ He also says: “The finest and sweetest water is every day carried up and is dissolved into vapour and rises to the upper region, where it is condensed again by the cold and so returns to the earth.” A physico-chemical phase change can be distinguished from a chemical transformation by its reproducible reversibility by simply heating and cooling, the transition temperature being an important stuff characteristic. Boiling point and freezing point may be considered transcultural pre-scientific universals (without already thinking of measuring the temperature). However, whether liquid water and ice is the same substance or two substances cannot be answered without making far-reaching assumptions, although the answer of modern science is unambiguous: the (pure) substance concept is phase-invariant (see Compounds and Mixtures in Section 3 of this Volume).⁶Metereologica, 354b28-30, 348a1. On the intricacies of Aristotle's notions of element and mixture see [Needham, 2006]. According to Düring [1944, 16] Aristotle only knew two states of aggregation, solid and liquid. “His theories on the nature of gases are still vague and confused.” Düring also stresses that Aristotle's writings on these matters are often unclear because of “the permanent conflict between the purely speculative schematism and real observation.”Substances can typically be decomposed into other substances. Following the terminological proposal of Paneth, as elucidated by Ruthenberg [2009], a simple substance is one which, when isolated does not allow further decomposition or separation (under “normal” conditions). ⁷ A substance that is not a simple substance is a compound (substance). A simple substance cannot occur as the sole item at the beginning of an (analytic) decomposition or at the end of a (synthetic) composition process. To establish which are the simple substances careful measurement and the assumption of mass conservation is required. The history of chemistry is usually written in terms of the emergence of “the modern concept of purity in the sense of single, stoichiometric compounds of single chemical elements” [Klein, 2008,27]. This may be the most rational approach when (theorising about) isolating and synthesising various substances. However, in this review this particular conceptual notion (of stoichiometric compounds) is down-played and treated as a special case (see in particular § 3 and § 12).⁷Simple substances correspond with basic substances, the latter referring to the ultimate constituents of matter whatever they may be, assumed to be invariant during physical and chemical transformations. Ruthenberg has suggested that one motivation for Paneth's proposals might be to suggest that the reality of protons scores better than that of molecules or atoms. The status of protons, neutrons, and electrons seems more clear cut than that of molecule, even though the relation of a proton to its three constituting quarks is far from clear.Chemical (microscopic) species are hypothetical constituents of substances, such as electrons, nuclei, ions, radicals, molecules, oligomolecular and supramolecular aggregates, atoms in ordered or disordered solid structures, etc., held together by various types of “interactions”, ionic, covalent, metallic bonds; hydrogen bonding, ion pairing, metal-to-ligand binding, spin-spin interaction, van der Waals attractive forces, etc, resulting in theoretically embedded geometrical, topological features of the “arrangement” of nuclei and electrons.

2. Phenomenology of Everyday Stuff (and Things)

Material stuff of one kind or another (iron, brass, water, bread, butter, nylon) is to be distinguished from other concrete “things”. ⁸ When a thing is destroyed, the stuff of which it is constituted often survives, but the stuff a thing consists of is not the same as its parts. Visual perception is thing-oriented. Smell and taste (as well as hearing) are more characteristic of stuff. Often, a stuff can be immediately identified as what it is and in this sense doesn't depend on things for its existence. The same stuff can be found in different places at the same time, which is connected to their tendencies (which is not the same as saying that it is a scattered individual). Quantities of stuff are form-indifferent, fusible, and dispersable. Every portion of stuff displays structures or forms that are characteristic of it.⁸This section is based on [Hacker, 2004] and [Soentgen, 2008].Stuffs can be divided into portions, but there is a borderline where it cannot be divided any further, becoming “dust” or “moist” when very small. Everyday stuffs don't coincide with chemical substances. For example, only in special circumstances is air a stuff (e.g. air bubbles in water); most of the time it is a medium. A medium is “immaterial” (like a flash of lightning); a stuff is material. But there is no sharp line between everyday and scientific observations, an example being the observation that coloured substances often keep their colour in solution.There seems to be a consensus that stuff discourse is not about things, but about properties of things. Stuff words name a collection of thing-properties applicable to things that are homogeneous with respect to some of these properties. Substance properties, including the notion of being pure, apply to quantities (see article on ‘Mereology’ later in this Part). Substance predicates indicate that the quantity of matter has a cluster of properties (actualities and potentialities, states, dispositions, capacities, affections, etc.).

3. High-Entropy Bulk Glassy Metals and Other Mixts

Philosophers of chemistry still disagree on the question as to whether there is salt in the sea [[Earley, 2005] and [Needham, 2008a]]. It might be suggested that the underlying issue goes back to an issue already discussed at some length by Aristotle, who took homogeneity to be the criterion of being a single substance and criticised believers in atomism for not being able to explain how substances are comprised of constituents. As Bogaard [2006] remarks on behalf of Aristotle, a substance must be more than a heap, a mixture of a special kind. In a range of publications Needham has shown the relevance of ancient views of mixtures for contemporary debates (see the chapters ‘Modality, Mereology, Substance’ and ‘Compounds and Mixtures’ later in this Part).Bensaude-Vincent has remarked that an essential tension remains intrinsic to chemistry between the two conceptual frameworks of the Aristotelian notion of mixt and the Lavoisieran notion of compound. She has argued that composite materials, which replace natural material resources by synthetic ones, invite a return to Aristotle's four causes and his notion of mixt [Bensaude-Vincent, 1998, 18]. For the greater part of modern chemistry, by focussing on stoichiometric compounds, “the enigma of the true mixt was simply discarded” [Bensaude-Vincent, 2008,54].Composites can be seen as true Aristotelian mixts with properties which are “more” than the properties of their constituents, whereas many old and new composites don't claim to be “pure substances.” A modern cutting tool may consist of a multilayered ceramic structure on a tungsten carbide substrate, containing various non-stoichiometric phases; a dental amalgam may consist of an alloy (solution) of two (intermetallic) compounds. ⁹ An alloy can be a conglomerate, solution, intermetallic or chemical compound, or a complex conglomerate of some (or many) of these. ¹⁰ Hence, although pure substances or species are needed to create order (chemical space, writing and performing chemical reactions, etc.), it would be wrong to think that the all-embracing goal is the pure substance. In designing multicomponent high-entropy alloys (such as AlCoCrFeNiTi_x), one may prefer the formation of simple solid solution phases, i.e. mixtures, instead of (stoichiometric) intermetallic compound phases, i.e. pure substances. ¹¹ In the history of making hard steel, austenite, lederburite and cementite were identified as different mixtures with rather different characteristic properties, long before “theory” told us that lederburite is a eutectic mixture of the phases austenite (a saturated solid solution of the component C in the component Fe) and pure cementite (Fe₃C), a compound [Findlay, 1951, 200-205].⁹Intermetallics are compounds of metals whose crystal structures are different from those of the constituent metals. Intermetallics form because the strength of bonding between the respective unlike atoms is larger than that between like atoms (e.g. Ni₃Al, TiAl, NbAl₃, and also Mg₂Si). Intermetallic compounds are to be distinguished from (chemical) interstitial compounds (Fe₃C, Cr₄C).¹⁰Alloys are homogeneous mixtures of metals. The general strategy for making alloys is to select one or two principal components for primary properties and add other minor components for the acquisition of a definite microstructure and other properties.¹¹A high entropy of mixing lowers the tendency to order and segregate, making a random solid solution more easily formed and more stable than the formation of intermetallics or other ordered phases [Zhou et al., 2007].Traditional, modern, and hypermodern practices all aim at producing both pure and composite substances, although there is a tendency for chemists to focus on pure substances or a couple of molecules, leaving the real stuff, whether pure or not, to the materials sciences.

4. Protochemistry

The protoscience approach in the philosophy of chemistry sees the aim of science not as the description of nature, but as the theoretical-instrumental support of “poietic practices”, i.e. practices that aim at the production of material goods (in the case of chemistry: brewing, dye-making, metallurgy, and so forth). Only after the chemist has distinguished “chemical stuff-properties” from other properties, does it make sense to introduce talk of atoms and molecules [Janich, 1994]. At a first level of reflection, simple observations are made such as: if two arbitrarily cut parts of a thing display the same “substantial” properties then it is “substantially” uniform or homogeneous. Choosing homogeneity as the defining feature of substances entails that ice, water, and steam are three distinct substances. The world is full of substances that cannot survive a phase change. Substances for which the phase interconvertibility criterion holds are called chemical substances. Along such operational lines further definitions can be given of chemical reaction, chemical composition, and so on. ¹²¹²For detailed expositions of this approach see [[Hanekamp, 1997] and [Psarros, 1999]] in German; in English: [[Psarros, 1995] and [Psarros, 1998]].On this view, the most basic laws are those that state the existence of particular substances and their properties, that is the reproducible identification and synthesis of substances and the reproducible measurement of their properties. The well-known “laws” of chemistry are “unpacked” in terms of a mix of following rules or norms. Proust's experiments in support of the law of constant proportions already presuppose this law [Psarros, 1994]. In order for a transformation to qualify as a chemical reaction, the mass of the reaction products must equal the mass of the raw materials. It must be possible to isolate the products of a transformation as pure chemical substances with constant composition. These norms regulate practice, somewhat similarly to the way conservation laws regulate physical practice. If practice seems to go against the norm, it is assumed that something has been overlooked or should be fitted in elsewhere (as happened with radioactivity).

5. Stuff Perspective

Schummer [1998; 2008] has argued that since Thales' suggestion that all things are made of water, there has been a subsequent Entstofflichung (literally: “de-stuffing”) of western philosophy, giving utter priority to form over substance or stuff. A mere form has by definition no dispositional or dynamic properties, to say nothing about the transformative dispositional relations of chemical properties. Therefore, Schummer argues, it is impossible to explain stuff properties merely by form properties for logical reasons alone.The recurrent patterns of specific stuff properties allow the building of the ontological category of stuff kinds, which we use when we claim that two objects consist of the same stuff. ¹³ The building blocks are the pure substances that retain their identity during phase transition and purification. Two objects are chemically identical if and only if they are found at the same place in chemical space. Chemical space contains all possible substances. Seen as a (nonlinear) network, chemical space consists of pure substances at the nodes; the relationships between the nodes are chemical reactions correlated to experimental practice (including reactions with as yet non-existing substances). This “forms the chemical core of experimental chemistry” [1998, 135]. ¹⁴¹³In special circumstances material properties may vary with the shape of particles; e.g. nanoparticles [Schummer, 2008, 16-17].¹⁴Its relational structure can be described in terms of the operational definitions of element, chemical mass equivalent and chemical reactivity [Schummer, 1996, 182-223].Formation of concepts and models in chemistry is based on the assumption of distinct pure substances because chemical properties are relational. Characterising the entities in a chemical reaction (and giving an operational definition of the latter) requires pure substances as “ideal limit” reference points. The properties of a solution or conglomerate are described in terms of its constituents, understood as pure substances. It is not possible to use “quasi-molecular” species as the nodes in the chemical network because this would be to give up the distinction between homogeneous and pure substance. Quasi-molecular species are identified independently of their environment, whether it is a solution or a pure substance, and their defining structure varies with circumstances.Somewhat similarly to the protochemistry approach and earlier Bachelard (see the article on ‘Bachelard’ in Part 1 of this Volume), Schummer argues that chemistry is governed by an action-related conception of knowledge as distinct from emphasis on formalisation and mathematisation, as in physics. The chemical praxis of making new things (new stuffs) is different from that of making careful measurements or carrying out crucial experiments. Therefore there is a greater affinity of chemistry to technology or art than to physics. The fact that chemistry is constantly enlarging the world it studies by making new stuffs makes the interaction of the cognitive and material praxis of chemistry very different from that of physics (and biology as well).

6. Natural Kinds

Criteria for identifying and dividing natural kinds include similarities variously defined (appearance, projectible predicates, microstructurally defined “essences”, etc.) and also origin (causal-historical criteria). But it is not obvious that such criteria will distinguish natural from other kinds [van Brakel, 1992]. The review below is restricted to chemical kinds, for which history, apart from hysteresis as a disturbing parameter, is not considered relevant. ¹⁵ A sample is the same chemical kind if it features the same features (although this may not always be true in mineralogy). Bhushan [2006] has argued that the category of natural kind is suspect to begin with, because chemical synthesis shows that for chemistry distinguishing natural and artificial kinds makes little sense. ¹⁶ Many new chemical substances have been synthesized, but they don't differ in some fundamental way from naturally occurring candidates.¹⁵For a discussion of natural kinds in which chemical kinds figure rather prominently see [LaPorte, 2004]. Hendry [2008, 117] has suggested using “relevant to the epistemic practice of chemists” as a criterion for separating chemical kinds from other natural kinds.¹⁶A satisfying definition of the distinction between natural and artificial kinds does not exist. For the purpose of this article it is assumed that chemical kinds are natural kinds, taking the latter in the rather wide sense of “subject of detailed scientific investigation”, including glass, air, and even toothpaste, to mention three substances LaPorte [2004, 18-32] would not include among natural kinds.Although there are many higher-order classifications in chemistry such as functional groups or types of carbon bonding in organic chemistry, as a rule they cannot be combined hierarchically. There may even be competing classifications of the same feature, as in the case of acids. Should we call Brønsted-Lowry and Lewis acids two models of one natural kind or two “alternative” natural kinds having their own context of application?One of the dominant intuitions in the history of western ideas is that “macrophysical properties are asymmetrically dependent on microphysical structures” [Kim, 1990, 14]. Unfortunately, almost all philosophers of the past 50 years who have drawn on this intuition for their metaphysical proposals suffer from the “domesticated science syndrome”, that is to say, they either ignore science where it is most relevant or use outdated science, typically drawing on intuitive or folk pictures of structural composition (and causation). ¹⁷ For example, too often philosophers still think that a pure substance can be defined as “a collection of molecules of the same type.” However, this definition only applies in rare cases if ever; pure liquid water does not consist exclusively of one type of molecule. The definition doesn't work for metals and numerous other chemical substances. ¹⁸ The most well-known variant of pseudo-scientific microstructuralism, the one popularised by Kripke and Putnam, will not be reviewed here given its lack of scientific sophistication. ¹⁹¹⁷See [Ladyman and Ross, 2007, ch.1] for the notions of domesticated, neo-scholastic and pseudo-scientific metaphysics.¹⁸For details see [van Brakel, 2000, chs. 3 and 4].¹⁹For chemistry-based critiques of the micro-essentialism of Kripke and Putnam see [[van Brakel, 1986] and [van Brakel, 2005]] and [[Needham, 2002], [Needham, 2008a] and [Needham, 2009]], for a critique of Kim's micro-essentialism see [Needham, 2009], and for Ellis' micro-essentialism see [VandeWall, 2007].

7. Molar Definition of “Pure Substance”

By taking the notion of “phase” as their starting point, in the period 1890-1930 a number of (physical) chemists developed precise operational definitions of “pure substance”. ²⁰ In the terminology of Timmermans [1928], not much different from that of either Wald and Ostwald [Ruthenberg, 2008], or contemporary practice [Earley, 2005, 91ff], a material is homogeneous if it cannot be separated into different materials by external (or capillary) forces. ²¹ Methods using thermal energy (distillation, crystallising, melting) can be used to divide mixtures and pure substances. Mixtures can be divided into (homogeneous) solutions (of the gas/gas, liquid/liquid, and solid/solid sort), ²² addition compounds (see § 13), and (heterogeneous) conglomerates (aggregates of different crystals; emulsions, colloids, smokes). Methods using energy or pressure at higher levels can be used to divide pure substances into compounds and simple substances. Substances always have a limited range of existence. For example, at temperatures above 500°C water vapour dissociates partially and its two gaseous constituents oxygen and hydrogen can be separated. Using the terminology introduced by Ostwald [1907, 166-170], if the properties of two coexisting phases remain invariant during a phase change, the system is called hylotropic; if not it is a solution (phase of variable composition). If it is hylotropic over a range of temperatures as the pressure varies, it is a pure chemical substance. If it is hylotropic only at a particular temperature and pressure, it is a special kind of solution, for example an azeotropic mixture. If it is hylotropic over all pressures and temperatures, except the most extreme ones, it is a simple substance; if not it is a compound.²⁰For critical but sympathetic discussion of their proposals see [Psarros, 1999, 133n151; Schummer, 1996, 185n11], and the article on ‘Wald’ in Part 2 of this Volume.²¹Of course such criteria break down if we move to the nanometer scale. Nevertheless it is assumed that a hybrid crystal in which two polymorphs coexist allows for separation in principle.²²Although the term “solution” originated with the observation that many liquids may form homogeneous mixtures, it is now also used for the solution of solids in one another. An ideal solution (of which there are few if any) has properties linear in composition. Save under the most exceptional circumstances, there can be only one gas/vapour phase.Against this operational background a pure substance can be defined as a substance of which properties such as density do not change during a phase conversion (as in boiling a liquid or melting a solid), which takes place at one constant temperature. Thus the melting point of a pure substance is a constant of nature, ²³ even though it refers to one substance only and there are many such unique constants characterising (pure) chemical substances.²³Assuming the existence of absolutely pure substances (which may not exist).Operationally a substance is pure if it is perfectly homogeneous after being subjected to successive modes of fractionating which are as different as possible and when attempts at further purification produce no further change in properties. As already noted in the first section, later refinements may show that what was once thought to be a pure substance is, after all, not pure. As noted by many writers in the field, purity is a matter of negotiation or consensus. Different applications require different standards of purification. Different separation techniques (crystallisation, electrophoresis, and so on) set different standards of purity. ²⁴ The ideal pure substance would pass all types of ideal purification tests, i.e. tests with unlimited resolution.²⁴Modern separation methods in down-stream processing include: high-speed counter-current chromatography, supercritical fluid extraction, nanotube membrane ultrafiltration. But the general principles of separation methods (unit operations) have been the same for the past century (see the article Chemical Engineering Science in Part 6).Phase transition is a fundamental defining characteristic of this approach to pure substances. Hence substances that exist in one phase only, easily decompose, only occur in solution, etc. can only be included by analogy. Timmermans [1928, 23-53] lists the following potentially difficult cases for the molar approach of substance definition as summarised in this section: azeotropic mixtures, dissociative compounds in equilibrium, enantiomers and racemates (§ 18), certain types of mixed crystals or other polymorphic compounds, polymers (§ 16), many biochemical compounds, and systems that are not in thermodynamic equilibrium.

8. Polymorphs

There are various reasons why two substances might not seem to be the same substance despite having the same chemical composition. Isomerism is often mentioned as the reason why specifying chemical composition alone is not sufficient for identifying sameness of substance. But there are also cases where what may seem to be different substances will be considered the same substance. ²⁵ Polymorphs are two forms of the same substance, which are connected by phase transitions. ²⁶ Because metastable phases are common (§ 11), the actual connecting phase transitions may not be easily accessible. Like other (compound) substances, a simple substance may occur in a variety of (allotropic, polymorphic) forms of one or more of its phases, usually the solid phase. Simple substances such as carbon, oxygen and sulphur have many allotropes. Ostwald noted that the allotropy of simple substances is just a special case of the phenomenon of polymorphism known for compounds, and proposed that the terms ‘allotrope’ and ‘allotropy’ be abandoned. This advice was repeated by various authors of otherwise influential publications, but not followed.²⁵Sometimes ‘isomer’ is used to refer to polymorphy of intermediate compounds. For example, it is said H₂SO₄·(H₂O)₂ has two isomers [Couling et al., 2003]. For this system, also molecular to ionic interconversion is referred to as “isomerizing.”²⁶It has turned out quite difficult (or impossible) to give a satisfactory definition of polymorphy, distinguishing it from dynamic isomers (see § 19) and “conformational polymorphs”, “pseudopolymorphs”, etc. For a (confusing) review of the confusion caused by the many different definitions see [Bernstein, 2002, ch. 1].Some substances exist in more than one liquid form: liquid crystals or anisotropic liquids, forming various types of mesomorphic phases (nematic, smectic, cholesteric). ²⁷ They share properties normally associated with both liquids and crystals. Sodium soaps have numerous mesomorphic states. Amorphous (metastable) phases of a substance may also display polymorphism. ²⁸ However, often the term ‘polymorph’ is explicitly restricted to the solid phase and polymorphic transformations are restricted to transformations involving phases with different crystal structures which are part of a single component system; ²⁹ such solid polymorphs form an identical liquid on melting. ³⁰ For example, phosphorous has many solid allotropes, which all revert to the same P₄ form when melted to the liquid state. In contrast, for isomers the difference between them persists in the liquid phase. Polymorphs (allotropes) can be combined with other substances to give the same compounds; isomers give different reaction products.²⁷A molecule is said to be mesogenic if it is able to form liquid crystalline phases.²⁸Highly densified vitreous silica may be an example of an “amorphous polymorph.” Recently the amorphous to amorphous (pressure induced high-density) reversible phase transition has received much attention in connection with the development of bulk metal glasses such as La₆₈Al₂₀Cu₂₀CO₂ [Liu and Hong, 2007]. Amorphous fluid phases are also possible.²⁹Although traditionally associated with inorganic crystals, polymorphism is not unknown for high polymers. Polytetrafluoroethylene has four different crystalline phases [Koningsveld et al., 2001,11].³⁰The allotropic γ- to α-transformation of iron is perhaps the most important allotropic transformation for modern culture as it is responsible for the unusual high strength of steel (Fe-C alloys).Earley [2005, 94] has suggested, with reference to the criterion of interconversion rate, that “the vast majority of chemists” would consider allotropes (such as graphite and diamond) to be “different chemical substances”, just like isomers are considered different substances. In both cases the difference is found in the ways atoms are connected and interconversion is slow (compared with “ordinary” phase transitions). Most writers in the philosophy of chemistry, whether of macroscopic or microstructural inclination, consider allotropes to be different forms of one basic/simple substance. But clear definitions covering all cases are difficult to find. Oxygen and ozone are sometimes referred to as two allotropes of the element (basic substance) oxygen, but they are also said to be two different substances in a way that diamond and graphite are not, because each survives its “own” phase transitions while preserving its individuality. Oxygen and ozone each have their own polymorphs and molecular species. ³¹ For example, of the many solid phases in the oxygen phase diagram, η-oxygen is said to have the structure of crystalline (O₂)₄ complexes or “molecular units” (Lundegaard et al. 2006), whereas ozone dimers (or [O₃·O₃] complexes) “belong to” ozone. ³²³¹The liquids (of ozone and oxygen) are said to be immiscible in the range of 9-91 percent ozone [Brown et al., 1955].³²Apparently [O₃·O₃] is not (O₂)₃. But what about [O₃·O], a complex of what is referred to as an “ozone molecule” or “ozone monomer” and an “oxygen atom” [Sivaraman et al., 2007]?The underlying issue (as to whether ozone and oxygen are or are not allotropes or polymorphs) seems to be the question how to situate various physico-chemical (phase) transitions relative to chemical transformations on the one hand and physical transitions on the other. ³³³³In addition to chemical transformation and (physico-chemical) phase transition, there are other physical transitions, e.g. a magnetic transformation (of, say, α-ferrite to β-ferrite). The latter can be distinguished from phase (or polymorphic) transitions because they don't take place at a definite temperature.

9. The Phase Rule

Although phase theory seems to be phased out of undergraduate chemistry programs in some places, it plays a central role in many cutting edge developments and discussions. For example the phase rule and phase diagrams feature in discussions on equilibrium models for the evolution of homochirality, ³⁴ in the selection of enantiomeric enrichment processes for new drugs, in studying phase equilibrium in supercritical fluids systems, in developing “high-tech” ceramics, to establish the stability ranges of nitric acid and sulphuric acid hydrates in solid particles in polar stratosphere clouds, and so on.³⁴Homochirality is a term used to refer to a group of molecules that possess the same sense of chirality (cf. § 18). Models for the evolution of homochirality is a hot issue, because the origin of life would be predicated on the question of the origin of molecular chirality. The question is how the single chirality of biological molecules developed from a presumably racemic prebiotic world.Gibbs (see the article on ‘Gibbs’ in Part 2 and the article on ‘thermodynamics in Part 5 of this Volume) introduced the notion of phase as follows: ³⁵³⁵Although Gibbs stressed that thermodynamics is independent of atomistic interpretations, it is perhaps not a coincidence that he uses the same word ‘phase’ in his publications on statistical mechanics. In chemistry, thermodynamic equilibrium is not mandatory for a phase. This is OK because in the usual pressure-temperature range chemists work, metastable phases can undergo reversible phase transformations with all the equilibrium thermodynamic laws observed [Brazhkin, 2006]. Nevertheless, because of the possibility of metastable phases (with indefinite lifetime), operational criteria for equilibrium are needed; for example including: “the same condition is reached no matter from which side approached.”In considering the different homogeneous bodies, which can be formed out of any set of component substances, it is convenient to have a term which shall refer solely to the composition and thermodynamic state of any such body without regard to its size. The word phase has been chosen for this purpose. Such bodies as differ in composition or state are called different phases of the matter considered, all bodies which differ only in size and form being regarded as different examples of the same phase. [Gibbs, 1931 [1876-78], 359; cf. 96]

Although, following Gibbs, a phase is often defined as “one of the uniform, homogeneous, not necessarily continuous, physically distinct and mechanically separable portions of a system in dynamic, heterogeneous equilibrium” [Findlay, 1951, 5], ³⁶ there are many ambiguous cases. For example, it is a common complaint that text books on the phase rule never give a clear answer to the question as to whether a 0.1% sol of clay in water is one phase or two phases. ³⁷Ricci [1951, 3] remarks that the criterion of homogeneity is secondary in the definition of a phase, in the sense that systems can be heterogeneous, provided heterogeneity is randomly distributed and/or properties only show continuous variation. ³⁸ Van der Waals deliberately did not include “homogeneity” in his definition of a phase, because “complexes of bodies” may differ not only by spatial but also by other non-thermodynamic properties (as in the case of optical isomers; see § 18). ³⁹³⁶Different authors may choose slightly different wording; cf. in the older literature: [Ostwald, 1907,117; Wald, 2004/1918, 112]. An important variation is to substitute “separated from other parts by a definite bounding surface” for “mechanically separable portions” [Rao, 2001, 508]. Van der Waals defines homogeneity macrostructurally as systems that do not show any parts or layers [1927, 12], perhaps reflecting the writings of Wald and Ostwald. If an observable boundary between phases is assumed (“the geometrical surface over which different phases are visibly in contact” [Buchdahl 1966, 119]), at which transition there is an abrupt change of properties, this requires operational, pragmatic criteria, in particular because of the possibility of metastable situations [Schummer, 1996, 172]. It does occur that one sees discontinuities, but that separation achieves no effect, or that a phase transition only takes place during a separation operation.³⁷In a sol (viscous fluid) or a gel (elastic solid), solvent and solute form a colloidal solution.³⁸In a homogeneous phase there may be statistical fluctuations due to isotope effects or crystal defects. These statistical effects may even cause visible effects as in critical opalescence. Strictly speaking a phase is not homogeneous in the direction of the field of gravity [Guggenheim, 1957, 327].³⁹Van der Waals gives the following elaborate definition of a phase: “consisting of n components is to be characterised by the nature of these components and, further, by the relations, first, which establish explicitly how the thermodynamic potentials and other thermodynamic quantities of the complex of bodies belonging to this phase depend on the temperature, pressure and the composition of the complex, and, secondly, and conversely, how the fixed values of the pressure, temperature and the (n−1) thermodynamic potentials of (n−1) components determine explicitly the appropriate values of the composition and other thermodynamic quantities” [1927, 11]. Both requirements are needed because each alone is necessary but not sufficient [Kipnis et al., 1996, 251].Thermodynamic criteria of equilibrium expressed in terms of the intensive properties pressure, temperature and chemical potential lead directly to Gibbs' phase rule (see the article Compounds and Mixtures in Part 3), relating the number of components (in the sense of the phase rule), the number of phases, and the number of degrees of freedom of the system (number of independent variables). ⁴⁰ Because of electrolytic and chemical equilibria the number of components of a system cannot be set equal to the number of chemical species (or to the number of substances, except in special cases—see below). This has led to trying to give more precise definitions of “component” and/or adding “special restraints variables” to the equation, thus undermining the “universality” of the phase rule. ⁴¹ In practice the “correct” application of the phase rule (in particular for systems of two or more components) can benefit enormously from prior knowledge concerning possible chemical species equilibria in the system [[Vemualapalli, 2008] and [Ricci, 1951], 125].⁴⁰There have been subtle debates discussing whether negative degrees of freedom are possible. The answer seems to be: yes! Starting from an invariant equilibrium, one may calculate which properties an additional phase must have in order to fit in with the equilibrium conditions [Oonk, 1981,47].⁴¹Depending on the context (e.g. presence of reactive components) additional constraints or variables have been introduced such as: “the number of independently variable constituents necessary for the statement of the composition of all its phases,” “number of independent chemical equilibria,” “number of stoichiometric constraints,” “number of constituent elements not present as elements,” and, if there weren't already enough parameters, “special constraints/restraints”. For examples see [Smith and Van Ness, 1975, 416; Rao, 2001, 517; Oonk, 1981, 42; Findlay, 1951, 9]. Also see note 45.By identifying “one component” with “pure substance” a definition of the latter is obtained. As Wald put it: “Phases are chemically identical when they coexist and correspond to the phase rule for one component. All other bodies are chemically different” [Wald, 1897, 647f; emphasis original]. To highlight that a prototypical pure substance may occur in several phases, we may choose its invariant triple point as its defining feature. In the prototypical case this will be the triple point of solid, liquid, and vapour (as in the case of water), but a substance that is not stable at higher temperatures, thus not forming a gas phase, may still have a characteristic triple point; for example polyethylene has a triple point at which one liquid and two crystalline phases are in equilibrium. A special invariant point is the critical point, not further discussed here. ⁴²⁴²The vapour-liquid critical point is non-variant because for this unique state there is an extra condition to be added to the phase rule derivation, namely the complete identity of the two phases.In binary systems other invariant points may occur (e.g. for azeotropic or eutectic mixtures). However, in general, by varying the pressure or collecting a larger part of the phase diagram the one component (or “unary”) system is easily separated from the binary ones. Eutectic points, at which a solution behaves as if it is a pure substance (one component), may be used to advantage. There are numerous applications of eutectic alloys which simulate being a pure substance by melting or freezing at a single sharp temperature, for example the (Sn-Ag)_eut+Cu soldering material.Congruent melting points in complex systems may correspond to the presence of intermediate compounds. ⁴³ For example, the system BaO-TiO₂ contains important intermediate compounds, which exist in their own right (having specific applications), for example BaTi₄O₉ is a useful material in the context of microwave frequency communication. ⁴⁴ Typically, intermediate compounds only occur in the solid phase and dissociate in the liquid phase. As the example just given illustrates, intermediate compounds often have a chemical composition that cannot be said to be a proportion of “small” numbers. Furthermore, intermediate compounds often show non-stoichiometric behaviour (cf. §§ 12 & 13).⁴³If the identity of the compositions of the phases during transition is independent of pressure and temperature, it is called a congruent transition.⁴⁴The system BaO-TiO₂ has two eutectic and four peritectic points in the temperature range 1000-1500 K (Chiang et al. 1997, 285), including the intermediate compounds BaTi₄O₉, BaTi₃O₇, BaTi₂O₅, BaTiO₃ (two polymorphs), Ba₂TiO₄.Although occasionally papers appear speaking of the “inapplicability of Gibbs phase rule” [Li, 1994, 13] or “beyond the Gibbs phase rule” [Mladek et al., 2007], this invariably means no more than that one of the ceteris paribus conditions Gibbs already mentioned is not fulfilled; for example, the phase rule doesn't cover systems in which rigid semi-permeable walls allow the development of pressure differences in the system. Gibbs explicitly allows for the possible presence of other thermodynamic “fields.” An extended phase rule has been proposed for, inter alia, capillary systems (in which the number and curvature of interfaces/phases play a role), ⁴⁵ multicomponent multiphase systems for which relative phase sizes are relevant [Van Poolen, 1990], colloid systems (for which, even if in equilibrium, it is not always easy to say how many phases are present), ⁴⁶ unusual crystalline materials, ⁴⁷ and more.⁴⁵Although still squabbled about in recent literature (Godek, Gaberscek and Jamnik 2009), the precise derivation of the phase rule for capillary systems was already given by Defay in his Brussels' Thesis of 1932 [Defay and Priogine, 1966/1951, 76], taking into account the presence of surfaces and related equilibria.⁴⁶It is generally assumed that colloidal supermolecular aggregates should obey phase-rule principles; for example when it is said that in the system water/oleate there are ten non-variant three-phase equilibria.⁴⁷Examples include hypercrystals and quantum Hall effect bubble solids. At finite temperatures all crystals contain point defects. Usually the effect of this for “simple” crystals can be neglected, but in the case of unusual crystalline materials the effect of an unit cell chemical potential can be dramatic (Mladek, Charbonneau and Frenkel 2007).Although these complications do not bear directly on a phase-rule definition of pure substance, there would still seem to be hidden presuppositions in connecting (pure) substance and component (in the sense of the phase rule). ⁴⁸Ricci [1951] has argued that the phase rule cannot provide a characterisation for chemical compounds, if only because the very meaning of the phase rule depends on the meaning of ‘phase’ and ‘component’, which cannot be given independent definitions (4). Further, according to Ricci, although a single substance must behave as a one-component system, unary behaviour does not guarantee that a material is a single substance (9, 165). Finally, the phase rule approach doesn't always make a clear-cut distinction between unary and binary systems or between compound and solution (125).⁴⁸Following a suggestion of van der Waals [1927, 227; cf. 14]: van Brakel [2000, 87; 2008, 160] has emphasised that the scientific (thermodynamic) notion of pure substance is grounded in a pre-scientific notion of pure substance; van der Waals wrote: “the considerations [concerning the phase rule] rest on the assumption that we know how to understand these words” (i.e. the words “chemically pure substance” and “chemical individual”). Gibbs (and all accounts based on his seminal publication) presuppose the notion of “pure substance” (as Wald already remarked).

10. Phase and Substance Properties

In a range of publications Needham has defended the view that, at least as far as the classical concept of (pure) substance is concerned, the application of the phase rule allows for the identification of the characteristic behaviour of a quantity comprising just one substance, thus determining which microstructures belong to one and the same substance. Using detailed accounts of the properties of water, he has definitively shown the meaningless/incorrectness of simplistic microstructuralist assumptions [Needham, 2010]. Microscopic principles complement macroscopic theory in an integrated whole, with no presumption of primacy of the one over the other. Gibbs' contribution can be seen in this light, as showing that finer distinctions are needed in the mechanical property of mass, recognising that a given quantity of matter may be divided into matter of distinct substances and blurring the distinction between physical and chemical properties [2008a].Proposing new mereological definitions, Needham has also given detailed specifications of microscopic dynamic and macroscopic thermodynamic equilibria for chemical substances and the time dependence of phase and substance properties. ⁴⁹ The distinction between features that are both homogeneous and intensive and those which are homogeneous but not intensive serves to distinguish phase properties from substance predicates. Predicates satisfying unrestricted distributivity and cumulativity are intensive and those satisfying spatial distributivity and cumulativity are homogeneous. Solutions are spatially homogeneous but not intensive. ⁵⁰⁴⁹In his work Needham presupposes the notion of “material” as a quantity that makes up a phase at a given time.⁵⁰Earley [2005, 90] has objected that “the conclusion that solution properties are not intensive does not seem consistent with the understanding and practice of contemporary chemists” and that therefore Needham is wrong to claim there is salt in the sea.Equilibrium implies that the macroscopic state of the matter is stable, not that there is no underlying change. In particular, there is a continual exchange of matter between the various phases of a heterogeneous quantity of matter. Accordingly, phase predicates like “is liquid” are not, in general, distributive (don't hold of all the parts for all the subintervals of a quantity and a time that they relate). This lends support to the thesis that a water molecule is not water. A substance predicate “water” is applicable to sufficiently large quantities for sufficiently long times over which the microscopic fluctuations are smoothed out, and not to what microdescriptions are true of for much shorter times. ⁵¹⁵¹For the last two paragraphs see Needham [2007, forthc.] and the article on ‘Mereology’ in Part 3 of this Volume.

11. Metastable and Other Esoteric Phases

Metastable phases are found in nature and are produced by traditional and novel material processing techniques. For example, a metallic glass golf club head with a strength and hardness twice that of cast stainless steel or titanium but of lower modulus and intermediate density, made by bulk undercooling prior to solidification, is a metastable phase. Some metastable modifications, e.g. amorphous phases, have no stability region anywhere, the traditional example being glasses. ⁵² The actual properties of such unstable material will depend on its history. Ricci [1951] argues that glass is strictly speaking neither a supercooled liquid nor a true solid. It is in fact not a “phase” at all since it is not in reversible equilibrium with any other phase (44). But in practice such phases are included in discussions of phase diagrams and metastable phases also claim to be substances. In more recent literature various proposals have been made suggesting that there may be underlying equilibrium considerations for the glass transition, considering it “as a second-order or third-order thermodynamic transition of the order/disorder type” [Koningsveld et al., 2001, 218].⁵²Glasses or amorphous solids lack long-range periodicity in their atomic arrangements. Solidification takes place at the glass transition temperature that lies below the melting temperature and a metastable glassy solid is formed, rather than an equilibrium crystalline intermediate compound. In recent literature it has been suggested that glassy (vitreous) and amorphous phases, at some level of description, do have structure.From the physicist's point of view, chemists work in a region of predominantly metastable molecular phases [Brazhkin, 2006]. ⁵³ Because the chemist works in a pressure-temperature range of quasi-equilibrium, the researcher may forget about metastability. For example, oxygen and hydrogen are metastable relative to water, although in “normal” circumstances they behave like three independent components.⁵³Most simple substances studied by physics qualify as stable phases.Special phases may arise because of special phase transitions, as in the case of second order phase transition from normal to superfluid. ⁵⁴ Another type of special situation occurs in so called block polymers in which microphase morphology is observed at the level of several tenths of nanometers. These segregated local domains at the microlevel are bound together, so that macroscopically the sample behaves as one single phase. ⁵⁵⁵⁴See § 15 on helium.⁵⁵These systems have been called “mesophases”. It would be incorrect to speak in this case of a multiphase system, because no separation on the macro-scale is possible.Stimulated by the proliferation of more and more polymorphisms of well-known substances, ⁵⁶ as well as the proliferation of new substances existing in one phase only, there is an increasing tendency to divide the material world into phases, the traditional role of (pure) substances being relegated to introductions to more complex systems, and one may speak of an intermediate phase La₇Ni₃ and an ordered phase FeNi₃ in the system Fe-La-Ni, of stable non-stoichiometric phases in the system Sr_1−xBi_2+2x/3Ta₂O₉(x=0-0.5), of an amorphous phase interlaying crystalline lamellae in nylon 6, of amorphous Ni₆₄Zr_36−xM_x membranes, ⁵⁷ and so on, without bothering to say which (how many) substances or species are “behind” the composite formula. Further, particular phases may only form in particular environments. Micron-sized particles or membranes may show different phase behaviour from the bulk because of surface effects, for example the formation of a glassy phase that doesn't form in the bulk. ⁵⁸⁵⁶For example, at high pressures solid oxygen changes into a metallic state. At very low temperatures it changes into a superconducting state.⁵⁷M = Ti, Nb, Mo, Hf, Ta, or W.⁵⁸This also leads to blurring the distinction between stuff and thing. Films, layers, membranes, vesicles, micelles (as well as mesomorphic phases and solid state structures) are formed by spontaneous association of a large undefined number of components into a specific phase having more or less well-defined microscopic organization and macroscopic characteristics. Other special phases may form as inclusions (occlusions) in crystal structures of hosts (see § 13).Plasmas, often described as the fourth state of matter, ⁵⁹ have macroscopic appearances, even structure, but don't fit in with prototypical substances. ⁶⁰Earley [2006] has discussed the features of “chemical coherences” (dissipative structures, self-organising collections, open systems) and considers them chemical “substances”, but not “chemical substances”. It has been said that plasma beam techniques can transcend not only kinetic, but also thermodynamic barriers; that is to say: they provide access to non-equilibrium regimes and exotic intermediate phases. ⁶¹⁵⁹The number of states of matter is not well-defined; compare a sharp transition such as melting with the gradual transition of liquid/vapour above the critical point. Sometimes superconductors and superfluids are considered different states of matter from the “ordinary” states, sometimes not.⁶⁰A plasma is a partially ionized gas, in which a certain proportion of electrons are free rather than being bound to an atom or molecule. Like gas, a plasma does not have a definite shape or a definite volume unless enclosed in a container; unlike gas, in a magnetic field, it may form structures such as filaments, beams and double layers. It is the most common phase of matter in the universe (the sun, stars, interstellar nebulae, fire, lightning, etc.)⁶¹In a typical plasma jet, the reactants are heated to their melting temperatures, causing a fraction of the particles to vaporize and react with each other. The atomic and molecular product clusters are then quenched. This happens in a few milliseconds, which doesn't allow enough time for the atomic rearrangement required for the formation of crystalline structures, resulting in new amorphous and other metastable phases.

12. Non-Stoichiometric Compounds

The norms of definite, multiple, and reciprocal proportions are neither a necessary nor a sufficient condition for being a (pure) substance, because of the existence of isomers and because of the existence of non-stoichiometric compounds. Over the years the latter phrase has been used for a range of more or less similar groups of substances, which have in common that they (allegedly) violate the law of definite proportions. Although the precise meaning of “non-stoichiometric compounds” is contested, the omnipresence of non-stoichiometric compositional formula in various parts of inorganic chemistry (including metallurgy, ceramics, soil science, etc.) cannot be doubted, for example: Cu_1.7−2.0S, K_0.62Si_3.51Al_2.03Mg_0.19Fe(III)_0.29O₁₀(OH)₂ [illite], or (Pb_1−xLa_2/3x)Nb₂O₆, or ZrH_0.05–1.36. Many non-stoichiometric compounds are important in solid state chemistry, and have applications in ceramics and as superconductors. Many oxide phases present problems for a “simple” rule of “simple” proportions in two ways. For example, in a phase diagram for the system titanium-oxygen, between TiO and TiO₂, there are not only many phases associated with oxides such as Ti₃O₅ (Ti₂O₃·TiO₂?) and Ti₁₀O₁₉, but there is also a homogeneity range of a “non-stoichiometric” or “oxygen-deficient” phase labelled TiO_{1.983–2.000}.There are also more formal arguments undermining the universality of the “law” (cf. § 4) of simple/multiple proportions [Christie, 1994]. For example, because of the different relative proportions of end groups in polymers such as nylon and polystyrene, they have slightly different composition, as well as differences in physical properties. In practice, nylon is considered one compound, although strictly speaking molecules with a different chain length (different molecular weight) would represent a different substance according to the law of constant proportions and strictly speaking it is not a case of simple multiple proportions (because of the end groups). ⁶²⁶²In practice mixtures of polymers in small-molecule solvents can be treated as pseudo-binary systems. Polymorphy of semi-crystalline polymers is highly complex and depends on molecular weight.In the context of supramolecular chemistry, ⁶³ the composition and structure of a non-stoichiometric compound can be considered in terms of the geometrical disposition of its constituents, which somehow display a guest-host structure. Under this broad definition, even a solution (interstitial or substitution crystals) can then be considered a non-stoichiometric compound or an adsorbed layer of gas on a solid surface can be considered as a separate phase [Mandelcorn, 1964]. This guest-host structure image can be extended to channel/tunnel and cage structures (see next section).⁶³Definitions of supramolecular chemistry vary. Schummer [2006] has suggested supramolecular chemistry deals with the associations of two or more species insofar as they constitute a system that can perform certain functions and in this sense it is the predecessor of nanotechnology.Perhaps the most common use of the term “non-stoichiometric” is for solids that contain crystallographic point or extended defects. ⁶⁴ Thermodynamically an ideal crystal is not possible. Hence, it has been said that all inorganic compounds show non-stoichiometry [Kosuge, 1993]. The requirements of crystalline stability may or may not be satisfied with ordinary stoichiometric proportions.⁶⁴The concentrations of point defects in thermal equilibrium can be calculated for many types of defect structures by statistical thermodynamics. Stoichiometry is determined by the interior of crystals; the surfaces of crystals often do not follow the stoichiometry of the bulk.Some interstitial compounds are variously referred to as non-stoichiometric compounds or variable composition compounds of the berthollide type. ⁶⁵ In this context Earley [2005, 91-2] has argued that the distinction between non-stoichiometric compounds and solutions is one of convenience. For some applications titanium hydride is best considered a compound, for others a solution. But it may be added that “convenience” is guided by the features observed in the Ti-H phase diagram. Perhaps at low concentrations it may be considered a solution, whereas at higher concentrations non-stoichiometric compound phases may form, all the way up to the “true compound” TiH₂, whereas in specific circumstances metastable phases may pop up. ⁶⁶⁶⁵Such as: TiC_xN_y, Ti_1−x−yO_yC_x, VC_0.88, NbC_0.95. The phrase “compounds of variable composition” is used to refer to the situation in which a compound exists over a narrow strip of composition in the phase diagram in which range it is the sole stable phase.⁶⁶Titanium hydride is a non-stoichiometric interstitial compound occluding a large amount of hydrogen (atoms distributed randomly along lattice sides) and is reversible in adsorption and liberation of hydrogen. Because hydrogen either in solution or as metal hydride causes embrittlement of metals and alloys, primarily what counts is the presence of hydrogen and only secondarily its phase properties. If hydrogen is forced into the titanium lattice, one may simply consider it a conglomerate of two different substances.Whether to admit non-stoichiometric compounds goes back to the argument between Proust and Berthollet. After a detailed investigation of intermetallic compounds, Kurnakov [1914] found that the maximum or minimum of the melting point or electric resistivity does not necessarily appear at the st

Date: 2015-02-16; view: 2149