thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...

, the limit of local stability with respect to small fluctuations is clearly defined by the condition that the

second derivative In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...

Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature an ...

is zero. :

0

The

locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...

of these points (the inflection point within a G-x or G-c curve,

as a function of

composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...

) is known as the spinodal curve.G. Astarita: ''Thermodynamics: An Advanced Textbook for Chemical Engineers'' (Springer 1990), chaps 4, 8, 9, 12.Sandler S. I., ''Chemical and Engineering Thermodynamics''. 1999 John Wiley & Sons, Inc., p 571.Koningsveld K., Stockmayer W. H., Nies, E., ''Polymer Phase Diagrams: A Textbook''. 2001 Oxford, p 12. For compositions within this curve, infinitesimally small fluctuations in composition and density will lead to phase separation via

spinodal decomposition Spinodal decomposition is a mechanism by which a single thermodynamic phase spontaneously separates into two phases (without nucleation). Decomposition occurs when there is no thermodynamic barrier to phase separation. As a result, phase separation ...

. Outside of the curve, the solution will be at least metastable with respect to fluctuations. In other words, outside the spinodal curve some careful process may obtain a single phase system. Inside it, only processes far from

thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...

, such as

physical vapor deposition Physical vapor deposition (PVD), sometimes called physical vapor transport (PVT), describes a variety of vacuum deposition methods which can be used to produce thin films and coatings on substrates including metals, ceramics, glass, and polym ...

, will enable one to prepare single phase compositions. The local points of coexisting compositions, defined by the common tangent construction, are known as a binodal coexistence curve, which denotes the minimum-energy equilibrium state of the system. Increasing temperature results in a decreasing difference between mixing entropy and mixing enthalpy, and thus the coexisting compositions come closer. The binodal curve forms the basis for the

miscibility gap A miscibility gap is a region in a phase diagram for a mixture of components where the mixture exists as two or more phases – any region of composition of mixtures where the constituents are not completely miscible. The IUPAC Gold Book defines ...

in a phase diagram. The free energy of mixing changes with temperature and concentration, and the binodal and spinodal meet at the critical or consolute temperature and composition.

Criterion

For binary solutions, the thermodynamic criterion which defines the spinodal curve is that the second derivative of free energy with respect to density or some composition variable is zero.Sandler S. I., ''Chemical and Engineering Thermodynamics''. 1999 John Wiley & Sons, Inc., p 557.

Critical point

Extrema of the spinodal in a temperature vs composition plot coincide with those of the binodal curve, and are known as critical points.Koningsveld K., Stockmayer W. H., Nies, E., ''Polymer Phase Diagrams: A Textbook''. 2001 Oxford, pp 46-47.

Isothermal liquid-liquid equilibria

In the case of ternary isothermal liquid-liquid equilibria, the spinodal curve (obtained from the Hessian matrix) and the corresponding critical point can be used to help the experimental data correlation process.

References

{{states of matter Thermodynamics