Kapitza resistance

Around 1936, the existence of a very considerable thermal resistance at the interface between liquid helium and solids was discovered (Kapitza resistance Rk) [50], A similar effect (contact resistance Rc) is present in any contact between two materials. In the presence of a heat flux across the boundary, this thermal resistance causes a temperature discontinuity (see Fig. 4.2). 

A review of the measurements below 100 mK has been done by Harrison in 1979 [58], whereas a study of the heat transfer between helium and sintered metals is due to Rutherford et al. in 1984 [59], A measurement of Kapitza resistance between Mylar and helium around 2K is reported in ref. [60],... 

The study of the contact resistance Rc between solids can be carried out over a temperature range (from milli-kelvin up to hundreds kelvin) much larger than in the case of the Kapitza resistance Rk (below 4K). It is hence possible to evidence effects, such as dispersion and inelastic phonon scattering, which are absent at low temperatures. 

Debye phonon velocity) and lower in the case of very dissimilar materials. For example, the estimated Kapitza resistance is smaller by about an order of magnitude due to the great difference in the characteristics of helium and any solid. On the other hand, for a solid-solid interface, the estimated resistance is quite close (30%) to the value given by the mismatch model. The agreement with experimental data is not the best in many cases. This is probably due to many phenomena such as surface irregularities, presence of oxides and bulk disorder close to the surfaces. Since the physical condition of a contact is hardly reproducible, measurements give, in the best case, the temperature dependence of Rc. 

In this section, we will describe the so-called continuous heat exchangers which are used down to about 1K. For lower temperatures, due to the increasing importance of the Kapitza resistance (see Section 4.3) step exchangers are used. They will be described in Chapter 6 in connection with the dilution refrigerator. 

From Fig. 10.13, we see the latter condition is fulfilled in the first three cases, but not in the fourth case. The most stable situation is obtained with Rx. The choice R = RcosL is however usually adopted when the power supplied to the resistor must be measured. The control of temperature in the real (dynamic) case is much more complex. The problem is similar to that encountered in electronic or mechanical systems. The advantage in the cryogenic case is the absence of thermal inductors . Nevertheless, the heat capacities and heat resistances often show a steep dependence on temperature (i.e. 1 /T3 of Kapitza resistance) which makes the temperature control quite difficult. Moreover, some parameters vary from run to run for example, the cooling power of a dilution refrigerator depends on the residual pressure in the vacuum enclosure, on the quantity and ratio of 3He/4He mixture, etc. 

The diameter of the copper slabs is about a centimetre smaller than the inner diameter of the stainless steel housing, leaving a channel for the helium flow. The top of housing has a conical shape, to reduce the amount of 3He necessary to have the phase separation in the right position. The total Kapitza resistance at 20 mK is about 45K/W (e.g. a heat leak of 50 xW on the cold plate of the mixing chamber would give a temperature difference of about 2 mK between the liquid and the cold plate). 

The c ontrol o f p honon t emperature i n e lectron-phonon c oupling measurements i s critical for a correct estimation of the electron-phonon coupling constant. In our experiment an additional electrically isolated S-Sm-S thermometer was placed near the Si film. Below IK the electron-phonon thermal resistance in silicon is considerably larger than the Kapitza resistance between Si film and the silicon oxide layer, and therefore the S-Sm-S thermometer next to the silicon film was assumed to be at approximately the same temperature as the phonon system in the silicon film. 

At low temperatures, Ac, may fall even below that of the matrix. The cause is a thermal boundary resistance between the filler and the matrix, which is a pl enomenon of phonon mismatch. This resistance, the Kapitza resistance, varies as T and is dominant at low temperatures. If the dominant phonon wavelength --T ) becomes larger than the particle diameter, this effect disappears. Whether or not Ac is increased or decreased, depends on the predominance of the Kapitza resistance and the thermal shortcut in the filler particles. At a fixed temperature, this is a function of the filler diameter, as shown in Fig. 12. Small particles can be used to reduce Ac below that of the matrix. Illustrative examples [ are shown in... 

Fig. 12. Effects of Kapitza resistance, RkiT), on temperature gradient. AT = Q Rk(T) Rk T. ...

P = probability for fatigue life survival Pirr = irreversible deformation power per unit of volume Q = heat flux Rk = Kapitza resistance S = cross-link distance or compliance tensor T = temperature U = internal energy Us = stored stress energy... 

Contact resistance between matrix and particles (known as Kapitza resistance) was investigated by Hasselman (1987), who developed formulas for spherical, cylindrical, and flat-plate geometry to estimate the effective thermal conductivity of polymer composites with interfacial thermal barrier resistance—equation (11.9). 

This thermal resistance, called Kapitza resistance, per filler-matrix boundary area, can be calculated approximately at low temperatures, ie, <20 K ... 

Kapitza resistance, on the other hand, can be applied to materials of extremely low thermal conductivity, ie, good insulators. For a polymeric particulate composite with small ceramic particles, ie, 0.1-1 ixm. in diameter, the thermal conductivity at 2 K is about 50 times lower than that of epoxy resins (53). The plot in Figure 9 shows the variation of thermal conductivities of composites. 

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