Residual entropy
Residual entropy is small amount of entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

 which is present even after a substance is cooled arbitrarily close to absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

. It occurs if a material can exist in many different microscopic states when cooled to absolute zero. This can occur if it has many different ground states with exactly the same zero-point energy
Zero-point energy
Zero-point energy is the lowest possible energy that a quantum mechanical physical system may have; it is the energy of its ground state. All quantum mechanical systems undergo fluctuations even in their ground state and have an associated zero-point energy, a consequence of their wave-like nature...

; it can also occur if the "ground states" have slightly different energies, but the system is prevented from finding and settling into the true lowest-energy state. It is seen most often in substances which have very weak tendencies to align into their energetic ground state, and/or which are cooled quickly.

A common example is the case of carbon monoxide
Carbon monoxide
Carbon monoxide , also called carbonous oxide, is a colorless, odorless, and tasteless gas that is slightly lighter than air. It is highly toxic to humans and animals in higher quantities, although it is also produced in normal animal metabolism in low quantities, and is thought to have some normal...

, which has a very small dipole moment. As the carbon monoxide crystal is cooled to absolute zero, few of the carbon monoxide molecules have enough time to align themselves into a perfect crystal
Perfect crystal
Crystalline materials are made up of solid regions of ordered matter . These regions are known as crystals. A perfect crystal is one that contains no point, linear, or planar imperfections...

, (with all of the carbon monoxide molecules oriented in the same direction). Because of this, the crystal is locked into a state with different corresponding microstates
Microstate (statistical mechanics)
In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...

, giving a residual entropy of , rather than zero.

Another example is any amorphous solid (glass
Glass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is soda-lime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...

). These have residual entropy, because the atom-by-atom microscopic structure can be arranged in a huge number of different ways across a macroscopic system.


One of the first examples of residual entropy was pointed out by Pauling
Linus Pauling
Linus Carl Pauling was an American chemist, biochemist, peace activist, author, and educator. He was one of the most influential chemists in history and ranks among the most important scientists of the 20th century...

 to describe water ice
Ice Ih
thumb|Photograph showing details of an ice cube under magnification. Ice Ih is the form of ice commonly seen on earth.Ice Ih is the hexagonal crystal form of ordinary ice, or frozen water. Virtually all ice in the biosphere is ice Ih, with the exception only of a small amount of ice Ic which is...

. In water, each oxygen atom is bonded to two hydrogen atoms. However, when water freezes it forms a tetragonal structure where each oxygen atom has four hydrogen neighbors (due to neighboring water molecules). The hydrogen atoms sitting between the oxygen atoms have some degree of freedom as long as each oxygen atom has two hydrogen atoms that are 'nearby', thus forming the traditional H2O water molecule. However, it turns out that for a large number of water molecules in this configuration, the hydrogen atoms have a large number of possible configurations that meet the 2-in 2-out rule (each oxygen atom must have two 'near' (or 'in') hydrogen atoms, and two far (or 'out') hydrogen atoms). This freedom exists down to absolute zero, which was previously seen as an absolute one-of-a-kind configuration. The existence of these multiple configurations that meet the rules of absolute zero amounts to randomness, or in other words, entropy. Thus systems that can take multiple configurations at or near absolute zero are said to have residual entropy.

Although water ice was the first material for which residual entropy was proposed, it is generally very difficult to prepare pure defect-free crystals of water ice for studying. A great deal of research has thus been undertaken into finding other systems that exhibit residual entropy. Geometrically frustrated
Geometrical frustration
In condensed matter physics, the term geometrical frustration means a phenomenon in which the geometrical properties of the crystal lattice or the presence of conflicting atomic forces forbid simultaneous minimization of the interaction energies acting at a given site.This may lead to highly...

 systems in particular often exhibit residual entropy. An important example is spin ice
Spin ice
A spin ice is a substance that is similar to water ice in that it can never be completely frozen. This is because it does not have a single minimal-energy state. A spin ice has "spin" degrees of freedom , with frustrated interactions which prevent it freezing...

, which is a geometrically frustrated magnetic material where the magnetic moments of the magnetic atoms have Ising-like magnetic spins and lie on the corners of network of corner-sharing tetrahedra. This material is thus analogous to water ice, with the exception that the spins on the corners of the tetrahedra can point into or out of the tetrahedra, thereby producing the same 2-in, 2-out rule as in water ice, and therefore the same residual entropy. One of the interesting properties of geometrically frustrated magnetic materials such as spin ice is that the level of residual entropy can be controlled by the application of an external magnetic field. This property can be used to create one-shot refrigeration systems.

See also

  • Proton disorder in ice
  • Geometrical frustration
    Geometrical frustration
    In condensed matter physics, the term geometrical frustration means a phenomenon in which the geometrical properties of the crystal lattice or the presence of conflicting atomic forces forbid simultaneous minimization of the interaction energies acting at a given site.This may lead to highly...

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