Magnetocaloric Effect

The magnetocaloric effect, first discovered in iron by Emil Warburg (1881), describes the resulting change in temperature of a material due to the application of an external magnetic field. In 1926-27, Debye and Giauque each independently explained the process and described procedures to reach very low temperatures through adiabadic demagnetization. With the construction of working refrigerators in the mid 1930's, the magnetocaloric effect (MCE) became the first method of reaching temperatures below 0.3 Kelvin. [1, 2]

While MCE has provided a means of reaching ultracold temperatures for some time, recent interest has considerably expanded with the prospect of using the MCE near room temperature as an alternative to current refrigeration and air-conditioning technology. A practical MCE refrigerator for use outside the laboratory would require a high effect under the relatively low field attainable by permanent magnets, with an ordering temperature near room temperature. As such, research has focused on attaining these goals through proper choice of materials, determination of optimum ratio, and doping. In 1997, a "proof of concept" refrigerator demonstrated that a MCE device could offer 30% improvement in energy efficiency [3]. The same year, the giant magnetocaloric effect (GMCE) was discovered, with a gain of up to 200%, depending on composition of the alloy [4]. Aside from the obvious cost savings of improved efficiency, the elimination of the compression cycle in standard devices makes MCE refrigeration attractive from an environmental point of view. Hazardous gasses currently used could be replaced by simple water, eliminating CFC and ozone depleting emissions. Also, without compressors, MCE devices would run almost silently and require less maintenance.

## Process

Comparison of MCE refrigeration with conventional method. Figure taken from [2]

In class, we discussed the various contributions to the heat capacity of a system, including the lattice and electrons. To describe the MCE, we must take into account the magnetic order of a system and its contribution to the total entropy. By applying or removing a magnetic field, we can alter the magnetic entropy of a system. In an effort to conserve the total entropy of the system, the material may heat or cool in response. The cycle by which this process may be effectively used is known as the Carnot cycle.

In the conventional process, a fluid (typically a gas) is taken through phase changes that absorb or emit heat at various stages in the cycle. This is shown on the right hand side of the figure above. The simplified version is as follows. First, the fluid is compressed adiabatically, raising its temperature. The compressed fluid can be brought back to ambient temperature by cooling with a fan or other means. The now ambient temperature fluid is then allowed to expand adiabatically, in doing so cooling below the desired refrigeration temperature. When the fluid is brought into thermal contact with the area to be cooled, heat is absorbed. The fluid is then transported back to the compressor, taking heat away from the cold area, and the cycle is repeated [5].

In the MCE process, the compressor is replaced with the application of a magnetic field, as shown on the left side of the figure. The MCE material to be used as a refrigerant is initially in a paramagnetic state, with magnetic spins randomly aligned, at ambient temperature. Application of a field forces the spins to align, reducing magnetic entropy and therefore heat capacity. Since the material is insulated such that energy cannot be lost, it heats up. With the magnetic field held constant, the material is then cooled to ambient temperature. Once again insulated, the field is lowered. The magnetic dipoles respond by realigning themselves randomly, absorbing heat to do so and thus lowering the temperature below the desired refrigeration point. The material can then be used to absorb heat from the area to be cooled, and the process repeats [2].

## Entropy

The MCE is a consequence of the ability of a magnetic field to effect the magnetic order, and thereby the magnetic part of the total entropy, of a material. We can define the total entropy as the sum of the entropy due to the lattice, SL, the electrons, SE, and the magnetic order, SM. At constant pressure, this would be a function of both temperature (T) and applied magnetic field (H) [1].

(1)
$$S(T,H)=S_M(T,H)+S_L(T)+S_E(T)$$

Pecharsky et al [1] plot this function for a material under no external field (H0) and a non-zero field (H1).

Entropy vs Temperature for a MCE material in zero (S0,T0) and non-zero (S1,T1) magnetic field. The solid linge gives the total entropy, while the dotted line shows the entropy excluding the magnetic component, and the dashed line shows the magnetic entropy alone [1].

It is clear from the figure that applying a field while keeping the entropy constant (adiabatically) would raise the temperature from T0 to T1. Similarly, applying the field at constant temperature would decrease the entropy from S0 to S1. We therefore have two standards of measure for the size of the effect, $\Delta T_a_d$ and $\Delta S_M$. Note that both depend on the field strength and initial values of S0 and T0.

We can solve for these characteristics by noting the following Maxwell relations where pressure (p) is held constant
[6].

(2)
\begin{align} \left(\frac{\delta S}{\delta H}\right)_T=\left(\frac{\delta M}{\delta T}\right)_H \end{align}
(3)
\begin{align} \left(\frac{\delta S}{\delta p}\right)_{T,H}=-\left(\frac{\delta V}{\delta T}\right)_H \end{align}
(4)
\begin{align} \left(\frac{\delta S}{\delta M}\right)_T=-\left(\frac{\delta H}{\delta T}\right)_M \end{align}

We can use the total derivative of SM to solve for $\Delta S_M$ and $\Delta T_a_d$.

(5)
\begin{align} dS_M= \left(\frac{\delta S}{\delta T}\right)_H dT + \left(\frac{\delta S}{\delta H}\right)_T dH + \left(\frac{\delta S}{\delta p}\right)_{T,H} dp \end{align}

Also, the specific heat at constant pressure, C, is defined as:

(6)
\begin{align} C=T \frac{\delta S}{\delta T} \end{align}

We are assuming constant pressure, which is the case in most MCE processes. Then, dp = 0, and the last term vanishes. We can easily calculate $\Delta S_M$, since in that case T is held constant, and dT = 0 also. Then, eq. (5) is just:

(7)
\begin{align} dS = \left(\frac{\delta S}{\delta H}\right)_T dH \end{align}

Which, using eq. (2), is:

(8)
\begin{align} dS = \left(\frac{\delta M}{\delta T}\right)_H dH \end{align}

Integrating both sides,

(9)
\begin{align} \Delta S_M(T,\Delta H) = - \int_{H_0}^{H_1} \left(\frac{\delta M(T,H)}{\delta T}\right)_H dH \end{align}

We see the change in magnetic entropy is dependent on the rate of change of magnetization with temperature and the change in field.

Now, consider the adiabatic case, in which temperature changes and entropy is constant (dS = 0 ). Then, eq. (5) becomes:

(10)
\begin{align} 0 = \left(\frac{\delta S}{\delta T}\right)_H dT + \left(\frac{\delta S}{\delta H}\right)_T dH \end{align}

Making use of eq.'s (2) and (6), and rearranging:

(11)
\begin{align} \frac{C}{T} dT = - \left(\frac{\delta M}{\delta T}\right)_H dH \end{align}
(12)
\begin{align} dT = - \frac{T}{C} \left(\frac{\delta M}{\delta T}\right)_H dH \end{align}

integrating both sides…

(13)
\begin{align} \Delta T_a_d(T,\Delta H) = - \int_{H_0}^{H_1} \left(\frac{T}{C(T,H)}\right)_H \left(\frac{\delta M(T,H)}{\delta T}\right)_H dH \end{align}

Note the quantity $(\frac{\delta M}{\delta T} )$. For the case of simple paramagnets and ferromagnets, the magnetization at constant field decreases with temperature, making $\Delta S$ negative and $\Delta T$ positive. For an appreciable effect, $(\frac{\delta M}{\delta T} )$ should be as large as possible in magnitude. For ferromagnets, this occurs at TC, the Currie temperature, below which the material orders magnetically. The TC is then an important quality to consider when selecting a ferromagnetic MCE material, as it gives an indication of the working temperture range. Proper doping and slight alterations of the ratio of materials in an alloy can shift TC to a more desirable range. Furthermore, the change in heat capacity near TC also serves to increase $\Delta T_a_d$, making ferromagnets the most useful material for room temperature MCE. For more information on ferromagnets and their properties, see Jialan's page, ferromagnetic-materials.

## Giant Magnetocaloric Effect

The change in ordering descibed above for the MCE is a magnetic disorder-order transition, in that the orientation of the spins goes from essentially random to aligned with the field. This is a secon-order transition, which is to say it occurs gradually as the spins come into alignment. In a first-order transition, the change occurs suddenly, with the material transitioning discontinuously from one state to another. Such is the case of some order-order type transitions. Consequently, the heat capacity, entropy, and magnetization display a jump at the point of transition [6]. The result can be a much larger MCE, with a change in $\Delta S_M$ of up to 600% [4]. Depending on the size of the effect, these materials are said to display giant [7]or colossal [8] MCE (GMCE and CMCE, respectively).

Some examples of exceptionaly large MCE given in ref. [1] are FeRh, Gd5(SixGe1-x)4, and LaMnO3. FeRh shows a large negative MCE as is transitions from antiferromagnetic to ferromagnetic order. Gd5(SixGe1-x)4 alloys are the most popular for MCE investigations because their transition temperature and the size of the effect can be tuned by choice of x or the addition of dopants. At $0 \le x \le 0.5$, this material shows a giant effect due to the simultaneous phase transition of magnetic order and structure. LaMnO3 represents a class of perovskite-type materials. These materials are the subject of much research because the structure is closely tied to the magnetic and electronic properties of the crystal. For example, work on perovskites was reported in Ching-Chang's article ferroelectrics. A large MCE is seen when La is substiuted with Y, Ca, Li, and/or Na, and Ti is partially exchanged for Mn.

## Materials

The materials mentioned above are only a small sample of those displaying MCE properties. Although Warburg first noted the effect in ferromagnetic iron, MCE was first succesfully used in paramagnetic salts. Including perovskites, a wide range of materials display MCE. To illustrate this point, consider this chart from ref. [3].

This chart compares the advantages (+) and disadvantages (-) of promising MCE materials with Gd based materials.

Each of the categories shown above represents a range of materials. Clearly, an exhaustive list of all MCE materials would be beyond the scope of this project. Instead, I will very briefly discuss the main classes of materials as per ref. [3], a comprehensive survey of the topic.

##### Gd5(SixGe1-x)4

GMCE was discovered in 1997 in Gd5Si2Ge2. These materials are of the general form R5T4, where R is a rare-earth element and T is Si, Ge, or Sn. Besides MCE, these compunds can also exhibit large changes in their volume (magnetostriction) and/or conduction (magnetoresistance) under application of a magnetic field. All such effects are the result of the coupling between magnetic and structural orders discussed earlier. At TC, these compounds undergo both a magnetic disorder-order transition, from paramagnetic to ferromagnetic order, and a crystallographic order-order transition from monoclinic to orthorhombic structure. The exact details of the transition and MCE are highly dependedt on choice of x, and can deviate from the usual sequence outlined above. This results in the largest $\Delta S_M$ near 300K, making these materials the most promising to date for room temperature refrigeration applications. The dependence on x is also useful in selecting the most desirable properties, as is substitution of Gd by another rare-earth. Currently, these are by far the most studied class of MCE materials.

##### Mn-based compounds

Mn based compounds, like those based on MnAs or MnFeP0.45As0.55 can also exhibit large MCE. Some, like Mn3GaC, show strong negative MCE. In the negative MCE, the application of a magnetic field increases the entropy. In some cases, magnetic and structural transitions occur at slightly seperated temperatures. The MCE properties of MnAs1-xSbx are very promising for room temperature refrigeration, but the material is difficult to prepare in large quantities, not to mention As is a regulated poison. Of course, a large MCE is not enough to be successful in applications. Some of the practical considerations for refrigeration technology will be discussed below.

##### La(Fe13-xMx)

The MCE in these materials is modest compared to those listed above, but the magnetic transitions are sharp and can occur at relatively low field changes of ~20kOe. They show a first order magnetic transition and, while no change in crystalographic order occurs, a large (~1%) change in volume is seen at TC. While many of these compounds have a large $\Delta S_M$, they do not necessarily have large $\Delta T_a_d$. In fact, for TC > 220K, the Gd compounds have $\Delta T_a_d$ values 25-50% larger for the same field.

##### Manganites

The rare-earth manganites have stimulated much interest since their structure is so closely tied to their magnetic and electric properties. Manganites, usually of perovskite form, exhibit large magnetoresistance, charge ordering, and ferroelectric effects in addition to MCE. Their $\Delta S_M$ values are comparable to those of the Gd compounds, though usually smaller. In the context of magnetic refrigeration, the manganites are difficult to work with. They have complicated phase diagrams, and the high dependence on even slight changes in composition make measurements of their properties difficult. Because of this, measurements of the same material by different groups often vary, making comparison of behavior challenging. For the same reasons, consistent fabrication of materials is not straightforward. Moreover, preparation of the compounds is an elaborate process.

## Refrigeration

In the above description of some representative MCE materials, I have tried to point out a few of the difficulties in selecting a substance for workable refrigeration applications. A large MCE is necessary, but is not itself enough to ensure a functioning or plausable device. Beyond the size of the effect and the ordering temperature, magnetic hysterisis, thermal conductivity, the ability to work the material into useful wires or shapes, toxicity, and cost must all be considered. The construction of a successfull refrigeration system is in fact its own area of research, and is of course quite complicated. Here, I will give only a brief outline of the application of MCE to the actual construction of refrigerators.

Practicallity is the first major concern. Materials must be cost effective and operate under small magnetic fields, preferably from permanent magnets. A material requiring a 7 tesla change in field would be hard to accommodate, not to mention the liquid helium used to cool the superconducting magnets would more than meet any refrigeration needs. Many of the MCE materials are composed of poisonous compounds that would require special handeling, and any candidate must be easily prepared in large quantities. To enable the best interaction with the fluid, the material should also be easily formed into spheres, wires, foils, plates, or porous mediums. The ability of the material to easily tranfer its heat to the fluid and hysterisis are further concerns. To date, the Gd compounds are best suited to applications.

In an oversimplified model, one can imagine a wheel to which the MCE material is attached. As that wheel rotates, it could take the material through the approriate fields and insulated regions, transfering heat from one region to another. In practice, the system would be much more complicated. For example, it may be advantageous to move the magnets rather than the material. Also, the material itself would not be the ideal heat transmitting medium. Instead, a fluid could echange heat with the material. Since the fluid is not required to undergo a phase change itself, this could be a simple alchohol/water mixture.

Any real refrigerator would require the following main parts: a magnetic refrigerant (the MCE material), a method to apply magnetic field, hot and cold heat exchangers, and a heat transfer fluid (with a way to control its flow) [6]. The fluid would transfer heat to and from the material, such that heat is absorbed at the cold exchange and discharged at the hot exchange.

Reference [3] details a number of sucessful devices working near room temperature constructed since 1997. Their cooling powers range from 8.8 to 600 Watts, in field changed from 7.6 to 50 kOe. Research at Ames Laboratory by Pecharsky and Gschneidner, leaders in the field of MCE, have constructed a room-temperature, permanent-magnet refrigerator using Gd-type powder and rare-earth magnets. Their device uses a rotary design, and overcomes two main hurdles. The researchers designed an array of permanet magnets which doubles the attainable field. Furthermore, they have developed a process to produce large amounts of high purity Gd5(Si2Ge2) from inexpensive commercial-grade Gd [9]. Some examples of other devices are shown below.

Examples of rotary designs taken from [3]. (a) A 14 kOe field is produced by permanent magnet, and the material itself is moved through the various stages. The refrigerator operates near room temperature with a 20°C temperature change and a cooling power of 95W. (b) A 7.6 kOe permanent magnet rotates between for MCE beds, stoping to allow fluid flow, with a cooling power of 40W.

A regenerator is a thermal device which assists in heath transfer between parts of the refrigeration cycle [6]. In fact, MCE materials are often used as regenerators to increase the efficiency and temperature span of standard gas refrigerators by absorbing heat from the gas at high temperature and pressure and returning heat at low temperature and pressure. In active magnetic regenerator (AMR) refrigerators, the MCE material works as both refrigerant and regenerator. It is this type of refrigerator that is most successful near room temperature. The process is outlined below.

Diagram of the active magnetic regenerator cycle, taken from ref. [1]. (a) Magnetization, (b)Flow from cold to hot, (c) demagnetization, (d) flow from hot to cold.

In the diagram, a MCE material is in contact with a cold sink (left side) and hot sink (right side), bridging the gap between them. The material bed is typically a bed of spheres or some other porous form allowing fluid to flow between the particles. Because the bed is in contact with both temperature extremes, a temperature gradient exists across the bed, shown at the chart in the figure.

In part (a), a magnetic field is applied. As a result, particles in the bed heat up, taking the temperature profile from the dotted line to the solid. Notice the temperature of the right hand side is now higher than that of the heat sink. Next (b), fluid flows from the cold end to the hot. The bed is cooled by the flow, going from the dashed line to the solid, and the fluid is warmed by the bed, emerging at a higher temperature than the heat sink. Thus heat is removed from the fluid at the hot end. (c) The flow is stopped, and the magnetic field removed. The bed cools in response, from the dashed to solid line. Fluid then flows (d) from the hot to cold end, heating the bed and cooling the fluid. The fluid emerges at a lower temperature than the cold sink and removes heat as it passes through a cold heat exchanger.

The AMR cycle has several advantages. Each particle of the bed contributes to the change in temperature profile. As such the temperature span can of each stage can exceed that of the MCE material. Heat transfer is optimized since the bed acts as its own regenerator as solid particles interact directly with the fluid. Finally, individual particles do not see the entire temperature span. The bed may therefore be made in layers, with each layer composed of a MCE material tuned to the particular temperature range encountered [1].

## Conclusion

Research continues into the magnetocaloric effect and its applications. Working room temperature refrigerators have been constructed and have been shown to work with considerable gains in efficiency. Traditional vapor technology has undergone more than 100 years of development. (In fact, Einstien held an early refrigeration patent.) In order to compete, much work remains to be done in streamlining the MCE process and bringing the technology to mass market in a cost effiective way. With the possible savings in expense, environmental impact, and preformance, MCE refrigeration offers promising future applications.

Bibliography
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9. Magnetic refrigerator successfully tested http://www.eurekalert.org/features/doe/2001-11/dl-mrs062802.php