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Indeed, many different systems have critical points, and large fluctuations of some thermodynamic parameter are a universal feature. Because critical fluctuations become macroscopic in size and involve enormous numbers of molecules, many features of critical-point behavior are controlled by the statistical behavior of the fluctuations, so that many types of systems exhibit the same behavior near the critical point.
Light scattering from critical fluctuations in a fluid is a simple and accurate technique to measure one fundamental property of the fluctuations: their decay rates (the inverse of their lifetimes). However, near the critical point, the fluid becomes highly compressible, so that just the weight of the fluid itself causes severe density gradients in the sample, distorting these measurements in a terrestrial laboratory.
The goal of the Zeno experiment was to build a precision, light-scattering spectrometer to measure the fundamental fluctuation decay rates in a sample of xenon, which would operate in the low-gravity environment of the Space Shuttle, thus removing the gravity-induced distortion of the measurements. The instrument performed beautifully on its first mission, and we were able to make our projected measurements with 1% precision to within 100 microK of the critical temperature (16.7 degrees C).
The light-scattering spectrometer is shown schematically in Fig. 1 (browser: figure 1). Light from the low-power, Helium-Neon laser was directed into the sample along one of two beam paths determined by the beamsplitter B1; the path was chosen by shutters (not shown). Light scattered by the sample (located in the cylindrical thermostat at the center of the figure) was then collected by the apertured photomultiplier tubes (PMT 1 and PMT 2), the primary signal which was processed to obtain the decay rate information. In addition, photodiodes (PD 1 and PD 2) placed behind partially-reflecting mirrors (M 2 and M 3) measured the intensity of the light entering and exiting the sample. These two signals were then ratioed electronically to give us information about the turbidity of the fluid, itself an important measurent and our primary method of locating the critical temperature of the sample.
The collection angles for the scattered light (determined by apertures A1, A2, A3, and A4) were precisely aligned so that the two scattering angles were supplements. This accomplished two things: it simplified data analysis, and it allowed us to measure decay rates at the same pair of widely-spaced angles regardless of the beam path through the sample.
The instrument was housed in two separate containers attached to the USMP carrier. The Optics Module contained the light-scattering spectrometer, built on a stiff, optical bench, in addition to temperature-sensitive signal amplifiers and ratio transformers for the thermostat's temperature control and measurement system. All addition electronics, including the instrument's computer system, were housed in the Elecronics Module.
During MET Day 0 the instrument was primarily in a warm-up and initialization phase. Then, just prior to the start of MET Day 1 the sample was warmed to 1 K, and the first scan to locate Tc began. The scan was followed by two warming periods. The slow cooling that spans MET Days 2 and 3 represents a zig-zag search for Tc, which extended into MET Day 4, followed by another warming procedure.
The first period of data collection began midway through MET Day 4. Decay rate measurements were collected at a geometrically-spaced set of temperatures progressively nearer Tc, the first one starting at 100 mK. The last measurements of this set, during MET Day 7, were at temperature within 100 microK of Tc, and actually contained some measurements at temperatures just below Tc.
MET Day 8 began with another warming cycle, followed by another period of data collection at various temperatures between 560 mK and 30 microK above Tc. Between MET Days 9 and 12 we took some additional measurements at temperatures futher from Tc, as well as repeating selected measurements from earlier in the mission. Near the beginning of MET Day 11 we began a series of 100 microK cooling steps which would again take the sample below Tc, recording decay rate measurements during the process with the hope of getting measurements which would show the effects of crossing the critical point into the two-phase region at temperatures at which it is impossible to make measurements on earth.
In flight, the thermostat required a stable thermal environment to do its job: the temperature contol of the Optics Module by radiative heat leak and control heaters was excellent. Because of the length of the mission and the long periods of constant Shuttle attitude, we were able to maximize the temperature control. Thus, the thermostat was able to achieve a routine control level of better than +/- 3 microK rms noise for periods of three hours or longer. Figure 3 (browser: figure 3) shows one period of four hours when the sample cell temperature was controlled to 10 microK peak-to-peak (with a 5-second time constant in the measurement) and 1.7 microK rms. This level of control easily met the requirements for the Zeno experiment.
To change the temperature of the sample, the set points of all three control shells of the thermostat were moved. Figure 4 (browser: figure 4) shows an example of a simple temperature change (cooling) of 2 mK; the temperature of the sample cell follows with a time constant of about 30 minutes.
For smaller temperature changes, we were able to make "fast steps" where the set points of the control shells were temporarily moved well beyond the target temperature and then returned just as the sample cell reached the desired temperature. Figure 5 (browser: figure 5) illustrates the fast change, which is finished in a matter of minutes.
The photomultiplier tubes were operated in photon-counting mode and the two signals were passed to the dual inputs of an ALV-5000 digital correlator, adapted specifically for this instrument. With this correlator we were able to make simultaneous measurements of forward-scattered and backscattered light. The correlator is a special-purpose, digital signal processor, which produces an accurate autocorrelation function of the stream of photocounts from the photomultipliers. This correlator is designed to cover a broad range in delay times by calculating the autocorrelation function on a geometrically expanding time scale so it is natural to plot the resulting correlograms with a logarithmic time scale. Figure 6 (browser: figure 6) shows a sample correlogram; note that this shape represents an exponential decay of the autocorrelation function.
The experiment was performed with very low light levels. The windows of the sample cell absorb small amounts of the incident light at their surface (measured to be about 2 ppm). Because the thermal expansion coefficient of the sample diverges at the critical point, the increase in temperature of the window surface due to absorbed light can lead to unwelcome perturbations in the density of the fluid, and measurements would no longer represent the behavior of the fluid at the temperature we measured for the bulk (see bibliography).
To avoid these problems, we reduced the power of the laser light by placing filters in the beam paths, selecting two different power levels that allowed us to cover two temperature ranges with density perturbations of less than 1%. A power level of 17 microW was used between 1 K and 1 mK from Tc, and the lower level of 1.7 microW was used at temperatures closer than 1 mK, with overlapping measurements made with both powers near 1 mK. With this combination of incident light powers we were able to cover a wide range of temperature with enough signal to reach our precision requirement of 1\ in the decay-rate measurements within the length of the mission.
In general, locating Tc involved moving the sample to a region very near the critical temperature and then adjusting its temperature in a controlled way and watching for signs that the critical point has been crossed. The signal which we monitored during the searches measured the turbidity of the sample, given by the logarithm of the ratio of the intensity of the light transmitted by the sample to the intensity of the incident beam. We attempted two techniques with different expected levels of precision; one worked, and one didn't.
The zig-zag technique, which we used to try to increase our level of precision for the value of Tc and which didn't work successfully. is a technique that we had previously used to locate the critical-mixing temperature of density-matched mixture of two fluids. In a density-matched critical mixture gravity has little effect and the technique gave clear indications, with 10 microK precision, whenever Tc was crossed. However, we were not able to try the technique with a pure-fluid system on earth because the dynamics of density stratification were too rapid. Our single zig-zag search on orbit was the first time that this technique had been tried in a pure fluid.
The zig-zag search started only 200 microK above the critical temperature. Figure 7 (browser: figure 7) shows the manner in which the temperature of the sample changed, alternately cooling by 20 microK and then warming by 10 mircorK; Fig. 8 (browser: figure 8) shows the response of the turbidity signal during the same portion of the search. Although it is clear that the sample is responding to the changes in temperature, we never detected a clear hysterisis in the turbidity which we expected to indicate that we had crossed Tc. We believe now that several unexpected effects in the turbidity signal (see the next secton) combined to obscure the transition. The entire period of the zig-zag search is shown in Fig. 9 (browser: figure 9). At some point just past MET Day 3, hour 12, the sample first crossed Tc and began the early stages of spinodal decomposition, indicated by the increasing amplitude of the variations in the signal. However, we will not be able to extract more precise information until we understand all of the effects in the turbidity response better.
Our second technique for locating Tc, which was successful, is a simple scanning procedure. The set point of the thermostat's control shells was changed in small steps of 50 microK every 6 minutes (Fig. 10) (browser: figure 10), which caused the temperature of the sample to change at a uniform rate of 500 microK/hour (Fig. 11) (browser: figure 11).
During the scan we monitored two signals as the sample crossed Tc: the turbidity of the sample, and the photon count rates of the light scattered by the sample. Figure 12 (browser: figure 12) shows the turbidity signal during the first of the three scans done during the mission. Between hour 4 and hour 6 the signal increased steadily as the critical temperature was approached. At a few minutes past hour 6 there was a small break in the increase, followed by a swift rise, then decline, in the signal. The break marked the sample's crossing Tc; the large peak was caused by the sudden increase in density fluctuations as the fluid began spinodal decomposition, the initial stages of phase separation in which the fluctuations first becomes unstable and begin to grow in size. This increase passed as the fluid began to resolve itself into distinct gas and liquid phases, and the turbidity signal dropped.
That this rather unassuming feature in the turbidity signal marked the phase boundary was confirmed by the intensity of the forward-scattered light, shown over the same time period in Fig. 13 (browser: figure 13). The light scattered at this angle increased as the critical point was approached, and then dropped as fluctuations at this wavelength lost stability and the fluid began phase separation. The position of the peak in the scattering-intensity curve matched that of the feature in the turbidity curve.
For comparison, Fig. 14 (browser: figure 14) shows a scan done on the instrument during pre-flight testing on the ground, at the same rate as the one in Fig. 12. There are two notable differences in the curves, both related to the effects of gravity. In the second hour of the terrestrial scan the turbidity appeared to be decreasing rather than increasing steadily. This was caused by the slowly-forming density gradient in the highly compressible sample: as the gradient became more severe, the amount of fluid near enough to its critical density to show increasing turbidity decreased, and the net effect was a decrease in the turbidity signal. In addition, the small feature immediately preceeding the phase-decomposition peak that marked the phase transition is not even visible. At this level of resolution, gravity had broadened the critical point to such an extent that it effectively disappears from the turbidity response.
During the mission the collection of three scans and the zig-zag Tc determinations seemed to show a drift in the critical temperature of 800 microK, substantially larger than we had seen during all testing with this instrument and with the functionally identical engineering model. However, we now believe that this may have been caused by small density rarifications in the beam path that resulted from repeated crossings into the two-phase region that were not completely removed by the warming procedures which we used to rehomogenize the sample. Subsequent ground scans with the instrument, along with the first on-orbit scan and previous ground measurements now tell us unambiguously that the critical temperature of the sample had indeed been stable during the mission to within the precision of our measurement of +/- 50 microK. This gives an upper bound of 120 microK/year on the stability of this sample, a drift rate spectacularly better than any previously experienced.
The turbidity of a fluid increases as the fluid is taken nearer to its critical point; measuring the turbidity gives information about the average size (the correlation length) of critical fluctuations. The turbidity measurements which we made during the mission have proved to be the biggest challenge to our understanding of the flight data. During the mission the turbidity signal showed small, long-time drifts at constant temperatures, sometimes to smaller values, sometimes to larger values. We believe that this was related to the small density inhomogeneities which we conjecture caused the apparent drifts in Tc. However, the analysis of these data are far from complete at this point.
Extracting values which actually represent the turbidity of the sample is complicated by two known effects which make difficult a direct, simple turbidity measurement. We have modelled these effects extensively and we now have a better understanding of their contribution. The first has to do with the optical properties of the sample cell itself; the second is a manifestation of the thermal response of a fluid near its critical point.
The interior of the sample cell created a 100-micron thick disk of xenon, the portion of the sample through which the laser beam passed, by bounding the region with two polished, parallel, fused-quartz windows. Reflections of the beam from the window surfaces can interfere, changing the intensity of the transmitted beam depending on the spacing of the windows. When the temperature of the sample cell was changed, thermal expansion or contraction of its components altered the window spacing and produced a measurable change in the turbidity signal, independent of the turbidity of the xenon itself.
Figure 15 (browser: figure 15) shows the turbidity signal responding to 100 mK steps in the temperature of the sample cell, which warmed the cell from 1 K to 5 K above the critical temperature of the xenon; this curve is from pre-flight testing of the instrument. At these temperatures the actual turbidity of the fluid is negligible and we are viewing the interferometric behavior of the sample cell only.
The large, sinusoidal variations in the curve were the result of the interference effect, showing that the separation of the windows changed by about 0.75 orders of interference per Kelvin, at approximately 400 orders of interference. This corresponded to a change in the window separation of 0.28 microns/K, a rather small effect. Nevertheless, it is clear from the magnitude of the change in Fig. 15 that in order to make good use of our turbidity measurements it is vital that we know where on this interference curve the measurements were made.
We were able to fit curves like this one with ample precision to make the necessary correction to the turbidity measurements. Important questions about how this characteristic of the instrument changes with time are now settled as well. We have found, from analysis of testing both before and after the mission, that the amplitude of this curve was stable and predictable. However, the phase of the interference showed a drift with time which suggests that the sample cell exhibits a small mechanical creep of its longitudinal spacing of about 10 nm/month. With this information we can be confident that we have placed the flight measurements at the appropriate place on the interference curve.
There is a second feature evident in Fig. 15 which is related to the fluid behavior itself. With each temperature step the turbidity signal, rather than simply moving to a new value appropriate to that temperature, overshot and then returned to the expected value; all of these temperature steps were of the fast-change variety. Although the fluid at these temperatures showed no critical enhancement to its turbidity, it was still near enough to the critical point that it was highly compressible. We have shown in previous work (see bibliography) that highly-compressible critical fluids can make very fast changes in their density via an adiabatic heat-transfer mechanism; this phenomenon is a manifestation of the adiabatic mechanism.
As the temperature change began, heat entered the outer walls of the sample cell and quickly caused a large thermal expansion of a thin layer (order 10 micron) of fluid near the wall. As a result, because of high compressibility, the bulk of the fluid was subject to adiabatic compression, which caused the bulk to cool rapidly. Significantly, the density of the bulk had changed, changing its index of refraction, hence the optical path length of the laser beam passing through that part of the sample. Thus, during the temperature change itself, the fluid in the optical path of the cell executed density changes which affected the interference characteristics of the cell. Note that this transient response depends on the temperature of the fluid and decreased as the sample was moved further from Tc.
We have been modelling this phenomenon extensively, as well as performing controlled experiments with our engineering model of the instrument. These analyses have now converged in a satisfying way.
Figure 16 (browser: figure 16) demonstrates the adiabatic effect, using data from the engineering model. The sample cell was subjected to a fast change of 100 mK, far enough from Tc that the actual turbidity of the xenon is negligible. Recorded in the lower curve is the response in the measured turbidity signal showing the overshoot and then return to its final, equilibrium value. It is significant that the final value is reached at precisely the same time as the temperature change is completed.
The upper curve in Fig. 16 shows data from our calibration of the apparent change in turbidity, represented as a change in phase along the interference curve, due solely to the interference effects of cell expansion. The difference between these two curves, shown in Fig. 17 (browser: figure 17), shows the effect due solely to a change in the fluid density from adiabatic heating effects. Since the adiabatic effects came into play only in response to a changing temeprature, we can verify this relationship by comparing the adiabatic response in Fig. 17 to Fig. 18 (browser: figure 18), which is the derivative of the sample-cell temperature measured during the temperature change: they have exactly the same shape. Note that the temperature is measured at the cell wall, approximately 1.5 cm from the region of fluid probed by the laser, confirming the long-range action of the adiabatic effect.
All of the mysteries of the flight turbidity data have not been settled. Figure 19 (browser: figure 19) shows the transient response of the turbidity signal when the sample cell temperature was changed by 70 mK (starting from Tc + 0.1 K). The adiabatic contribution that we have measured on earth is evident here in the initial sudden decrease in the signal as the temperature change began. However, this was soon overwhelmed by a much larger change in the opposite direction, followed by a slower settling to the final value. We know that this response is apparent because gravity has been removed: we had never seen it in data taken during ground testing. We believe that it is related to later stages in an adiabatic response that is quenched by convection in 1 g, but we have not yet completed the modelling of the behavior.
The next several figures show representative correlograms, illustrating the range of temperatures and incident-beam intensities. Figure 20 (browser: figure 20) shows a pair of correlograms taken 560 mK from Tc, the furthest from the critical temperature at which we made measurements. As in all the correlograms, the upper curve with the slower decay is the forward-scattering correlogram. At this temperature the intensity of the scattered light was at its lowest, although we were using the higher power beam path. The decays of the two curves, particularly the backscattering curve, are barely discernable, although the measurements are statistically meaningful.
Figure 21 (browser: figure 21) shows a pair taken 1 mK above the critical temperature, using the higher-intensity beam path. At this temperature the critical fluctuations were well developed, the intensity of the scattered light was relatively high in both directions, and the correlograms appeared virtually noiseless. However, a new feature is evident here in the backscattering correlogram: it shows two decays. The second decay, which has the same time scale as the forward scattering correlogram, is actually forward scattered light.
The forward-scattering cross section had become large enough, 100 times the backscattering cross section, that the small amount of light reflected in the backward direction at the exit window of the sample cell contributed a forward-scattering correlation to this signal. It was for this reason that the two scattering angles were chosen to be precise supplements: results from the analysis of the forward-scattering correlograms could be used to remove this contribution from the backscattering correlograms.
For comparison, Fig. 22 (browser: figure 22) shows a pair of correlograms at the same temperature as in Fig. 21, but at the lower beam-intensity of 1.7 microW. The backscattering correlogram shows an increased amount of noise (due to the statistical process of the light detection), but the decay rate obtained from it is still precisely determined. Finally, Fig. 23 (browser: figure 23) shows correlograms collected at Tc + 100 microK. These measurements are well inside a region in which, on earth, the observation of fluctuation correlations is completely distorted by gravity.
The data sets were collected for a total amount of time which we had predicted would result in fluctuation decay-rate measurements with a precision of 1% or better. We achieved this precision with all of the measurements except for the backscattering measurement at the warmest temperature, where the extremely low intensity of the scattered light precluded it. Each data set was made up of 10--15 separate correlograms so that we could examine our results statistically.
The decay rate values were extracted from the correlograms by non-linear, least-squares fitting. The forward-scattering correlograms were fit to a simple decaying exponential; for the backscattering correlograms the forward-scattering contribution was first removed where necessary, and then a simple decaying exponential was used.
The results of the fluctuation decay-rate measurements are shown in Fig. 24, (browser: figure 24), the backscattering decay rates, and Fig. 25, (browser: figure 25), the forward-scattering decay rates. In both figures we have included data from ground testing on the instrument (in both cases the upper curve of symbols), as well as a line representing the best available theoretical predictions.
Quite evident in these graphs is the effect of removing gravity and the distortions it creates in the density of the sample, already apparent as far as 30 mK from the critical temperature. The limiting values of these curves is a significant measurement, and it is clear from these graphs that it is not possible to make accurate measurements of the limits on earth, whereas we have been able to penetrate well into the limiting region without the distortions due to gravity. As measurements were made closer yet to Tc the density gradient in the earth-bound sample, as probed by the 100-micron diameter of our focused laser beam, led to errors of 100%, doubling the apparent decay rate in the terrestrial measurements compared with the measurements in microgravity.
We believe that the limiting values of the fluctuation decay rates from our data set will be the key contributions from this flight of the Zeno instrument. It is true that some questions remain about the homogeneity and precise density of the xenon at each temperature, questions raised by the drifts in the turbidity signal. Nevertheless, we have made measurements so close to the critical temperature that we have been able to make accurate measurements of the limiting values of the decay rate, accurate regardless of the possibly small variations in density. These limits are beyond the predictive capabilities of current theories and represent a challenge and target for future calculations of transport properties in near-critical systems.
To conclude, we present our results for the limits. From the backscattering, the limiting decay rate is 51200 +/- 400 rad/s at a scattering angle in the fluid of 169.5636 degrees. From the forward scattering, the limiting decay rate is 36.8 +/- 0.3 rad/s at a scattering angle inside the fluid of 10.4478 degrees.
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