Dilution refrigerators are widely used in the fields of condensed matter physics and quantum technology. The condensation-driven dilution refrigerator uses a condenser operating at temperatures below the Still to liquefy the 3He vapor and achieve the circulation of 3He, which has the advantages of compact structure, lightweight, low cost, and low vibration. The published research primarily focuses on the system architecture and performance, and the research and analysis on the thermodynamic cycle are still incomplete. In this paper, the condensation-driven dilution refrigeration cycle has been clarified as a thermally driven refrigeration cycle, and the thermodynamic performances including the figure of merit and thermodynamic perfectibility are investigated. Optimizations have been made to the key components of the previous prototype, and the lowest no-load temperature of 68 mK and a cooling power of 4 μW at 100 mK were achieved. Compared with the previous prototype, the thermodynamic perfectibility of the system increased from 7.63% to 17.83%. The impact of the Still heating strategies on the system was analyzed, and a start-up strategy was proposed to speed up the cooldown process.

Sub-Kelvin temperature offers an extreme experimental environment for many important applications, such as condensed matter physics research and quantum devices. The methods to achieve sub-Kelvin temperatures primarily include an adsorption refrigerator (AR), an adiabatic demagnetization refrigerator (ADR), and a dilution refrigerator (DR). Among them, the DR stands out as the most popular choice due to its characteristics including continuous cooling and a relatively large cooling power.

In a DR, the 3He atoms in the concentrated phase diffuse across the phase boundary into the dilute phase, absorbing the heat and generating cooling.1 According to the types of circulating pumps, DRs can be divided into conventional DRs using ambient mechanical pumps2 and cold-cycle DRs with cryogenic adsorption pumps or condensation pumps.3 The condensation-driven dilution refrigerator (CDR) uses a condenser operating at temperatures below the Still to liquefy the 3He vapor and achieve the circulation of 3He, which has the advantages of compact structure, lightweight, low cost, and low vibration. Due to these characteristics, the CDR can meet the requirements of specific applications, such as the forthcoming generation of Cosmic Microwave Background observatories (Simons Observatory), which requires a cryostat to cool the background-limited detectors below 100 mK.4 In addition, the astrophysics balloon missions require a compact and convenient-to-operate cryocooler to cool the x-ray and γ-ray detectors.5 

The CDR was first proposed by Edel'man et al. in 1972.6 They built a wet-type CDR precooled by the liquid 4He dewar and 3He cryostat, and the 3He circulation flow rate depends on the cooling power of the pre-cooling refrigerator. Early CDR with the 1 K liquid helium bath and the sintered heat exchanger even can achieve a minimum temperature of 12.1 mK and provide a cooling power of 36.9 μW at 95 mK.7 However, this structure did not make full use of the advantages of the CDR in terms of compactness and other aspects. The advances in 4 K GM cryocoolers led to the development of the cryogen-free CDR (or dry CDR), which has become the mainstream of research now. The lowest temperature achieved is 48 mK8 and generally provides μW-class cooling power at 100 mK. The published research primarily focuses on the system architecture and performance, and the research and analysis on the thermodynamic cycle are still incomplete.

In 2023, we built a CDR with the lowest no-load temperature of 84.4 mK,9 and the cooling performances need to be further improved. Meanwhile, a deeper analysis on cycle thermodynamic characteristics and the start-up dynamics is yet to be conducted. In the following, the thermodynamic process and experiments of the CDR are introduced and the performances are analyzed.

Figure 1 illustrates the schematic diagram of the CDR and the corresponding setup in our lab. The system consists of a condensation pump (CP), heat exchangers (HEX), a mixing chamber (MC), and a Still (S). Compared with a conventional DR, the CDR uses a condensation pump (∼400 mK) at a temperature lower than the Still (∼700 mK) to liquefy the 3He vapor, and then, the liquid drops down, flows through the HEX, and enters the MC. The CDR has a height of 300 mm. The Still is positioned 50 mm above the cold plate at the bottom of the mixing chamber. The length of the continuous tube-in-tube heat exchanger is 583 mm. Other details of the component dimensions can be referred to in Ref. 9.

FIG. 1.

The schematic (left) and experimental setup (right) of the CDR.

FIG. 1.

The schematic (left) and experimental setup (right) of the CDR.

Close modal

One of the issues affecting the efficiency of the prototype is the insufficient heat exchange area in the MC. A fin-type configuration with a total heat transfer area of 41 cm2 was used in the previous MC.9 The analysis therein shows that a significant temperature difference (19–25 mK, depending on the cooling power) may exist between the dilute phase and the cold plate in the MC (Ref. 9) as shown in Fig. 2, which is strongly related to the Kapitza resistance10 between the dilute phase and the cold plate. With a decrease in the cooling temperature, the entropy generation resulting from the limited heat transfer between the dilute phase and the cold plate increases sharply, becoming one of the important factors affecting the thermodynamic efficiency of the system.

FIG. 2.

The comparison between the calculated dilute liquid temperature and the measured value of cold plate temperature in the MC.

FIG. 2.

The comparison between the calculated dilute liquid temperature and the measured value of cold plate temperature in the MC.

Close modal

Bearing this heat transfer inefficiency in mind, a sinter silver powder heat exchanger with a diameter of D = 19 mm, height of h = 2 mm, a specific surface area of 0.4 m2/g, and a total surface area of A ≈ 1 m2 was developed for the cold plate, which is significantly larger than that of the prototype. This change has brought a significant improvement, as will be introduced later.

The experiment uses a GM-type pulse tube cryocooler (SRP-082B) from Sumitomo Heavy Industries11 to provide pre-cooling at 4 K and a two-stage AR (CRC07046) from Chase Cryogenics12 to provide pre-cooling for the CP. The holding time of the cycle and the 3He circulation flow rate are limited by the cooling power of the AR. In the experiments and prototype machine tests in this article, the same pre-cooling refrigerators were used.

The CDR system utilized ruthenium oxide resistance temperature sensors (Lakeshore RX-102A13) as thermometers, with temperature measurements conducted using a Lakeshore 372 resistance bridge.14 The measurement errors included electronic accuracy, calibration accuracy, self-heating errors, measurement resolution, and interpolation errors. Lakeshore provided an estimated total error of about 4 mK at 100 mK.14 

The thermodynamic cycle of the CDR is shown in Fig. 3; from a fundamental thermodynamic perspective, the cycle here is a very interesting one. Based on numerical simulations, the complete thermodynamic process of the CDR is like a butterfly in a T–S diagram as shown in Fig. 3(b). The dashed line represents the saturation liquid line and saturation vapor line of 3He. Dilution refrigeration uses a mixture of 3He and 4He, but for an ideal dilution refrigeration process, only 3He circulates within the system, so the state points described in the T–S diagram represent the state of 3He. It should be noted that 3He in the dilute phase is generally considered as a weakly interacting Fermi gas. Therefore, in the T–S diagram, points 2, 3, and 4 represent the states of 3He in the dilute phase. The entire refrigeration cycle can be divided mainly into five phases: dilution refrigeration, heat absorption in dilute stream, evaporation in the Still, condensation, and heat release in condensed stream.

FIG. 3.

Illustration of the state point of the CDR thermodynamic cycle and the numerically obtained T–S figure of the cold cycle. (a) The schematic of the CDR (light blue: dilute phase, dark blue: concentrated phase). (b) The T–S diagram of the CDR cycle: (1) MC inlet; (2) MC outlet; (3) outlet of the dilute channel of the HEX; (4) Still liquid; (5) Still gas; (5′) gas in the gas tube; (6) saturated vapor in the CP; (6′) saturated liquid in the CP; (7) outlet liquid of the CP; (8) inlets of the concentrated channel of the HEX.

FIG. 3.

Illustration of the state point of the CDR thermodynamic cycle and the numerically obtained T–S figure of the cold cycle. (a) The schematic of the CDR (light blue: dilute phase, dark blue: concentrated phase). (b) The T–S diagram of the CDR cycle: (1) MC inlet; (2) MC outlet; (3) outlet of the dilute channel of the HEX; (4) Still liquid; (5) Still gas; (5′) gas in the gas tube; (6) saturated vapor in the CP; (6′) saturated liquid in the CP; (7) outlet liquid of the CP; (8) inlets of the concentrated channel of the HEX.

Close modal

The 3He in the concentrated phase enters the dilute phase through the phase boundary in the MC, absorbing the heat, and generating the cooling.

The 3He in the dilute phase enters the HEX for heat absorption from the condensed stream, with both temperature and entropy increasing.

3He in the dilute phase enters the Still after passing through the HEX, where it is heated by the driving heat source in the Still, and at a certain pressure, the superheated 3He gas at state five precipitates.

The superheated 3He gas from the Still enters the CP through the gas tube, where, at constant pressure, the unsaturated gas condensed into saturated liquid 6′, and then, the liquid is further cooled to CP temperature to reach subcooled state 7. After state 7, 3He passes through the adiabatic tube under gravity to reach state 8, and the temperature and entropy remain constant.

The concentrated phase enters the HEX for heat release to the dilute stream, with both temperature and entropy decreasing.

Previously, the CP temperature and MC temperature are directly used to evaluate the thermodynamic second law efficiency.15 However, a deeper look on the butterfly shape actually reveals two processes: a refrigeration process between the CP temperature and the MC temperature, and a thermal engine process between the Still temperature and the CP temperature. In other words, this system should be viewed as a thermally driven refrigeration cycle.

Under this circumstance, an ideal figure of merit FOMmax is proposed for a thermally driven refrigeration cycle, where T is the temperature with the subscripts denoting corresponding parts,
F O M max = T s T cp T s T mc T cp T mc ,
(1)
The practical FOM and the thermodynamic perfectibility of the system ζII are
F O M = Q mc Q cp Q mc = Q mc Q s ,
(2)
ζ II = F O M F O M max ,
(3)
where Qmc is the cooling power, Qcp is the dissipated power of the condensation pump, and Qs is the heating power of the Still. For a typical operating condition in the previous prototype9 (Qs = 280 μW, Qmc = 1.91 μW, Tmc = 100 mK Ts = 660 mK, Tcp = 452 mK), the FOM is 0.0068 and the corresponding ζII is 7.63%, which indicates that there is a significant margin for improving the thermodynamic efficiency.

A typical run of the dilution refrigerator is shown in Fig. 4, with a charge volume of 4.8 L [at standard temperature and pressure (STP), 3He concentration is 36%] to ensure that the gas–liquid phase boundary is inside the Still. After testing, the cycle can be divided into the following three stages.

FIG. 4.

Temperature curve of the lowest no-load temperature experiment of the CDR.

FIG. 4.

Temperature curve of the lowest no-load temperature experiment of the CDR.

Close modal

The entire system cools from room temperature to 4 K in about 30 h. Subsequently, the 4He stage cooling of the AR is activated, and the dilution refrigeration system cools to below 900 mK after approximately 2 h. Finally, the 3He stage cooling of the AR is activated.

The Still reaches the phase separation temperature before the MC. At this point, the Still is heated to maintain a temperature above 0.7 K. As the 3He stage of the AR continues pre-cooling the MC, the temperature of the MC decreases below the phase separation temperature. When the phase boundary enters the MC, the cooling rate of the MC increases significantly, marking the formal start of dilution refrigeration.

The temperature of the MC rapidly drops to 100 mK. As the cooling temperature decreases, the cooling power decreases, and the influence of heat leakage gradually increases, leading to a decrease in the cooling rate. When the cooling power of AR is fully consumed, the cycle ends, and the MC reaches the lowest no-load temperature of 68 mK.

In practice, dynamic control of the start-up process may apparently affect the CDR performance. Normally, in the previous strategy, we start to heat the Still until the temperature of both the MC and the Still falls below 400 mK. The dashed lines in Fig. 5 show how the MC temperature changes during this process. We applied heating to the Still at time = 3.8 h; however, the MC did not generate the expected cooling effect, and the temperature of the MC increased instead. This phenomenon may imply that a phase boundary may form in either the Still or the HEX tube, which has also been mentioned by Herrmann et al.16. Bearing this guess in mind, the control strategy was adjusted by maintaining the temperature of the Still at above 700 mK during the whole start-up process.

FIG. 5.

Temperature curve of the lowest no-load temperature experiment with different heating strategies of the CDR (the temperature variation with the time of the previous strategy is represented by a dashed line).

FIG. 5.

Temperature curve of the lowest no-load temperature experiment with different heating strategies of the CDR (the temperature variation with the time of the previous strategy is represented by a dashed line).

Close modal

The results show that precooling time has been reduced and MC temperature can reach 0.1 K more quickly. Compared to the previous strategy, this new strategy can reduce the cooling time by 6 h (from 0.9 to 0.1 K), as shown in Fig. 5. The CDR can operate at 100 mK for approximately 2 h (depending on the cooling power), only limited by the cooling power of the single-shot AR.

With the new MC configuration and new control strategy, the lowest no-load temperature of the MC eventually reaches 68 mK with a mixture amount of 4.8 L. Figure 6(a) further shows the cooling power under different MC temperatures. Typically, with 320 μW applied to the Still, a cooling power of 4 μW at 100 mK can be obtained with Ts = 660 mK, Tcp = 480 mK, and the corresponding FOM and ζII are 0.0123 and 17.83%, respectively.

FIG. 6.

Experimental results at different temperatures: (a) the cooling power; (b) FOM and ζII.

FIG. 6.

Experimental results at different temperatures: (a) the cooling power; (b) FOM and ζII.

Close modal

Based on the cooling power test results, the thermodynamic efficiency and thermodynamic perfection of the condensation-driven dilution refrigeration cycle were evaluated, as shown in Fig. 6(b). When the cooling temperature increases, both the thermodynamic efficiency and thermodynamic perfection increase. This is mainly due to the decrease in irreversible losses caused by viscous dissipation and non-isothermal heat transfer with increasing cooling temperature.

This paper also conducted experiments with different amounts of mixture charged, increasing the amount from 4.8 to 5.28 L, and found that the cooling power of the MC was not sensitive to this amount change. In theory, the amount of charge may not affect the dynamic characteristics of the system, as long as there is enough height difference between the concentrated and dilute liquid columns, and a gas–liquid phase boundary in the Still and a concentrated-dilute phase boundary in the MC. Certainly, an unnecessary extra amount leads to a higher cost.

The calculations on the no-load experimental conditions were conducted using the existing mathematical model.15 The comparison between the calculations and experimental results is shown in Table I. Qs, Tmc, Ts, and Tcp represent the measured data from experiments, and n3 is the 3He circulation flow rate, which is calculated based on the energy conservation of the Still. Qmc,cal is the calculated cooling power of the MC, which should be totally balanced by the heat leakage under no load. The heat leakage within the MC includes the thermal conduction of the support structure and lead wires (Qhl,con), as well as the radiative heat from the high-temperature components (Qhl,rad). Qhl,con was calculated with the thermal conductivity of the material and the temperature difference. Qhl,rad represents the heat leakage caused by radiation. As shown in Table I, Qhl,rad are very close under five different operating conditions. Considering that the external environment and temperature of the mixing chamber was comparable in these five experiments, this implies similar heat leakages. According to the simulation, the lowest temperature of 32 mK is achieved without any heat leakage in our system. That means reducing heat leakage is a viable strategy for enhancing system performance.

TABLE I.

The experimental and simulation results of our CDR.

Qs (μW)Tmc (mK)Ts (mK)Tcp (mK)n3 (μmol/s)Qmc,cal (μW)Qhl,con (μW)Qhl,rad (μW)
240 71.5 602.7 449.6 7.66 2.48 1.46 1.02 
280 70.7 632.0 462.0 8.78 2.67 1.58 1.09 
320 68.1 658.3 479.6 9.87 2.63 1.68 0.94 
360 67.2 681.1 495.4 10.95 2.70 1.79 0.91 
400 67.0 695.8 505.1 12.07 2.84 1.86 0.98 
Qs (μW)Tmc (mK)Ts (mK)Tcp (mK)n3 (μmol/s)Qmc,cal (μW)Qhl,con (μW)Qhl,rad (μW)
240 71.5 602.7 449.6 7.66 2.48 1.46 1.02 
280 70.7 632.0 462.0 8.78 2.67 1.58 1.09 
320 68.1 658.3 479.6 9.87 2.63 1.68 0.94 
360 67.2 681.1 495.4 10.95 2.70 1.79 0.91 
400 67.0 695.8 505.1 12.07 2.84 1.86 0.98 

In conclusion, the CDR is shown based on a thermally driven refrigeration cycle and we developed a CDR that achieved the lowest no-load temperature of 68 mK, with a cooling power of 4 μW at 100 mK. The thermodynamic perfectibility of the system increased from 7.63% to 17.83%.

In the future, a sintered heat exchanger will be added to the setup for the purpose of reaching a lower temperature. This compact CDR can easily integrated with the existing higher temperature platform of many laboratories and support the development of high-end equipment such as astronomical telescopes, quantum devices, single photon meters, and x-ray detectors.

This work was financially supported by the National Key Research and Development Program of China (Grant No. 2021YFC2203303) and the National Natural Science Foundation of China (Grant No. 52176027). We thank A.T.A.M. de Waele from the Netherlands for valuable discussion.

The authors have no conflicts to disclose.

Weijun Cheng: Data curation (equal); Investigation (lead); Writing – original draft (lead). Hongye Zu: Formal analysis (equal); Methodology (lead); Writing – original draft (equal). Zhiheng Li: Investigation (equal). Yanan Wang: Investigation (equal); Project administration (lead); Writing – review & editing (supporting). Wei Dai: Conceptualization (lead); Funding acquisition (lead); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
F.
Pobell
,
Matter and Methods at Low Temperatures
, 3rd edition (
Springer
,
Berlin
,
2007
).
2.
J. G.
Brisson
, “
Cold-cycle dilution refrigeration
,”
J. Low Temp. Phys.
111
,
181
199
(
1998
).
3.
H.
Zu
,
W.
Dai
, and
A. T. A. M.
de Waele
, “
Development of dilution refrigerators—A review
,”
Cryogenics
121
,
103390
(
2022
).
4.
A. J.
May
, “
Sub-kelvin cryogenics for experimental cosmology
,”
Ph.D. thesis
(
The University of Manchester
,
2019
).
5.
R.
Snodgrass
,
J.
Ullom
, and
S.
Backhaus
, “
Performance of a miniature, closed-cycle dilution refrigerator at tilt angles between 0 and 30 degrees”
in
Proceedings of the 22th International Cryocooler Conference
,
Boulder
(International Cryocooler Conference, Inc.,
2022
).
6.
V. S.
Edel'man
, “
A dilution refrigerator with condensation pump
,”
Cryogenics
12
,
385
387
(
1972
).
7.
V. E.
Sivokon
,
V. V.
Dotsenko
,
L. A.
Pogorelov
, and
V. I.
Sobolev
, “
Dilution refrigerator with condensation pumping
,”
Cryogenics
32
,
207
210
(
1992
).
8.
G.
Teleberg
,
S. T.
Chase
, and
L.
Piccirillo
, “
A miniature dilution refrigerator for sub-kelvin detector arrays
,”
SPIE
6275
,
6275OD-6271–6279
(
2006
).
9.
H.
Zu
,
W.
Cheng
,
Y.
Wang
, and
W.
Dai
, “
Experimental and numerical study of a condensation-driven dilution refrigerator
,”
Cryogenics
135
,
103746
(
2023
).
10.
Z.
Ji
,
J.
Fan
,
J.
Dong
et al, “
Development of a cryogen-free dilution refrigerator
,”
Chin. Phys. B
31
,
102703
(
2022
).
11.
Industries SH
, see http://www.shicryogenics.com for the information about the GM type pulse tube cryocooler.
12.
Chase ST
, see https://www.chasecryogenics.com/products for “Chase research cryogenics.”
15.
H.
Zu
,
W.
Cheng
,
Y.
Wang
et al, “
Numerical investigation of a condensation-driven dilution refrigerator
,”
Int. J. Refrig.
151
, 224–240 (
2023
).
16.
R.
Herrmann
,
A. V.
Ofitserov
,
I. N.
Khlyustikov
, and
V. S.
Edel'man
, “
A portable dilution refrigerator
,”
Instrum. Exp. Tech.
48
,
693
702
(
2005
).