Preston Bohm
cryogenics thermodynamics

Dilution Refrigeration: Cooling with the Entropy of Mixing

Overly simplified dilution refrigeration cycle
pump 300 K, ³He gas Q out, compression at 300 K condenser ~1.3 K Q out to 1 K stage still 0.7 K ³He evaporates preferentially counterflow heat exchangers warm return precooled by cold dilute stream mixing chamber 10 mK concentrated phase, nearly pure ³He dilute phase, 6.6% ³He in superfluid ⁴He crossing absorbs Q = TΔS Q in, heat from the experiment temperature of the circulating ³He
³He atom, colored by its temperature (scale below schematic) superfluid ⁴He background
Follow the ³He stream: It leaves the pump as warm gas, condenses, is precooled in the still and the exchangers, and enters the concentrated phase of the mixing chamber. When it crosses the phase boundary into the dilute phase it gains entropy, and the heat for that comes from whatever the chamber is bolted to (the Q in arrow). Osmotic pressure then drives it up the dilute channel to the still, where it evaporates preferentially, and the pump recompresses it, ejecting the entropy at room temperature (the Q out arrow). The atom's color tracks its temperature on the logarithmic scale at the bottom of the schematic; the scale ends at 100 K, so the room temperature stretch near the pump simply shows the maximum color.

Entropy of mixing enables cooling

Dilution refrigeration works through the phase-separation behavior of helium-3 and helium-4. Below about 0.87 K the liquid separates into a concentrated ³He-rich phase and a dilute phase where ³He stays dissolved in superfluid ⁴He. Forcing ³He atoms to cross from the concentrated into the dilute phase moves them into a higher-entropy state, and the heat for that entropy increase is drawn from the mixing chamber, cooling whatever is attached to it. This makes dilution refrigeration the workhorse continuous cooling technology below about 50 mK.

Why evaporation runs out

The classic way to cool a liquid is to pump on it: evaporation removes the most energetic atoms and the latent heat comes out of the bath. The catch is the vapor pressure. For liquid ³He the number of atoms available to evaporate falls off as

\[ p \;\propto\; e^{-L/k_B T} \]

where \( L \) is the latent heat per atom, with \( L/k_B \approx 2.5\ \mathrm{K} \) for ³He. Below about 300 mK the exponential wins: the vapor pressure becomes tiny, the pump has almost nothing left to remove, and the cooling power collapses. A pumped ³He pot stalls near 250–300 mK no matter how large the pump. To go further you need an "evaporation-like" process whose carrier population does not vanish as \( T \to 0 \). That is what the helium-3/helium-4 mixture provides.

The helium phase diagram

Figure 1. Phase diagram of liquid ³He and ⁴He mixtures at saturated vapor pressure. The lambda line separates normal and superfluid mixtures and ends at the tricritical point near 0.87 K. Below it the liquid separates into two phases, and the dilute branch of the coexistence curve does not reach zero concentration: even at absolute zero, 6.6 percent of ³He remains dissolved in superfluid ⁴He.

Cool a mixture of the two stable helium isotopes below about 0.87 K and it spontaneously separates, like oil and water, into two liquid phases. The separation is not chemical: both atoms are helium. ⁴He has an even number of particles and is a boson, so at low temperature it can condense into a superfluid with many atoms in the same ground state. ³He has an odd number and is a fermion, so identical ³He atoms cannot share a state; they fill momentum states and form a Fermi liquid with a finite Fermi pressure. Because the isotopes differ in quantum statistics, mass, and zero-point energy, the mixture lowers its free energy by separating into a lighter concentrated phase, essentially pure ³He, floating on a dilute phase of mostly superfluid ⁴He with some ³He dissolved in it.

The remarkable feature, discovered in the 1960s [3], sits at the bottom left of Figure 1: as temperature goes to zero, the dilute branch of the coexistence curve does not collapse to zero concentration but saturates at a ³He fraction of about 6.6%.

\[ x_{d}(T \to 0) \;\approx\; 6.6\% \]

This finite solubility at absolute zero is what makes dilution refrigeration to such low temperatures possible. A ³He atom is lighter than a ⁴He atom, so it has more zero-point motion and binds more strongly to dense liquid ⁴He than to liquid ³He. Atoms keep dissolving into the ⁴He until the dissolved ³He, a degenerate Fermi gas inside the inert superfluid, builds up enough Fermi energy to balance that binding advantage, landing at 6.6 percent. The superfluid ⁴He carries almost no entropy below 0.5 K; it acts as a vacuum through which the ³He quasiparticles move.

It is worth pausing on the third law. Classically, the ideal entropy of mixing of a solution,

\[ \Delta S_{\mathrm{mix}} \;=\; -R\,\bigl[\,x\ln x + (1-x)\ln(1-x)\,\bigr] \]

would survive to \( T = 0 \), violating the requirement that entropy vanish there. Most solutions escape by unmixing as they cool; the helium mixture escapes differently. The dissolved ³He atoms are identical fermions that settle into a unique degenerate ground state, so the mixture reaches zero entropy while staying mixed. The configurational entropy is not frozen in but absorbed into the quantum ground state of the Fermi system.

The mixing chamber is an evaporator that never dries out

Now place the phase boundary inside a small chamber and force ³He across it, concentrated to dilute. The dissolved ³He in the dilute phase has a much smaller Fermi temperature than pure liquid ³He (the atoms are farther apart and their effective mass higher), so at the same temperature it holds far more entropy. Both phases are Fermi liquids with entropy linear in temperature, but with very different slopes:

\[ S_{c}(T) \;\approx\; 22\,T \quad\text{and}\quad S_{d}(T) \;\approx\; 107\,T \qquad \mathrm{J\,mol^{-1}\,K^{-1}} \]
Figure 2. Molar entropy of ³He in the two coexisting phases at low temperature. Each mole that crosses from the concentrated to the dilute side must gain the entropy difference, and the heat \( Q = T\,(S_d - S_c) \) is extracted from the mixing chamber and its load.

Crossing the boundary is thermodynamically identical to evaporation: the atom moves from a low-entropy "liquid" into a high-entropy "gas", except the gas is a dilute solution whose density is fixed by the phase diagram rather than by an exponentially dying vapor pressure. Each mole that crosses at temperature \( T \) absorbs

\[ Q \;=\; T\,\bigl(S_d - S_c\bigr) \;\approx\; 85\,T^{2} \ \ \mathrm{J\,mol^{-1}} \]

and a continuously circulated ³He flow \( \dot n_3 \) gives the famous cooling power of an ideal machine [4], [5]:

\[ \dot Q \;\approx\; 84\,\dot n_3\,T^{2} \]
Figure 3. Cooling power versus temperature for a circulation rate of 100 µmol/s, on logarithmic axes. Evaporative cooling dies exponentially with the vapor pressure, while dilution cooling falls only as \( T^2 \). A power law always beats a decaying exponential, which is why dilution refrigerators reach a few millikelvin while pumped ³He stalls near 300 mK.

The quadratic looks punishing, and in absolute terms it is: a typical machine circulating a few hundred µmol/s delivers hundreds of microwatts at 100 mK but only a fraction of a microwatt at 10 mK. The point of Figure 3 is the comparison: the evaporative competitor is not falling as a power law but off a cliff, and below roughly 300 mK dilution is the only continuous game in town. Practical base temperatures sit near 5 to 10 mK, the best machines reaching about 2 mK, limited by heat exchanger performance and residual heat leaks rather than the principle itself.

Closing the loop

The animation at the top shows how the crossing is sustained in steady state. The ³He removed into the dilute phase must leave the mixing chamber or the process stops. It travels up a column of dilute solution to the still, held near 0.7 K, where the vapor above the dilute liquid is almost pure ³He, because there the vapor pressure of ³He exceeds that of ⁴He by orders of magnitude. Pumping on the still therefore distills ³He selectively out of the solution. The resulting deficit of dissolved ³He lowers the still's osmotic pressure, and the osmotic pressure difference along the column, roughly \( \pi \approx n_3 k_B T \) for the dilute Fermi gas, pushes fresh ³He up from the mixing chamber. The pumped gas is recompressed, condensed near 1 K, precooled in the still and in counterflow heat exchangers against the cold outgoing dilute stream, and injected back into the concentrated phase. The superfluid, entropically inert ⁴He barely participates: it is scaffolding through which the ³He cycle runs.

Seen from a distance the machine is an entropy pump with no moving parts at the cold end. Heat enters the mixing chamber as ³He climbs the entropy ladder from concentrated to dilute, and the same entropy is ejected at the still and the room-temperature pump. The principle was proposed by London, Clarke, and Mendoza in 1962 [1], demonstrated in Leiden in 1965 [2], and refined for six decades since [5], [6]. The engine at its heart has not changed: two quantum liquids, one phase boundary, and the entropy of mixing doing the work evaporation no longer can.

References

  1. H. London, G. R. Clarke, and E. Mendoza, "Osmotic pressure of He³ in liquid He⁴, with proposals for a refrigerator to work below 1°K," Phys. Rev., vol. 128, no. 5, pp. 1992–2005, Dec. 1962, doi: 10.1103/PhysRev.128.1992.
  2. P. Das, R. de Bruyn Ouboter, and K. W. Taconis, "A realization of a London-Clarke-Mendoza type refrigerator," in Proc. 9th Int. Conf. Low Temperature Physics, Columbus, OH, USA, 1965, pp. 1253–1255.
  3. D. O. Edwards, D. F. Brewer, P. Seligman, M. Skertic, and M. Yaqub, "Solubility of He³ in liquid He⁴ at 0°K," Phys. Rev. Lett., vol. 15, no. 20, pp. 773–775, Nov. 1965, doi: 10.1103/PhysRevLett.15.773.
  4. O. V. Lounasmaa, Experimental Principles and Methods Below 1 K. London, U.K.: Academic Press, 1974.
  5. F. Pobell, Matter and Methods at Low Temperatures, 3rd ed. Berlin, Germany: Springer, 2007, doi: 10.1007/978-3-540-46360-3.
  6. H. Zu, W. Dai, and A. T. A. M. de Waele, "Development of dilution refrigerators: A review," Cryogenics, vol. 121, art. no. 103390, Jan. 2022, doi: 10.1016/j.cryogenics.2021.103390.