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Josef TOMISKA |
ABSTRACT. The Knudsen method of molecular effusion is employed successfully to determine small vapor pressures (< 20 mbar) - even at temperatures higher than 1500 K. The various techniques are rewiewed briefly. Special emphasis is placed (i) on two newer automatic apparatus for pressure measurement by the TORKER-method (Torkometer and Computer-operated dynamic torsion pendulum) as well as (ii) on the newer PENKER-method (Pendulum electronically balanced Knudsen-effusion recoil).
1. INTRODUCTION
The gas phase of a substance in thermodynamic interaction (exchange of energy and mass) with a condensed phase (liquid or solid) of the same substance may be called "vapor phase". Over each condensed sample exists a vapor phase which contains pressures of the vapors of each component of the sample. The partial pressure of each component vapor changes therefore with temperature and composition of the sample, maintaining the vapor-condensed phase equilibrium. The density of vapors (gases) may be characterized i) either by the pressures under which they are standing or ii) by means of the "mean free path" of the vapor molecules. Many interesting substances - specifically metal alloys - show at actual temperatures vapor pressures lower than 20 Pa. Those pressures are called "small", and they correspond to mean free paths greater than 1 cm. Such lengths of the mean free path are already comparable with the characteristic dimensions of experimental equipments. "Small" vapor pressures implicate therefore that the probability of a molecule impinging upon a wall is considerably higher than the probability of collision between two molecules. With the logical consequence that these "dilute" (also: "very dilute") vapors (gases) will not obey any more the laws of hydromechanics, and their behaviour must be described by means of the kinetic theory [1]. The measurement of small vapor pressures require therefore experimental techniques different to those employed for determing "common" vapor pressures. Several effectful ways are known to determine small vapor pressures at lower temperatures (< 1000 K), but there is a tremenduous lack of convenient methods applicable in high temperature chemistry [2-4]. The experimental difficulties of vapor pressure measurements increase considerably with the temperature of the condensed samples: E.g., at temperatures higher than 1500 K materials are limited to make inert, but still compact cell liners. Also applicable sensors are rare, as well as suitable materials for convenient supporting-, handling- and protecting systems.
2. KNUDSEN EFFUSION METHOD
One of the few methods, which are still applicable in high temperature chemistry is the molecular effusion after Knudsen [5]. Following this method, small vapor pressures are determined by means of the effusion of vaporized sample out of an isothermal vessel which is called "Knudsen cell". This is a (cylindrical) crucible with a small knife-edge shaped orifice (0.5 - 1.5 mm diameter) in the lid. Effusion through the orifice gives a molecular beam which spreads out in isotropic distribution over a sphere ("cosine law"). If inside the Knudsen cell thermodynamic equilibrium is established between the condensed sample and its vapor phase, then the pressure of the escaping molecular beam can be calculated from the equation for steady-state effusion of dilute gases [5],
pj = dmj/dt (1/Ao) sqrt(2 kB T/Mj) (1)
where pj = equilibrium partial vapor pressure of the species j, dmj/dt = mass rate of effusion of the species j, Ao orifice area, kB = Boltzmanns constant, T = temperature in K, Mj = mass of a molecule of the species j in the vapor. Vapor pressure measurements based on the Knudsen effusion can be performed in three different ways (compare figure 1).
Figure 1. Block diagram of the Knudsen-effusion techniques.
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2.1. Mass-loss technique
If the mass M of the molecules in the vapor is known, the direct determination of the mass rate of effusion dm/dt allows the evaluation of the equilibrium vapor pressures of the actual sample by Eq. 1 [5]. Newer application employs collection methods, however, most sensitive analytical techniques like neutron activation, microprobe, radiochemical counting, etc. are required. Herzig and co-workers [6,7] employ Knudsen twin cells and condensation on a collecting plate. The ratio of the vapor pressure of pure species (e.g., Au) and of the partial pressure of this species over the alloy (e.g., Au-Pd, Au-Ag, Au-Ag-Pd) is then determined by analysing the decay rate of 2 radioisotopes of this species (e.g., 195Au, 198Au) in an intrinsic germanium well-type detector. This method yields directly the partial mixing functions of the investigated component (e.g., Au) of the investigated alloy.
2.2. Knudsen cell mass spectrometry
Following this technique, the Knudsen cell is employed as the "gas source" of a high-temperature mass spectrometer, and the effusing molecular beam is directed into the ionisation chamber of the connected mass spectrometer [compare 8,9]. Detecting the ionized vapor beam by means of an electron multiplier yields then the intensities of the composing species of the vaporized sample - may be the most convenient technique to determine the composition of vapor phases. Thermodynamic evaluation can then be based upon the relation between the partial vapor pressure pj of the species j (inside the Knudsen cell) and the ion current intensities Jlj of the isotopes l of this species j [2,8-11]:
pj(xj,T) = Jlj(xj,T) T / (C sigma-j gamma-j Nlj) (2)
where xj = mole fraction of the component j; C = sensitivity factor ; sigma-j = ionisation cross section; gamma-j = multiplier gain; Nlj = abundance of isotope l of the species j.
Modern Knudsen cell mass spectrometry is a well established method for high-temperature investigations of the thermodynamic properties of both gaseous and condensed phases and is employed specifically in alloy thermodynamics with great success. However, direct determination of the absolute values of vapor pressures are beset with difficulties associated with indeterminate changes in the sensitivity factor C (for detailed formula see [8]) when sample is changed. Employing the A.I.R.-(algebraic intensity-ratio) method makes independent of the troublesome sensitivity factor C, and yields directly the values of the thermodynamic mixing functions of binary or ternary alloy systems [11-13].
2.3. Momentum sensors techniques
Due to the fact that the pressure of a gas may be defined as the average
force per unit area, caused by the momentum which is transferred by the
gas to the unit surface per unit time, vapor pressures can be determined
by means of suitable impact momentum or recoil momentum sensors. Impact momentum
sensors have been used rather infrequently, because in general they are not
sufficiently sensitive. For vapor pressure measurements they have, when compared
with recoil momentum sensors, the considerable disadvantage that the fate
of the impinging molecules, i.e., whether they condense on, or revaporize
from, the target, must be known [2]. In literature three different techniques
have been described for vapor pressure measurements based on the recoil momentum
of the molecular beam effusing the Knudsen cell (see figure 1).
3. MIKER-TECHNIQUE
The microbalance - inverted Knudsen effusion - recoil-technique (MIKER) was suggested by Margrave [14,15]. This author has pointed out that it is possible to determine both vapor pressure and the average molecular weight if an inverted Knudsen-cell, with an orifice pointing downward, is suspended from a vacuum microbalance. The effusing vapor beam is directed away from the cell suspension and the mass of the cell is de-termined not only with but also without effusion either by measuring the cell mass both when hot and when cold or by opening and closing the orifice while the cell is on the microbalance [2].
4. TORKER-METHOD
Since Volmer [16] and his co-workers [17] introduced the "torsion - Knudsen effusion - recoil" (TORKER) technique for vapor pressure measurements this method has been used by many groups [2,4]. In the TORKER method the measured vapor pressure is independent of the molecular weights of the effusing molecules. Detailed descriptions are given in Margrave [2] by Carter and Freeman and in Rapp [18] by Cater. The Knudsen cells applied in the TORKER-technique are often called "TORKER cells" and show not one but 2n orifices (n=1,2,...), with pairs pointing in opposite directions (compare figure 2). The recoil of the 2n antiparallel escaping vapor beams causes the momentum of the TORKER cell, which twists the supporting torsion fibre until it is counteracted by the elastic torsion momentum of the fibre (see figure 2). The deflection angle in rad is measured by means of a light beam, and the vapor pressure p is given by the equation
p = alpha MT F2n, (3)
where MT is the torsion constant in N.m/rad, and the TORKER cell factor F2n is defined by
2n F2n = (0.25 pi SUM[1/(di.di) li fi)], (4) i=1
where di = the effusion orifice diameter in m; li = the moment arm of di in m; fi = the correction factor after Freeman and Searcy [19]. For description of TORKER apparatus of the simple first generation see [2,4]. Although the principle of the TORKER technique is very simple, in actual application many difficulties can arise [2,20].
4.1. Torsion elements
Measuring range and sensitivity of TORKER apparatus depend basicly on the mechanical quality of the torsion fibres, and on the length of the light beam used. The most common fibres are made from quartz, phosphor bronze or tungsten [2,21], and can be used without greater difficulties for measurements of vapor pressures not lower than 0.1 Pa, a typical TORKER arrangement supposed (TORKER cell: 2 orifices, cell factor F2 = 124*106 m-3 (d = 8*10-4 m, l = 0.01 m, f = 0.8); length of the light beam: 2 m). Replacement of the torsion fibres by Pt - Ni tension ribbons allows measurements of vapor pressures of 0.03 Pa and, for TORKER arrangements not heavier than 15 g, even as low as 0.003 Pa [20,22].
4.2. Traditional damping and compensation systems
Unfortunately small vibrations of the torsion system cannot be prevented. Most authors use magnetic damping to control these unavoidable oscillations, whereas Spencer and Pratt [23] and later Cunat et al. [24] successfully damped the vibrations by means of a pair of small paddles partially immersed in vacuum oil contained in a shallow angular trough. Both kinds of damping have the disadvantage that they must always be re-moved for calibration of the torsion element, if the method of undamped oscillations is to be used. Compensation of the deflection angle due to effusion with mechanical or electromagnetic forces can be done with less complicated and less expensive experimental equipment than is required for precise angular measurements over large deflections, and long light beams are possible. Precise measurement of torsion pendulum angle is necessary only at the null position. By using an optical lever of 2 m, one can achieve a precision of angular measurement of 1 * 10-4 rad with simple and inexpensive equipment. Another advantage of a compensation technique is the much broader range of vapor pressure measurements without changing torsion elements. Munir and Searcy [25] report a mechanical null method (figure 2-A). These authors made use of a goniometer to compensate manually for the angle of deflection due to effusion. The disadvantages of such a mechanical drawing back are evident (see figure 2-A): (i) The goniometer must be in an easy-to-handle position, (ii) Determination of lower vapor pressures demand expensive high vacuum rotary transmission leadthroughs (with respect to the requirements to maintain a sufficient high working vacuum). (iii) Adjusting of the TORKER cell in the high temperature furnace is beset with difficulties with respect to the fixed rotary transmission leadthrough. (iv) Resolution and sensitivity of TORKER arrangements are limited. Following the traditional electromagnetical compensation techniques [23,26,27], a magnet is rigidly attached to the suspension system, and surrounded by suitable coils (figure 2-B). The torsion momentum due to the effusion is exactly counter-balanced by current passing through the coils needed to maintain null rotation. But attachment of a magnet to the torsion fibre, as cited above, limits the sensitivity of TORKER measurements: Light magnets are too weak for compensation and heavier ones require stronger and therefore less sensitive torsion fibres [20].
4.3. Newer TORKER arrangements
4.3.1. Torkometer.
This device is a simple, inexpensive null method TORKER apparatus to measure
a broad range of vapor pressures without changing the torsion fibre [20,22].
The compensation system of the "Torkometer" can be operated manually or automatically
and is based on an industrial core-magnet moving-coil galvanometer with two
windings. A meter movement of a core-magnet moving-coil galvanometers [20]
is schematised in figure 3: In contrast to common moving-coil galvanometer
the permanent magnet (8) of a core-magnet meter movement is cylindrical, connected
rigidly to the support system (2), and surrounded by the mo-ving coil. The
winding support (5) of the moving coil is attached to the mirror (10) by
the suspension rod (9), and to the lower end of the upper tension ribbon
(3). The tension ribbons (3,11) are connected to the sup-port system (2)
via curved plate springs (1). The weight of the entire moving coil (including
the winding support, both windings (7), and the connection pieces (4,6)) is
less than o.5 g [20]. A meter movement such as the one described can be easily
added to TORKER arrangements (compare figures 2-B and 3): Enlarge suitably
the upper tension ribbon (figure 3, (3)), cut off the lower parts of the
supporting system (figure 3, (2)), and replace the lower tension ribbon (figure
3, (11)) by the bayonet socket(figure 2-B, (7)) of the suspension rod (figure
2-B, (8)) of a high temperature TORKER cell. Connecting the adapted meter
movement to a precise adjusting system yields then the "Torkometer" [20].
For a suitable precise adjusting system see [22]. Either winding of the moving
coil can be used to compensate for the recoil momentum caused by the molecular
beams escaping the Knudsen cell, returning the mirror (figure 3, (10)) to
its original position. The null point is sensed by means of a 2-meter light
beam and the mirror attached to the Knudsen cell suspension. Using a laser
beam and 3 photodetectors, the current necessary to maintain null rotation
can be automatically regulated via 2 operational amplifiers [20]. The vapor
pressure can then be calculated from the current measured by an ampmeter,
either manually or automatically by means of a simple microprocessor. Short
circuit of the winding not used for actual compensation allows magnetic damping
of unavoidable vibrations of the torsion system with the advantage of cut-ting
out the damping at any moment desired. This enables calibration of the torsion
fibre under high vacuum with improved accuracy (0.1 %). In the null method,
where the torque due to the vapor is counteracted by electromagnetic interaction,
the vapor pressure is related to the current i through the moving coil necessary
to maintain zero rotation, and Eq.(3) must be substituted by
p = i Cw MT F2n, (5)
where Cw is the winding calibration factor (Cw = / i ). The low re-sistance
winding I may be employd to determine small vapor pressures whereas the
high resistance winding II allows the measurement of higher pressures without
using excessive current. By means of such a Torkometer it is possible to
determine continuously vapor pressures within the range 0.005 Pa to the upper
limit of Knudsen effusion conditions (about 20 Pa). Resolution is limited
by the current measurement and is about 0.6%-0.9% of the picoammeter range.
The significant advantages of the Torkometer are: (a) there is con-stant
sensitivity over the entire pressure range of Knudsen effusion conditions;
(b) there is negligible mass load of the torsion fibre (max. 0.5 g); (c)
calibration of the torsion element can be done under high vacuum with improved
accuracy (0.1%).
Figure 3.
4.3.2. Computer-operated dynamic torsion pendulum.
Edwards and co-worker [28,29] have developed a sophisticated computer automated
data aquisition system for a dynamic Knudsen torsion pendulum applied to vapor
pressure measurements with a precision of +- 0.0035 rad angular deflection.
The objective of the automatic data acquisition system is to observe the times
at which the oscillating torsion pendulum reaches preset positions, to record
those data as matched position versus time pairs, and to use those data to
obtain the integral equation of motion of the torsion pendulum. Following
this technique (figure 4-A), laser beams (1) reflected from mirrors (3) on
the torsion pendulum are detected by a bank of photodetectors (2) which, in
turn, transmit signals representing pendulum positions through an encoder
and interface to a personal computer with an internal clock. The computer
uses torsion pendulum position and time data with stored information to solve
the equation of motion of the torsion pendulum and to calculate vapor pressures.
The data acquisition system operates undamped in the dynamic mode, the swing
is alternately enhanced and retarded by the recoil momentum of the effusing
vapor. The sources of the light beams are four inexpensive, low-power, He-Ne
lasers on adjustable mounts which allow them to be directed individually during
alignment of the system. The beam detection system consists of a hemicylindrical
(radius 0.3 m) bank of 144 phototransistors arranged in four levels, with
36 on each level, to correspond with
Figure 4. Computer-operated dynamic torsion pendulum [28,29]. A: Top view
of one level of the photodetector assembly: 1, laser beam; 2, phototransistor;
3, mirror; 4, vacuum wall (glass); B: Mirror mount. the 4 mirrors and the
4 laser beams. The beams are introduced radially along the symmetric bisector
of the bank of detectors; thus only the acutely reflected beam falls on
the detectors. The detectors are set in 36 vertical mounting posts, each
containing 4 detectors, one on each level. The system does not interact physically
or through any field with the torsion pendulum; thus no additional torque
or damping required compensation [28]. The data acquisition system may be
added to existing TORKER apparatus with only minor modification being required.
E.g., taking the TORKER arrangement on figure 2-B, the magnet (12) and the
solenoids (13) must be omitted, the mirror (6) substituted by the 4-mirror
mount (figure 4-B), and a glass vacuum wall must be incooperated. For vapor
pressure measurements at high temperatures the equation of motion when the
TORKER cell is at the experimental temperature is compared with the equation
of motion when the cell is at room temperature to determine the torque due
to effusing vapor and thence the vapor pressure inside the TORKER cell [28].
4.4. Discussion of TORKER technique
The important advantage of the TORKER method lies in the high sensitivity and in the possibility of quick measurements of the total vapor pressures at different temperatures. But this is balanced by difficulties, discussed in detail by Freeman in [2]. Perhaps the most important problem of the TORKER technique arises from the unsymmetrical furnace-TORKER cell configuration which causes thermomolecular flow forces especially at high temperatures [2]. Czanderna in [30], and Thomas and Poulis in [31] estimated that this effect can falsify the torque caused by the torsion momentum which is produced by the effusing vapor beams. Reduction of uncertainties in vapor pressure measurements due to insufficient vacuum surrounding the TORKER cell, unstable furnace temperatures, and temperature gradients is no longer a question of technical knowhow, but only of available money and resources.
5. PENKER-METHOD
"PENKER" stands for Pendulum electronically balanced Knudsen effusion -
recoil and represents a new recoil momentum sensor technique [4,32]. As figure
5 shows the PENKER-method is an automated null method employing a linear
pendulum, suspended near the centre of mass, instead of the torsion pendulum
which is used in the TORKER method. Following the PENKER-method [32] the
determination of vapor pressures is also independent of the molecular weights
of the effusing molecules: the displacement of a linear pendulum, caused
by the recoil of the escaping molecular beam which is automatically counteracted
by an electromagnetic compensation system which is electronically controlled.
Using a linear pendulum for vapor pressure measurements, the effusion orifice
must be drilled into the side-wall of the Knudsen cell - mnemonically called
"PENKER cell" (see sector drawing in figure 5).
Figure 5.
To double or multiply the force due to the escaping vapor beam two or more PENKER cells can be used together (compare figure 5) [32]. A PENKER apparatus is schematised in figure 5. The pendulum (3) is supported by a sapphire knife-edge (4) and an agate edge bearing. Therefore the bearing friction loss is negligible small. The upper arm of the pendulum consists of an alumina capillary with a thermocouple (3). The PENKER cell (2) is connected to the upper arm of the pendulum (3) via a bayonet socket and is in the centre of the high temperature furnace (1).
The temperature is measured directly below the PENKER cell (2) by the thermocouple or by an optical pyrometer (not in figure 5). When sample is changed, the sensitive pendulum system must be protected. This requires a robust, but precise adjusting and catching system (5): A catching-motor lifts the pendulum from the edge bearing until it is caught within a conical guide (not in figure 5). The adjusting system (5) makes sure that the knife-edge (4) will be positioned reproducibly [32].
5.1. Control of the pendulum position
The control of the pendulum position together with precise counteracting of pendulum displacement is the most sensitive part of PENKER apparatus. Convenient devices may be obtained by simple adapting of linear susceptibility meter [32]. It is only necessary, to replace the electromagnet and sample arrangement with the high temperature furnace and the PENKER cell. Fully automated systems with high resolution (about 2 * 10-13 SI units) have been reported by several authors [33-35]. Test measurements [32] on cadmium and on lead proved the efficiency of the compensation device constructed in cooperation between Sobczak and the A.PAAR KG, Graz (Austria) [34]: The deflection of the linear pendulum is determined capacitively by means of a plate condensor (figure 5; (8)). One plate of the condensor is rigidly attached to the lo-wer part of the pendulum (figure 5; (3)), the second plate is rigidly attached to the outer framework (not in figure 5). The actual capacity of the plate condensor (figure 5; (8)) is determined electronically and converted into the corresponding intensity of the compensation current traversing the compensation coil (figure 5; (7)). The required current intensity is proportional to the vapor pressure and is indicated by the four 1/2 digits evaluation unit with various ranges of sensitivity [32]. With the counterweight (figure 5; (6)) the sensitivity of the balance can be raised to a desired value. PENKER apparatus may also be based on the compensation system de-veloped by Müller and Güntherodt [35]: The spot of a laser beam, which is reflected from a mirror fixed to the frame of the pendulum gives rise to an error signal on a photopotentiometer. This signal is amplified and controls a current source. The controller increases the current through the compensation coil until the pendulum is returned to null position.
5.2. Discussion of PENKER technique
The difficulties arising from the application of the TORKER method, as described in point 4.4., can be dealt with by employing the PENKER technique. At present PENKER apparatus are less sensitive the Torkometer, nevertheless the PENKER method has clear advantages over the TORKER technique: (a) The symmetrical furnace-cell configuration together with the fact that the PENKER cell is rigidly attached to the arm of a linear pendulum, prohibit electromagnetically induced cell rotations. (b) Neither oscillations of the pendulum nor a null point shift may be expected - even at high temperatures (compare [32] and [35]). (c) The cylindrical PENKER cells enable smaller diameter of the cylindrical furnace, and the necessary electric input is clearly reduced too. Another clear advantage over the TORKER method is (d) the better mechanical stability of the used linear pendulum. (e) PENKER cells are simpler manufactured than TORKER cells, and the same cells can be used for measuring either the vapor pressure of one sample or the difference in vapor pressure of two samples: The effusion orifices must only be turned either to the same direction or to opposite ones.
Acknowledgments
Grateful acknowledgment is made to the financial support of the "Fonds zur Förderung der wissenschaftlichen Forschung in Österreich".
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