No interaction = no phase transition.

Unless you are thinking of something like buckling (http://en.wikipedia.org/wiki/Buckling) but I don't think the concept of first or second order applies to that situation. I would also suggest reading http://en.wikipedia.org/wiki/Phase_transition. Look under "Ehrenfest classification" and "modern classifications". This was based on observations.

Are you considering interactions between the pieces?

Hi Noe, I was thinking about the situation where there are interactions between different pieces, but it would also be interesting for me to know what would happen if there are no interactions. Thanks!

No interaction = no phase transition.

Unless you are thinking of something like buckling (http://en.wikipedia.org/wiki/Buckling) but I don't think the concept of first or second order applies to that situation. I would also suggest reading http://en.wikipedia.org/wiki/Phase_transition. Look under "Ehrenfest classification" and "modern classifications". This was based on observations.

"is it possible to determine the order of a phase transition simply based on experimental observation?"

There are a variety of methods of determining the coexistance of phases, though this will depend on the system you are actually looking at. I think I can give a more exact answer if i know the specific system you are looking at.

Dear Gert and Jeffrey,

Thanks for your suggestions!

I think one of my major confusions right now is: if we experimentally observed the coexistence of two phases (in this case curved films and flat ones), can we use that as a criterion to say we observed a first order phase transition?

The examples in the Wikipedia page seem to support the foregoing argument. For example, the coexistence of water and vapor in a first order transition and the inability to distinguish gas and liquid beyond the critical point, which corresponds to 2nd order phase transition. I'm wondering if this (phase coexistence) is a universal feature in all first order transitions or not.

For the buckling phenomenon, is it true that the buckling state is not a thermodynamically stable state, and thus no order can be assigned to the transition?

For an experimental system, I was thinking about the tubulation process of a piece of membrane bilayer, as happens in a lot of cellular transport processes. If we can experimentally observe the process of a plat membrane growing into tubes, most likely through the action of curvature sensing proteins. Then, is it possible to experimentally determine the order of this transition? (from flat membrane to curved membrane)

Thanks again for your help!

Zheng

Zheng Shi: I understand your explanation. But I think that you will need a microscopic model to "emulate" the physics of your problem (with interactions between the "pieces" of your system). Then, using statistical mechanics tools (theory or simulation) it will be possible to study the (possible) phase transition(s).

It is not obvious that curved films are a different phase than flat films. As Gert said, it could maybe apply to buckling. That is your scenario 1. I guess you need to understand WHY only some of them change. Is the transition driven by thermal fluctuations close to the transition regime? This is what happens in creep fracture, but it has nothing to do with phase transitions. In which case, the transition time can vary. Each object is waiting for a sufficiently large (thermal) kick. You can imagine it with red blood cells flipping to a spherical shape under, say, enzyme treatment.

There is a huge literature on shape transformations of lipid membranes - see all the work based on the Helfrich equation. I've just checked, I also have some very hard papers by Udo Seifert. E.g. Phys. Rev. A, 44, 1182, 1991. This one does discuss phase transitions in vesicles.

I hope this helps - A bit disorganized...

Dear Zheng Shi,

The order of phase transition according to the Ehrenfest scheme is somewhat tricky to apply in practice and besides, it is not so meaningful in more complex cases, see the attached paper where this is discussed.

Best regards

John

Hello Zheng. Your problem is similar to the coil-globule transition in polymers: usually polymers in solvent form a coil. But if the solvent quality changes (say the temperature drops down), and becomes unfavorable to the polymer, it will form a compact globule form to minimise contact with the solvent. This transition is a second order transition, and you can have phase coexistence between low density coil phase and high density globule phase -in which case it would be first order, just like liquid gas transition. In your case, your flat sheets would correspond to the coil form and the curved sheet to the globule form. This comparison would hold only as a thought experiment -as I understand it from your initial query. In practice you have to explain in more details to what actual physical conditions your scenario apply.

Dear Zheng,

Please find attached a paper I published last year. You may find some useful information about a transition from first-order to second-order phase transition in it.

Also have a look to references. The external field in my work is stress.

As far as I know there are many experimental ways to characterize the nature of phase transition (Hysteresis, latent heat, a detectable two-phase coexistence, heat anomalies, jump in order parameter, elastic wave propagation (detectable from acoustic emission). I have found a gradual change from first-order to second-order phase transition with grain size reduction to nanoscale in superelastic NiTi undergoing a first-order phase transition at coarse grain case.

Best

Aslan

Moreover, I do not think the change in curvature of a film due to application of stress even falls into category of phase transition so you can define its order parameter unless you have some physical phenomena inside your film responsible for shape changes. I think you cannot deal with this problem based on macroscopic observations.

Is there any structural changes inside the film responsible for shape changes? If yes, then i guess there should be a correspondence between the macroscopic response and underlying phenomena.

I have been working on this issue within the past two years. We have designed some experiments to prove it.

Hi, Zheng,

I think to determine whether it is first order or second order, you need to see whether the entropy is continuously change or not. As you explained, the first case, the entropy suddenly jumps (the pieces suddenly curved), it means the phase transition is first order, while in second case, the entropy continuously increase (C increased from 0 to C0), so it is second order.

Hope my understanding helps.

I agree with Jeffrey Brender. The best way to determine the phase transition order is through experimental findings. However, In the present case, it is very hard to have experimental data. As I understood from the Zheng Shi, he is dealing with a theoretical case. Therefore, A correct answer to this request needs a more precise thermodynamic data such as that of the Helmholtz free energy from which you can determine other properties.

First order phase transition is accompanied by a thermal effect, at this transition abruptly changed the first derivatives of the thermodynamic potentials. At phase transitions of the second order the thermal effect is absent; during such transitions abruptly changed the second derivatives of the thermodynamic potentials (thermal expansion coefficient etc.).

Dear Zheng Shi. I would like to mention, that phase transition takes place in systems, where nearly infinitely large number of microstates is possible. If your system is macroscopic, and there are not so many layers, the concept of phase transition may fail. Another point is according to suggestion of Aurelien Perera. Order of phase transition can be determined by the probability distribution of microstates. For example, you have a system, where are an initial state A, a final state C and an intermediate state B. In the first order transition you can normally observe microstates A, and C, but not B, because there is an energy barrier between A and C. This takes place in coil-to-globule transition of some globular protein. If you are heating it up, initially you have only globules, finally only coils, and at the moment of transition the mixture of these two, but not other structures. In the second order transition you will observe mixture of A, B, and C, where initially dominates A, than B, and finally C. But there should be some sharp change of heat capacity or something like this, anyway. Otherwise it is not clear is this a true phase transition.

Phase coexistence can be used as an evidence for first order phase transition, as you rightly mention. The other checks for a first order phase transition are based on (1) hysteresis in the transition temperature/pressure/any other intensive variable, (2) latent heat of transition through DSC measurements ( but you are using force! ) and (3) discontinuous change in the order parameter like curvature in your case, I guess. All three features and phase coexistence are routinely used for characterising a temperature induced first order phase transition.

Dear All!

As is pointed out in the article I attached in my earlier response there is a lot of confusion that leads to misunderstanding using the Ehrenfest scheme based on the order of a phase transition. This scheme should be avoided. It is based on the behaviour of the free energy for an equilibrium system when temperature is varied slow enough to maintain the equilibrium. For a 1-component system 1 first order transition is also a heterogeneous transition but that is not the case when there are more components. So, please do not use the terms 1:st or 2:nd order transitions when you mean heterogeneous or homogeneous transitions.

Hello Zheng, to me it looks like a stress induced transition. For case 1 you apply some external force say pressure and slowly change the curvature of the film. When you are changing the curvature there is an internal stress developed in the system. The problem is similar to the ferroelectric systems. If you take the case of PbTiO3, the ferroelectric to paraelectric transition happens at a high temperature. It is a first order transition as the spontaneous strain suddenly disappears. However if one increases pressure the transition temperature comes down and at a certain critical pressure it comes to room temperature. Since pressure reduces spontaneous strain, it slowly reduces with application of pressure. Therefore the transition becomes a second order type. Here the order parameter one can think of to be the spontaneous strain. A similar situation may apply to your thought experiment as by changing the curvature you will be changing the internal strain of the thin film. Therefore to define your transition type in your system you may need to define a proper order parameter - which may be spontaneous strain.

Dear John Agren,

I had read your paper. This is really a good paper.

I think you are right such classification should be avoided. But if a phase transition happens homogeneously it has many characteristics of second-order phase transitions. Very much correlated.

Dear Dhananjai Pandey,

We have recently developed a technique for measuring latent heat when force/stress is the deriving force. The paper is attached above.

Aslan

Thanks Aslan. Prof Manosa also has developed a calorimeter for measurements of heat of transition etc under pressure and stress for barocaloric and elastocaloric materials.

1-st pathway we can consider as a two-tier system with the transition type of Schottky.

http://www.wikiwand.com/en/Schottky_anomaly

Hi,

One way to determine experimentally the orde of a phase transition is to look at specific heat either Cv, Cp or the relevant thermodynamic variable in your case. The existence of a maximum in the Cv (Cp) curve versus temperature, say, indicates there is a first order pahse transition behind. The reason is that according to Ehrenfest classification, first order phase transitions correspond to discontinuos changes in the first derivatives of free energy. Then, a peak in the Cv indicates an abrupt change in the entropy S of the system and S, is the first derivative of F in terms of temperature.

Hope this may help.

By the way. There are phase transitions even in the absence of interactions!, the ideal quantum Bose gas transition from normal to Bose condensate state is a remarkable example ! :)

I disagree with what Luis Olivares-Quiroz writes above regarding the maximum in Cv (Cp). A maximun in the heat capacities can occur at first order (discontinuous) phase transitions, at continous phase transitions, and even without any phase transition.

Dear Zheng Shi,

First order phase transitions coincide with those where at the PT point, the order parameter presents a discontinuity. They are also accompanied by an hysteresis. For second order phase transitions the evolution of order parameter is continuous, and is reversible (no hysteresis). Besides, the order parameter transforms like an irreducible representation of the high symmetry group of the system, which determines the form of thermodynamic potential expansion.

If transitions occur versus temperature, PTs are accompanied by anomalies on entropy, leading to :

- first order PTs : latent heat at transition, that appears as a peak in a DSC experiment

- second order PTs : jump of specific heat, but if the order parameter fluctuations are strong, a sharp increase occurs at critical temperature (it is why it may be difficult to differentiate the PTs on the basis of DSC measurements).

In the case you present, it seems to me that the order parameter is unclear, because the problem definition is not enough accurate. In particular it should be clear what is the relative disposition of films with respect to others. Do you see them as molecules in crystals for instance (case a) (namely films of finite size constituting the motif, repeating in a lattice), or as an arrangement in a one-dimensional crystal (case b), or distributed randomly in space ?

In case a, a symmetry lowering occurs between 3D crystallographic groups, induced by the curvature of the motif that reduces the point symmetry, in case b you have to consider 1D groups (relatively to translation symmetry), and if you have a random distribution in three directions of space, you have to specify if all films are initially parallel or if there is also an orientational disorder. We may think also to other more exotic orderings, and in each case the situation will be different with respect to the order parameter.

From the better definition of your object, the nature of the order parameter of the PT in your system will be clarified.

Dear Zheng Shi,

I generally agree with Pierre Saint-Gregoire. Your problem is not defined precisly.

One answer to the question you raised might be that you don't have any phase transition. This might be merely elastic deformation of various crystals (thin films as you call them). As each crystal has a probability of having different orientation towards the force, and as the elastic coeffcients in any direction are not equal, you might get a picture of different bending (differnt curvature) in those ctystals. So please define your question more clearly.

There are software programs: PANDAT 8.1, Thermo Calc, COST 531, CALPHAD (Calculation of Phase Diagram) which based on database optimized thermodynamic parameters for the constituent binary systems, calculate and determine the thermodynamic parameters for ternary systems.

Dear Zheng,

I see the problem as Gerd: No interaction, no phase transition. For the interaction you could start with the Landau expansion for an order-disorder phase transition. The order parameter is the mean spontaneous curvature c00.

F=(1/2)*A*c00^2+(1/3)*B*c00^3+(1/4)*C*c00^4+...

The linear term drops out since we want to describe a phase transition. The coefficient A, B and C are temperature dependent for example.

In the next step the symmetry of the phase transition has to be considered. If the system is symmetric where +c00 equals -c00 then the B=0. In this case a second order phase transition is expected. In the other case (+c00=/=-c00) a first order phase transition. We have experiments to your problem: 6. T. J. Scheffer, H. Gruler, G. Meier: Periodic Free-Surface Disclination at the Nematic to Smectic Phase Transition. Solid State Comm. 11, 253 - 257 (1972)

Order of a phase transition depends on how the system properties change during the transition. In a first order phase transition, the properties at the transition temperature change discontinuously – e.g. freezing of water is accompanied by an abrupt change in volume (and some other properties, enthalpy, entropy); the heat capacity at the transition temperature is infinite. The first derivative of chemical potential discontinuously changes, that is the slope of the dependence of the chemical potential on temperature suddenly changes. In the second order phase transitions the first derivative of chemical potential is continuous, but its second order changes discontinuously. The order of the transition does not depend on path.

Dear Zheng,

you already received many answer. Most of them contain correct general consideration on phase transition. However, let me add a few comments in a quite different direction and ask a few questions.

Ok, there is a classification of phase transitions, thermodynamic signatures, order parameters, specific heat, ... but where is the thermodynamic system in your problem ?

Certainly there are cases such that thermodynamic language may be applied to situations where there is no obvious thermodynamic but it is important, to avoid misunderstandings, to identify very clearly, from the beginning, if you are working with an equilibrium thermodynamic system or not.

If your system is an equilibrium thermodynamic system, which variables describe its equilibrium states ?

Provided you have a reasonable macroscopic description of equilibrium states, is the resulting thermodynamic valid for your system ? (for example, is the energy a homogeneous function of its extensive variables ?

Honestly, even if I think your system made by biological membranes, It is not obvious to me that you are facing a true thermodynamic system.

In such a case, you can still use a thermodynamic-like terminology, but you need to use something different from thermodynamics or statistical mechanics.

Maybe Catastrophe Theory. In that theory you can still use terms like "first-order/second-order transitions", but with a quite different meaning.

Dear Zheng Shi,

I agree with professors John Ågren and Giorgio Pastore.

“You can still use terms like "first-order/second-order transitions", but with a quite different meaning”.

Dear Zheng Shi,

I think your basic line of reasoning is (mostly= correct. I.e. phase coexistence is characteristic for a first order transition. Also there is an interaction, because you need to apply a force to go from a flat to a curved state. Bit in my opinion this is exacty the flaw in your model: It is not clear if the force needed for a given change in in curvature depends on the curvature itself. If this is the case (and the dependence is big enough) then you would see a "first order transition" else not.

Jochen, I don't know if going from flat to curved state under influence of a force would qualify as a phase transition. For instance, if I take a rubber band and stretch it I would not consider that a phase transition between an unstretched and a stretched state either, even if the rubber has non-linear behavior.

Gert, for a single rubber band this may be true. But the flaw in reasoning here is to mix a single object behaviour with an ensemble average (equiv to a thermodynamiv quantity).

There is a very interesting example of "simultaneous inflation" of a group of rubber baloons:

http://books.google.de/books?id=5h_3MEd3QeMC&pg=PA84&lpg=PA84&dq=rubber+stretching+phase+transition&source=bl&ots=Ya-bH8kiX0&sig=AsLhSzYWBU1QqhAoHo7-Gk0x7Xo&hl=de&sa=X&ei=zAI0U92PAYSHswbZ_oDoDA&ved=0CD0Q6AEwAQ#v=onepage&q=rubber%20stretching%20phase%20transition&f=false

And the result is very close to the question IMHO.

For the more mathematically inclined people, this my be interesting

http://arxiv.org/abs/1307.3868

Luis, the Bose gas is not a gas of independent particles. Although I agree that there is no interaction of the usual type (by way of forces acting between them), don't you think that the symmetry of the wavefunction puts a restriction on the collective behavior of bosons, and that this can be viewed as a form of interaction? However, I admit that I had not thought of this example, and now I just want to save my original statement by arguing that bosons "feel" each other's presence through the symmetry requirement.

Jochen, aren't the balloons connected by tubes? See p. 72 of your link.

@Zheng For an experimental system, I was thinking about the tubulation process of a piece of membrane bilayer, as happens in a lot of cellular transport processes. If we can experimentally observe the process of a plat membrane growing into tubes, most likely through the action of curvature sensing proteins.

This is probably a good beginning for the specific problem you are looking at:

Coordination of Kinesin Motors Pulling on Fluid Membranes

1. Note that in any case tubulation does not correspond to a phase transition of any type. There are transitions in membranes in lipid membranes involving curvature changes that more closely correspond to actual phase transitions for example the gel to rippled gel phase transition (loosely speaking a buckled to flat membrane phase transition at low temperatures) and inverted hexagonal phases with some lipids (flat to cylindrical however the membrane is effectively inside out with the headgroups facing in).

2. First order phase transitions during tubulation may be possible in multiple component membrane systems due to lipid demixing by curvature induced sorting. See for example

Dynamic sorting of lipids and proteins in membrane

tubes with a moving phase boundary

PNAS 107 7208–7213, doi: 10.1073/pnas.0913997107

3. there are other mechanisms for inducing tubulation for example by particles inserting into the outer leaflet only of the membrane with a repulsive interaction between particles (or another mechanism that minimizes lipid exchange between bilayers). Tubules produced by the peptides alpha-synuclein and IAPP are an example of this.

http://www.cell.com/biophysj/fulltext/S0006-3495%2808%2970364-3

If F(x) is not continues, first order. If F'(x) is not continues and F(x) is continues second order.

Dear Zheng Shi,

1) In your question, your “two pathways” only involve different shapes but not different phases, there is only one phase that existed in different shapes.

Is there another parameter to denote that the different curved states of the thin films correspond to different phases? Maybe you had not presented your question completely and caused misunderstanding. If yes, you should revise your question.

2) If you want to describe your thin films by thermodynamic approach, the best way is to write the equation of the process, as you mentioned, I think it is a coupling process, involves two or more forces of thermodynamics, i.e., the gradients Δμ, ΔY (the 'force' you applied -the tension), etc. From the beginning to write the driving forces of the process, it seems to be easier to obtain the equation.

Please, consider the notes of Giorgio Pastore and Jhon Agren.

I agree with John Ågren and Giorgio Pastore. But there seem to be something interesting, such as “curvature induced sorting”. Maybe there is no contradiction between Zheng Shi and others.

Apply a 'force' to “the pieces of thin films” to change its curvatures, I think the process can be described by thermodynamic approach.

Dear Zheng Shi:

This is an elementary cosideration and you probably know: At a 1st order transition the order parameter ( for instance Polarization or its derivative, Dielectric constant) changes discontinously and shows thermal histeresis; at a 2nd order transition the order parameter change contunuously and there is no thermal histeresis; in between both it is a "tricritical poit transition " (characterized by a different set of critical exponents) with vanishing thermal histeresis.

For instance, solid solutions of Triglycine Selenate (1-x) - Deuterated Triglicine Selenate(x) show a continuous change from quasi tricritical ( no thermal histeresis) to 1st order, with increasing thermal histeresis for increasing (x).

Under pressure a phase transition can change character from 1st to 2nd order.

If you are sufficiently interested, you can take a look to my book "Effective Field Approach to Phase Transitios and Some applications to Ferroelectrics" 2nd ed.

( World Scientific: Singapore,2006)

I think the "thought" experiment that you have started with is not clear. Let us take an analogy. You can heat water and convert it into vapour. This is a first order phase transformation. But, the heat supplied is distributed uniformly over the body of water during the transformation. In your case, you seem to imagine that the thin films are independent and force can be applied to individual films. You are getting two different order of transformation through this assumption. Any how, this raises an interesting question. Can this happen in real systems?

Dear Zheng Shi,

The problem with the order of phase transitions is that there is a huge amount of (quite interesting) theoretical literature, but rather sparse experimental results really non-ambiguous. I often stressed this point in my lectures on material science in Orsay (France). I have no experience about the special experimental study that you report, so I can only give you some general experimental considerations. My own experience is rather connected with alloys or minerals. I think there are three non-ambiguous experimental observations that allow to determine the first, or second, order of a transition : For a second order transition,

1) there is no latent heat (the enthalpy of the transformation is zero);

2) the transformation is very spread in temperature, about several hundred of degrees (in fact, the transition "point" is rather the end of the transformation; this is appearent on the specific heat, or on the order parameter, if you can define one...) (a case in point is the order-disorder transition in beta CuZn);

3) there never coexistence of two phases (it is a continuous transformation : the whole of the phase is progressively transformed).

I think the last criterium is the easier to observe...

Michel

Dear Michel Guymont & Zeng Shi:

I agree one hundred percent with Michel comments.

Let me ask him a simple question: Are there first order (discontinuous) transitions in alloys undergoing order-disorder?

There are many such first order transitions in ferroelectrics ( for instance Ba Ti O_3) and ferroelastics ( for instance an A B O_3 a perovskite with equal valence for A and B which which I do not remember at this time)

Thermal histeresis , or coexistence of two phases , or latent heat , are good indicators all of them of first order character.

May be Professor E. Salje, University of Cambridge, could give us his opinion about 1st and 2nd order transitions.

Dear Julio,

Most of order-disorder transitions are first order. I agree that, among all phase transitions the "order-disorder" ones are those with the smallest latent heat. Because in such sort of transitions, there is not much rearragement and/or disruption of the structure (contrary, for instance, to melting, vaporisation, or sublimation). At the limit, the latent heat (at constant pressure) becomes vanishingly small : and the transformation becomes second order. I can name (after having myself experimentally studying these) two second order transitions in alloys : beta CuZn, and AlFe3.

Michel

I would not dare to say that most order-disorder transformations are first order. Actually many in the BBC structures are second order. In the classical FCC example Au-Cu they are first order with a two-phase field in the phase diagram, but from a strictly thermodynamic reasoning (looking at derivatives of Gibbs energy with respect to temperature) they become second order as I discuss in my paper. Ther are of course transitions from second to first order in some systems marked by a tri-critical point.

Thank you Michael and John for your informativeanswers.

The ferroelastic perovskite I meant was Pr Al O_3 but i was wrong it looks second order.

Globally your problem seems in connection with « thermodynamics of small systems » which is not yet sufficiently developed since the initial works of Hill and more particularly with the study of relaxation processes towards meso or nano equilibrium.

The concept of 'shape transition' seems more appropriate than 'phase transition' to refer to a body change of shape caused by applied stresses. This type of mechanical behaviour of a body is usually referred as 'buckling' if associated to structural failure of a thin-shell. Phase transition usually refers to a specific response of a substance (rather than a body) to heat transfer. For a discussion on the glass transition, which has been interpreted as resembling a second order phase transition ─ according to Ehrenfest classification ─ under idealized circumstances that can not be realized in practice, see:

Carlos A. Queiroz and Jaroslav Šesták, "Aspects of the non-crystalline state", Physics and Chemistry of Glasses - European Journal of Glass Science and Technology Part B, Vol. 51, Nº 3, 2010, 165-172.

Article Aspects of the non-crystalline state

Dear all,

It seems to me that we did not progress really on this problem because it is still not correctly expressed (see my remarks three weeks ago, and other remarks in the discussion). General properties of phase transitions are known, that is not the problem, what you have to face here is : (i) to define your system in an unambiguous way, (ii) to define your order parameter.

I agree with Pierre Saint-Gregoire.

About thought experiment and the order of a phase transition

From "Surely You're Joking, Mr. Feynman!"

by Richard P. Feynman

"The next day I gave my talk and explained all about liquid helium. At

the end, I complained that there was still something I hadn't been able to

figure out: that is, whether the transition between one phase and the other

phase of liquid helium was first-order (like when a solid melts or a liquid

boils -- the temperature is constant) or second-order (like you see

sometimes in magnetism, in which the temperature keeps changing).

Then Professor Onsager got up and said in a dour voice, "Well,

Professor Feynman is new in our field, and I think he needs to be educated.

There's something he ought to know, and we should tell him."

I thought, "Geesus! What did I do wrong?"

Onsager said, "We should tell Feynman that nobody has ever figured out

the order of any transition correctly from first principles... so the fact

that his theory does not allow him to work out the order correctly does not

mean that he hasn't understood all the other aspects of liquid helium

satisfactorily." It turned out to be a compliment, but from the way he

started out, I thought I was really going to get it!"

So, I "should tell" Mr. Zheng Shi "that nobody has ever figured out

the order of any transition correctly from" intuition or logical pathways.

Sorry, but in theory you need to have an exact solution of the problem.

Thanks Vladimir for this wonderful anecdote! I think it emphasizes what I have tried to say a couple of times in this discussion, namely that the concept of the order of a transition is neither a useful concept nor is it easy to apply to a practical case. The main reason is that the order of the transition (applying the formal definition) depends on the experimental conditions rather than the transition itself. E.g. melting of a pure substance is a first-order transition when temperature is controlled but second order if the amount of added energy is controlled. In addition it is difficult to evaluate from experimental information if a derivative changes continuously or discontiniously because what you see is a spike and the question is if it is more like a gaussian or like a dirac delta function.

Of course, I think Lars Onsager was right :

Nobody, as far as i know, has figured out the order of a phase transition "a priori".

But still I think it is a useful concept:

No matter how slowly you change temperature up and then down (or viceversa), in a first order transition ·you find some thermal histéresis.

In a second order transition you find none.

At least this is what I have found in ferroelectric transitions.

To add a comment in the same direction as Julio Gonzalo, I would say that you can never be sure of the pure second order character of a given phase transition : if you don't see a jump of the order parameter at the phase transition point, or if you don't see a thermal hysteresis, therefore if your observations are in agreement with the theoretical background for a second order PT, you cannot exclude the existence of an OP jump, or of a thermal hysteresis, that would both be too small with respect to the accuracy of your measurements. However, if you observe a jump and a thermal hysteresis, for sure your PT is of first order.

Other indications concern the thermal evolution of the OP : from data of OP versus temperature, you can guess if you are close to a tricritical point, where in the phase diagram, the phase transition changes from first order to second order, or if you are far from it.

It is important to define properly thermodynamical parameters, p,T, K-mean curvature.

If you can follow each particle of the system let You make the statistics of microscopic quantities and calculate distributions of them. Then by maximizing entropy you will derive equilibrium state equation properly describing "phase transitions" and their orders. For reading I would suggest the following paper: An Experimental Introduction to Statistical Thermodynamics, K. Sokalski, Physics and Applications (Bratislava), 11(1984)103 and there in.