REVIEW OF SCIENTIFIC INSTRUMENTS
VOLUME 73, NUMBER 9
SEPTEMBER 2002
High-temperature contactless viscosity measurements by the gas–film levitation technique: Application to oxide and metallic glasses Paul-Henri Haumesser, Jacky Bancillon, and Michel Daniel Commissariat a` l’Energie Atomique, DTEN/SMP/LESA, CENG, 17 av des Martyrs, 38054 Grenoble, France
Michel Perez INSA de Lyon GEMPPM, 25, Avenue Jean Capelle, 69 621 Villeurbanne Cedex, France
Jean-Paul Garandet a) Commissariat a` l’Energie Atomique, DTEN/SMP/LESA, CENG, 17 av des Martyrs, 38054 Grenoble, France
共Received 7 November 2001; accepted for publication 13 June 2002兲 In the field of thermophysical characterization of materials at high temperature, a crucial issue is to limit the effect of chemical or physical phenomena occurring at the interface between the sample and the container. Therefore, contactless techniques are well adapted to high-temperature measurements. The gas–film levitation method has recently proved to be applicable to viscosity measurements. Our purpose in this article is to derive viscosity values by the observation of the dynamical response of a perturbed levitating drop. We present here recent improvements in this technique, with particular attention paid to measurement accuracy issues and temperature calibration problems. Viscosity measurements performed on oxide and metallic glasses reveal that a gas–film levitation based viscometer is able to provide measurements with good accuracy 共⫾10%兲 in a wide viscosity range, from very viscous 共up to several kPa s, aperiodic relaxation of the drop兲 to fluid liquids 共a few mPa s, damped oscillation of the drop兲. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1499756兴
I. INTRODUCTION
both relaxation regimes, viscosity is related to a characteristic relaxation time constant. Section IV is devoted to the description of the experimental procedures to determine this time constant, with particular emphasis on measurement accuracy issues. To complete the measurement procedures, the temperature calibration of the device is described in Sec. V. At last, experimental results are reported in the case of a soda lime 共aperiodic regime兲 and a metallic glass 共damped oscillation regime兲.
Thermophysical properties of liquid materials at high temperature are difficult to evaluate by standard techniques, mainly due to chemical or physical phenomena occurring at the interface between the sample and the crucible, such as contamination or wall solidification. To overcome this difficulty, contactless experiments under microgravitational conditions have been carried out.1 Levitation techniques were developed as well, which make use of electrostatic,2 electromagnetic,3 aerodynamic,4 or aeroacoustic5 phenomena. Another method first described by Granier and Potard6 is the gas–film levitation technique, based on the gas–film lubrication theory. This method consists of making a liquid drop float on a thin gas film and is applicable to a wide range of materials. First developed for defect-free materials synthesis, this technique was proposed by Papoular and Parayre7 to measure viscosities at high temperature, by a dynamical study of the response of the drop to an imposed perturbation. This method has proved to be efficient to measure hightemperature viscosity of oxide glasses.8 The aim of this article is to present the recent improvements and latest results concerning viscosity characterization of oxide and metallic glasses. After a brief description of the experimental setup, the dynamical response of a drop to a small perturbation will be studied, which can be either aperiodic or oscillating depending on the liquid viscosity. In
II. EXPERIMENTAL SETUP
The apparatus is designed to perform the deformationrelaxation process presented in Fig. 1 at high temperature. The block diagram of the device and a closeup on the sample chamber are presented in Figs. 2共a兲 and 2共b兲, respectively. The sample 共typically a few hundred mg兲 is placed between two graphite membranes, namely the bearing and crushing diffusers. The sample levitation is obtained by the forcing of pressurized argon through the bearing porous membrane; the sample floats on a thin gas film 共⬍100 m兲. The bearing diffuser is curved to laterally stabilize the sample 共Fig. 1兲. The gas flow is regulated by controlling the pressure difference across the porous membranes 共typically 1.5 bar兲, ensuring a good mechanical stability of the sample. The device is induction heated by direct coupling on the graphite diffusers using a 50 kW and 100 kHz Celes radio frequency generator. Temperatures as high as 1700 °C can be reached with a good homogeneity 共see Sec. V兲 around the sample using an appropriate thermal insulator surrounding the hot parts. The device temperature is measured by a ther-
a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
0034-6748/2002/73(9)/3275/11/$19.00
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FIG. 1. Drop deformation and relaxation in the gas–film levitation process. The levitation is obtained by a gas flow through a porous element. The levitating sessile drop 共a兲 is crushed using a second diffusing element 共b兲. After releasing the deformation, the drop recovers its equilibrium shape 共c兲.
mocouple placed in contact with the inner face of the bearing membrane, opposite to the sample. The temperature is monitored using an Eurotherm controller. The sample is heated up to the working temperature, and thermal equilibrium is reached within a few minutes. This duration can be compared to a characteristic heat transport time in graphite defined by
th⫽
L C2
␣C
⫽
L C2 C C ,C KC
,
共1兲
where ␣ C is the thermal diffusivity, C the mass density ( C ⫽1550 kg/m3 ), K C the thermal conductivity (K C ⫽76 W/m/K), C p,C the specific heat (C p,C ⫽1250 J/kg/K) of porous graphite, and L C a characteristic length of the dif-
fuser 共its diameter L C ⫽0.03 m兲. The so-calculated characteristic time is 23 s, in good agreement with the few minutes experimentally required to reach thermal equilibrium. The deformation of the drop is performed by approaching the crushing diffuser, using a Schneeberger motion controller to assess the perturbation speed and amplitude. Typical values are an amplitude of 300 m 共about 10%–15% of the typical size兲 at a 100 m/min rate. The gas flow is obviously maintained during the whole process to avoid any contact between the sample and the graphite. The perturbation is suddenly released 共within a few ms兲. When the drop relaxation is aperiodic, it is recorded using a Sony high resolution color video camera and a Sony U-matic video recorder at a 50 fps rate. Typical records cover a 10 time period, and up to about 500 frames, processed whenever possible 共i.e., when 10 ⬎10 s兲. In the case of a damped oscillation, a Weinberger camera was used at a 500 fps rate with a 2000 frame capacity. The temperature can then be modified, and a new crushing-relaxation process engaged. It should be noted that when exploring temperature ranges where the sample viscosity is high 共with an associated aperiodic relaxation time exceeding a minute兲, it is necessary to crush the drop at higher temperature, where the sample is fluid enough to accommodate the imposed perturbation 共a few hundred m in a couple of minutes兲. The device can then be brought to a working temperature and the relaxation recorded. III. RELAXATION OF PERTURBED VISCOUS DROPS A. Shape factor and Chandrasekhar’s theory
Let us consider a drop of mass m and density levitating on a thin gas film, as described in Fig. 1共a兲. Its shape results from the contradictory effects of gravity 共flat drop兲 and surface tension 共spherical drop兲. To quantify the respective influence of both phenomena, the shape factor is introduced: f⫽
R eq , lc
共2兲
where R eq is the radius of the equivalent spherical drop, given by
冉 冊
3 m R eq⫽ 4
1/3
共3兲
and l c is the capillary length, defined by l c⫽
FIG. 2. Block diagram of the gas–film levitation viscometer 共a兲 and detail of the sample chamber 共b兲. 1—Vessel; 2—crushing diffuser; 3—bearing diffuser; 4—thermal insulator; 5—regulation thermocouple; 6—control thermocouple; 7—light source; 8—translation controller; 9—hydraulic jack; 10—vacuum unit; 11—CCD high-resolution camera; 12—pressure regulation unit; 13—video recorder; 14—control computer; and 15— radiofrequency generator.
冑
. g
共4兲
If f Ⰶ1, the capillary effect takes over, and the drop is spherical, whereas if f Ⰷ1, the drop is flat. For practical purposes, a shape factor close to unity is desirable. The response of a spherical drop in the absence of gravity to a small perturbation was described by Chandrasekhar.9 Two relaxation regimes are possible, depending on the liquid properties and on the drop radius R. A critical radius may be defined as follows: R *⫽
2 ,
共5兲
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Rev. Sci. Instrum., Vol. 73, No. 9, September 2002
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where S(R p ) is the ellipsoid surface. The condition E/ R p ⫽0 is used to determine the polar radius at equilibrium R p0 . The drop is considered to behave like a viscoelastic element 关Fig. 3共b兲兴 whose stiffness constant is equal to K⫽
冏
2E R 2p
共10兲
. R p ⫽R p0
The motion equation of the drop can be modeled as10 FIG. 3. Modeling of the drop in the variational approach. The drop shape is assumed to be an ellipsoid 共a兲, and the drop of mass m behaves like a viscoelastic element of stiffness K and dissipative coefficient 共b兲.
where is the dynamic viscosity. If R eq⬍R * 共which is typically observed when ⬎1 Pa s兲, the relaxation is aperiodic, the drop recovering its equilibrium shape exponentially, with a characteristic time given by
a,Ch⫽
19 R eq . 20
共6兲
If R eq⬎R * 共for values typically lower than 1 Pa s兲, the relaxation is a damped oscillation with a damping time constant p,Ch given by
p,Ch⫽
1 5
2 R eq
.
共7兲
In both cases, the assumption is made that the only Kelvin mode involved in the relaxation process is the l⫽2 mode.8 This hypothesis is reasonable as this mode is the best adapted to the imposed deformation geometry; the excitation is therefore quite selective, the l⫽2 mode being preferentially excited. In addition, higher modes, if ever present, have shorter decaying times.
B. Variational approach
Chandrasekhar’s theory is only valid in the absence of gravity, considering a symmetrical deformation of the drop. In the actual device, however, none of these conditions is fulfilled 共Fig. 1兲; the drop is placed in earth’s gravity field, and nonsymmetrically deformed, as its south pole is fixed. To evaluate the influence of these effects, the same variational approach, as described elsewhere in the case of forced oscillations,10 can be used. This will lead us to introduce two corrective factors, namely, C a and C p in the aperiodic and periodic modes, respectively. In the variational approach, the drop shape at equilibrium is approximated by an ellipsoid of volume: 3 , V⫽ 43 R p R 2e ⫽ 34 R eq
共8兲
where R p and R e are the polar and equatorial radii, respectively 关Fig. 3共a兲兴. As this volume is constant and equal to the volume of the actual drop, this ellipsoid is completely defined by its polar radius R p . In this description, the energy of the drop is given by E 共 R p 兲 ⫽ S 共 R p 兲 ⫹ VgR p ,
共9兲
⌬R¨ p ⫹2⌬R˙ p ⫹
K ⌬R p ⫽0, m
共11兲
where ⌬R p is the instantaneous small deformation around R p0 and is a dissipative coefficient. The response of the drop to an initial deformation ⌬R p0 can proceed through two regimes, as in Chandrasekhar’s theory. If ⬍ 冑 (K/m), the relaxation is a damped oscillation of the drop, with a pulsation and a damping characteristic time p :
冉 冊
R p ⫽R p0 ⫹⌬R p0 exp ⫺
t cos共 t⫹ 兲 p
共12兲
with
⫽
冑
K ⫺ 2 . m
An aperiodic relaxation is observed when ⬎ 冑 (K/m), which is characterized by a relaxation time a :
冉 冊
R p ⫽R p0 ⫺⌬R p0 exp ⫺
t . a
共13兲
The energy dissipated during both relaxations processes is given by E diss⫽ 21 K⌬R 2p0 .
共14兲
Another way of determining this energy is to evaluate the displacement field within the drop, considering that the polar radius varies from R p0 ⫺⌬R p0 to R p0 . This approach leads to the following expression of the dissipated energy in the aperiodic regime: E diss⫽
3 V ⌬R 2p0 . 2 a R 2p0
共15兲
Comparison of Eqs. 共14兲 and 共15兲 leads to the following expression of viscosity:
⫽
1 R 2p0 3 K a 共 aperiodic regime 兲 . 4 R eq
共16兲
Similar calculations performed considering a damped periodic oscillation leads to much more complicated expressions of the displacement field dissipated energy. However, the comparison with Eq. 共14兲 before integration is straightforward, leading to the following expression for viscosity: 2 R 2p0 共 damped oscillation regime兲 . 3 p
⫽
共17兲
To sum things up, the influence of nonideal conditions with respect to Chandrasekhar’s formalism 共namely gravita-
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tional deformation and asymmetric relaxation兲 can be accounted for through the ratio / Ch in both relaxation modes: 1 38 R 2p0 K ⫽ C a⫽ 共 aperiodic regime兲 , 2 Ch 4 40 R eq
⌬
⫽
⫹
共18兲
10 R 2p0 C p⫽ ⫽ 2 共 damped oscillation regime 兲 . 共19兲 Ch 3 R eq
⌬
⫽
R p0 , R eq
共20兲
˜f r2 ⫽
3 K . 32
共21兲
Both coefficients only depend on the shape factor f. An approximated relationship is given with the following polynomial expression with an accuracy better than 1% in the 0 ⭐ f ⭐2 range: ˜R p0 ⫽0.0679f 3 ⫺0.1949f 2 ⫺0.1277f ⫹1.0127,
共22兲
˜R p0˜f r ⫽⫺0.003 707 34f 6 ⫹0.035 7917f 5 ⫺0.062 550 07f 4 ⫺0.073 349 42f 3 ⫹0.264 157 32f 2 ⫺0.029 605 84f ⫹0.550 402 45.
共23兲
Finally, C a 共 f 兲 ⫽ 共 ˜R p0˜f r 兲 2 共 aperiodic regime兲 ,
共24兲
C p 共 f 兲 ⫽ 103 共 ˜R p0 兲 2 共 damped oscillation regime兲 .
共25兲
38 15
Thus, considering the influence of the actual experimental conditions on the relaxation processes leads to the following corrections on the expressions given by Chandrasekhar:
⫽
20 a C 共 f 兲 共 aperiodic regime兲 , 19 R eq a
2 1 R eq ⫽ C 共 f 兲 共 damped oscillation regime兲 . 5 p p
共26兲 共27兲
As a conclusion, when considering a damped oscillation relaxation, the corrective coefficient C p ( f ) varies in the range of 2.4 to 1.6 when f varies from 0.7 to 1.2. Such a correction obviously needs to be taken into account. In the case of an aperiodic relaxation, the corrective factor C a ( f ) varies between 1 to 1.3 within the same f values range. As will be shown below, the viscosity measurement uncertainty in the case of an exponential decay of the deformation can be as low as ⫾10%. This is why the correction introduced here to account for the actual experimental conditions has been considered to derive all the results presented below. C. Viscosity measurement accuracy
A crucial issue in the development of a measuring tool is the assessment of the measurement accuracy. From Eqs. 共26兲 and 共27兲, this uncertainty is given by
⌬C a 共 f 兲 共 aperiodic regime兲 , C a共 f 兲
共28兲
⌬p ⌬ ⌬R eq ⫹ ⫹2 p R eq ⫹
These coefficients can be estimated setting ˜R p0 ⫽
⌬ a ⌬ ⌬R eq ⫹ ⫹ a R eq
⌬C p 共 f 兲 共 damped oscillation regime兲 . C p共 f 兲
共29兲
The viscosity determination requires the evaluation of several parameters, namely, the relaxation time a or p , the surface tension , the density , and the mass m of the drop. It should be mentioned that our experimental device allows to measure surface tension and mass density independently from viscosity, by a fit of the equilibrium shape of the drop. The procedure 共step by step integration of Laplace’s law兲 as well as the related uncertainties is comprehensively discussed elsewhere.11 The relaxation time measurements will be detailed below. Let us assume the values of , , and m, as well as the corresponding uncertainties ⫾⌬, ⫾⌬, and ⫾⌬m are known. The equivalent radius uncertainty is determined from Eq. 共3兲 to be
冉
冊
⌬R eq 1 ⌬ ⌬m 1 ⌬ ⫽ ⫹ ⬇ R eq 3 m 3
共30兲
as ⌬m/m usually is negligible 共⌬m/m⬇0.1% whereas ⌬ / ⬇2% in most cases兲. To determine ⌬C( f )/C( f ) 关with C( f )⫽C a ( f ) or C p ( f ) depending on the relaxation regime兴, we have to calculate ⌬ f / f . According to Eq. 共2兲, this quantity is given by
冉
冊
⌬ f 1 ⌬ ⌬ ⌬R eq 5 ⌬ 1 ⌬ ⫽ ⫹ ⫹ ⬇ ⫹ . f 2 R eq 6 2
共31兲
The uncertainty associated to C( f ) is then determined by derivation of the polynomial expressions 关Eqs. 共22兲 and 共23兲兴: ⌬C a 共 f 兲 ⫽2⌬ f ⫻ 兩 ⫺6⫻0.003 707 34f 5 ⫹5 ⫻0.035 7917f 7 ⫺4⫻0.062 550 07f 3 ⫺3 ⫻0.073 349 42f ⫹0.264 157 32兩 共 aperiodic regime兲 ,
共32兲
⌬C p 共 f 兲 ⫽2⌬ f ⫻ 兩 3⫻0.0679f 2 ⫺2⫻0.1949f ⫺0.1277兩 共 damped oscillation regime兲 . 共33兲 Typical values are ⌬ / ⫽3%, ⌬ / ⫽2% which leads to ⌬C( f )/C( f )⫽1.5% in both relaxation regimes. Isolating the uncertainty on the estimation of the decay time, the overall measurement uncertainty can be written as ⌬
⫽
⌬a ⫹5.2% 共 aperiodic regime兲 , a
共34兲
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FIG. 4. Relaxation curve of the REF710a soda-lime-silica standard glass at 1079 °C in natural coordinates 共a兲 and logarithm of the normalized deformation 共b, 194 experimental points兲. Influence on the measurement uncertainty and determination of the k value 共k is chosen to give the 2 value closest to 192兲.
⌬
⫽
⌬p ⫹4.8% 共 damped oscillation regime兲 . 共35兲 p
For some industrial applications, the viscometer should provide results with an accuracy of say ⫾10%. In that case, the relaxation measurement is a crucial step in the procedure: uncertainties lower than ⫾5% have to be achieved considering either aperiodic or damped oscillation relaxations. Section IV will be devoted to this particular issue.
where h ⬁ and h 0 correspond to the drop height at equilibrium and at maximal deformation, respectively. An equivalent formulation introduces the normalized deformation of the drop (t): 共 t 兲⫽
h ⬁ ⫺h 共 t 兲 ⫽exp共 ⫺t/ a 兲 . h ⬁ ⫺h 0
It is useful to linearize this expression taking the logarithm of the deformation: ln关 共 t 兲兴 ⫽⫺
IV. EXPERIMENTAL PROCEDURES A. Determination of the relaxation time and measurement accuracy in the aperiodic regime
The exponential relaxation of the drop 共see Sec. III兲 leads to the following variation of drop height vs time: h 共 t 兲 ⫽h ⬁ ⫺ 共 h ⬁ ⫺h 0 兲 exp共 ⫺t/ a 兲 ,
共36兲
共37兲
1 t. a
共38兲
Following this pattern, a procedure has been developed to exploit the experimental data. Each frame of the relaxation record is numerically processed to detect the position of the top of the drop h i (t i ) 关Fig. 4共a兲兴. The determination of h 0 is straightforward 共which determines the time origin t 0 ⫽0兲, and a good estimate of the relaxation time est can be deduced from the raw data, which will be useful in the subsequent steps. The evaluation of h ⬁ has to be as accurate as
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possible to make sure that the deformation actually converges towards zero. To achieve a subpixel accuracy, the h ⬁ value is taken to be the average value of h i (t i ) with t i ⬎6 est . At this stage, it is possible to calculate for each experimental point the logarithm of the normalized deformation: y i ⫽ln共 i 兲 ⫽ln
冉
冊
h ⬁ ⫺h i , h ⬁ ⫺h 0
冉
冊
1 ⌬ i 1 ⬇k ⫹ , i h ⬁ ⫺h i h ⬁ ⫺h 0
共40兲
where k is an arbitrary constant, to be determined later. Additional assumptions in the derivation of Eq. 共40兲 are a ⫾1 pixel uncertainty on both h i and h 0 values, and to suppose that the uncertainty on h ⬁ can be neglected. The linear regression itself consists in minimizing the 2 value given by N
2⫽
兺
i⫽1
冉
y i ⫺b⫺at i i
冊
2
共41兲
,
where a and b are the slope and ordinate at origin of the fitting straight line, respectively. This problem has an analytical solution involving the following sums:12
冉 冊 兺冉 冊
N
S⫽
兺 i⫽1
N
S xx ⫽
i⫽1
1 i
N
2
S x⫽
,
ti i
兺 i⫽1 i
2,
N
2
,
N
ti
S xy ⫽
兺 i⫽1
tiy i
2i
S y⫽
y
兺 2i , i⫽1 i
,
and ⌬⫽SS xx ⫺S 2x .
共42兲
The values of a and b are then given by a⫽
SS xy ⫺S x S y S xx S y ⫺S x S xy b⫽ ⌬ ⌬
共43兲
and a ⫽⫺1/a. It is clear from these equations that knowing the i ’s with an unknown proportionality coefficient 关see Eq. 共40兲兴 does not affect the determination of a and b. On the contrary, the standard deviation associated with a depends on the k value, as
共44兲
To obtain a valid estimate of the i ’s leading to a correct value of a , we make use of the goodness-of-fit parameter Q, given by12
共39兲
where the time variable was omitted for clarity. The experimental points are arranged in ‘‘stairs,’’ the gap between two successive steps corresponding to one pixel 关Fig. 4共b兲兴. The spatial resolution of the video system is typically 10 m per pixel. Considering a 300 m initial deformation, this resolution corresponds to about 3% of this perturbation. Therefore, experimental points recorded after the drop has recovered 97% of the deformation cannot be taken into account. This is why the experimental curve is cut off as soon as y i ⬍⫺3.5. By this procedure N data points are obtained to be considered in the following linear regression. A prerequisite to the linear regression is the determination of the standard deviation i of each experimental point y i . It is generally particularly delicate to evaluate experimental uncertainties. However, if the actual values of the i ’s are not directly accessible, their overall variation can be estimated as
i ⫽k⌬y i ⫽k
S 2a ⫽ 2a ⫽ ⬀k. ⌬
Q⫽Q
冉
冊
2 N⫺2 ⌫ 共 2 /2;N⫺2/2兲 ; ⫽ , 2 2 ⌫ 共 2 /2兲
共45兲
where ⌫ is the standard gamma function. As 2 is proportional to 1/k 2 , the Q value is k-dependent. If Q⬇1, the experimental standard deviations i are overestimated. On the contrary, if Q⬇0, the i ’s are underestimated. It is possible to adjust k in a reasonable range 共between 0.3 to 0.7兲 to obtain a correct Q value, say 0.001⬍Q⬍0.7. It should be noted that this adjustment would fail in cases where the linear model is not valid 共in other words, the relaxation process is not purely exponential兲. In such cases, Q values close to zero are obtained even with large k values (kⰇ1). Another useful criterion is that 2 ⬇N⫺2 for a correct estimate of the experimental standard deviations. The latter criterion is used to discriminate k values leading to a correct Q parameter. This procedure thus gives access to a good estimate of the a standard deviation. The corresponding relative uncertainty on a is then considered here to be ⌬a 2a , ⫽ a a
共46兲
which corresponds to a 95% confidence interval. A typical example is presented in Fig. 4共b兲, proving the influence of k on the result accuracy. Both criteria on Q and 2 lead to a k value of 0.4, leading to an uncertainty of ⫾1.4% only on the result, which is very good in comparison with the 5% required 共see the previous section兲. Changes in k have little influence on the final uncertainty, whereas Q varies a lot, which indicates that this uncertainty is reliable. In some occasions a lateral oscillation of the drop on the curved bearing membrane may occur. Due to the curvature of the bearing membrane, this lateral oscillation results in a parasitic periodic variation in the position of the top of the drop. This perturbation is visible on the logarithmic relaxation curve 共Fig. 5兲 and leads to larger k values (k⬎1). This can be understood without questioning the linear model if one considers that this parasitic motion introduces more uncertainty in the determination of the actual drop height, increasing the i ’s. However, even in that unfavorable case, the uncertainty related to a remains acceptable with regard to the ⫾5% required to obtain an overall viscosity uncertainty of ⫾10%. In favorable cases, this procedure leads even to much better results, as accuracies of ⫾1%–2% are reachable in the determination of the relaxation time. By this technique, viscosities can thus be measured in a wide range with relative uncertainties of a few percent only.
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FIG. 5. Linear regression of the relaxation curve of the keralite precursor glass at 1435 °C 共147 experimental points兲, influence of lateral oscillation on the relaxation curve and on the linear regression.
B. Determination of the relaxation time and measurement accuracy in the damped oscillation regime
In the case of oscillating drops, a similar procedure was developed for simplicity, that lead to surprisingly encouraging results, as will be shown here. A typical relaxation in the damped oscillation regime is recorded in Fig. 6共a兲. The drop height at equilibrium h ⬁ is calculated to be the mean value of all experimental points. The h 0 value is taken to be the drop height at the largest deformation observed and determines the time origin. The relaxation curve is then processed taking the logarithm of the absolute value of the normalized deformation:
冉冏
冏冊
h ⬁ ⫺h . y⫽ln共 兩 兩 兲 ⫽ln h ⬁ ⫺h 0
共47兲
The maxima of this processed relaxation curve are then detected. Every experimental point situated at a local maximum is taken into account. If two successive points of the same value compose this maximum 共which is likely to occur due to the finite spatial resolution of the video system兲, both experimental points are considered. This procedure generates a collection of N experimental data y i (t i ), which are processed through a linear regression algorithm, exactly like an aperiodic relaxation 关Fig. 6共b兲兴. The only difference concerns the evaluation of the experimental standard deviations i ’s. As in the aperiodic case, they are taken to be
i ⫽k⌬y i ⫽k
⌬ i , i
共48兲
where k is an arbitrary constant, again to be determined during the process from consistency arguments. The evaluation of ⌬ i is not as straightforward as in the aperiodic case, due to the superimposition of an oscillation to the exponential relaxation. Basically, the problem arises from the fact that the drop oscillation frequency ⫽ /2 共a few tens of Hz兲 is not completely negligible in comparison with the sampling rate F 共500 Hz兲. Therefore, a deviation is likely to occur between the experimental point taken as the local maximum
of deformation and the actual maximum, as shown in Fig. 7. In the worst case, two successive images are captured that symmetrically surround the actual maximum max at t max . Considering that /FⰆ1, this leads to the maximal deviation ⌬ osc given by
冏
冉 冉
⌬ osc⫽ cos共 2 t max⫹ 兲 ⫺cos 2 t max⫹
冊冏 冉 冊
⫹ ⬇
1 2 F
1 2F
冊
2
共49兲
.
This uncertainty is integrated in the ⌬y i value considering
冉 冊
i ⫽exp ⫺
冉 冊
ti t max cos共 2 t i ⫹ 兲 ⬇exp ⫺ p p
共50兲
and differentiating this equation
冉 冊
⌬ i ⬇exp ⫺
冉 冊
t max t max ⌬ osc⫹⌬ exp ⫺ , p p
we obtain the desired ⌬y i value:
⌬y i ⫽
冉 冊
⌬ i 1 ⬇ i 2 F
冉 冊 冉 冊
⌬ exp ⫺
2
⫹
exp ⫺
t max p
t max p
,
共51兲
共52兲
where the second term is given by Eq. 共40兲. The same arguments as in the case of aperiodic relaxations lead to the determination of the uncertainty related to p 关Fig. 6共b兲兴. In fact, this procedure proves to be very efficient, as uncertainties as low as a few percent can be obtained. In particular, k values in the same range as in the previous case are generated. This good accuracy fits again with the ⫾5% required to reach an overall uncertainty of about ⫾10%. In conclusion, this study evidences that our viscometer is suitable to explore a very wide viscosity range, from several 1000 Pa s to a few mPa s, with an accuracy compatible with industrial requirements.
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FIG. 6. Relaxation curve of the Pd40Ni10Cu30P20 glass at 700 °C in natural coordinates 共a兲 and logarithm of the normalized deformation 共b, 283 experimental points兲. Influence on the measurement uncertainty and determination of the k value 共k is chosen to give the 2 value closest to 281兲.
V. TEMPERATURE CALIBRATION OF THE DEVICE AND EXPERIMENTAL VALIDATION OF THE TECHNIQUE
At such an accuracy level, the large viscosity variation of glasses with temperature is far from being negligible. A
difference of a few °C only between the measured and the real temperature is sufficient to largely over- or underestimate viscosity. Such an error is likely to occur in our experimental setup, as the thermocouple is not in direct contact with the sample 关Fig. 2共b兲兴. Therefore, an accurate temperature calibration of the viscometer is necessary. A natural way to achieve this calibration is to melt pure bodies to assess the deviation between the measured temperature and the actual melting point of the samples. Such
TABLE I. Temperature calibration of the device by melting pure metals.
FIG. 7. Uncertainty in the determination of a local maximum of deformation in the case of a damped oscillation due to the sampling of the relaxation curve.
Metal
Melting point 共°C兲
Meas. temperature 共°C兲
Difference 共°C兲
Cu Au Ag Ni Fe Zn
1083 1064 962 1453 1535 419
1044 1029 925 1427 1510 405
39 35 37 26 25 14
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Rev. Sci. Instrum., Vol. 73, No. 9, September 2002
Gas film levitation viscosity measurement
3283
TABLE II. Relevant data concerning the studied glasses.
共mN/m兲
共kg/m3兲
m 共mg兲
f
VFT
Ref710a
357⫾9
2360⫾40
261.2⫾0.1
0.76⫾0.02
Keralite precursor
480⫾20
2370⫾40
175.3⫾0.1
0.57⫾0.02
4560 共 T °C⫺240.8兲 7490.3 log10共 兲 ⫽⫺2.805⫹ 共 T °C⫺224.6兲
BL soda lime Pd40Ni10Cu30P20
264⫾8 730⫾20
2390⫾50 9000⫾200
137.1⫾0.1 233.5⫾0.1
0.72⫾0.02 0.63⫾0.03
Sample
experiments were carried out using pure metals with melting points covering the temperature range of interest. The samples were levitated in the same conditions as the working samples, and heated up at a very slow rate 共3 °C min兲 until melting. The results are gathered in Table I. It appears that the displayed temperature underestimates the actual sample temperature. This temperature difference can be large 共up to 40 °C兲 and varies with temperature. However, an accurate temperature calibration would require more experimental points. Moreover, iron and nickel, which were used to explore the high temperatures, present a chemical reactivity with graphite. In each case this resulted in a catastrophic wetting of the sample on the bearing diffuser, which strongly limits the practical use of these metals. To overcome these difficulties and complete the temperature calibration in a more efficient and practical way, we took advantage of the existence of tabulated viscosity versus temperature variations for some well-known glasses to ‘‘measure’’ the actual sample temperature. In a first step, we used the NIST Ref710a standard glass whose viscosity versus temperature curve is accurately tabulated through a Vogel–Fulcher–Tamman 共VFT兲 law.13 Hence, measuring the sample viscosity should give a good estimate of its real temperature. This procedure has the great advantage to easily provide numerous experimental points in a wide temperature
log10共 兲 ⫽⫺1.729⫹
range. Furthermore, such experiments will be useful to validate our viscosity measurement technique. This standard glass is quite viscous, and the drops relax in the aperiodic regime. However, viscosity of this glass becomes quite weak at high temperature 共⬍20 Pa s at 1400 °C兲, which limits the calibration accuracy. To explore this temperature range, another ‘‘standard’’ oxide glass provided by Corning SA with full characteristics including the VFT law, was used. It is a keralite precursor which remains viscous enough at high temperature and provides easy to process aperiodic relaxations.14 Useful data concerning both glasses are gathered in Table II. Experimental calibration points, resulting from viscosity measurements with both glasses as well as data obtained by melting metallic samples, are gathered in Fig. 8, where the temperature difference variation with working temperature is plotted. Both types of experiment led to consistent results, which gives evidence of the relevance of the theoretical developments presented above. Indeed, all experimental points are arranged along a line evidencing two different regimes. At lower temperatures 共⬍1100 °C兲, the temperature difference linearly increases with temperature. At high temperature, on the contrary, this difference decreases. This behavior can be related to the heat transfers in the device as a whole, the sample chamber being the hot part
FIG. 8. Temperature calibration curve of the viscometer.
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Rev. Sci. Instrum., Vol. 73, No. 9, September 2002
FIG. 9. Experimental viscosity vs temperature curve of the BL131295 glass.
关Fig. 2共b兲兴 and the cooled carriers 共at constant temperature兲 evacuating heat. At lower temperatures, thermal exchanges are conduction dominated. In such a regime, temperature gradients are proportional to the working temperature; the temperature difference linearly increases with temperature. At high temperatures, radiative heat exchanges take over. Within the sample chamber, temperatures are more homogeneous. As the thermocouple is contained within the cavity formed by the thermal insulator, it is thus not surprising to observe a decrease in the temperature difference. From a practical standpoint, these experiments enable us to accurately control the sample temperature during viscosity measurements. The dotted line plotted in Fig. 8 gives access to the actual sample temperature with an accuracy of ⫾5 °C only 共this estimate corresponds to the scattering of the experimental calibration points兲. It should be noted that this uncertainty is in good agreement with estimated values of the temperature gradients inside the drop.15 In combination with the improved viscosity measurement accuracy, this enables us to confidently record viscosity versus temperature curves of glasses. VI. APPLICATION TO OXIDE AND METALLIC GLASSES
from 930 to 1450 °C. The obtained values cover a large range of more than three orders of magnitude, with accuracies as low as ⫾7% for the higher viscosities. At low viscosities, the relaxation time becomes quite short 共about 0.1 s兲, leading to larger but still reasonable uncertainties 共up to ⫾20%兲. As expected, these measurements reveal a continuous decrease in viscosity when increasing temperature, with a remarkably limited scattering of the experimental points.
B. In the damped oscillation regime: Viscosity of the Pd40Ni10Cu30P20 metallic glass
The second example illustrates the ability to measure very low viscosities by the gas–film levitation technique. The material selected here is a metallic glass of chemical formula Pd40Ni10Cu30P20 provided by DLR Cologne. The data concerning this material are gathered in Table II. In the liquid state, metallic glasses usually present viscosities in the range of several mPa s, which make them good candidates to explore the damped oscillation regime. A typical relaxation curve is presented in Fig. 6, and all experimental results are plotted in Fig. 10, for temperatures ranging from 620 to
A. In the aperiodic regime: Viscosity of the BL131295 soda lime glass
A first example of the application of this viscosity measurement technique concerns the BL131295 soda lime glass provided by Corning SA. As other oxide glasses, this material is quite viscous in the temperature range reachable in our experimental setup, leading to relaxations in the aperiodic regime. All relevant data concerning this glass are summarized in Table II. Viscosity measurements were carried out following the procedures described in Sec. IV A and the temperature corrected according to the calibration curve represented in Fig. 8. The results are presented in Fig. 9. Viscosity of the BL131295 glass was measured for temperatures ranging
FIG. 10. Experimental viscosity Pd40Ni10Cu30P20 metallic glass.
vs
temperature
curve
of
the
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Rev. Sci. Instrum., Vol. 73, No. 9, September 2002
940 °C. In this temperature range, viscosity varies from 50 to 6 mPa s, which is comparable to data reported within the same temperature range for the Pd40Ni40P20 metallic glass.16 ACKNOWLEDGMENTS
This work has been supported by the European Commission 共Craft project Visco G6ST-CT-2000-50048兲 and by Corning SA. The authors wish to thank GBX Instruments for technical support. Fruitful discussions with Dr. Jean-Charles Barbe´ and Dr. Nelly Kernevez are also gratefully acknowledged. 1
R. W. Hyers, G. Trapaga, and M. C. Flemings, Proceedings of Solidification 1999, edited by W. Hofmeister, J. Rogers, N. Singh, S. Marsh, and P. Vorhees 共TMS, Warrendale, 1999兲. 2 W. K. Rhim and K. Osaka, J. Cryst. Growth 208, 313 共2000兲. 3 I. Egry and S. Sauerland, Mater. Sci. Eng., A 178, 73 共1994兲. 4 F. Babin, J. M. Gagne´, P. F. Paradis, J. P. Couture, and J. C. Rifflet,
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Microgravity Sci. Technol. VII, 283 共1995兲. P. C. Nordine and J. K. R. Weber, Microgravity Sci. Technol. VII, 279 共1995兲. 6 J. Granier and C. Potard, Proceedings of the 6th European Symposium Material Science and Microgravity, Bordeaux, France, 1987, ESA SP-256. 7 M. Papoular and C. Parayre, Phys. Rev. Lett. 78, 2120 共1997兲. 8 J. C. Barbe´, C. Parayre, M. Daniel, M. Papoular, and N. Kernevez, Int. J. Thermophys. 20, 1071 共1999兲. 9 S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability 共Dover, New York, 1990兲. 10 M. Perez, L. Salvo, M. Sue´ry, Y. Bre´chet, and M. Papoular, Phys. Rev. E 61, 2669 共2000兲. 11 P. H. Haumesser, J. Bancillon, M. Daniel, and J. P. Garandet, Int. J. Thermophys. 共to be published兲. 12 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C 共Cambridge University Press, Cambridge, 1992兲. 13 Certificate Standard Material 710a, NIST 共1991兲. 14 Corning SA 共private communication兲. 15 J. C. Barbe´, PhD. thesis, Insitut National Polytechnique de Grenoble, 2000. 16 K. H. Tsang, S. K. Lee, and H. W. Kui, J. Appl. Phys. 70, 4837 共1991兲. 5
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