simple A demonstration of a metastable state
Narayanan Menon
Department of Physics and Astronomy, University of Massachusetts, Amherst, Massachusetts 01003-3720
Received 31 May 1999; accepted 4 July 1999 A system whose macroscopic properties appear to be unchanging in time may not be in a state of minimum free energy. A common example of such a metastable state is a supercooled liquid. Liquid sodium acetate is a system in which the passage of a supercooled liquid into its stable, crystalline form is readily demonstrated. 1999 American Association of Physics Teachers.
In introductory courses on thermodynamics, it is customary to finish the discussion of the concept of equilibrium with a few remarks on the hazards of metastable states. Our intuitive concept of equilibrium is based on our experience that a system subjected to a set of constraints tends to evolve to a state in which its macroscopic behavior becomes unchanging in time and independent of its history. One task of equilibrium thermodynamics is to describe this equilibrium state in terms of a small number of variables such as compositional variables and mechanical constraints . However, in almost all real-life laboratory situations, there is no true equilibrium state in which a system will remain without change. Long-lived states that appear to be equilibrium states to even the most patient observers will inevitably transform into states of lower free energy. This transformation may occur simply with the passage of time, but in many situations such a transformation has to be provoked by nudging the system down appropriate pathways. For instance, a mixture of hydrogen and oxygen gases held in a closed vessel at standard temperature and pressure will remain in the gaseous state for a long time, but will react very rapidly to form water in the presence of a spark. In this article, I describe a simple demonstration that can be used to illustrate a metastable state that appears to be an equilibrium state, but is in fact, not a state of minimum free energy. The system is an enclosed bag of hydrated sodium acetate, which is sold as a hand-warmer in many sports shops and outdoor shops, and is also available from scientific suppliers.1 The sodium acetate is initially in a liquid phase. It can remain in this phase indefinitely, even when squeezed, shaken, or subjected to variations in room temperature. A thin, metal button floats in the liquid. The button is flexed in one of two positions of opposite curvature. When pressed between the fingers, the button snaps from one position to another, launching a sound wave into the liquid. This perturbation locally nucleates a crystalline droplet, which immediately starts growing radially outward until all of the liquid has crystallized see Fig. 1 . The crystallization process is accompanied by the release of a considerable amount of heat, corresponding to the enthalpy of formation of crystalline NaCOOCH33H2O. Of course, it is this release of heat from which the product derives its utility as a hand-warmer. Because the process is exothermic, it is immediately obvious that the crystal phase is lower in free energy than the liquid. The observed stability of the liquid phase leads to a misleading conclusion regarding the nature of the equilibrium state; what we are tempted to label as the equilibrium state is merely a deeply metastable state. Supercooled liquids form a large and important class of metastable states, of which the sodium acetate system is a
1109 Am. J. Phys. 67 12 , December 1999
specific instance. Almost any liquid can be cooled to a metastable state by cooling sufficiently rapidly through the freezing point of the substance. Sodium acetate is a particularly convenient example: it may be cooled relatively slowly into the supercooled state, and the supercooled liquid is very stable on human time scales. The metastable liquid phase is relatively stable in these liquids because they become more and more viscous on cooling, and it can take extremely long times for fluctuations to create a crystalline nucleus that is large enough to be stable and grow. In extreme cases, crystallization is very difficult to initiate. For arsenic sulfide, a chalcogenide glass-former, there is no known laboratory protocol to produce the crystal. The crystal phase is known only because it exists as the mineral orpiment, prepared over geological time scales in the ovens of the earth's depths. It is straightforward to do a demonstration similar to that of the hand-warmer in which a supercooled liquid immediately freezes when brought into contact with a small seed of its own crystal. The sodium acetate system is suitable for such a demonstration, as are other more carefully studied glass-forming liquids such as orthoterphenyl.2 The picture conventional of the stability of a glass-former involves an energy landscape such as the one shown in Fig. 2 b .3 Phase space is almost entirely taken up by the liquid state, which encompasses an enormous number of minima. The crystal state is essentially a single point in this phase space, and it can take a macroscopically long time for the system to find its way to this deep minimum. We are more accustomed to thinking of metastability in terms of energy barriers such as the one sketched in Fig. 2 a . However, in the situation sketched in Fig. 2 b , there need not be a substantial barrier to nucleation.4 The central point illustrated by the sodium acetate demonstration is that seemingly immutable states can be metastable
Fig. 1. The growth of needle-like crystals of sodium acetate is shown approximately 1 s after crystallization has been initiated at the white button in the left half of the picture. The crystal nucleus grows radially and fills the bag containing the fluid within seconds. 1999 American Association of Physics Teachers 1109
all, we have no means of identifying when a truly final state is reached with the arguable exception of some instances in which microscopic calculations may be made and almost any system on which we make thermodynamic measurements may turn out to be in a metastable state. Thus, metastable states are not to be viewed as hazards to be avoided, but rather the typical situation in which we make measurements. From this point of view, the question of whether a system is in equilibrium depends on the measurement one is interested in making: if the system remains unchanged for a time that is much greater than the duration of the measurement, then for all practical purposes, the system is in equilibrium. ACKNOWLEDGMENTS My thanks to S. R. Nagel, who first showed me this demonstration and introduced me to the field of liquids and glasses.
1
Fig. 2. a An energy landscape showing two minima, a global minimum and a local, metastable minimum. A system can stay in the metastable minimum until a thermal fluctuation or some other perturbation helps it surmount the energy barrier separating it from the global minimum. The situation shown in b better typifies the situation in supercooled liquids, where there is a large entropic barrier to falling into the unique excluding permutations of particles crystalline minimum. Phase space is filled with local minima that scale exponentially in number with the number of particles.
with respect to some other more stable state. The pragmatic lesson to be drawn from the demonstration is that the machinery of thermodynamics is not to be held in abeyance until some mythical, true equilibrium state is achieved. After
Central Scientific Company, 3300 Cenco Parkway, Franklin Park, IL 60131. As described on p. 149 of the 1998 catalog, flexing the disk ``activate...