Static and Dynamic Compressibility of Lubricants Under High Pressures

Abstract

This paper presents an experimental investigation conducted on several synthetic lubricants of various chemical nature: silicone fluids, polyalphaolefins and a diester. Volume losses are first reported at different temperatures. Then, isothermal secant and tangent compressibility modulus values are deduced. They are compared to isentropic bulk moduli, calculated from ultrasonic measurements. Finally, a discussion on the correlation between the relative value of these moduli and the lubricants physical state completes the paper.

Full text

STATIC AND DYNAMIC COMPRESSIBILITY OF LUBRICANTS UNDER HIGH
PRESSURES
P. Vergne
Laboratoire de Mécanique des Contacts, URA CNRS 856
INSA, Bâtiment 113, 20 avenue Einstein, 69621 Villeurbanne cedex, France
This paper presents an experimental investigation conducted on several synthetic lubricants of various
chemical nature: silicone fluids, polyalphaolefins and a diester. Volume losses are first reported at different
temperatures. Then, isothermal secant and tangent compressibility modulus values are deduced. They are
compared to isentropic bulk moduli, calculated from ultrasonic measurements. Finally, a discussion on the
correlation between the relative value of these moduli and the lubricants physical state completes the paper.
1. INTRODUCTION
During the last decades, the lubricants
compressibility under high pressure has been the
rheological parameter which has certainly been less
studied. Nevertheless, the compressibility role
appears to be significant in many circumstances, as
for instance:
- in highly loaded lubricated contacts as those
occurring between gear teeth,
- and also in most of lubricated metal forming
processes where small amounts of lubricant are
compressed between the tool and the metal sheet,
trapped by surface asperities.
The compressibility contribution in the running
of E.H.D. contacts has been first described by
Dowson and Higginson [1] and recently developed
by Jacobson et al. [2,3] for instance.
The reference work in the fluid compressibility
domain has been brought by Hayward who proposed
at first [4] an adequate terminology and methods of
expressing results. His contribution has concerned
also compressibility equations [4,5], testing methods
[4,6,7], experimental investigations in various
liquids [4,8].
To our knowledge, most of compressibility
equations are empirical relationships which have
frequently been experimentally verified in narrow
pressure ranges. Hayward [5] and later on Yusa,
Mathur and Stager [9] have reviewed and tested
some of them, but due to the experimental limits
already mentioned, any clear tendency emerges.
In lubrication, the more frequently used density-
pressure equation found in literature is the Dowson
and Higginson [1] relationship which is written as
follows:
ρ/ρ
0
= 1 + 0.6P/( 1 + 1.7P ) (1)
where ρ is the density under pressure P,
and ρ
0
is the density at ambient pressure.
Concerning experimental results on lubricants,
two different approaches could be mentioned. Very
accurate experiments have been run in a restricted
domain (typically 700 bars) and therefore values for
higher pressures are extrapolated. This method has
been applied by Klaus and O'Brien [10], Wright [11]
and Tichy and Winer [12] on various kind of fluids:
mineral base oils, polyphenyl ether, silicone fluids
and diesters.
The second approach consists to conduct
experiments in a pressure domain as large as
possible. The limit generally corresponds to the
maximum admissible stress of the high pressure cell
material or to the appearance of leakage problems.
Since Galvin et al.[13] who reach 350 MPa, special
devices have been developed to work under 1 GPa
or more[2,3,14,15].

DOS SIL1 SIL2 POL1 POL2
Chemical
Structure
Di 2 ethyl hexyl
sebacate dimethyl
polysiloxane methylchlorophe
nyl polysiloxane
polyalphaolefin polyalphaolefin
Pour Point °C
-60°C -50°C -75°C -40°C -57°C
Refractive
Index
1.449
1.497
1.416
1.470
Density at
20°C
0.913
1.066
1.016
0.848
0.848
Kin. Viscosity
at 40°C
11.7 cst
77 cst
52 cst
408 cst
33 cst
Kin. Viscosity
at 100°C
3.5 cst
22.2 cst
20.9 cst
41.6 cst
7 cst
VI
E
199 314 414 154 181
α in 1/GPa
12.5 at 20°C 18.2 at 60°C 12.8 at 60°C 12.6 at 60°C 12.6 at 60°C
α in 1/GPa
12.1 at 25°C 10.4 at 150°C 10. at 150°C 9.7 at 150°C 8.7 at 150°C
Table 1 : Lubricants properties (α is the secant pressure-viscosity coefficient, calculated at 400 MPa).
2. TESTED MATERIALS AND DEVICE
2.1. Tested materials
Five products have been studied: one diester, two
silicone fluids and two polyalphaolefins respectively
named DOS, SIL1, SIL2, POL1 and POL2. Most of
these lubricants are pure synthetic fluids: the
exception is POL2 which contains an additive
package. Their properties are summarised in Table
1: apart from pour point temperatures, the results
reported in Table 1 have been collected on our own
devices.
-Refractive index has been measured according to
the A.S.T.M. D1218 standard.
-Density variations versus temperature have been
determined with a Mettler density kit set up on an
electronic balance.
-Kinematic viscosity changes versus temperature
have been deduced from dynamic viscosity results
obtained with a Couette viscometer and from density
data.
-VI
E
has been determined according to the A.S.T.M.
D2270 standard.
-Pressure-viscosity coefficients have been calculated
from experiments run on our high pressure falling
body viscometer. We have deliberately chosen to
give the secant coefficients computed for a 400 MPa
pressure increase instead of usual coefficients which
serve for EHD film thickness calculations. Here, the
aim is to report a parameter which represents the
viscosity increase in a wide pressure domain.
The fluids are all characterised by low pour
points, low viscosity at ambient pressure and
temperature (except for POL1) and high viscosity
index. SIL2 exhibits the greater stability: pressure
and temperature influence is very weak compared to
the other lubricants. This is the consequence of its
chemical structure which contains phenyl groups and
chlorine atoms.
2.2. Experimental device
Reliable bulk moduli data are difficult to obtain
on a fluid [7,11]: following the advice of previous
authors, a maximum of caution has been taken
during the experimental phase.
The measurements have been performed in the
high pressure (up to 700 MPa), wide temperature
range (from -25°C to 160°C) vessel of our falling
body viscometer. To avoid dispersion due to un-
homogeneous strains, the high pressure densimeter
works under perfect hydrostatic pressure and
temperature. It is made of a cylindrical stainless steel
cell filled by the sample: its top end is closed by the

ceramic disk of an ultrasonic transducer and its
bottom end is sealed by a metallic piston supported
by o-rings. This feature allows to accommodate
volume changes in the fluid column by a translating
motion of the piston. Consequently the distance
between the ceramic disk and the piston changes as a
function of pressure and temperature.
These distance variations are recorded by means
of an ultrasonic technique which requires the
preliminary measurement of the sound velocity in the
sample. This previous work is run at similar ceramic-
reflector distance to avoid possible linearity
fluctuations of the electronic unit. We use an
impulsion method (3 periods maximum by pulse)
with longitudinal waves: the transducer works as an
emitter/receiver. To limit energy dissipation in the
fluid under pressure, we have chosen a high
ultrasonic frequency (1 MHz) and a low emission
one (125 Hz): this means that the ratio of the time
during fluid molecules undergo ultrasonic waves to
the real time is negligible.
The experiments are computer controlled:
recorded data are directly converted in length and
thus in volume. Elastic deformations caused by
hydrostatic pressure and temperature fields are
directly taken into account: their influence is very
weak.
To avoid uncertainties in the low pressure
domain, the first pressure step is 25 MPa for DOS
and 50 MPa for the other fluids. Hayward [7] has
cleverly suggested that during compressibility
experiments, the first pressure step should be at least
one tenth of the maximum pressure to minimize the
errors, especially on the moduli.
To purge the air entrapped in the fluid, the
densimeter is placed in a tank filled with an excess
sample volume, to allow the handling. Then all the
parts (metallic surfaces + sample volume) are
degassed under vacuum (pressure < 10
-1
mbar ).
Then the densimeter is closed, always in the tank,
filled this time by air-free lubricant.
As it has been reported above, compressibility
measurements are very difficult to conduct because
of the low volume changes with pressure: a high
accuracy is required. In our case pressure
measurements are run with a precision of +/- 0.25%
and the temperature is known to within +/- 0.5 %.
These values suggest that the main cause of potential
deviation in our results comes mostly from volume
measurements. Taking into account our experience
in the field, we consider that our volume losses are
obtained with an accuracy of +/- 3 %.
The best means to validate this point is to
compare our results to previously published data.
Here we have chosen di 2 ethyl hexyl sebacate (DOS
in Table 1) results from the A.S.M.E. Pressure
Report [16]. The comparison at 25°C is presented
Figure 1, where the relative volume reduction (-∆
V/V
0
) is plotted versus pressure. We observe a good
agreement in the low and in the high pressure
domains: nevertheless a sensible discrepancy (close
to 7%) is noted between 2 and 3 kbar. Even if this
kind of deviation could appear acceptable for
volume losses, it should be still considered with care
for bulk moduli determination, especially for
tangent bulk moduli.
The comparison with Dowson and Higginson
relationship [1] shows a good agreement at low and
at high pressure. However the analytical curve is
flatter than experimental ones which implies
significant differences in the moduli. The discussion
on the validity of this relationship is presented in the
next section.
3. VOLUME LOSSES RESULTS

All the results obtained at 60, 100 and 150°C are
reported in Figures 2 to 5, respectively for SIL1,
SIL2, POL1 and POL2.
Most of the results concerns a maximum pressure
increase of 500 MPa, except for SIL1: its viscosity
shows a strong pressure dependence at 60°C.
Consequently compressibility tests are impossible to
run under this condition. Normally the pressure steps
are equal to 50 MPa but due to ultrasonic problems
(signal fluctuations), some points have been
voluntary omitted.
After the initial pressure steps, SIL1 curves
suggest a linear increase of volume losses with
pressure as usually volume reduction found at the

highest pressure levels tends to decrease. Although
data at 60°C are close to those given by equation (1),
the gap between experimental and calculated values
increases continuously. At 150°C, the experimental
curve is very different from the analytical one.
SIL2 behavior is very specific: it shows a large
compressibility as the volume losses reach or exceed
20% at 500 MPa at all tested temperatures.
Furthermore, we observe a very large volume
reduction since the first pressure steps: at 150 MPa
and 60°C, 10% is reached as in SIL1, this reduction
is found at 250 MPa. In the highest pressures domain
(typically when P > 300 MPa), we discern the same
tendency that for SIL1. The total volume loss
increases linearly with pressure (up to 500 MPa): the
slopes found with SIL2 are lower than those
observed on SIL1.
Polyalphaolefins show results in better agreement
with usual tendencies. POL1 data at 60°C and
equation (1) are very close. The maximum volume
reduction attains 16% at 150°C as it was greater than
22% for SIL2. At 150°C, we have obtained very
similar results for POL1 and POL2 as at lower
temperatures, the curves differ.
To illustrate the above mentioned tendencies, we
have reported all the volume losses measured at
100°C in Figure 6. The curves confirm that SIL2 is
more compressible than the other fluids. It is also
shown that the behavior of each chemical structure is
typical: polyalphaolefin curves tend to an horizontal
asymptote as silicone curves suggest the continuation
of a linear increase.
Concerning the validity of equation (1), our
results show that the fluid chemical nature, pressure
and temperature govern the lubricants compression
behavior. Correlation with diester results near
ambient temperature is acceptable and comparison
with polyalphaolefins data has shown that a shift
occurs during the first pressure steps and after the
deviation remains constant. Even for SIL2, we can
consider that the data could be fitted by a similar
equation as the curvatures are very close. All these
observations suggest that equation (1) could be used
on a wide experimental domain if the two constants
(0.6 and 1.7) are changed by more appropriate ones.
Temperature influence is more important with
SIL2 and the less significant with POL2. Volume
reductions found on POL2 are more temperature
dependent than those obtained with SIL1.
A detailed analysis of the curves shows that
temperature influence takes place essentially during
the two first pressure steps. Beyond 100 MPa, we
could consider for each lubricant that the curves
obtained at 60, 100 and 150°C are roughly parallel.
These observations on the temperature influence are
consistent with literature [10-13].
Comparisons with previously published result is
not obvious. From our bibliographical analysis, it
appears that silicone fluids are typically very
compressible [4,10,12,13], compared to mineral oils
for instance. Klaus and O'Brien [10] have compared
predicted and measured (from A.S.M.E. [16])
isothermal secant bulk moduli for a similar fluid that
SIL1. At 100°C, they found 8.2% and 11.1% at 152
and 242 MPa and our more approaching data are
respectively equal to 7.7% and 10.9% at 150 and
250 MPa.
Galvin et al. [13] have reported data at 100°C on
a silicone fluid which fits very well our results on
SIL2. Concerning polyalphaolefins, Jacobson and
Vinet [2,3] have reported some data but only for
pressures greater than 425 MPa, making the
comparison impossible.
4. BULK MODULI RESULTS

4.1. Isothermal secant bulk moduli
Isothermal secant bulk moduli (K
S
) are directly
derived from volume losses by the following
equation:
K
S
= ∆P / (-∆V/V
0
) T = constant (2)
Equation (2) means that K
S
refers to the
volumetric change with pressure from atmospheric
pressure references. Measurements are run under
steady state pressure and temperature conditions.
The results are reported in Figure 7 for silicone
fluids and in Figure 8 for polyalphaolefins. The
initial value at P = 0.1 MPa has been determined
following an analytical method presented in the next
section.
In our pressure domain, we note that the
isothermal secant bulk moduli increase roughly
linearly with pressure. In agreement with the remarks
reported on silicone fluids volume losses, the
correlations between secant bulk moduli variations
and linear fittings is very good. For these fluids, the
results differ essentially in their initial value, i. e.
their slopes are almost equal and not temperature
dependent.
Table 2 presents the linear fitting results of the
isothermal secant bulk moduli variations versus
pressure for SIL1 and SIL2.
K
S
= K
S0
+ A
S
. P (3)
where A
S
is the slope,
K
S0
is the initial isothermal bulk modulus (in MPa),
and P the pressure (in MPa).
F
luid
SIL1 SIL2
T
°C
60 100 150 60 100 150
K
S0
1774
1612
1349
953 776 585
A
S
2.62 2.57 2.73 3.25 3.33 3.36
Table 2 : Results of linear regressions on K
S
(in
Mpa) versus pressure for SIL1 and SIL2.
Concerning POL1 and POL2 isothermal secant
bulk moduli, the tendencies are more temperature
dependent. At 60, 100 and 150°C for POL1 and at
60°C for POL2 a second degree polynomial gives a
better fitting than a simple linear regression. Table 3
presents the values of this correlation:
K
S
= K
S0
+ B
S
. P + 10
-3
.C
S
. P
2
(4)
where
B
S
and
C
S
are the polynomial coefficients,
K
S0
is the initial isothermal bulk modulus (in MPa),
and P the pressure (in MPa).
F
luid
POL1 POL2
T
°C
60 100 150 60 100 150

K
S0
1663
1285
1103
1580
1256
1023
B
S
2.16 2.99 3.12 1.98 3.15 3.62
C
S
4.70 2.13 1.18 2.52 0.630
0.147
Table 3 : Results of degree 2 curve fitting on the
isothermal secant bulk moduli for POL1 and POL2.
4.2. Isothermal tangent bulk moduli
This parameter is the thermodynamically correct
isothermal bulk modulus. It represents the true rate
of volume change at the pressure of interest. The
main difference with K
S
is that the tangent modulus
does not refer to the initial volume V
0
. When P tends
to the atmospheric pressure, K
S
and K
T
become
equal. K
T
is given by:
K
T
= - V . ( δ P/ δV ) at T = constant (4)
As V(P) is known point by point, the K
T
determination is not direct. At first, the volume
losses have been fitted by a polynomial function.
Then these functions have been derived and K
T
values have been deduced.
Figures 9 and 10 present the isothermal tangent
bulk moduli found respectively in silicone fluids and
in polyalphaolefins. These moduli are always greater
than secant ones.
Concerning SIL1 and SIL2 (Figure 9), the effects
previously mentioned conduct to weakly curved
slopes.
The variations are more significant with POL1
and POL2 (Figure 10). The curves shows that high
values (> 10 GPa) have been reached at 400 or 500
MPa. It is also expected that under higher pressures,
the moduli can be multiplied by a factor of 10 at
least. This means that volume losses should be very
limited, actually close to zero.
4.3. Isentropic tangent bulk moduli
It is the volumetric tangent modulus of elasticity
under conditions of constant entropy. Under normal
conditions it is greater than the isothermal tangent
bulk modulus by the ratio of the fluid specific heats.
When data are available, this parameter is used
under conditions where pressure changes are rapid,
with little opportunity for the temperature to come in
equilibrium. This definition underlines the interest to
use this parameter in lubrication problems where
dynamic high pressures occur.
One of the methods developed to run isentropic
compressibility experiments is based on ultrasonic
velocity measurements. It is supported by the
following equation:
k
T
= - V. ( δ P/δV ) = ρ . c
2
at S = constant (5)
where k
T
is the isentropic tangent bulk modulus,
ρ is the density,

c is velocity of sound and S the entropy.
The validity of equation (5) has been verified by
Hayward [6] and Noonan [17]. Due to our peculiar
device which requires the ultrasonic velocity
measurements before running compressibility tests,
we have been able to determined k
T
under the same
conditions as the other moduli.
The results are plotted on Figure 11 for silicone
fluids and on Figure 12 for polyalphaolefins. The
curves are almost strait lines: a weak curvature
towards the pressure axis is observed. This implies
that isentropic tangent bulk moduli increase almost
linearly with pressure. Apart for initial values, a
linear regression performed on these results gives an
acceptable agreement (see Table 4).
k
T
= k
T0
+ D
T
. P (3)
where D
T
is the slope,
k
T0
is the initial isentropic bulk modulus (in MPa),
and P the pressure (in MPa).
Fluid T °C k
T0
D
T
SIL1 60 1479 9.95
SIL1 100 1314 8.80
SIL1 150 1038 8.76
SIL2 60 1098 9.32
SIL2 100 921 9.08
SIL2 150 734 8.73
POL1 60 1591 9.56
POL1 100 1377 9.00
POL1 150 1112 8.78
POL2 60 1999 11.15
POL2 100 1652 11.15
POL2 150 1244 11.05
Table 4 : Results of linear fitting on k
T
.
At the opposite of isothermal tangent bulk
moduli, all the fluids give similar curves: they differ
by the initial value at P = 0.1 MPa as their slopes are
very close.
The explanation could come from the fluids
viscoelastic behavior in compression. First of all, it
is important to remind that longitudinal waves could
be splitted in pure compression waves plus pure
shear waves and that longitudinal velocities are also
frequency dependent [18]. The limit is the infinite
frequency velocity which is 20 to 40% greater than
the ultrasonic one, in the liquid domain. If we apply
time (frequency) pressure equivalence principle,
there any reason for the ultrasonic velocity to
increase suddenly in our experimental conditions. As
in parallel the density increase is continuous, this

could not lead to discontinuous results or to very
large variations.
From the lubricant physical state point of view,
we could consider that the material is not in
thermodynamically equilibrium when submitted to
ultrasonic waves. As the output analysis is also made
during a very short time, only a part of the fluid
response could appear during this kind of
experiment.
Concerning the ratio of the isentropic tangent
modulus to the isothermal one, our results confirm
the usual rule i. e. isentropic values are greater, but
only in the first half of our pressure domain. For
higher pressures, isothermal tangent moduli becomes
greater. Consequently this is true also for heat
capacities.
5. CONCLUSIONS
Compressibility experiments have been
conducted under high pressure on several lubricants
of various chemical composition.
Volumes losses have been found to vary as a
function of pressure, temperature and of the fluids
chemical nature. Among the tested lubricants,
silicone fluids have been found more compressible
than polyalphaolefins. Dowson and Higginson [1]
relationship should be used to fit compressibility
results if its parameters are determined before.
Isothermal secant bulk moduli increase almost
linearly as a function of pressure, with slopes weakly
temperature dependent. Isothermal tangent bulk
moduli are greater than secant ones. Their increase
with pressure is more pronounced, especially under
the highest pressures where they can reach or exceed
10 GPa.
Isentropic tangent bulk moduli have been
deduced from both compressibility and ultrasonic
measurements. We have observed a nearly linear
increase with pressure. This means that under high
pressures, isentropic tangent bulk moduli becomes
lower than isothermal ones, in opposition to the
comments found in literature. This discrepancy has
been explained by the fluids viscoelastic behavior in
compression.
ACKNOWLEDGMENTS
This research was partly supported by the C.E.A.
(Centre du Ripault). G. Roche from L.M.C. is thanks
for his contribution to the experiments.
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