simulations of a thermal conductivity cell filled with normal 4he and dilute mixtures of 3he in...

Journal of Low Temperature Physics, Vol. 114. Nos. 3/4. 1999

Simulations of a Thermal Conductivity Cell Filledwith Normal 4He and Dilute Mixtures of 3He

in Superfluid 4He

Daniel Murphy*

Department of Physics, Duke University, Durham, North Carolina 27708-0305, USA

(Received by F. Pobell July 17, 1998; revised September 29, 1998)

We have simulated the thermal response of a cylindrical thermal conductivitycell filled with liquid helium to AC and DC heat fluxes. The conductivity cellin these simulations is realistic in that it includes sidewalls and gaps, whichcannot be treated analytically or in a one-dimensional simulation. Oursimulations are to able to account quantitatively for the apparent departureof the effective thermal conductivity, Keff, of dilute mixtures of 3He in super-fluid 4He from theoretical predictions. We have recently demonstratedexperimentally that this departure is due to the presence of gaps in previousthermal conductivity cells. These simulations also show that the additionalphase lag in the response of normal 4He to an AC heat flux, measured byOlafsen and Behringer, is due to gaps in the heated plate.

I. INTRODUCTION

Measurements of thermal transport properties in any medium arecomplicated by the interaction of the experimental apparatus and themedium in question. The goal experimentally is to reduce these interactionsto the point where their effect on the measurement is inconsequential, butit is often difficult or impossible to do so; it is necessary in these cases toknow precisely what effect the experimental apparatus has on the result inquestion. One tool which is useful in analyzing experimental design issimulation. For example, Duncan et al.1-3 have shown that numericalmodels of sidewall heat flow in thermal conductivity cells containing liquidhelium provide valuable information for interpreting experiments.

Over the past two years, we have shown that several anomalousresults of transport measurements in liquid helium, in both its superfluid

* Present address: Quantum Institute, University of California, Santa Barbara, California93106, USA.

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0022-2291/99/0200-0389$16.00/0 © 1999 Plenum Publishing Corporation

and normal fluid phases, were due to the design of the thermal conductivitycells used. In particular, the presence of a gap between the copper endplatesof a thermal conductivity cell and the low-conductivity sidewall createdproblems in measurements of the effective thermal conductivities of dilutemixtures of 3He in superfluid 4He,4 boundary resistance measurements insuperfluid 4He near TA ,5 , 6 and time-dependent thermal measurements innormal 4He.7 In addition, Li observed that modifying similar gaps affectedthe measured boundary resistance in superfluid 4He near TL.8 The effectsof these gaps were demonstrated by building a cell in which they were sup-pressed;4-6 however, we were unable to completely eliminate the gaps, andresidual effects were observed.

As a result, we have simulated two of the above experiments, namelyDC measurements in dilute superfluid mixtures and AC measurements innormal 4He, in order to show that gaps would in fact lead to the previouslyobserved effects. Also, we hope that these calculations will enable us todetermine what effect residual gaps might have on future experiments. Wehave not studied the AC response of a thermal conductivity cell filled withdilute superfluid mixtures, nor the effect of the thermal conductivity cell onmeasurements of the boundary resistance in the superfluid phase near TL.Modelling the AC response of dilute superfluid mixtures requires the solutionof a complicated set of coupled differential equations, while the boundaryresistance problem is non-linear, and because of their increased complexitysuch simulations have not yet been performed.

II. BACKGROUND

The two experiments discussed in this paper measure heat transportthrough a fluid layer, and as a result were performed in thermal conduc-tivity cells which share a similar geometry, shown schematically in Fig. 1.The cell is cylindrically symmetrical, and is composed of two copperendplates, shown at the top and bottom of the figure. The endplates areseparated by a thin stainless steel sidewall which both contains the fluidand thermally insulates the endplates from each other. (The copper used inthe endplates has a large thermal conductivity, while stainless steel is apoor thermal conductor.) Fluid fills the space between the endplates, but italso fills the gaps between the sidewall and the endplates. Either an AC orDC heat flux Q is applied to one end of the thermal conductivity cell, andis removed at the other end, which is kept at a constant temperature. Thetemperature difference AT between the top and bottom of the cell ismeasured, and from AT the transport properties of the fluid are deduced;this task is complicated by the geometry of the thermal conductivity cell.

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Although the experiments described in this paper share a commonexperimental technique, the microscopic mechanisms by which heat iscarried are quite different in the two systems, with several important conse-quences. One result is that we cannot simulate the AC behavior of thedilute mixtures of 3He in superfluid 4He in the same way we can simulatethe DC behavior of these mixtures and the AC behavior of normal 4He,because the underlying equations are different. A second effect is that thethermal transport coefficients such as the thermal conductivity differgreatly in magnitude, and so the interactions of fluid and measurementapparatus are quite different for the mixtures and the normal fluid. Also,the quantities of interest in the two experiments are not the same. Wedescribe the two systems in more detail below.

A. Dilute Mixtures of 3He in Superfluid 4He

Superfluid 4He is well-characterized by the two-fluid model, in whichthe fluid is described as being composed of two parts, a normal and super-fluid component whose relative fraction varies with temperature.9 Heat iscarried in a superfluid by a process known as counterflow; the normal fluidcomponent carries heat from the heated side of the fluid layer to the tem-perature-regulated side, while the superfluid fraction flows in the reversedirection in order to conserve mass.10 Counterflow is a dissipationless pro-cess, since the normal and super components do not interact, which meansthat a temperature difference cannot be supported across a layer of super-fluid, even in the presence of a heat flux.

The addition of 3He impurities to superfluid 4He changes the behaviorof the system.10,11 Heat is still carried by counterflow in these mixtures, butthe impurities interact with the normal component, and are carried alongwith the heat flow. Unlike the normal component, however, the 3Heimpurities cannot take part in the mass-conserving superfluid current, anda 3He concentration gradient is established across the fluid layer, leadingto a chemical potential gradient which balances the drag force onthe impurities by the normal component. The 3He impurities thereforerestrict the normal fluid current, leading to thermal dissipation in thesystem.

Despite the fact that heat is transported by counterflow rather thanthermal diffusion, the DC behavior of a dilute superfluid mixture is stilldescribed by Fourier's law:

Simulations of a Thermal Conductivity Cell 391

diverging as X goes to zero. Dilute superfluid mixtures have extremely largeeffective thermal conductivities.

Another important factor in experiments on dilute superfluid mixturesis the Kapitza, or boundary, resistance RK. When heat flows across aninterface between materials with different heat transport coefficients, therewill be a temperature difference between them. The coefficient whichcharacterizes this temperature drop is known as the Kapitza resistance, anddepends on the properties of the two substances. In most materials the tem-perature drop is too small to measure, but in pure 4He Keff is effectivelyinfinite, and the Kapitza resistance between 4He and the copper comprisingthe conductivity cell is much larger than the thermal resistance of the fluidlayer. In dilute superfluid mixtures, Keft is still very large, and the Kapitzaresistance must be taken into account when determining the temperaturedifference across the fluid layer.

Recently, Murphy and Meyer4 showed that previous measurementswhich suggested a departure of Keff from the expected X-1 concentrationdependence were the result of the design of the experimental cells used.(For a review of the earlier results, please refer to Ref. 4.) As the molarconcentration of the mixtures is increased from a nominally pure state( X = 2 . 4 X 10- 9) to X= 1 x 10-3, the effective conductivity of the fluid goesfrom being much higher than that of copper to much lower, with the resultthat at low concentration very small channels filled with a superfluidmixture will have a smaller thermal resistance than much larger blocks ofcopper. At the lowest concentrations, the measured thermal resistance inthe thermal conductivity cell described above will be only that of the cop-per endplates and the Kapitza resistance. At intermediate concentrations,the measured thermal resistance of the conductivity cell must be correctedfor the resistance of the copper and the Kapitza resistance, which is takento be the resistance of the thermal conductivity cell measured whenX = 2 4 X10-9. However, the presence of gaps between the sidewall and theendplates in the conductivity cell results in a mismeasurement of thiscorrection factor for higher concentrations. In nominally pure 4He, thesidewall gap carries a large fraction of the heat; in the cell used by Murphyand Meyer, the gap carried over half of the total heat current.5 By contrast,

392 Daniel Murphy

The transport coefficient which characterizes counterflow is referred to asthe effective thermal conductivity, Keff, rather than the thermal conduc-tivity, K. In pure 4He, Keff is nearly infinite, but with the addition of 3He Keff

becomes finite. The effective thermal conductivity varies as the inverse ofthe molar concentration of 3He impurities, X,

once X is increased to 10 6 the thermal resistance of the fluid in the gapsbecomes larger than that of the copper, and most of the heat flows throughthe endplate.

Murphy and Meyer4 constructed a thermal conductivity cell whichnearly eliminated the gaps between the endplates and the sidewall, dramati-cally improving agreement between theory and experiment; however, theauthors were unable to completely suppress this gap. We have simulatedthe case of a conductivity cell with gaps in order to show that the physicalprocesses known to occur in dilute superfluid mixtures will lead to theeffects observed in earlier research, and that a new mechanism need not beinvoked to explain the experimental results.

B. AC Response of Normal 4He Near TL

The second set of experiments we simulated were originally performedby Olafsen and Behringer.12,13 In these experiments, a thermal conductivitycell, with a geometry similar to that described above, was filled with nor-mal fluid helium, which behaves as classical fluid described by the thermaldiffusion equation:

where q = h ( w / D T ) 1 / 2 exp( — iP/4) and R0 is the zero-frequency thermalresistance of the layer. The response is measured as a function of the fre-quency of the applied heat, and in general has both an amplitude andphase, where the phase is measured relative to that of the applied heat flux.Olafsen and Behringer12,13 subsequently measured the response of a layerof normal fluid 4He near TL to an AC heat flux, and found that theexperimentally determined response was different from that predicted. In


where p is the density, CP is the specific heat, and K is the thermal conduc-tivity. One end of the thermal conductivity cell was maintained at a fixedtemperature, while a sinusuoidal heat flux was applied to the other end ofthe cell, and the temperature difference across the cell, AT, was recorded asa function of time.

Behringer14 has shown that for a simple fluid layer of thickness h andthermal diffusivity DT = K/pCP, the thermal response to an AC heat flux ofthe form Q0 exp iwt is given by

particular, the response lagged in phase much further behind the appliedheat flux than expected.

Murphy and Meyer7 and Olafsen and Behringer13 showed that theseresults could be explained by the presence of the gap between the heatedendplate and the stainless steel sidewall, which increased the thermal massof the heated plate. At low temperatures, the specific heat of copper ismuch smaller than that of 4He, and, although the mass of the copper platewas much greater than that of the fluid, the thermal loading due to thehelium in the gap is dominant. Murphy and Meyer7 showed that relaxationtime measurements in normal fluid 4He were affected by this thermal massas well. (Note that this effect is very different from that observed in dilutesuperfluid mixtures. In the case of superfluid mixtures the thermal conduc-tivity of the helium in the gap is much greater than that of the copper,while in this case the thermal conductivity of the fluid is small comparedto the copper. However, the heat capacity of the fluid is much greater thanthat of the copper.)

Olafsen and Behringer proposed using the high-frequency end of theAC response to measure the Kapitza resistance in the normal phase of 4He,although they recognized that their data was not in the asymptotic high-frequency limit.13 However, their analysis of high-frequency data reliedupon the model which includes the thermal mass of the fluid in the gap asan additive correction to the thermal mass of the warm plate. It is notobvious that this model can be used to correct for the effect of the gaps orthe sidewall quantitatively, despite its qualitative success; the model isquasi-one-dimensional, while the real problem is three dimensional.

We therefore simulated the response to an AC heat flux of a 0.1 cmthick cylindrical layer of normal fluid helium both with and without theexperimental cell at a reduced temperature E = l x 1 0 - 3 , where E =(T—TL) /TL In these simulations, we show that the cell has a significanteffect on the response of the entire system to an AC heat flux. In particular,the phase of the response of the warm plate is altered, which implies thatthe apparent relaxation time of the fluid layer will be different from thatexpected on the basis of the fluid layer alone.

III. SIMULATION

Both DC measurements of Keff in dilute mixtures of 3He in superfluid4He and AC measurements in normal 4He can be modelled by the Fouriertransform of the diffusion equation:

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For the dilute superfluid mixtures, the zero-frequency form of this equationis Laplace's equation, which is obeyed both in the superfluid mixture andalso in the copper and stainless steel which makes up the thermal conduc-tivity cell. Equation (5) has several advantages for the normal 4He simula-tion. First, this form for the diffusion equation converts a time-dependentproblem into a boundary-value problem, at the cost of going from realvariables to complex ones. Another advantage is that the AC experimentsof Olafsen and Behringer13 are a realization of this equation, and theexperimental results can be directly compared to solutions of Eq. (5). Weincluded the Kapitza resistance in our simulations of both dilute superfluidmixtures and normal 4He.

An important simplification made in our simulations is that weassumed both dilute superfluid mixtures and normal 4He behaved linearly.That is, we assumed that the properties of the fluid layer and the thermalconductivity cell do not depend on temperature. In superfluid mixtures, theconcentration is coupled to the temperature gradient, because the 3Heimpurities are dragged along with the normal motion. Since Keff dependsupon X, this concentration gradient should in principle be taken intoaccount. In normal fluid 4He, the properties of the fluid vary greatly withtemperature near the superfluid transition temperature TL, and a tem-perature gradient in the fluid layer can change the local transport coef-ficients. However, in both superfluid mixtures and normal helium, theexperimental data to which the simulations will be compared were acquiredin the linear regime, at small heat fluxes where non-linear effects are negli-gible, and therefore we will not consider the more difficult non-linearproblems.

The cylindrical thermal conductivity cell we simulated has the samefeatures as that shown in Fig. 1. However, we have simplified the geometryslightly, retaining the sidewall and the gaps between the sidewall and thecopper endplates while omitting some of the structure associated with thesidewall spacer. Also, because the cell is cylindrically symmetric, the three-dimensional problem is mapped to a two-dimensional simulation by anappropriate change of coordinates. Figure 2a shows the two-dimensionalgeometry used in our simulations. The temperature regulated plate islocated at z = 0, and a radially-uniform heat flux is applied in the — z direc-tion at z = l; the applied flux either varies sinusoidally with time or isconstant. The center of the cell is located at r = 0, and has a zero heat fluxcondition in the radial direction, while the boundary at r = 1 also has azero radial heat flux condition because it is exposed to vacuum. The copperis assumed to have a thermal conductivity of 2.0 W/cmK,15 and the stain-less steel has a conductivity of 1.3 x 10 - 3 W/cmK.16 The Kapitza resistancebetween the metals and the fluid is 1.0 cm2K/W.5


396 Daniel Murphy

Fig. 1. A schematic of the thermal conductivitycell used in experiments with dilute mixtures of3He in superfluid 4He and AC measurements innormal fluid 4He. The figure shows the stainlesssteel sidewalls as well as the gaps in the cell.

A uniform square grid, with 127 x 127 grid points, was applied to thegeometry shown in Fig. 2a. None of the grid points lie on the boundaries,so the boundary conditions were absorbed into the finite-difference equa-tions, which were solved using a complex multi-grid method. For clarity,the surface plots shown later in this paper display every other grid point.A multigrid solver was used rather than a relaxation method because thetransport coefficients change discontinuously at the boundaries between thefluid and the metals. The thermal diffusivities of the metal and the liquiddiffered by several orders of magnitude, greatly slowing convergence, andrequiring the use of fast solvers. For an introduction to finite-differencesolutions to differential equations, see for instance Ref. 17, while informa-tion on multigrid methods can be found in Refs. 18 and 19.

Fig. 2. The thermal conductivity cell, showing (a) the uniform grid spacingof the cell and (b) the non-uniform real-space dimensions used in most ofour simulations. All real-space dimensions are in centimeters.

The real-space r and Z dimensions are shown in Fig. 2b. The real-spacecoordinates are non-uniform in order to increase the number of grid pointsin the gap between the sidewall and the endplates, which is very narrow,and in main sample space. This real-space non-uniformity requires that thethermal conductivity and heat capacity, as well as the spatial derivatives,be scaled by the changes in the r and z dimensions so that the simulationcorresponds to the original physical problem. The resulting differentialequations have anisotropic coefficients, which result in anisotropic coef-ficients in the finite difference equations. Our simulation routine automati-cally accounted for these dimensional considerations. Also, while for mostsimulations we used the values indicated in Fig. 2b, which are the dimen-sions of a thermal conductivity cell used by Murphy and Meyer in theirdilute superfluid mixture experiments, the dimensions were varied in someof the simulations described below; see the text for details.

IV. RESULTS

A. Dilute Mixtures of 3He in Superfluid 4He

In dilute superfluid mixtures, the quantity of interest is the effectivethermal conductivity, Keff. (At the temperature used in the simulationsbelow, Keff=0.011 X-1 mW/(cmK).) The experimentally accessible quan-tity is the observed thermal conductivity, Kobs, defined as h/R f lu id, whereRfluid is the thermal resistance of the fluid layer. In our experiments we donot directly measure Rfluid, but rather Rfluid + 2Rb, where Rb is the sum ofthe Kapitza resistance RK and the resistance of the copper, RCu. In thepast, we experimentally determined Rb by measuring the thermal resistanceR of our thermal conductivity cell filled with nearly pure 4He(X=2A x 1 0 - 9 ) as discussed above; pure 4He has a conductivity which forour purposes is infinite, and we assumed that a measurement of the thermalresistance with this mixture would constitute a measurement of Rb. That is,our observed conductivity would be

However, a significant fraction of the heat applied to the warm end ofthe cell flowed through the gaps in the cell filled with pure 4He, eventhough these gaps were very narrow. Figure 3a shows the results of oursimulation for X = 2 .4x10 - 9 , demonstrating that the fluid in the gaps,


398 Daniel Murphy


which are only 0.003 cm wide and 1 cm long, is isothermal with the fluidbetween the plates. (The vertical axis in these plots is the temperature at aparticular location in the cell as indicated by the r and z coordinates. Thegeometry of the cell is the same as that shown in Fig. 2, and the labels inthe three-dimensional plot can be compared with the structures in this two-dimensional figure.) When the concentration is increased to X=1 x 10~6 ,as shown in Fig. 3b, the temperature of the fluid in the gap is slaved to thatof the copper plate, and much less heat flows through the gap, and muchmore through the copper. This effect becomes even more pronounced asthe concentration is increased further to X= 1 x 10~3, as shown in Fig. 3c.Therefore we overestimated Rfluid for the larger concentrations, in which farless of the heat flowed through the gaps than in pure 4He.

We calculated Kobs from our simulations in exactly the same way wedid in our experiments, and show the results in Fig. 4. The effective thermalconductivity of dilute superfluid mixtures is predicted to vary as X-1, asindicated by Eq. (2), where X is the 3He molar concentration; this behavioris indicated by the solid black line in the figure. The dashed lines are theresults of our simulations, while the symbols indicate the data measured fordifferent plate separations h. The quantitative agreement between oursimulations and the data demonstrates that the gaps are responsible for thediscrepancy between Kobs and Keff.

Fig. 4. The effective and observed thermal conductivity for dilutesuperfluid mixtures plotted versus X at E= 1 x 1 0 - 3 . The solid lineis the theoretical prediction, the dashed lines are the results of oursimulations, and the symbols are the data of Murphy and Meyer.7

B. AC Response of Normal 4He Near TL

Several aspects of a thermal conductivity cell can affect the response ofa layer of normal 4He to an AC heat flux, including the sidewall, theboundary resistance, and the gap between the sidewall and the endplate. Inorder to separate the effects of these different components from each other,we have simulated the following five situations: a fluid layer alone; a fluidlayer with copper endplates on either side; a fluid layer with copper end-plates and a stainless steel sidewall; a fluid layer with copper endplates,a stainless steel sidewall, and a 0.003 cm thick gap between the sidewalland the plates; a fluid layer with copper endplates, a stainless steel sidewall,and a 0.008 cm gap. The simulation of the fluid layer alone can be com-pared with Behringer's analytical solution. All calculations were done at areduced temperature E= 1 x 10~3 above TL unless otherwise noted.

We have also solved the case with endplates and the fluid layeranalytically, allowing us to compare simulations to analytical solutions.The amplitude and phase of the temperature at the warm end of the cell,which are the experimentally accessible quantities, are plotted versus q inFig. 5a and b. (In this figure the amplitude of temperature oscillations hasbeen normalized to one.)

The first thing to notice in these figures is that for the cases whereanalytical solutions are possible (the fluid, and fluid and endplates), thesimulations agree very well with the analytical solutions, particularly forthe amplitude, where there is no discernible difference between the two.There is a slight discrepancy in the phase at higher frequency (higher q),but the agreement is still quite good.

We can compare the zero-frequency simulation including the sidewallwith the model in which the fluid layer and the sidewall are treated asparallel conductors. This model is only accurate when the temperaturedifference across the fluid and the sidewall are the same, so that the tem-perature gradients are the same. Such a model is not accurate when aspatially uniform heat flux is applied to one end of the fluid and sidewall.In our case, the thick copper plates at the ends of the cell ensure that theformer condition is satisfied, as shown in Fig. 6a where we plot theamplitude of oscillation in arbitrary units versus r and z. We find that thezero-frequency temperature difference across the cell, with endplates andsidewalls, agrees quantitatively with the expectation from the parallelresistance model. (We note that if the plate were made too thin, the fluidand sidewall would not act as parallel conductors.)

At frequencies other than zero, though, the case with a sidewall is dif-ficult to solve analytically, because the response becomes two dimensional.This difference in the radial dependence of amplitude is most obvious if we

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Fig. 5. The (a) amplitude and (b) phase of the response of normalfluid 4He plotted versus q for several cell configurations, asexplained in the text.

examine a plot at non-zero q (in this case q = 8) for the situation both withand without a sidewall, shown in Figs. 6b and 6c, respectively.

Our simulations agree quantitatively with analytical solutions in allcases where such solutions are possible, so we believe that our simulationsare reliable. Returning to Figs. 5a and 5b, the plots of amplitude and phaseversus q, we examine the results of our simulation for the cases where wehave a sidewall, and when we have gaps. The sidewall reduces the zero-fre-quency amplitude as expected because the sidewall provides a parallel path

402 Daniel Murphy


for heat flow. However, as q increases the results of the simulation for theamplitude join with the curves for the cases without the wall. The phase forthe simulations with the sidewall tracks closely the phase of those without.Overall, the addition of the sidewall has only a minor effect on the phase.

The addition of a gap as narrow as 0.003 cm changes the resultsdramatically, though. The zero-frequency amplitude and phase are slightlyaffected by the gap, but as the frequency increases the amplitude neverrejoins the calculations for the case without the gaps. Also, the phasebegins to lag significantly at larger q, which is precisely the behavior obser-ved by Olafsen and Behringer. In Figs. 7a and 7b we compare the results

Fig. 7. The (a) amplitude and (b) phase of the response of a cellwith the same geometry used by Olafsen and Behringer, comparedwith their data.13

of a simulation of Olafsen's and Behringer's thermal conductivity cell, asdescribed by Gao et al.,20 with their data for a fluid layer thicknessh = 0.0996 cm and a reduced temperature e = 0.103;13 both simulations anddata are in real units. At this reduced temperature the thermal conductivityK of normal 4He is 1.46 x 10~2 W/(mK), the density p = 0.145 gm/cm3, andthe heat capacity CP = 9.6 J/(moleK).21 If we use the stated value for thewidth of the gap at the bottom plate (t = 0.0005 cm), the agreementbetween our simulations and the data is only qualitative. However, if weassume that the gap is wider by a factor of five than claimed, our simula-tions are in good quantitative agreement with Olafsen's and Behringer'sdata. Olafsen recalculated the gap width for their cell, and found that itwas 1.1 x 10 ~3 cm, or twice the value reported by Gao et al., which sup-ports the idea that the gap at the bottom of the cell used by Olafsen andBehringer is wider than first reported.22 Recalling that Figs. 5a and 5bshow that a sidewall will not introduce an appreciable phase lag, we cantherefore say that the gap between the sidewall and the copper plate isresponsible for Olafsen's and Behringer's AC results, and that the sidewallby itself plays a relatively minor role.

Another interesting effect is evident in the simulation results plottedversus q for two different gap widths, also shown in Figs. 5a and 5b. Noticethat for q > 8, the phase begins to increase for the 0.008 cm gap, but notthe 0.003 cm gap. A comparison of Fig. 8a and 8b, in which the amplitudeis plotted versus r and z at large q (q= 10) for both gaps, shows why.When the gap is 0.003 cm wide, the amplitude of temperature oscillationsin the gap is the same as that of the copper endplates. When it is 0.008 cmwide, the amplitude of temperature oscillations is smaller in the gap than inthe warm plate. This means that the fluid in the wider gap decouples ther-mally from the plate when the frequency is large, while it is slaved to it whenthe frequency is small. The penentration depth L of the thermal oscillationsis approximately equal to ( D T / w ) l / 2 = h/q; for q = 10, L~0.01 cm. It isprecisely when L approaches the gap width that the temperature of the fluidin the gap decouples from that of the warm endplate, and so over ourwhole frequency range the 0.003 cm gap is coupled, while at high frequencythe 0.008 cm gap decouples.

This result has implications for high frequency measurements; if themodel used to extract the boundary resistance, Rb, at large q does notaccount for the decoupling of the fluid in the gap, the resulting boundaryresistance will be underestimated. Therefore, only cells in which gaps havebeen suppressed will yield meaningful results for Rb if an AC method isused. (Measurements of the boundary resistance in the normal phase whichdo not rely on this AC method, such as those by Lipa and Li,23 are unaf-fected by our analysis.)

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V. CONCLUSIONS

We have shown that simulations of thermal transport experimentscan reproduce complex experimental results if sufficient detail about theapparatus are included. In particular, the presence of small gaps in thermalconductivity cells filled with high-conductivity fluids such as dilute mixturesof 3He in superfluid 4He can affect the apparent conductivities of suchfluids, and our simulations reproduce these effects quantitatively despite thelarge discontinuities in transport coefficients. Our simulations demonstratethat such gaps affect normal fluid AC heat transport measurements as well,leading to a mismeasurement of the relaxation times and hence the thermaldiffusivity of the fluid layer. We have shown that experiments whichmeasure Rb in the normal phase using high-frequency AC heat fluxes, asproposed by Olafsen and Behringer,13 are only possible in cells withoutgaps between the sidewall and endplates, because the fluid in the gaps willthermally decouple from the endplates at high frequencies.

ACKNOWLEDGMENTS

The author would like to thank Horst Meyer for his support while thiswork was being done, and Andrei Kogan for many valuable discussions.The author would also like to thank Horst Meyer, Jeffrey S. Olafsen, andRobert P. Behringer for their comments on a draft of this paper.

REFERENCES

1. R. Duncan, R. Akau, S. Gianoulakis, U. Israelsson, and T. C. Chui, Physica B 194-196,603 (1994).

2. R. Duncan, S. Boyd, W. Moeur, S. Robinson, R. Akau, and S. Gianoulakis, Int. J.Thermophys. 17, 631 (1996).

3. M. J. Adriaans, W. A. Moeur, S. T. P. Boyd, D. M. Strayer, and R. V. Duncan, Cryogenics36, 787 (1996).

4. D. Murphy and H. Meyer, J. Low Temp. Phys. 107, 175 (1997).5. D. Murphy and H. Meyer, J. Low Temp. Phys. 105, 185 (1996).6. D. Murphy and H. Meyer, J. Low Temp. Phys. 109, 801 (1997).7. D. Murphy and H. Meyer, J. Low Temp. Phys. 99, 745 (1995).8. Q. Li, Ph.D. thesis, Stanford University (1990) (unpublished).9. See, for instance, I. M. Khalatnikov, An Introduction to the Theory of Superfluidity, W. A.

Benjamin, Inc., New York (1965).10. F. London, Superfluids, Vol. II, Dover Publications, New York (1954), p. 192.11. I. M. Khalatnikov and V. N. Zharkov, Zh. Eksp. Teor. Fiz. 32, 1108 (1957) [Sov. Phys.

JETP 5, 905 (1957)].12. J. S. Olafsen and R. P. Behringer, Physica B 194-196, 595 (1994).13. J. S. Olafsen and R. P. Behringer, J. Low Temp. Phys. 106, 673 (1997).14. R. P. Behringer, J. Low Temp. Phys. 81, 1 (1990).

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15. R. V. Duncan and G. Ahlers, Phys. Rev. B 43, 7707 ( 1 9 9 1 ) .16. M. Dingus, F. Zhong, and H. Meyer, J. Low Temp. Phys. 65, 185 (1986).17. William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling,

Numerical Recipes in C, Cambridge University Press, Cambridge (1988).18. Stephen McCormick (ed.), Multigrid Methods, Society for Industrial and Applied Mathe-

matics, Philadelphia (1987).19. W. Hackbusch, Multigrid Methods and Applications, Springer-Verlag, Berlin (1985) .20. H. Gao, G. Metcalfe, T. Jung, and R. P. Behringer, J. Fluid Mech. 174, 209 (1987) .21. C. F. Barenghi, P. G. J. Lucas, and R. J. Donnelly, J. Low Temp. Phys. 44, 491 ( 1 9 8 1 ) .22. J. S. Olafsen, Ph.D. thesis. Duke University (1994) (unpublished).23. J. A. Lipa and Q. Li, Czech. J. Phys . 46-S1, 185 (1996).


simulations of a thermal conductivity cell filled with normal 4he and dilute mixtures of 3he in...

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