# Phases and Phase Diagrams

## Introduction

In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties. A straightforward way to describe phase is "a state of matter which is chemically uniform, physically distinct, and (often) mechanically separable." Ice cubes floating on water are a clear example of two phases of water at equilibrium. In general, two different states of a system are in different phases if there is an abrupt change in their physical properties while transforming from one state to the other. Conversely, two states are in the same phase if they can be transformed into one another without any abrupt changes. There are, however, exceptions to this statement, such as the liquid-gas critical point. Moreover, a phase diagram is a type of graph used to show the equilibrium conditions between the thermodynamically-distinct phases. Common components of a phase diagram are lines of equilibrium or phase boundaries, which refer to the lines that demarcate where phase transitions occur. A triple point is, in a pressure-temperature phase diagram, the unique intersection of the lines of equilibrium between three states of matter, usually solid, liquid, and gas. The solidus is the temperature below which the substance is stable in the solid state. The liquidus is the temperature above which the substance is stable in a liquid state. There may be a gap between the solidus and liquidus; within the gap, the substance consists of a mixture of crystals and liquid. Phase diagrams are useful for material engineering and material applications. With their aid, scientists and engineers understand the behavior of a system which may contain more than one component.

## Definitions

Concept Definition
Bimodal curves In the phase diagram, the compositions which are in equilibrium with each other as a function of temperature.
Coexisting compositions Different compositions in phase diagram that are at local energy potentials and have the same free energy.
Equilibrium state Reversible
Glass Transition Below a transition temperature, individual polymers lose mobility and form a solid lacking a crystalline structure.
Gradient energy coefficient Has the dimension of length squared. The length in question must be related to the range of intermolecular forces involved.

where k is the gradient energy coefficient (Jones p. 34)

Helmholtz free energy $F=U-TS$, also commonly denoted $A$. The Helmholtz free energy is a thermodynamic potential relating to the amount of useful work one may extract from a system that is isothermic (no temperature change) and isochoric (no volume change). We say "useful" work to mean work that can be extracted from the system, and not work done to change the system itself. An equivalent result is that the Helmholtz energy is minimized at equilibrium for an isothermic and isochoric system.

If a reversible process is performed on a system, the change in free energy is $\delta F=-P \delta V-S\delta T$.

Mean field approximation The interaction between molecules, even of different species is the same. The volumes of the molecules are the same.
Non-equilibrium state History dependent
Nucleation A large enough fluctuation of composition in the metastable regime to produce a region of lower chemical potential.
Nucleation – Homogeneous vs heterogeneous Nucleation normally occurs at nucleation sites on surfaces containing the liquid or vapor. Suspended particles or minute bubbles also provide nucleation sites. This is called heterogeneous nucleation. Nucleation without preferential nucleation sites is homogeneous nucleation. Homogeneous nucleation occurs spontaneously and randomly, but it requires superheating or supercooling of the medium. [[1]]
Order parameter Zero for disordered phase
Phase In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (density, crystal structure, index of refraction, etc.).[[2]]
Phase change In thermodynamics, a phase transition is the transformation of a thermodynamic system from one phase to another.[[3]]
Phase diagram A phase diagram In physical chemistry, mineralogy, and materials science is a type of graph used to show the equilibrium conditions between the thermodynamically-distinct phases [[4]]
Phase transition Phase change to non-equilibrium state
Phase transition, first order Order parameter changes discontinuously, e.g. melting of a crystal
Phase transition, second order Order parameter changes continuously, e.g. liquid-to-gas at critical point
Regular solution model A mean field theory to predict the free energy of mixing.\: The interaction between molecules, even of different species is the same. The volumes of the molecules are the same.
Self-assembly Spontaneous arrangement
Spinodal In the phase diagram, the locus of compositions where the second derive in the free energy is zero.
Spinodal decomposition Fluctuations in the unstable portion of the phase diagram grow continuously, and the
c d
c d

## Active "phase transitions" in biology

### Camouflage in cephalopods

Cephalopods, i.e. squids and octopuses, are amazing in their ability to hide from predators by practically vanishing from sight. They are able to control their appearance by a fancy mechanism in their dual layer skins. The active contraction and control of components in their skin layers can be considered as some form of phase transition, in which the optical properties of the tissue change drastically within fractions of a second.

Biologists at the Marine Biology Labs in Woodshole have been studying such amazing phenomena for a while now (Maethger and Hanlon, 2006). What they found is that:

1) one layer of the skin is formed of stacked plates of protein, which can be spaced at will. By adjusting the spacing to a value close to the wavelength of light, squids can produce bright bursts of light. They give their skin the property of irridescence.

2) this light, in turn, passes through the second layer of the skin, which contains sacks (the chromatophores) that can be contracted at will. By changing the properties of this layer, squids can change not only the brightness, but also the color of the light that passes through.

But the amazement does not stop here, as has been recently discovered ([5]). Camouflage is the ability to avoid being seen (by your predators). But how do you communicate visual signals to your friends under water while not being seen by your enemies? The answer, for squids, lies in light polarization. Their eyes are sensitive to light polarization, unlike ours. Moreover, the iridophores - the components of the first layer of skin - also cause the outgoing light to be polarized at will. So for example, if a squid decides to look blue to blend with the surrounding water color, it will be difficult for us to see it; but in the mean time, if it changes the polarization of the light bouncing off of it cyclically, then to other squids, it could look like a squid alternatively looking dark and then light and then dark and then light. All of this under our very eyes, but without us noticing anything!

You can find more information on this at the given references, most of them studies carried out by Maethger and Hanlon at the Marine Biological Labs at Woodshole; our Teaching Fellow Margaret Gardel spent an intensive summer of fun and work there, in case you're curious to know more about the place.

### Blood clotting

The amazing mechanical process of blood clot formation, wherein a viscous fluid very quickly arrests its flow and solidifies through the collaborative action of components such as platelets and fibrin, occurs in the following way:
The big picture description of blood clotting is that when one gets a cut (ie. there is damage to a blood vessel), blood platelets (which are cell fragments produced from megaaryocytes) are activated such that they adhere to the edges of the cut vessel. Chemicals are released to attract more platelets, which causes blood coagulation. Then, small molecules (clotting factors) cause fibrin to stick together and seal the inside of the wound. The end result is a blood clot protecting the cut vessel. Once the vessel heals, the blood clot dissolves.

More detail.
The thin, single layer of cells that line a blood vessel is called the endothelium. Capillaries are comprised of only endothelium. Under normal conditions, clots do not form in blood vessels, and there are a number of factors that ensure this is true. Some of these factors are: prostacyclin (which is synthesized by intact endothelium and prevents platelet activation. Nitric oxide is also released under these conditions, which keeps the blood vessels dilated); thrombomodulin (which binds thrombin and protein C, which activates protein C, which in turn inactivates important clotting factors in the blood); and others.

However, when there is a break in the endothelium, platelets are allowed to contact collagen and the other factors activate platelets. Platelets adhere to one another and to subendothelial tissue via fibrinogen and VWF receptors. This coagulation of the blood occurs more rapidly since tissue factor is now exposed, and the surface of activated platelets provides the environment for the activation of the cascade that ultimately converts prothrombin to thrombin. By now, the developing clots consist of interlaced fibrin fibrils and activated platelets. This general clotting process also has several positive feedback loops that quickly magnify a tiny initial event to stop bleeding. Once sufficient healing has occurred, clots are dissolved. Plasma containing plasminogen binds to the fibrin molecules, and then nearby healthy cells release issue plasminogen activator, which also binds to fibrin, and activates plasmin. Plasmin digests firbin, thus dissolving the clot.

Hemophelia A, B and C are conditions where there is a lack of clotting factor that leads to uncontrolled bleeding.

## Interesting Tidbits

### How caffeine is removed from coffee

Source: Molecular Driving Forces: Statistical Thermodynamics in Chemistry & Biology by Ken A. Dill and Sarina Bromberg

The figure above shows how caffeine is removed from coffee. Coffee is mixed with carbon dioxide at a temperature and pressure above the critical point of the carbon dioxide. The supercritical carbon dioxide dissolves the caffeine (small black dots in the diagram), decaffeinating the coffee beans. The fluid mixture of carbon dioxide with caffeine then flows into a chamber where the pressure is lowered below the critical point so caffeine partitions into water. The carrier carbon dioxide is recaptured and the caffeine is dumped in the aqueous phase.

• Cates
• Chapter 7. Equilibrium and flow properties of surfactants in solution
• "A classical binary phase diagram (surfactant + water) exhibits a succession of isotropic, liquid crystalline, and crystalline phases as a function of temperature and composition." "Besides complicated phases at very low water concentration (which correspond to hydrated solids), the phase diagram contains three main phases: the lamellar phase which is a periodic stack of fluid membranes made of the surfactant molecules (lyotropic liquid crystals), the hexagonal phase made of infinite tubes placed on a two-dimensional triangular lattice (lyotropic liquid crystal) and the isotropic liquid phase made of spherical micelles. (Note that liquid crystal phases are called 'lyotropic' when controlled by concentration as well as temperature.)" There are also more complicated phases such as the cubic phases, the microemulsion phases, the sponge phase, and the phase of giant micelles. p 185
• Chapter 11. Aging in glassy systems: new experiments, simple models, and open question
• Chapter 12. Phase separation and aggregation in colloidal suspensions
• Davis
• Chapter 6. Phase Equilibria
• Chapter 14. Ternary/amphiphilic systems.
• Evans
• Chapter 10. Phase Equilibria, phase diagrams, and their application
• Koningsveld
• Part 3. Heterogeneous equilibrium in macromolecular systems
• Chapter 8. Introduction
• Chapter 9. Partial miscibility of strictly binary polymer systems
• Chapter 15. Miscellaneous phase transitions
• Laughlin;
• Chapter 1. Introduction
• "Like salt, surfactants and water form liquid, crystal, and gas phases. Unlike salt, surfactants and water may also interact to form other kinds of phases that are neither liquids nor crystals. These are usually termed 'liquid crystal phases', or alternatively 'mesomorophic phases' or 'mesophases'. Some may be appropriately described as 'plastic crystals'. The gas phase may coexist in equilibrium with an aqueous surfactant mixture, but since surfactants are typically nonvolatile this phase is ordinarily pure water." p. 4-5
• All the phase states found in other systems (gas liquid, crystal) also exist in surfactant systems. All the phase transformations encountered in other systems (melting, azeotropic, eutectic, peritectic, and polytectic phenomena,) are also encountered in surfactant systems. No phase transformation has been found in surfactant systems that is unique to these systems. Thus, no unique phase chemistry will be encountered in surfactant systems -- only unique phase structures." p. 5
• A quote by Sienko and Plane on the philosophy of research: "The acquisition of knowledge is simplified by seeking the principles that underly the field of study. In chemistry, we can consider the principles as being composed of the generalizing statements that summarize observed phenomena and of the theories, or explanations, proposed to account for the observations. Such principles of chemistry serve both as a convenient framework for remembering a large body of information and also as a firm basis from which to make predictions about the unknown." p. 10
• Chapter 5. The characteristic features of surfactant phase behavior
• Chapter 8. The structures and properties of surfactant phases