TECHNIQUE FOR VOLUMETRIC EXPANSION OF LIQUIDS
AND SOLIDS FROM 200-400K
_____________________________________________________________
Ernest G. Wolff
ABSTRACT
A new dilatometer for volumetric expansion measurements over the
temperature range 200-400K is described. It uses readily available fluids such as
ethylene glycol based mixtures and quartz capillaries with a demountable sample
container. Liquids and solids of any shape be measured. Equations permit a
complete parametric analysis for optimizing measurement resolution. It is shown
that results depend chiefly on knowledge of system dimensions and volumetric
CTE of the working fluid. A statistical F-test can be used to derive the volumetric
CTE of the sample for heating/cooling data. Measurements with variable sample
sizes of metallic test samples show excellent agreement with predictions, while
materials with higher volumetric CTE values would show proportional increases
in accuracy. Further studies with other working fluids, such as relatively pure
glycols, indicate further improvements in accuracy .
INTRODUCTION
Measurements of volumetric expansion, , have been of special interest to
studies of liquids, irregular material shapes, curing of adhesives, crystal and
particulate behavior. For an isotropic material, the linear thermal expansion
coefficient  = (1/3) . For an orthotropic material  = α1 + 2 + 3, so if two of
the linear coefficients are known the third can be deduced from a measure of the
volumetric expansion. Similar considerations apply to crystals with other types of
symmetry. Simple methods for accurate volumetric expansion measurements have
been sought for many years [1], but generally have had significant limitations,
such as temperature range, sample size and shape, leaking of liquid samples, or
had measurement problems in terms of reproducibility, accuracy, or ease of setup
[1,2]. Changes in density can in principle be used to derive volumetric CTE but
normal methods such as pycnometers [3,4], densitometers [5], hydrometers [6],
and gas volume changes [7] are limited by resolution and/or temperature ranges.
A method based on measuring the period of oscillation of a vibrating U-shaped
tube filled with liquid samples gave accuracy of  1 x 10-5 g/cc over the range 2535oC [8]. Dilatometers based on mercury require handling precautions [2,9], and
in any case mercury freezes below about 235K.
________________
Precision Measurements and Instruments Corporation, Corvallis OR 97333
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The use of capillary tubes to follow a liquid expansion has been common [1012]. In cases involving direct liquid measurements, one method required sealing
of the quartz capillary and filling with the aid of a water jet [11] or centrifuge
[12]. Microscopes, telescopes and/or electronic calipers can be readily used to
measure the expansion of a liquid in a capillary.
THEORY
The present system is similar to that described in [2] but is simpler to
construct and use. Figure 1 illustrates the basic experimental scheme consisting of
a cylindrical cup containing the sample and covered by a lid with exiting capillary
tube. The working fluid fills the cup and a small section of the lower capillary
tube. Using symbols:
Vcap = volume of capillary from lid to height of fluid = hod2cap (/4)
Vfl = total volume of fluid (inside cup and capillary)
hc = initial height of cup
dc = initial internal diameter of cup
Vc = internal volume of cup = hc d2c (/4) for cylinder
hs = initial height of cylindrical sample
ds = initial diameter of cylindrical sample
Vs = sample volume, (hs d2s (/4) for cylindrical sample)
dcap = internal diameter of capillary tube
ho = initial height of (working) fluid in capillary
hfl = final height of (working) fluid in capillary
h = hf- ho
fl = volumetric expansion coefficient of fluid
s = volumetric expansion coefficient for sample
c = volumetric expansion coefficient of cup
fl = density of the fluid
If we consider a temperature change without any leaks, the total mass (*V) of
the fluid remains constant. From Fig 1 this implies;
ofl [ Voc – Vos + Vocap ] = ´fl [ V´c – V´s + V´cap]
(1)
where the o-superscripts denote initial values and the primes denote the values
after heating or cooling. Since, at constant pressure,
´fl = M [Vo (1 + V/V) ]-1
and V/V = fl (T) T
(2)
[Vc – Vs + ho d2cap (/4) ] ( 1 + fl (T) T) =
Vc(1 + c (T) T) – Vs ( 1 + s (T) T) + hf d2cap (/4)
(3)
2
A small amount of fluid resides in the quartz capillary outside the heated/cooled
zone at all times and does not participate in the temperature changes. βfl, βc, and βs
are normally functions of temperature. For example, while the expansion of a
quartz cup may be taken as zero, an aluminum cup is desirable for better heat
transfer and to minimize changes in Δh. Use of an aluminum cup suggests the
polynomial :
Βc (T) = βAl (T) = 3{ Bo + B1 T + B2 T2) }/ (T – Tref)
(4)
where the coefficients B0 = -0.583 x 10-3, B1 = 23.107 x 10-6 and B2 = 8.024 x
10-9, Tref = 25 oC and T is measured in oC. [13]. The sample and cup volumetric
expansions may often be considered constant over small temperature ranges, or
averaged over the test range. Then, the predicted change in fluid level in the
capillary is calculated from Equation 3:
(T – Tref) [ Vc fl (T) – Vs fl (T) + Vs s – Vc c + ho d2cap(π/4)fl (T) ]
h (T) = ---------------------------------------------------------------------------------(/4) d2cap
(5)
When the fluid alone is being measured, dcap, ho and βc can be assumed to remain
constant:
fl (T)
=
Vcc + (Δh(T) / (T – Tref )) d2cap (/4)
-----------------------------------------------Vc + ho d2cap ( /4)
(6)
We can compute the sample volumetric CTE from measured Δh(T) data as;
s (T) = fl (T) + Vc (c(T) - fl (T)) + d2cap [ ( h (T) / (T- Tref) ) - ho fl (T) ]
Vs
4 Vs
(7)
EXPERIMENTAL APPARATUS
Figure 1 illustrates the essential features of the test system. The cylindrical
cup containing the working fluid can be any non-reactive solid whose volumetric
CTE is known, such as quartz or aluminum. It is surrounded by rubber gaskets
and/or “O”-rings which are compressed by the demountable fixture. This consists
of two plates with four holes for steel bolts and nuts. Springs (not shown) help to
maintain compression of the plates against the cup. Temperature measurement to
±0.1K was carried out by cementing a thermocouple (T.C.) junction in a hole
drilled part way in the upper lid, within a few millimeters of the capillary also
cemented through the lid.. An additional thermocouple site can be placed in the
fluid to monitor thermal gradients. These are kept to within ±1K by controlling
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the heating/cooling rate. When cooling with liquid nitrogen in the coils, it was
found that stirring the (antifreeze) cooling solution to promote uniform heat
transfer to the aluminum cup also stirred in the ice phase of the fluid, increasing
its viscosity and reducing the cooling rate. An unstirred solution allowed this
phase to precipitate on the coils and temperatures of < 200K could be reached
Figure 1 Schematic of volumetric CTE Apparatus. The fixture uses steel bolts to
compress the gaskets and the aluminum plates against the cup.
The working fluid in the cup can be any non-hazardous liquid which does not
decompose, vaporize, freeze or experience phase separation or excessive
viscosity. It should not react chemically with the sample. In addition , it should
have as small a volumetric CTE as possible, so that the measurement of relatively
small values of β (such as of metals) should not mean having to extract very
small changes due to the sample from large measured changes due to the working
fluid. Of 76 organic fluids listed in [14], less than 8 have β values below about
800 ppm/K; most are in the 1000-1500 ppm/K range. Ethylene glycol (C2H6O2)
and diethylene glycol (C4H10O3), the major constituents of common automobile
antifreeze have reported values of β of 626 and 635 ppm/K, respectively at about
293K. This suggests initial investigation of commonly obtained antifreeze
formulations. We note that the working fluid of the heating/cooling bath and in
the cup can be the same. However, the value of fl (T) must be measured first as
some antifreeze formulations contain additional corrosion inhibitors, water, etc.
Water is known to affect the density of ethylene glycol and its derivatives [15].
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EXPERIMENTAL PROCEDURE
The capillary is cleaned by attaching a vacuum hose to the lid end and
immersing the other in deionized water, isopropyl alcohol, and/or methanol. The
rubber gaskets around the cup are compressed by the lid and held by the steel
fixture after the fluid and sample have been inserted. The excess fluid rises in the
capillary (the initial ho plus inactive fluid) so that sealing of one end is
unnecessary to draw in the fluid. Heat sealing (e.g. [11]) may damage test
material [2] and the open end also allows isobaric expansion conditions. Trapped
air bubbles are avoided by an overflow of the working fluid over the cup and
gaskets prior to lid compression. The assembled fixture is then placed in the
heating/cooling bath with LN2 copper cooling coils in a beaker on a hot plate.
The temperature is recorded after the meniscus in the capillary passes the
horizontal crosshairs of the telescope. (See also [16]). A leak in the system
negates the theory of constant mass above. This is easily checked by watching
the meniscus in the capillary for a time prior to measurement.
ERROR ANALYSIS
Sources of error are identified as follows:
1) Capillary Diameter The capillary used had an internal diameter varying from
535.0 to 536.0 microns over a 10 m length [17]. This translates into 0.1
microns over a range of h of 100 cm giving an error in s of about 0.8%
2) Uncertainty in ho. Equation 3 assumes that the total height of fluid ho in the
capillary is subject to the temperature change T. Depending on the geometry
of the system, this is normally true only for a few cm above the heating bath.
A typical value for ho then is 5 cm. An error of 1 cm here translates to < 1%
error in  in a typical measurement of > 10oC.
3) Support system motion. Since the system is mounted on a hot plate or cooled
in a container, whereas the measuring telescope is not, the apparent h may
change due to the system motion relative to the measuring telescope. This
was checked by watching a mark on the capillary with the telescope
throughout a run. The maximum error over a 150oC excursion was 0.01cm.
4) Verticality of the capillary. It the telescope scale and capillary are not
mutually parallel, there is a cosine error due to any angular () deviation from
the line of the telescope scale. The true h = the apparent h / (cos ). This
is easily measured with a protractor across the beaker and generally varies
between 0 and 15o.
5) Thermal Gradients.
These are caused mainly by the finite thermal
conductivity of the working fluid (and possibly sample) and result in a slight
hysteresis when reversing the heating/cooling direction. For maximum
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accuracy, it is necessary to measure the fluid alone and the fluid plus sample
at the same fluid composition, heating or cooling rate and over the same
temperature range.
TYPICAL RESULTS
A series of tests was undertaken to assess the validity of the theory. The Al
cup dimensions were 2.26 cm internal diameter and 1.864 cm high. The ratio of
Vs/Vc was varied for a relatively low CTE material, namely steel, by measuring
different numbers of steel balls placed in the aluminum cup. Initially the
antifreeze fluid was measured. A linear regression of the data gave Δh = 1.0921
deg C –25.245 with correlation coefficient R2 = 0.9962. Use of Equation 6
converts this to βfl = 399.72 x 10-6/K.
Figure 2 Change in capillary meniscus level with temperature for working fluid
alone and for different relative volumes of a low carbon steel sample.
The solid lines in Figure 2 compare the experimental data with Δh predicted
from this linear regression curve and from Equation 5 assuming the volumetric
CTE of steel is 35.1 x 10-6/K. Agreement is seen to be excellent for all cases, so
that the Δh data can correspondingly be used to predict βs according to Equation
7. However, the βfl value is not constant to lower temperatures. A typical
polynomial fit of the data gave;
Δh (cm) = -7e-6 T4 –0.0004 T3 – 0.0123T2 + 1.3792 T – 18.38
(8)
in the region 210K < T < 283K with R2 = 0.9973. This trendline is superimposed
on the fluid data in Figure 3. Fluid data were then used to predict the results for
different Vs/Vc values on cooling and these curves are shown to lie closely over
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the data in Figure 3. In the case of Vs/Vc = 0.248, data were continued cooling
from higher temperatures and prediction using a fixed βfl of 399 x 10-6/K gave a
good prediction down to about 283K. Below that Equation 8 was used.
Figure 3 Cooling data for samples of Figure 2. A polynomial fit is made to the
fluid data which is then used to give the predicted (line) data for the steel samples.
DISCUSSION
A parametric analysis of Equation 7 suggests that for maximum resolution, Vs
must  Vc especially if s << fl. Eq.7 consists of three terms, the first and
third positive, he second negative. To maximize the effect of the Δh measurement
and to minimize the effect of the second term, Vs/Vc again should be as large as
possible. Equation 7 can be used to estimate the relative importance of fluid
expansion. For example, if we consider the parameters used in Figures 2 and 3,
with Vs/Vc = 0.489, ΔT = 50K, ho = 5 cm, dcap = 0.053cm, βfl = 399 x 10-6/K, and
βs = 35.1 x10-6/K (for steel), we predict a Δh of 28.47cm. However, if we
measure this value of Δh but the βfl were 401x10-6/K, we would deduce that the
steel had a βs of 34.75 x 10-6/K.
There are several ways in which the data can be analyzed for greater accuracy.
A fourth order polynomial such as Equation 8 has limitations in predicting the
meniscus level, e.g. when steel and brass cylinders are cooled. One possibility for
greater accuracy is to use an F-test (available on MS Excel) together with a
polynomial for the fluid. This calculates the one-tailed probability that the
variances in two data sets are not significantly different. Thus one can insert
different values of βs into Equation 5 and compare that data set with the
measured one. A maximum (0.9589) for the F-test occurs with βs equal to 36.1 x
10-6/K as compared to 0.9133 for βs = 35.1 x 10-6/K or 0.9534 for βs = 38.1x 10-
7
6
/K. The value of 36.1 x 10-6/K compares favorably with the low temperature
data of [18].
This method has the capability to measure the volumetric CTE of arbitrarily
shaped samples such as gels, waxes, and greases. However, the working fluid
must not react with the sample. Consequently, other fluids are being sought. We
have studied perfluoro compounds, water, synthetic motor oils, machine oils,
alcohol based compounds, and glycols. Perfluoro compounds with 12 carbons
such as the 3M “Fluorinert” series are inert with some gels and appear to have a
linear beta on the order of 1200 ppm/C. They also have the advantage to protect
many ceramic and optical crystals from reactions with moisture. Water has a low
value of βfl near room temperature but it increases rapidly with temperature and is
very non-linear. Low viscosity synthetic motor oils are less successful due to
poor wetting of the capillary. Some alcohol based fluids, e.g. brandy, leave
residues after moving up and down a capillary.
Relatively pure (99%+) ethylene and diethylene glycol show closely similar
behavior (Figure 4). Figure 5 shows the instantaneous volumetric expansion
coefficient as a function of temperature for ethylene glycol. The scatter in the data
points reflects the use of various heating/cooling rates. The relatively low β
values of the glycols suggested study of propylene glycol (C3H8O2), whose
advantages include greater safety and possible supercooling of the liquid to below
200K. It boils above 460K. However, we found considerable variability from test
to test, possibly because of the dipotassium phosphate and deionized water
contained in the “inhibited” version tested. Besides pure polypropylene glycol,
we are investigating silicone based fluids, such as polydiethylsiloxanes with
viscosities on the order of 10 centistokes and molecular weights on the order of
400 – 500 .These can also be used over the range 200-400K.
Figure 4 Relative displacements of meniscus in a 0.053 cm I.D. quartz capillary
with temperature for ethylene glycol (upper curve) and diethylene glycol.
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Figure 5 Instantaneous volumetric expansion coefficient for 99%+ ethylene
glycol versus temperature
SUMMARY
Major advantages and features of this inexpensive, completely reusable
dilatometer are summarized as follows:
1)
2)
3)
4)
5)
No heat sealing or mercury is required.
The dilatometer is demountable and the same system can be used for many
liquid, solid or mixed phase samples, including powders, gels, waxes, etc.
Accuracy and resolution are enhanced by use of statistical analyses of the
data, such as a statistical F-test.
The available temperature range is limited by the properties of the
working fluid and is about 200- 350K with common antifreeze
formulations at 1 atm pressure. Higher temperatures are possible with pure
glycols and possibly silicone based fluids.
Any size/shape sample can be measured as long as it fits into a cup with
accurately determinable internal volume and expansion coefficient. A high
CTE metal such as aluminum is easily machined, its expansion reduces
h for a given temperature range and a high thermal conductivity
promotes accurate sample temperature measurement.
ACKNOWLEDGMENTS
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The author is grateful to the staff of Precision Measurements and Instruments
Corporation for helpful suggestions and technical assistance, in particular,
Benjamin Nielsen, James Sharp and Don Schneider.
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