A phase, or state of matter, is a domain within a many-body system within which relevant physical properties are uniform. Relevant properties may include chemical composition, stoichiometry, and density, which do not reflect how the components are arranged in space. They also may include measures of order such as the translational correlation length and the orientational correlation length. Different domains with the same physical properties are said to be in the same phase even if they differ in such thermodynamically irrelevant parameters as orientation. Thus ice cubes in a glass of water are all in the crystalline phase. So also with magnetic domains in a ferromagnet.

For systems in thermodynamic equilibrium, what truly distinguishes two phases is if there is a discontinuity in the system's specific heat upon varying a physical parameter such as the temperature or pressure. In first-order phase transitions, this discontinuity takes the form of a jump in the specific heat, and clearly distinguishes the two phases. Second-order phase transitions are more subtle because they involve only changes in derivative of the specific heat with changes in the control parameter.

For systems out of equilibrium, the definition of phases and phase transitions is far more varied and complicated.

This is probably one of the better questions I have come across here at RG.

A very complex question...To get the disucssion going here is a more than likely an incomplete list of the various phases matter can take…add to it...and maybe we can all create a better definition of "phase" right here at RG. I know I'm looking forward to seeing what stems from the posed question. Hopefully the members of RG can help create a better all encompasing definition of Phase.

Chemical Properties that distinguish different chemical classifications

Exist in a host of various “phases”

A phase set of states within the chemical system being measured

can have similar bulk structural properties over a range of conditions such as pressure and temperature.

The phase of the matter being measured is defined by phase transition to which energy entered or taken out of system affects the structure of the system instead of changing bulk conditions:

Solid, Liquids, Gases

Solids break down (iron and others) (alpha, gamma, delta Phases)

Solid Solutions - Crystal phases

Vapor Phase

Mesophases

Aqueous Phases- Hydrophilic and hydrophobic differentiation

Immiscible phases - Emulsions and colloids

Multiphasic Liquids

Crystal Phases

Phase Transistions & Phase Boundary

Plasma Phases

Bose-Einstein Consdensates

Fermonic Condensates – Fermionic Particles

Paramagnetic

Ferromagnetic phases of magnetic materials

A phase, or state of matter, is a domain within a many-body system within which relevant physical properties are uniform. Relevant properties may include chemical composition, stoichiometry, and density, which do not reflect how the components are arranged in space. They also may include measures of order such as the translational correlation length and the orientational correlation length. Different domains with the same physical properties are said to be in the same phase even if they differ in such thermodynamically irrelevant parameters as orientation. Thus ice cubes in a glass of water are all in the crystalline phase. So also with magnetic domains in a ferromagnet.

For systems in thermodynamic equilibrium, what truly distinguishes two phases is if there is a discontinuity in the system's specific heat upon varying a physical parameter such as the temperature or pressure. In first-order phase transitions, this discontinuity takes the form of a jump in the specific heat, and clearly distinguishes the two phases. Second-order phase transitions are more subtle because they involve only changes in derivative of the specific heat with changes in the control parameter.

For systems out of equilibrium, the definition of phases and phase transitions is far more varied and complicated.

In trying to distinguish between phases, I'd take a look at the interaction between particles and at the internal energy, entropy, etc. It seems a sure but hard to measure way of characterising a phase.

Non-equlibrium or metastable systems may also be important - such as semicrystalline polymers, where one speaks of crystalline or amourphous phases.

Classically, but still more intuitive than rigorously, a phase is a region of the thermodynamic space, throughout which all physical properties of a material are essentially uniform.

One can see how broaden the definition is: all physical properties seems very strict, and if actually considered like this probably an indefinite number of phases would exist; essentially uniform, how much can a deviation from uniformity be accepted...

Fortunately or not, the term phase is also used as a synonym for other concepts as the state of matter, a set of equilibrium states demarcated in terms of state variables, etc...

This makes your question, really tough to be answered (at least by me)

I tend to think about phases in terms of potential energy landscape and its purely geometrical properties. A system under investigation is a single point wandering in configuration space (or in phase space, but this one has more dimensions and therefore is more complicated). It may be confined to a (local) potential well - then we speak about a single phase. When there is another potential well nearby and separated from the first by some barrier (a saddle point) then there is a possibility for a system to transit into this other well. Think about it as of egg cartoon filled with water. After the water level rises the available space slowly, but continuously increases until the water reaches the saddle point. At this moment we can observe discontinuous increase of the accessible volume what, in my opinion, may be interpreted as a phase transition. High water level means paramagnetic state - every location within egg cartoon is accessible. Lower level forces the system (it is a single point, remember?) to select exactly one of the now separated basins and consequently we have crossed the Curie point. The substance becomes magnetized in one particular direction, however not necessarily magnetically saturated because the width (area, perimeter) of a selected potential well is non-zero. But, as usually in egg cartoon, there are many potential wells to choose from, all nearly identical. Therefore our system under investigation may spontaneously divide into several parts and occupy several well from now on. It seems natural to consider each part to be a separate phase, namely, in case of ferromagnetic material to be a set of individual magnetic domains. The net magnetization is close to zero. Tilting the egg cartoon (applying external field) makes the wells no longer identical and those located at higher positions will no longer be occupied - net magnetization emerges.

Consequently, in this picture, phase transitions should be regarded as changes in number of simply connected components making altogether the accessible configuration space.

Let me repeat: it is the number of simply connected regions* in configuration (or phase) space what defines the current phase(s) (equilibrium?) state of a sample. Now it is easy to see that a phase transition always corresponds to some saddle point, but not conversely - a long standing mystery.

Let me know what do you think and whether this picture is applicable to other known phase transitions.

*) I take the liberty to reserve the rights to my original idea of simply connected components of phase space.

@David: I can't agree that orientation is 'thermodynamically irrelevant parameter', especially in magnetism. While it may be true for paramagnets it is certainly false for ferromagnets. You simply cannot disregard their hysteretic behavior and its relation to free energy and alike. Magnetism is fascinating due to its inherent nonlinearity, among other things.

@Marek: I'd certainly agree that orientation of the local magnetic moment is important in ferromagnetism. This need not, however, be tied to crystallographic orientation. A polycrystalline film can be in a single phase (i.e. crystalline), consisting of domains of different crystallographic orientation, and still have a uniform magnetization, which is a separate characteristic. In that case, the difference in crystallographic orientation from domain to domain does not signal a difference in the thermodynamic phase from domain to domain, nor does it distinguish the thermodynamic phase of the polycrystalline film from that of a single crystal. Polycrystallinity in this case reflects the kinetics of the phase transition, rather than the thermodynamics. In that sense, the crystallographic orientation of the domains is thermodynamically irrelevant.

@David: Your example is smart indeed. But let's imagine a simple test of observing the orientation of magnetization of such powdered ferromagnetic substance (with immobile grains, say hot pressed) in reaction to rotating external field with fixed magnitude. Observe the magnetic moment of the sample or just the torque. Then compare this result with the one obtained using a single crystal of exactly the same material. Select one with high magnetocrystalline anisotropy. But, please, don't apply very strong field, keep it below coercive field. Depending on history of a single crystal you may obtain quite different results. Namely, demagnetized sample may be very similar to the powdered one, but saturated sample will behave quite differently. This shouldn't happen if both samples were in a single phase state, right?

Hi Marek, so about the definition of phase, I learned that phase is a region of the same crystal structure and same composition. This differ one phase from other. Consequently the properties from them are different too.

@Alex: Certainly, the notion of 'phase' is coarse-grained, i.e. it is applicable to the set of many atoms or molecules, not just few of them. Suppose you observe the same crystalline sample at two different temperatures. Many physical quantities will be different: optical spectra, electrical conductivity, specific heat, speed of sound, and so on. Yet, in majority of cases the phase will be the same, unless there is a phase transition of the second kind (say structural) located somewhere between the two temperatures. On the other hand there is no change in crystal structure or composition when the magnetic sample passes over the Curie temperature, changing its phase from ferromagnetic to paramagnetic or reversely. The same applies to the normal state superconductor phase transition.

The notion of phase certainly needs to be defined more precisely than today. There should be no contradiction between chemist's and physicist's point of view. Maybe biologists should contribute too? How about live dead 'phase transition'?

A parallel to your suggestion about magnetic domains is piezoelectrics. We may consider different phases (in this case the convention defines them by the crystallographic structure, but not orientation) which change the unit cell and therefore the polarity of the structure as a whole. However, if twinning occurs or other defects that may create regions of opposing polarity, we don't discuss it in terms of macroscopic phase, but rather macroscopic success as a piezoelectric. As examples, if you want to use a magnet which has had its domains unsuccessfully aligned (as in the case of curie temperature "demagnetizing" a ferromagnetic material) or if you want an oscillator which responds effectively to an applied potential, these domains must be in our favor. However, if they are not, it does not redefine the structure of the unit cell, and therefore does not define the phase. However, discussing polymers throws this convention out the window, because the entire structure can have chemical properties (e.g. glass phase) which cannot be defined by any repeating structure. Perhaps it's semantics?

Looks like the term 'phase' is more or less obvious only for substances with short-ranged (crystals, liquids) or quite negligible interactions (ideal gas) between their molecules. When long-ranged interactions are present, like in ferromagnets, it is possible to have non-uniform phase (isn't it counterintuitive?) or rather a mixture of several phases? Superconductors of the first kind don't seem to folow this suggestion.

What a super discussion. I'm not very familiar with superconductivity, but I think the phase transition you're mentioning would rather resemble the type of phase transition we see in common bulk materials. If I'm correct, the breakdown of superconductivity is due to restructuring of the material in domains, which are different from magnetic domains in that the non-superconducting domains are changed fundamentally to a different phase than the phase of surrounding domains. The macroscopic result is then due to the change in behavior at the lattice level, not simply due to a canceling net field. Again, I'm not very knowledgeable about superconductors, but I don't think they're analogous.

I suggest all phases are described by matter's properties in its *ideal* case. The interesting question you raise seems to address two important aspects of how we describe a substance in the non-ideal examples of the real world.

Firstly - how closely the substance resembles our theoretical description of its perfect state (for instance a ferromagnet where every unpaired electron contributes magnetic field perfectly parallel to one another), and how the deviation alters our definition of the phase,

Secondly - how our definition of phase changes according to the scale we observe. In this consideration, long-range effects often have different properties from short-range even in an ideal phase. Consider plasma, for example - a phase which is defined by its conductivity, charge neutrality, opacity, and many other features on a centimeter scale. However, within the Debye sphere, its properties are entirely different, but in this case it defines the macroscopic condition at its boundary - if there were no microscopic charge separation, there would be no macroscopic conductivity. Non-newtonian fluids also are worth bringing into the discussion, since their long-range properties are determined by the molecular bonding.

It depends on your mean of phase. usually there are two, three phases in liquid simultaneously. If you want to reach a single phase you should to carry out some conditions. these conditions depend on nature of your material. for examplr if you want to reach a ferromagnetic order in iron. you should to apply a filed and lower the temperature of your sample and so on.

How about "the set of all microscopic states which confirm to a single equation of state"?

I think the thermodynamic phase a system can be defined as " the set of all microstates that conform to a single equation of state".

"State" is too broad, and "Equation of State" is just terminology describing the question here. I'm thinking that might end up circular logic.

What I meant was for a given phase a unique constraint relation between the thermodynamical variables of the system can be written, like in the case the ideal gas equation or van der Waals' gas. This I suppose should be possible for each phase of a system, at least in principle.

I don't think "Equation of State" is universally good idea. It is fine from the theoretical standpoint but seems less usable for experimentalists. We have (usually) no problem with sweet tea but things get obviously more complicated with ferrofluids, at least when they are examined at the macroscopic level. There is also a clear difference between nice crystal and exactly the same material in amorphous state. Well, this last example may be questioned by saying that amorphous alloys are not in thermodynamic equilibrium but rather in metastable state. So what, does it matter?

Should we treat core-shell magnetic nanoparticles in an amorphous matrix as a "phase"? Certainly not, but we are able to present some "Equations of State" for such a system.

It appears that the word phase is used in more than one sense, or rather just to mean a condition of material with some qualitative characteristics , which are useful in describing the material, such as when we talk about "fine" powder and "coarse powder". This is the case, I think, when we say some material is a "heterogeneous phase", which is obviously not a thermodynamical phase, but just a mixture of macroscopic chunks of different phases of one meterial or even different materials.

Surely, we may not be able to define this kind of use of the word phase!

But we might be able to define what do we mean by a thermodynamic phase. This I believe is applicable to only to a system in equilibrium (or at least that is how it is understood when we are talking about phase transitions). Then the phase has to be a set of thermodynamic states. Each of these states is uniquely defined by values of a minimum number of thermodynamical variables ( such as volume, temperature and density of a gas), a these values are independent of the history of the material. A phase persists over a certain ranges of these variables, giving us the phase diagram.

I don't think you will find a single definition that will satisfy all academic disciplines at the same time!

In physics, the term phase has two distinct meanings. The first is a property of waves. If we think of a wave as having peaks and valleys with a zero-crossing between them, the phase of the wave is defined as the distance between the first zero-crossing and the point in space defined as the origin. Two waves with the same frequency are "in phase" if they have the same phase and therefore line up everywhere. Waves with the same frequency but different phases are "out of phase." The term phase also refers to states of matter. For example, water can exist in liquid, solid, and gas phases. In each phase, the water molecules interact differently, and the aggregate of many molecules has distinct physical properties. Condensed matter systems can have interesting and exotic phases, such as superfluid, superconducting, and quantum critical phases. Quantum fields such as the Higgs field can also exist in different phases.

In chemistry, A phase of matter is characterized by having relatively uniform chemical and physical properties. Phases are different from states of matter. The states of matter (e.g., liquid, solid, gas) are phases, but matter can exist in different phases yet the same state of matter. For example, mixtures can exist in multiple phases, such as an oil phase and an aqueous phase.

Wikipedia define phase as "a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform."

I'm not fully convinced that the notion of "phase" is necessarily related to electromagnetism, no matter that my original question was about magnetic domains. Thermodynamics was successfully created before electromagnetic waves were discovered, even before the Maxwell equations.

I am also rather skeptic that water molecules interact differently in solid, liquid and gaseous state, at least in classical, i.e. not quantum mechanical picture. The difference will arise at extremely high temperature, when strongly vibrating water molecules fall apart - but we are no longer dealing with water then (is this a phase transition?). How to distinguish water being in the so called supercritical state from its liquid or gaseous phase as there are no borders on a phase diagram defining boundaries of of both? Maybe phase diagrams with coordinates like (p,T), (V,T), etc. are defective or simply don't deserve their name?

First, I think what Ahmed says is most relevant to the core of your question. It really is about equations of constraint. However, that definition is still complicated by what you're mentioning - equations of constraint depend on minimum variables. Can we choose variables such that certain properties are not considered? For instance, in your water example, ice is simply ice by an elementary definition of phase. But by another definition, supercooled water is quite different than slowly formed ice, and as a matter of fact there are a number of different ice structures that can form. Do we consider these different phases?

Second, I think the question of whether phase diagrams are defective is founded on questionable definitions. If I'm correct, water rising to the level of molecular breakdown is not a phase transition, but is instead classified as decomposition. But why that classification? Usually decomposition involves the cross into a more stable thermodynamic state, but water molecules - if energy leaves the system - will reversibly form again, making it closer to the definition of a phase transition to plasma. Surely, with that level of energy, plasma phase would likely accompany such breakdown. Either way, I don't think it makes our current model of phase transitions defective, but I am interested in how the current model can answer your questions.

The term phase in experimentally observable system should be related to degrees of freedom F=C-P+2 which means that in a pure substance when C=1 we may change 3-P variables until the properties of the system change abruptly and number of phases changes. Unfortunately in real system of more chemical compounds this may be used rather for determination of number of physico-chemically independent compounds. A very illustrative well known example of the later is distillation of spirits, where at azeotropic composition we have effectively only one component while having obviously two compounds. Exactly this relates to number of constraints in the system for which the equality of chemical potentials must be maintained. In real experiment namely the termy "abrupt" may be sometimes quite difficult to define which comes from the fact that chemical potentials / activities are notoriously difficult to measure.

The 'phase' of a wave is no way related to what we are discussing here. It is simply a case of same word being used to mean completely different things (in different contexts). Just like the 'phases' of moon!

Also, in my answers above I have used the word 'state' to mean a microstate, as the worfd used in Statistical Physics. Once again there is scope for confusion here: because, occasionally the word 'state' is used to mean phase, as in the common phrase "states of the matter". This I believe is common in school text books and popular science books.

The original question by marek concerns the thermodynamics phase. One characteristic of this phase (David pointed out in his first response) is uniformity. But this uniformity is over macroscopic length scales, or length scales that are large compared to the "particles" of the matter. For example, consider a colloid in liquid phase. The colloidal particles are large particles (nanometers to micrometers), made up of aphiphilic molecules and some solvent molecules. And these messy "particles" are dissolved in another solvent. Thus, if we look for uniformity over molecular length scales, we won't find it! But if we look at a portion of the solution (colloidal particles + solvent) which conatains millions of colloidal particles it will be uniform.

Nobody has mentioned yet the mysterious "order parameter", for which I am unable to find good definition, too. Some even say it may be a tensor, not just number. This "animal" is often correlated with phase boundaries where it is not constant. Order parameter and its changes are probably unrelated to Dever Norman's constraints.

@Dalibor: do you think the Gibbs phase rule is applicable to the case of magnetic domains? How?

If I remember correctly, this equation is derived from the free energy equation with only pressure, temperature, and chemical potential considered. If you want to find the influence of magnetic domains you need to include a generalized force term -(mu)H which I would think would bring it to F= C-P+3, but I haven't sat down to derive it.

According to T.B. Massalski and D.E. Laughlin (CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 3-7):

A phase is a physically distinct region of a system which has attained thermodynamic equilibrium, and which has a specific set of order parameters (n1; n2, ....) which specify its physical properties. Such parameters include composition, structure, atomic order, magnetic order, ferroelectric order, etc.

Therefore, a change of phase occurs when one or more of its order parameters changes discontinuously from one value to another or from zero to a non-zero value (or vice versa).

In physics a phase is defined by thermodynamics I would say. If the Gibbs-equation becomes unsteady at a point you will have a phase transition. Thus I think it is reasonable to define a steady part of the Gibbs energy function is a phase.

It may be noteworthy that plasma is not to be considered a "real" phase in physics because ionisation doesn't involve a phase transition.

So, according to Bogdan Pawłowski, two nearby magnetic domains have at least some chance to be considered as belonging to two different phases. Or am I going too far and they are simply two parts (subsystems) of the same ferromagnetic phase? Or, maybe, the thermodynamical definition simply fails when the surface (more precisely: interphase space, often equipped with fractal dimension) is no longer negligible so we are rather far apart from the "thermodynamic limit"?

Marek, you can look at magnetic materials with domains in two different ways. One - you can consider a single domain itself as your system. If the domain is large enough (say millions of spins), it is an instance of an ordered phase. In that case two nearby domains, are both in the same phase - they are two different "pieces" of a a material in a certain phase. In each one of the the order parameter(in this case magnetization M) will have a finite, non zero value. Thus a chunk of iron can be considered a a collection of domains in the same phase. In ordinary talk, we simply say the piece of iron is in "ferromagnetic phase", but what is meant that is it is a collection of a large number of smaller pieces, all of them in the same phase.

Now you can ask the question - what about the whole chunk of iron - what phase it is? If you consider the the piece of metal as collection of atomic spins, then the system is not spatially homogeneous, or "uniform" (because the direction/magnitude of magnetization changes from domain to domain), so it does not qualify as a single phase.

But you can look at the chunk as a collection of domains. Now a domain plays the role of an atomic spin (this analogy is only approximate - because the magnitude of the magnetization varies from domain to domain). This system is a disordered phase of little magnets( i.e. domains) all oriented randomly, a sort of paramagnet made of domains. Over the length scales large compared to domain size, this is a spatially homogeneous system in equilibrium, thus qualifies as a phase. Now if you place this chunk is a strong magnetic field, all the domains will orient in the same direction, and you will have a different, ordered phase.

(By the way, in the above, I suppose there is no "phase transition" in the usual sense of an abrupt change in the order at a sharply defined magnetic field.)

Firstly I would like to suggest the members of RG those who have the knowledge of the concerned topics should participate in that discussion. Sorry plz dont mind about this as this may give tension to the person who asks.

Dear Marek, as for my knowledge I think there may be of same phase with different orientaion.

My example of just two domains is not so exotic and artificial as one might think. In tapes made from amorphous material, roughly 50 micrometers thick and and few (3-6) millimeters wide, it is possible to create exactly two antiparallel domains, magnetized in plane, along the tape, and up to 1 meter long. What then? Is there any qualitative difference between this case and the more common situation?

Marek, in this example a tape with only two domains - it is just that: two seprate regions in the same magnetic phase. They are two different samples of the same phase, oriented differently, placed side by side.

Ahmed, so you think that the domain wall between two magnetic domains is not a phase boundary? Even if you don't like the idea of vector order parameter (magnetization), and consider only its absolute value as being meaningful, than, consequently, a domain wall has to be another phase. Do you really mean that? Hard to believe, but maybe only my imagination is limited.

Marek, as far as I understand, the term "phase boundary" means the interface between two DIFFERENT phases, like the interface between water and water vapour. Two adjacent magnetic domains are not in two different phases.

I have heard of the phrase "vector order parameter", and I am sure it is useful wherever it is used. But I do not think magnetization is an example of that. If you just flip a bar magnet you do not change its phase.

Ahmed,

Anyway, there must be some good reason for the very existence of domain walls in ferromagnetic materials and lack of similar structures between two samples of liquid water in a single glass. Besides: flipping my amorphous tape containing two magnetic domains doesn't change its phase (or phases) as well. But the domain wall is still there, separating something from something.

It is difficult to define a phase uniquely as it has different meaning in different aspects like phase for XRD analysis is not the sane for SAXS analysis beacuse for XRD the phases are crystalline and amorphuos defined by the arrangement of etities in a system, whwre as the electron density will define the phases for SAXS analysis

Marek,

The reason for the existence of domain boundaries is the stability - large domains are less stable than smaller ones, that is there is some optimal size distribution which makes the system stable. This stuff is discussed in introductory solid state physics textbooks.

phase and angle are synonymous,dsf crothers

See The New Collins Concise English Dictionary

The main feature of the phase is existence of boundary which distinct it from the another one. Certainly the boundary may have dispersional geometry

I think that a phase is a region of matter with uniform appearance and with uniform properties throughout the phase. Askeland (in the famous textbook "The Science and Engineering of Materials (3rd ed.)" defines it as "A material having the same composition, structure and properties everywhere under equilibrium conditions. A distinctive part of a microstructure."

I believe that this definition, although given in the 90's is still quite accurate and covers also electrical and magnetic properties, crystalline structure and chemistry.

Dear Rafael

I think that homogeneity of the phase properties is not crucial. For example, gas or liquid loss their homogeneity in, for example, gravitational field remaining the same phase up to moment of phase transition. In the phase transition point appears boundary border as main indication of the new phase appearing. Therefore the boundary existence is the main requirement of the phase distinction.

Phase is undefined if not specified.

@Dimo Uzunov: I really can't figure out what do you think about two nearby magnetic domains. Could you be a bit more specific? Are they two different phases or not?

@Dimo Uzunov: OK, let z-direction will be an easy direction (uniaxial case, no matter weak or strong, just say: non-negligible). What I'm asking for is the situation when you have one domain magnetized in "+z" direction and, nearby, another one magnetized in "-z " direction. Of course, there is a domain wall between them. Do those belong to the same or to two different phases? Hint: the Faraday or Kerr effect has different sign in both.

Within my own sort of every-day-working-understanding and -wording, I would consider them as belonging to the same phase.

Also, if I had another piece of that same kind of material and observe one of the two states for some piece of volume in there, I'd tend to say that this is of the same phase.

Definition of phase varies from experiment to experiment. For example for XRD we bother about crystallographic phase but in case of Small Angle X-ray Scattering (SAXS) the phase mean electron density variation.

A phase is something homogeneous on a (light-) microscopic scale (microns and a bit below, but of course not on the atomistic scales). It should be noted that often the term "phase" is used in equilibrium thermodynamics, e.g. considering Gibbs phase rule. Then one issue raised by several contributers is ruled out automatically: mixed crystals (= solid solutions) will not show concentration gradients (e.g. resulting from segregation during crystal growth), because unlimited time (= equilibrium) would level out all gradients inside one phase by diffusion. Imagine you are a tiny alien, jumping between the atoms: Then one phase is always indistinguishable for you. This holds e.g. for different grains, or even magnetic domains, of one and the same solid: From inside you see the same (-> single phase). But a eutectic is, of course, a mixture of several phases, at least 2. Again contrary, a cup of tea, if optically really clear, is single phase: you cannot see optically the sugar/caffeine/etc. molecules moving between the water molecules.

Dimo, according to your last comment, it should be quite naturally to see two neighboring magnetic domains in an uniaxial ferromagnet as belonging to two different phases. Why? They are subjected to weak (Earth) magnetic field, hence densities of their (magnetic) energy are evidently different, with domain wall between them acting as a phase boundary. Continuing, the H=0 should be considered to be a phase transition point. Here, however, we have a problem: at H=0 we still observe two domains, not one. By the way: if such two domains are in the same phase then how would you call the domain wall separating them? A defect??

@Detlef Klimm: your impressive alien jumper, after it crosses the domain wall, must see the world "upside down" but otherwise unchanged. If the alien itself was magnetized, then sudden change of the force acting on it (or, more precisely, the torque) wouldn't go unnoticed. At the same time a magnetic domain may be quite large, up to several mm^2 in thin layers, thus satisfying your requirement to be 'macroscopic'.

@Dimo, do I get this right, that what you say in the 1st paragraph is true in case that ferromagnet occupied the entire space since then the energy difference between +z and -z orientation is infinite at arbitrarily small (but finite) fields? Otherwise, at T>0, one would have to consider fluctuations, I guess. In principle, with a finite energy difference between +z and -z orientation at T>0 there is a finite probability of finding the higher energy configuration.

Good and bad are all relative in space - time scale that depends your understanding on that term how are you defining.

There is a very simple and rigorous definition of a phase of matter. It is where the free energy of the system (in the thermodynamic limit) is an analytic function of the control variables (coupling constants and external fields).

Phase transitions may thus occur at points or regions, where the free energy becomes non-analytic. These appear as singularities in the response functions, or some other related quantities.

Good definition of phase is defined only if specified for a certain property of the material system.

Dimo, I 'm a little confused by your answer. Here's how it goes for simple ferromagnets (Ising model):

H=0 is an invariant manifold, where above T_c (Curie temperature) the system is paramagnetic (disordered). At T_c there's a second-order phase transition to the ferromagnetic phase, which is two-fold degenerate (all + or all -). The system chooses one of these ground states and below T_c only one prevails (this is called ergodicity breaking). Below T_c the dominant thermal excitations above the ground state are flipped spins and domain walls. At T=0 the system has perfect long-range order.

For any finite H (no matter how small) the spins are always oriented by the field, and thus no phase transition exists. Along T=0 there's a (trivial) first-order phase-transition as a function of the field at H=0. This is called level crossing.

@Tapio Ala-Nissila: I hope your are aware that magnetic domain is by no means in thermodynamic limit since its size is finite and even quite small. I'm also surprised with your second statement that at T=0 there is perfect long-range order in ferromagnetic system. Well, in crystallographic sense - yes, but are you sure there are no domains at T=0? You seem to forgot that we are talking about uniaxial ferromagnet. The magnetization of such a substance simply cannot be parallel to arbitrarily weak magnetic field, unless this field itself is parallel to easy axis. Therefore, especially at low temperature, it is favorable to split into many domains, exactly because such an arrangement has lower energy. Even in absence of an external field one of the two neighboring domains interacts with magnetic field created by the other one. The lower the temperature the higher the energy barrier preventing the coalescence of the two. Only in very small samples, roughly 20 nm in diameter the single-domain state has lower energy than many-domain mixture.

Dimo, It is not my intent to limit the discussion to the domain wall problem. What I'm trying to get is a good definition of phase, that would be general enough to be applicable also to my problem, exactly as stated in original question. Until now I don't see a single convincing argument against my idea of ferromagnetic material (below Tc) being a many-phase mixture with domain walls as phase boundaries.

I hope you understand deeper consequences than just proper classification. Enough to say that something quite similar, namely a ferrofluid doesn't raise my doubts.

Marek, my comment is rigorously correct. I did not say anything about finite-size effects; that's a whole different discussion. I also assumed that the field is along the easy axis (by definition of the Ising model). The field direction is relevant in the Heisenberg model. Please be precise what model you're talking about.

Tapio, you are inclined to theory, including quantum mechanics, while I simply observe the world as it is. Here every single object has finite size. I'm trying to look at those magnetic domains with Stoner-Wohlfarth model in mind. In particular, I'm interested whether the magnetocaloric effect is possibly somehow related to the domain structure of an active material in magnetic refrigerator. It is not the same effect as adiabatic demagnetization of a paramagnet, I believe. If so, then not necessarily all domains will be equally responsible for the refrigeration process, and therefore the distinction between "active" and "passive" phase may be important.

Fair enough, Marek. Just keep in mind that Physics is not just about observations; it's about mathematical modeling and physical explanation of observations :).

Marek, I feel that after your very interesting theoretical discussion you do not satisfied yet regarding to practical problems arising with the phases consideration. I do not want to seek the absolutely justified definition. Instead I can propose a relevant algorithm of the phases consideration as follows.

1. Each system can be considered as born (practically or hypothetically) from some homogeneous parent one, if one provides the respective intensive parameters (conditions).

2. Returning to actual conditions, this system spontaneously subdivided to different regions which in turn surrounded by respective boundaries. These regions are named phases. The boundaries are characterized by the extensive properties jumping across it.

3. The above appeared phases may be in the state of the thermodynamic equilibrium or not. All equilibrium phases in the given system described by same chemical potentials(i.e. thermodynamic potentials as calculated for 1 mole of each molecular component). However different phases have to have distinction in their specific extensive parameters.

4. The non equilibrium phases have greater chemical potential than equilibrium ones. The more the chemical potential the less stable the non equilibrium phase from the thermodynamic point of view.

The non equilibrium phases are difficult to general consideration. Nevertheless, you can assume that different non equilibrium phases have different specific extensive parameters or their different distribution functions ( in non homogeneous case)

I personally think that the magnetic properties should not be included in this discussion, for a simple reason: historically, a phase was defined by appearance. If the appearance is homogeneous, and there are no distinguishable precipitates then we speak about a phase, whether it is crystalline, amorphous, liquid or gas. Add some sugar to warm water with some tea extract, you have a glass of sweet tea phase. If there is too much sugar it will precipitate on the bottom and you will have a two-phase mixture of sugar (precipitated as a liquid+solid) and liquid tea. You can pour the tea out of the glass and have it again as a single phase, liquid tea. Until some more sugar precipitates out and you can repeat the process...

You can now take the mushy-zone sugar+tea mixture that remained on the bottom of the glass, and dry it. This is a two-phase mixture of tea and sugar. Heat it up and water will turn into vapor and will leave dry single-phase sugar behind.

Perhaps this is simplistic, but it helps students understand the concept of phase.

Rafael, I like your simple, practical point of view. Yet, I think it is incorrect to exclude magnetic phenomena/properties from closer examination. Indeed, in early days the ferromagnets appeared homogeneous but now we know they aren't. We shouldn't ignore this well established fact. On the other hand we have something called phase transitions with many physical properties characterizing them, like the freezing point of platinum (long used for calibration of temperature measuring devices). From this point of view the sweetened water is something different than pure water - it freezes at different temperature, still remaining the "liquid phase", with the same uniform chemical composition in its every macroscopic part, and without any detectable phase boundaries. Wouldn't it be more proper to call the sweetened water a mixture rather than a phase?

Marek you have to take to account that state of the water and state of the sugar changed after their mutual dissolution. Mixture does not change the equation of state of its components. I sure that you remember that vapor pressure of water upon sugar solution is lower than upon pure vapor in their gas phase. The short order interaction (molecular range one) the thing which make difference between phase and mixture. More over, the insertion of phase term give multiplicity of useful solutions in all material science.

As I said earlier,phase in applied mathematics usually refers to angle,

as in plane polar coordinates or spherical polar coordinates,in particular

to polar angle and azimuthal angle respectively.

Of course phase also refers to liquid,gaseous and solid phases.

In other words in English 'phase' is a multi-meaning word.

Derrick Crothers

To my knowledge, phase is "a distinguishable state of a system". So "distinguishable" and "system" are very wide concepts that have to be specified in each case. In any case, you have to define what is your system and which are the properties that define it. Every system that follows the same definition and present the same characteristic is indistinguishable from each other and therefore define a phase.

For instance, when dealing with the classical states of matter, they are distinguishable from each other though properties like viscosity, elasticity, heat capacity, response to pressure, etc. Also, one might just say that they are visually distinguishable, what is much simpler. For "system", we say that it is a certain finite mass of material.

Now, think about ice: there are sixteen types of ice, from which fifteen are experimentally observed and one is still only theoretical. When you define what is the physical character that distinguish one from the other (crystal structure), then they are considered different phases of ice, even though they are not visually distinguishable.

The cup of sweetened tea you described, may be considered as a single phase when characterized by some chemical property that overlook mixture or saturation. But once you define it, you may see a phase change cooling it down as crystals of sugar will for on the bottom of the cup, due to saturation dependence on temperature. Now you may say that saturated sweetened with sugar crystals is a single phase, when characterized by some other physical/chemical property.

When talking about ferromagnetic materials, you may consider each magnetic domain as the "system" and then if something changes inside it, you may see a phase change. You may also consider a set of domains as the system, distinguishable from each other due to a given macroscopic property. In this case, it doesn't matter if a single domain changes, as long as the "system" still has the same property. An example is the transition that magnets undergo when heated above the Curie point (Tc). It is a different phase than the magnet (since it does not present a macroscopic magnetic field) but, for temperatures not so far from Tc (but above it) the magnetic domains are still there, and so if your systems are the domains, no phase change is distinguishable.

I hope it helps. This is the definition I use in class. Above everything, when teaching thermodynamics, we have to keep in mind that a state of matter may be a phase, but not all phases are states of matter.

Gustavo.

I saw some confusion about in a few answers above, but I believe that the question is about phase in the sense of physical phase (that was the premise to construct my answer), and not "angular phase" or "wave phase".

Gustavo, you are right when it comes to what "phase" I'm interested in.

At the occasion: the paper you requested (2006) is a bit outdated and pretty long but at least easy to read and follow (so I'm told). More recent and more general exposition is given in pair of papers "Breakthrough in interval ..", available both at RG and from arxiv (2009). Here I definitely improve the results presented in earlier publication: now standard deviations are very similar to those obtainable by more classical means. The presentation's versions of those two papers are here.

Gustavo, if you don't explain the concept of phase ("of matter") in terms of the analyticity of the free energy density you are misleading then students. Please take a look at https://noppa.aalto.fi/noppa/kurssi/tfy-0.3223/luennot

I apologize for the belated answer. To answer Marek, I want to provide the student with a simple explanation. The sweetened water in my example is uniform liquid phase that includes a second component according to its solubility limit at the temperature. I don't think sweet water is a mixture. When you put salt on the roads it is salt, then it is wet salt after absorbing water but it is still single-phase salt, with homogeneous appearance and a water content (again determined from the phase diagram).

Magnetic properties are of course important, but if you try to insert this property in the definition of a phase for first-year engineering (not physics or chemistry) students you would lose them...There is the Occam's razor rule here, and it says to keep things simple. The example is given for a basic understanding of what is a phase. It used to be a homogeneous appearance, first under naked eye, then under microscope, then under electron-microscope. So of course, with better visualization techniques you can expand the explanations to include properties that were not observable before. But still, a phase is something that has homogeneous appearance, chemistry and properties up to a phase boundary. The magnetic transition in iron is considered a second-order transition, not primary like the ferrite--austenite transformation.

I do agree with Rafael.

Rafael, I wish the things were so "obvious" as sweet tea or salty water. There are no evident phase boundaries after solids are dissolved in water, contrary to my example of two magnetic domains. Yet we have another case of nice freshly produced crystals. Careful X-ray investigators sometimes say: they are not a single phase objects. What do they really mean, when the unwanted phases are not inclusions?

The fact is that the concept of "phase" applies only for macroscopic systems. That is a statistical magnitude that only has a meaning for a system with a large number of particles. If we come to the interfaces or to the transition regions there is not a clear real boundary. This is the case for instance for the very simple case of liquid water and air with vapor. That is why we have to add terms like the surface tension to the total energy balance of a system to deal with.

If you are interested please have a look to our recent work:

J.L. Pérez-Díaz, M.A. Álvarez-Valenzuela, J.C. García-Prada, The effect of the partial

pressure of water vapour on the surface tension of the liquid water-air interface, Journal of Colloid and Interface

Science (2012), doi: http://dx.doi.org/10.1016/j.jcis.2012.05.034

@Marek

My example was of course intended to keep things simple, so that we can introduce first year students to the concept of phase. But there are phase boundaries in these systems: If you add too much sugar to tea, it would settle at the bottom of the glass and this is a two-phase situation. But if we add magnetic properties to any description, because they are not visible, it will complicate the definition and the understanding of the term by visual means...IN XRD one can distinguish between pure phases and phases with miscibility, because the lattice parameter is affected.

No doubt phase is a thermodynamic ally equilibrium system with homogeneity in all respects.So one has to select the term phase whenever required.

I would rather say that a phase is an analytical part of the Gibbs-energy (so that a discontinuity of the Gibbs-function in dependence of temperature of pressure in the n-th derivative characterises a phase-transition of n-th order).

I'd also like to emphasize that a plasma is, strictly thermodynamicalle speaking, NOT a phase of its own (as I read here several times before), because ionisation alone does not include a phase transition.

However, one can have phase transition WITHIN an existing plasma, which is for example described here:

M. Schlanges, M. Bonitz, A. Tschttschjan: Plasma Phase Transition in Fluid Hydrogen-Helium Mixtures, Contributions to Plasma Physics, Volume 35, Issue 2, pages 109–125, 1995

V. S. Filinov, V. E. Fortov, M. Bonitz, P. R. Levashov : Phase transition in strongly degenerate hydrogen plasma, Journal of Experimental and Theoretical Physics Letters, October 2001, Volume 74, Issue 7, pp 384-387

Rafael, following strictly your arguments, those concerning visibility, I would have to admit that ordinary air is the single phase material. While this perception is fine in many practical applications, we know it is false. Similarly, the change of lattice parameters, as determined by XRD, is an ordinary feature of the so called mixed crystals, made from two or more miscible substances. At the same time, the X-ray experts talk about the foreign phase when they see some extra peaks, corresponding to different symmetry, not merely the change in lattice parameters. Today, thanks to STM technique, we can even sometimes observe those foreign phases. Similarly, we CAN observe magnetic domains magnetized in different directions, by means of Faraday or Kerr effect, or even using X-rays - by magnetic circular dichroism (MCD).

Of course, we may treat any given sample/substance as being a single phase unless proved otherwise. Aren't magnetic domains in ferromagnetic bodies such a proof?

Johannes: your definition may, or may not, be good for theoreticians. The Gibbs energy is not a directly measured quantity, exactly like entropy. Therefore your definition seems useless for laboratory researchers.

On the other hand, your plasma example adds even more confusion, and I am grateful for mentioning this case. Not being a phase, plasma may nevertheless undergo phase transitions! Isn't it funny (or simply absurd)? Perhaps no, I can see some similarity with alloys, which can be melted. I guess that Raphael will tend to call an alloy a single phase material, while your categorical statement about plasma rises my sincere doubts.

Maybe we need more than one definition of "phase"?

@Marek: Being an experimentalist myself I would not agree that this definition holds just for theorists (in the end every good theory should somehow be measureable ;)) - but it depends on what you define as measurable - you can, for example, interpret the Gibbs-energy as enthalpy and this is measurable - you just keep the pressure of your system constant, measure the temperature and add some heat energy. At some point the temperature will stay constant over some time regardless of the heating, because you have a phase transition, say, from solid to liquid (but this is of course just one example).

But also if you google scholar you will find a vast amount of papers which are devoted to the measurement of the Gibbs energy.

My comment to plasma is not so confusing at the end (sorry, if I caused some trouble there), because "plasma" is merely a sort of "description" of a physical system (in some cases liquids and even solids may be described by the means of plasma physics). You may consider supraconductivity as an analogon: if you cool down a solid enough, it may become a superconductor, but it is also still a solid (so you have a phase transition from "normal" to superconductor within the phase "solid").

But you are perfectly right: at first glance this sounds somehow weird...

Marek, yes, if we base our definition on observations alone than air would be considered one gas phase, but you can separate its components by cooling...In gases it's difficult to define phase boundaries, but add water vapor however and it is definitely a second phase; add some fumes and they will also be distinguishable...

The precipitates in age-hardened Al-Cu alloys were not observed until the TEM was invented in 1947, so the properties of these alloys were a mystery. when the precipitates were observed the behaviour could be explained. And yes, this was a second coherent phase embedded in the alpha-Al phase. That's why I talked about the changes in definition with changes in observation techniques...

I'm glad to find an article mentioning explicitly "Reorientation phase transitions in magnetostatically coupled ultrathin multilayers", by Vitalii Zablotskii, Tatyana Polyakova, Marek Kisielewski, Alexandr Dejneka, Lubomir Jastrabik, Maria Takielak, and Andrzej Maziewski, physica status solidi (b), Volume 250, Issue 2, pages 382–386, February 2013, DOI: 10.1002/pssb.201248422, see: http://onlinelibrary.wiley.com/doi/10.1002/pssb.201248422/abstract

Oh, there's another one, in the same volume of phys.stat. sol.: "Anisotropic Curie temperature materials" by Jason N. Armstrong, Susan Z. Hua, and Harsh Deep Chopra, pages 387-395, DOI: 10.1002/pssb.201248186.

The effect has been found in monoclinic Fe7S8 (pyrrhotite) single crystal.

Phase is a region in which discontinuties in properties like surface tension or dielectric constant, occur only at the boundary.

Vishnampet vaidhyanathan

Don't forget the "interphase", the extended region where two phases in contact mix.

Good point, Christian! Well, it seems that maybe we should distinguish "chemical" phase from "physical" phase. But is this a real and well defined distinction? At first sight we have phase boundaries separating "physical" phases and possible (but by no means mandatory) interphase region in the second case. We may tend to think of phase boundaries as of purely mathematical objects, a 2D surfaces in 3D space, not necessarily simply connected, that is as of zero volume things. But in reality the volume of a phase boundary is non-zero; there are molecules contributing to surface tension effects. In other words, such regions are built from matter. While at macro scale ("in thermodynamic limit") it may be safely neglected, then in case of multilayer nanostructures we are in troubles. Pushing even further: would it be correct to speak of high-Tc superconducting (single!) crystals as of many-phase objects?! Certainly not from chemical point of view.