J. Phys. F: Met. Phys. 17 (1987) 593-604. Printed in the U K
Non-equilibrium molecular dynamics calculation of the shear
viscosity of liquid rubidium
P T Cummingsf and G P Morrisst
+ Department of Chemical Engineering, Thornton Hall, University
of Virginia.
Charlottesville, VA 22901, USA
Research School of Chemistry. Australian National University, GPO Box 4, Canberra,
A C T 260 I , Australia
Received 18 April 1986. in final form 25 June 1986
Abstract. We report non-equilibrium molecular dynamics simulations of liquid rubidium
using the isokinetic sllod algorithm for the dynamics and the Price potential for the
interactions. The three state points simulated are the same three chosen by Bansal and Bruns
who used a quite different non-equilibrium method. However. our results differ substantially
from theirs. Specifically, we find that the Price potential consistently predicts better results for
the shear viscosity. A more fundamental disagreement between our results and theirs is that
our results underestimate the experimental data where theirs overestimate and vice versa.
This suggests that the method used by Bansal and Bruns has systematic errors.
1. Introduction
Over the past decade, the non-equilibrium molecular dynamics (NEMD)technique has
developed into a powerful tool for the study of transport properties of both simple and
molecular fluids (Hoover and Ashurst 1975, Evans and Morriss 1984a, Evans 1985,
Evans and Hoover 1986). The most promising recent developments include the derivation
of synthetic techniques for calculating such properties as the shear viscosity, thermal
conductivity and self-diffusion and the use of Gauss's principle of least constraint as a tool
for constraining simulations in physically meaningful ways (yielding, for example,
isokinetic and/or isobaric ensemble simulations (Hoover et a1 1982, Evans and Morriss
1983)). Finally, the formal status of these and other NEMD techniques has been clarified
tremendously in recent years through linear response theory, so that previously raised
objections to the orthodoxy of the NEMD methods have now been effectively answered
(Evans and Morriss 1984a, Morriss and Evans 1985).
Of all the NEMD techniques, methods for calculating the shear viscosity have the
longest history and therefore have come under the greatest scrutiny. The methodsreviewed in $ 3 of Evans and Morriss (1984a)-amount
to setting up a steady state
planar Couette flow in a homogeneous system. The streaming velocity has a non-zero
component in the x direction, U,, which satisfies
dU,/d>x= y
(1)
where y is the constant strain rate. Equation (1) implies that there is a linear velocity profile
in the x direction, the characteristic of planar Couette flow (Bird et a1 1960). The J'S
0305-4608/87/030593 + 12 $02.50
0 1987 IOP Publishing Ltd
593
594
P T Cummings and G P Morriss
component, Pyx,of the pressure tensor P i s related to the velocity profile by the constitutive
relation
PYx=-~(d~,/d.v)=-~~
(2)
where for Newtonian fluids q, the shear viscosity, is dependent on the state variables
(pressure, temperature T and composition, for example). Non-Newtonian fluids are fluids
for which q depends on y in addition to the dependence on the thermodynamic state. The
pressure tensor P, which is of second rank, is calculated from the expression
Y
I=
.
Y
Y
I
where m is the mass of each molecule, U , is the velocity of molecule i, Vis the volume of the
system, r!, = r, - r, with rk being the position of the centre of mass of molecule k and F , is
the force exerted by molecule i on moleculej and so is given by the gradient of the pair
potential q ( r ) between the molecules. Notice that the velocities in equation (3) have the
streaming part U subtracted out and are thus peculiar velocities (Evans and Morriss 1984a).
The shear viscosity is therefore calculated from equation (2) using the time average of the
jix-component of P and the hydrostatic pressure is the time average of the trace of P.
In order to perform a molecular dynamics simulation of planar Couette flow, one
necessary modification of the usual formalism is that the images of the central simulation
cell located above and below it must move in time in opposite directions at a constant rate
determined by the strain rate 7, and the x-component of the velocity of a molecule leaving
either the top or the bottom of the central cell in the y direction must be modified to
be consistent with the overall velocity profile (the so-called ‘sliding brick’ boundary
conditions). These boundary conditions were first introduced by Lees and Edwards (1972)
and in their original NEMD algorithm the velocity profile was set up and maintained solely
by these moving boundaries, the dynamics of the particles in the simulation cell being given
by Newton‘s equations of motion,
dri - pi
dt m
dPi _
- Fi
dt
where pi is the laboratory momentum of molecule i and Fi is the force on molecule i due to
all the other molecules in the system. A convenient reformulation in terms of first-order
equations of motion (Evans and Morriss 1984b) is the sllod algorithm given by
dri
dt
-
Pi
m
dPi
-- Fi
dt
+ri.Vu
-pi
*
VU
combined with the Lees-Edwards ‘sliding brick’ boundary conditions. As far as the
trajectories r,(t) of the individual molecules are concerned, the sllod algorithm can be
shown to be equivalent to the Lees-Edwards algorithm except at time f = O since
elimination of pi from equations (5a) and (5b) leads to Newton’s equations of motion.
However, from equations (5) it is clear that the sllod equations embody the velocity profile
5 95
Shear riscositv of liquid rubidium
Table I . A comparison of predicted and experimental shear viscosities for liquid rubidium.
The theoretical predictions of this work and Bansal and Bruns (1984) are obtained using two
different non-equilibrium molecular dynamics techniques.
318.9
625.5
939.9
a
1S O I
1.323
1.1808
0.665
0.215
0.138
0.590
0.338
0.204
0.644
0.234
0.155
This work.
Bansal and Bruns (1984).
Landolt-Bornstein ( 1 969).
(since the gradient of the streaming velocity is added to the peculiar velocity pi/m). The
sllod algorithm is thus homogeneous and consistent with the boundary conditions. Evans
and Morriss (1984a) review the development of the sllod algorithm (as well as the closely
related Doll’s tensor method) and the reasons for their utility in non-equilibrium
simulations of planar Couette flow; we refer the reader to their paper for details.
Other methods for calculating the shear viscosity include the Green-Kubo relation
(which requires long equilibrium molecular dynamics simulations of large systems for
accuracy equivalent to the synthetic sllod algorithm (Holian and Evans 1983)), the
differential trajectory method (Ciccotti and Jacucci 1975, Ciccotti et a1 1976, 1979) and
the method of Singer and co-workers (Singer et a1 1980). It has recently been shown
(Paolini et a1 1986) that the differential trajectory method is more reliable when the
perturbation is a delta function rather than a step function (as used by Singer et a1 1980).
This last method was used by Bansal and Bruns (1984) to determine the shear viscosity of
liquid R b using the Price potential (Price et a1 1970, Price 1971) at three state points:
T=318.9 K, p = 1.501 g ~ m - ~T=625.5
,
K, p = 1.323 g ~ m -and
~ T=939.9 K, p =
1.1808 g cm-3 where T is the absolute temperature and p is the density. Their results
(shown in table 1) suggest that the Price potential is only moderately accurate in predicting
the transport properties of liquid Rb. In fact, according to Bansal and Bruns. the error in
the shear viscosity predicted by the Price potential ranges from 32 to 45%. Given the
success of the Price potential in predicting thermodynamic and static and dynamic
structural properties of liquid R b (Rahman 1974a, b, Mountain 1978, Arlinghaus and
Cummings 1987), this discrepancy seems somewhat large and prompted the reexamination of the predictions of the Price potential given in this paper. Using the best
available technique-that
is, Gaussian thermostatted sllod dynamics-we find the results
given in table 1, which indicate that the Price potential results range in error from 3 to
11%, a significant improvement over that of Bansal and Bruns (1984).
In 8 2 of this paper, we describe the method of calculation, the results obtained and
their analysis in more detail. Section 3 contains our conclusions.
2.
NEMD
simulation method
The potential used in the simulations reported in this paper is the Price potential calculated
according to the prescription given by Price and co-workers (Price et a1 1970, Price 1971).
As with all effective interionic potentials derived using pseudopotential theory (Harrison
1966), the Price potential is density dependent; thus the potential differs at each of the
596
P T Cutnmings and G P Morriss
three state points simulated due to the different densities involved. The
solved are the isokinetic sllod equations given by
NEMD
equations
where ,Iis given by
where 0 indicates the full contraction of two second-order tensors and N is the number of
particles (pseudoatoms in the current context) in the simulation cell. The formula for 1 is
dictated by requiring that the temperature T is held constant, which is equivalent to
demanding that EL I p: remain a constant of the motion.
All the simulations were performed using 108 rubidium ions. The calculations were
carried out on a Computer Signal Processing Inc. (CSPI) 6420 64-bit word array processor
in the Center for Computer Aided Engineering at the University of Virginia. Since the
potential and the force are not given analytically, they were discretised onto a grid of 900
points equally spaced in r 2 corresponding to the range O<r<3a where Y is the
interparticle separation. Both the force and potential were obtained by linear interpolation
of this grid. The potential and force were assumed to be zero for r > 3a and the potential
shifted to make it continuous at the cut-off 3a. Those simulation runs which start from a
crystalline configuration begin with a face-centred cubic lattice which is equilibrated over
20 000 time steps. At each state point, planar Couette flow was simulated at five values of
the shear rate y given in dimensionless form by y* = y a ( m / ~ ) ’ /where
~
a is the diameter of
an ion which can be defined to be the interionic separation at which the Price potential is
zero so that the potential is positive for all r < U (= 4.4 8, for the densities simulated), m is
the mass of the ions (= 14 1.9 17 x
g) and E is the depth of the Price potential (which
is taken to be c / k B= 4 0 0 K, kB being Boltzmann’s constant, at the three state points
simulated). In each simulation, the dimensionless time step used in the fourth-order
predictor-corrector integration routine (Evans and Morriss 1984a) is Ar* = Ar(c/ma2)1/2=
0.005 which corresponds to a real time step of 1.1 16 x
s. The values of )I* along with
the number of time steps per value of y* and the calculated values of the shear viscosity
are given in tables 2 to 4. Notice that at each state point the shear viscosity reduces with
Table 2. N E M D simulation results for liquid rubidium at a temperature of 318.9 K and
a density of 1.501 g cm-j. Least-squares fitting of the data to equation (9) gives
7=0.665 -0.371 y * ’ I 2 cP. The zero strain rate value of 0.665 CP compares with the
experimental value of 0.644 CP and the Bansal and Bruns (1984) value of 0.590.
80000
160000
240000
400000
3 60000
1 .o
0.64
0.36
0.16
0.1
*
0.3 I O 0.002
0.358 0.001
0.424 i 0.004
0.508 k 0.01
0.569 k 0.01
- 3.63 i 0.01
- 3.80
i 0.01
-3.91 *0.01
- 4.00 0.0 1
- 4.02
0.0 I
*
*
5.02 i 0.01
4.8 1 + 0.01
4.68 0.01
4.56 i 0.02
4.54 i 0.0 I
Shear riscositj' oj'liquid rubidium
597
Table 3. N E M D simulation results for liquid rubidium at a temperature of 625.5 K and
density 1.323 g ~ m - ~ Least-squares
,
fitting of the data to equation (9) gives
~ = 0 . 2 1 5-0.029
cP. The zero strain rate value of 0.215 C P compares with the
experimental value of 0.234 C P and the Bansal and Bruns (1984) value of 0.338.
20000
I20000
240000
3 20000
240000
1 .o
0.63
0.36
0.16
0.1
0. I85 i 0.002
0.192 z 0.005
0.199 i 0.004
0.203 + 0.01
0.205 + 0.003
~
3.086 i 0.004
- 3.140 i 0.002
- 3. I 7 2
i 0.004
i 0.002
- 3.193 i 0.003
- 3.188
4. I4 k 0.01
4.088 i 0.001
4.05 i 0. I
4.033 T 0.003
4.027 i 0.004
increasing shear rate-that is, rubidium exhibits shear thinning, and so is non-Newtonian
(Bird et a1 1960). in the range of strain rates used in the N E M D simulations. This is typical
of the NEMD technique. Because of the small system sizes involved in these simulations, the
strain rates must be quite large in order for there to be a discernible velocity profile across
the simulation sample (so that a value of P,, can be determined with small fluctuations).
For liquid rubidium, a reduced strain rate of y* = 1.0 corresponds to a strain rate in real
units of y=4.482 x l o i i s - ' which exceeds considerably strain rates currently accessible
to experiment. (Note, however, that for many systems-particularly colloidal suspensions
where the particles have a very high molecular mass-a
reduced strain rate of y* = 1.O
corresponds to strain rates in real units well within present laboratory capabilities.)
Clearly then, the viscosity of interest which must be compared with experiment is the
Newtonian viscosity obtained as the zero strain rate extrapolation of the simulation results
for 7. Theoretical considerations (Kawasaki and Gunton 1973, Yamada and Kawasaki
1975, Ernst et a / 1978) and extensive " I D simulation results (Evans 1979, 1981, Evans
and Hanley 1980) indicate that to leading order v is related to the strain rate by
where v o and vI are independent of 7. Consequently, we illustrate the simulation results for
the shear viscosity in figures 1 to 3 by plotting v against
From these figures, it is clear
that the simulation results correlate very well with equation (9). To calculate vo and vl, the
simulation results are least-squares fitted to equation (9) with the results given in tables 2 to
4. The zero-strain rate results are then reported in table 1.
As an aside, the error estimates given in tables 2 to 4 and illustrated as error bars in
figures 1 to 3 deserve some explanation. In the appendix, the individual simulation runs
which were used to obtain the means and error estimates given in tables 2 to 4 are
e.
Table 4. h E M D simulation results for liquid rubidium at a temperature of 939.9 K and
density 1.1808 g cm-3. Least-squares fitting of the data t o equation (9) gives
q=0.138-0.0099 y*'" cP. The zero strain rate value of 0.138 C P compares with the
experimental value of 0.155 C P and the Bansal and Bruns (1984) value of 0.204.
80000
80000
l20000
280000
280000
I .o
0.64
0.36
0.16
0. I
0.128 f 0.001
0.130 i 0.001
0.132 i 0.002
0.135 i 0.002
0. I 3 4 i 0.002
-2.571 kO.003
- 2.609 i 0.001
- 2.62 i 0.01
- 2.62 f 0.01
- 2.627 i 0.005
4.04 1 0.007
4.00 i 0.01
4.00 i 0.01
4.00 i 0.02
4.00 i 0.01
598
P T Cummings and G P Morriss
Figure 1. The shear viscosity q of liquid rubidium at 318.9 K and 1.501 g cm-3 as a
function of the square root of the reduced strain rate y*. The circles and the error bars
represent the simulation results and the estimated error respectively (see text for explanation).
The straight line is the least-squares fit of the experimental data to equation (9) and is given in
table 2.
described in detail. At most state points (defined by the density, temperature and strain
rate), several simulation runs were performed, each of which yielded a value (or
‘measurement’) for q, the pressure P and the configurational energy Uconf.The values
reported in tables 2 to 4 are then obtained as the statistical mean of the ‘measurements’
(weighted according to run length) and the errors represent one standard deviation. For
Figure 2. The shear viscosity q of liquid rubidium at 625.5 K and 1.323 g cm-3 as a
function of the square root of the reduced strain rate y*. The circles and the error bars
represent the simulation results and the estimated error respectively (see text for explanation).
The straight line is the least-squares fit of the experimental data to equation (9) and is given in
table 3.
599
Shear viscosity of liquid rubidium
0.120
0
0.2
0.6
0.4
0.8
1.o
112
Figure 3. The shear viscosity q of liquid rubidium at 939.9 K and 1.1808 g cm-3 as a
function of the square root of the reduced strain rate y*. The circles and the error bars
represent the simulation results and the estimated error respectively (see text for explanation).
The straight line is the least-squares fit of the experimental data to equation (9) and is given in
table 4.
those state points where only one long simulation run was performed, the errors have been
conjectured, the conjecture being based in part on the block averages of 2000 timesteps
performed throughout every simulation run. Thus, the reported values and error estimates
in tables 2 to 4 represent block size weighted block averages and standard deviations.
Recently, Straatsma et a1 (1986) have examined the formal correctness of estimating errors
based on standard deviations obtained from block averaging and they conclude that the
quality of the error estimate depends on the choice of the block size, which must be
neither too small (as this increases correlation between ‘measurements’) nor too large (as
this reduces the number of ‘measurements’). On the basis of their conclusions, it is possible
that the errors reported in this paper may be artificially low. In view of this, we have
included details of the individual simulation runs in the appendix for the reader to draw his
or her own conclusions about the quality of our results.
From the results in table 1, we conclude that the Price potential is quite accurate in
predicting the shear viscosity of liquid rubidium at the three state points shown. It is
interesting to note that at the high-density state point (which is very near the triple point),
the Price potential prediction overestimates the experimental result while it underestimates
it at the medium and lower density. Since the temperature is also varying at each state
point, it is difficult to draw a firm conclusion about where the Price potential might yield
exact viscosities, but it is clear that this region of exactitude of the Price potential is
bounded by the temperatures and densities of the medium- and high-density state points
simulated in this study.
That the Price potential is most accurate near the triple point and less accurate at
expanded (lower density) states is consistent with results obtained by Rahman (1974b) and
Mountain (1978) for the structure factor of liquid rubidium. Using equilibrium molecular
dynamics, Rahman found that the Price potential predicted the structure factor and
inelastic neutron scattering data for rubidium at the triple point very well while Mountain
found using Monte Carlo simulation that at expanded states the Price potential yielded less
accurate predictions for the structure factor. The source of the inaccuracy of the Price
600
P T Crrtntiiitigs and G P Morriss
potential at expanded states is not clear and will undoubtedly by the subject of further
investigation.
It is known (Kawasaki and Gunton 1973, Yamada and Kawasaki 1975, Ernst et al
1978; Evans 1979, 1981, Evans and Hanley 1980) that the pressure P depends on the
strain rate according to the equation
P=Po + P l y ?
(10)
Figure 4 shows the results obtained by plotting P against y*3'2 and clearly equation (10) is
satisfied by the simulation results. (Strictly speaking, the results presented in figure 4
correspond to the trace of the pressure tensor P given in equation (3). The full expression
for the pressure for liquid metals involves terms which take into account the electronic
contribution and the density dependence of the Price potential.)
According to Evans (1 983), the configurational part of the internal energy depends on
the strain rate according to
uconf
= LO
' + U ]y 3 2 .
( 1 1)
The results obtained in this simulation are consistent with this dependence and can be fitted
to equation ( I 1) a s indicated in figure 5.
The N E M D results at the highest density state point, given in table 2 , warrant further
comment. Of the three state points simulated, the results at this state point show the most
'f
42
4.0
30
1
0
02
04
06
T*
08
10
312
Figure 4. The dimensionless trace of the pressure tensor-see equation (3)-as a function of
the cube of the square root of the strain rate Y* for liquid rubidium at the three state points
T=318.9 K, p = 1.501 g cm-3 (labelled I), T = 6 2 5 . 5 K, p = 1.323 g C I T I - ~ (labelled 11) and
T=939.9 K . p = 1.1808 g ~ m ((labelled
- ~
111).
Shear riscositj- of liquid rubidium
601
deviation from the strain rate dependence for viscosity given in equations (9). In fact,
plotting v against y* on a log-log plot suggests that these results fit a strain rate
dependence of the form
V=Vo + V I f
(12)
where a is slightly less than i. It has been observed that a failure to satisfy the strain rate
dependence (9) for the shear viscosity over the whole range of simulated strain rates may
indicate that there are several distinct regions where (9) is valid with different values of v0
and q l for each region corresponding to Bingham plastic behaviour (Evans 1982). This
behaviour would indicate that at the density and temperature in question the simulated
fluid would, at zero strain rate, be a crystalline solid rather than a liquid and that the
imposition of a non-zero strain rate induces shear melting. However, there is no evidence in
the present simulations nor in that of Rahman (1947b) that at zero strain rate the Price
potential is predicting a solid rather than a liquid state. On the basis of the available
information, we therefore conclude that the deviation from equation (9) is not due to shear
melting effects. In view of the discussion above, we suspect that the errors in the results
tabulated in table 2 represent underestimates and that equation (9) is satisfied to within the
actual error of the simulation. T o confirm this, we are in the process of performing further
simulations at this state point using more values of y and evaluating more carefully the
dependence of the results on system size. (To date, our investigation of system size
dependence has been limited to relatively short calculations at this state point using 256
molecules for y* equal to 1.0 and 0.36 which differ from the results reported in table 1 by
less than 3%, suggesting that system size dependence of the results in table 2 is minimal.)
3. Conclusions
We have demonstrated the accuracy of the Price potential in calculating the shear viscosity
of liquid rubidium. The NEMD results presented in this paper represent errors of only
3- 11% when compared with the corresponding experimental data. These results contrast
602
P T Cummings and G P Morriss
with those obtained by Bansal and Bruns (1984) who found errors of 32-45% using a
somewhat different NEMD simulation technique. Moreover, as is clear from table 1, Bansal
and Bruns underestimate the experimental results where the results presented in this paper
overestimate and tiice i w s a . Thus there is a strong discrepancy between the results
presented here and those of Bansal and Bruns (1984) which must be attributed to the
simulation methodology since the same pair potentials are used in both studies. (Since
Bansal and Bruns also used the Price potential and the same size system (108 molecules),
apart from the difference in simulation methodology the only other possible source of the
discrepancy is the cut-off (the point beyond which the potential is assumed to be zero). In
this paper, as noted in 0 2 a cut-off of 3a was used while Bansal and Bruns utilised a cut-off
equal to half the box size which for the three densities 1.501 g, 1.323 g and 1.1808 g cm-3
correspond to cut-offs of 2.47, 2.57 and 2 . 6 7 ~respectively. To probe the importance of the
cut-off. we performed several short simulation runs with a cut-off of 2 . 5 ~at the
intermediate density state point. The results obtained for the shear viscosity lie within the
error estimates given in table 3. This eliminates the difference in cut-off as a source of the
discrepancy between the present results and those of Bansal and Bruns.)
Given the success of the NEMD technique used in this paper on many other systems
(both simple and molecular fluids), we are confident of the accuracy of the results
presented in this paper. Accepting our results and considering the agreement of the
simulation results presented in this paper with experimental data suggests that the Price
potential is capable of predicting shear viscosities of liquid rubidium with quantitative
accuracy. In future research, the accuracy of the Price potential for other transport and
thermodynamic properties of liquid rubidium, as well as other liquid metals, will be
examined.
Acknowledgments
PTC acknowledges the Camille and Henry Dreyfus Foundation for their support of this
research through the award of a Grant for Newly Appointed Faculty in the Chemical
Sciences. The simulations reported in this paper were performed on a CSPI 6420 64-bit
word array processor in the Center for Computer Aided engineering at the University of
Virginia. The authors are indebted to the Center for the provision of computational
facilities in support of the array processor. The support of the National Science Foundation
in providing matching funds (CPE-8405715) for the purchase of the array processor to
support research in statistical mechanics and chemical process dynamics is gratefully
acknowledged.
Appendix
Details of the individual simulation runs are provided in table A l . At each density and
temperature, the simulations began with strain rate y* = 1.O and the molecules initially in a
face-centred cubic crystalline configuration. For smaller values of the strain rate, a
melted configuration obtained from an earlier simulation run was used as the initial
configuration. At each value of the strain rate, a run of between 10000 and 2 0 0 0 0
timesteps was performed and discarded in order to reach a steady state. The discarded
runs are not recorded in table A 1.
603
Shear viscosity of liquid rubidium
Table A 1. Individual non-equilibrium molecular dynamics simulation runs performed to
obtain the results given in tables 1 to 4.
Timesteps
T(K)
p(gcm-’)
y’
II (CP)
20000
20000
20000
20000
80000
80000
80000
80000
80000
80000
80000
80000
80000
80000
80000
80000
80000
120000
6000
14000
40000
80000
80000
80000
40000
40000
80000
80000
80000
80000
80000
40000
120000
20000
20000
40000
40000
40000
l20000
120000
40000
120000
40000
80000
160000
318.9
3 18.9
3 18.9
318.9
3 18.9
318.9
318.9
318.9
318.9
3 18.9
318.9
3 18.9
3 18.9
318.9
318.9
318.9
318.9
318.9
625.5
625.5
625.5
625.5
625.5
625.5
625.5
625.5
625.5
625.5
625.5
625.5
625.5
625.5
625.5
939.9
939.9
939.9
939.9
939.9
939.9
939.9
939.9
939.9
939.9
939.9
939.9
1.501
1.501
1.501
1.501
1.o
0.306
0.310
0.3 14
0.309
0.359
0.357
0.420
0.429
0.423
0.497
0.503
0.528
0.498
0.5 12
0.560
0.583
0.563
0.569
0.181
0.186
0.191
0.192
0.199
0.204
0.196
0.194
0.202
0.21 I
0.187
0.2 12
0.208
0.206
0.202
0.129
0.128
0.127
0.129
0.130
0.132
0.134
0.136
0.135
0.134
0.133
0.134
1.501
1.501
1.501
1.501
1.501
1.501
1.501
1.501
1.501
1.501
1.501
1.501
1.501
1.501
1.323
1.323
1.323
1.323
1.323
1.323
1.323
1.323
1.323
1.323
1.323
1.323
1.323
1.323
1.323
1.1808
1.1808
1.1808
1.1808
1.1808
1.1808
1.1808
1.1808
1.1808
1.1808
1.1808
1.1808
I .o
I .O
1.o
0.64
0.64
0.36
0.36
0.36
0.16
0.16
0.16
0.16
0.16
0.1
0.1
0.1
0.1
I .o
1 .o
0.64
0.64
0.36
0.36
0.36
0.36
0.16
0.16
0.16
0.16
0.1
0.1
0.1
1 .o
1.o
1
.o
0.64
0.64
0.36
0. I6
0.16
0.16
0.1
0.1
0.1
Uconf/N& Pa3/&
- 3.628
- 3.643
-3.617
- 3.638
- 3.791
- 3.802
-3.915
-3.913
-3.910
- 4.000
- 3.989
- 4.008
- 3.998
- 4.006
- 4.020
- 4.023
-4.016
- 4.026
- 3.092
- 3.083
- 3.143
-3.138
-3.167
- 3.175
-3.171
-3.177
-3.190
-3.186
-3.186
-3.188
-3.189
-3.194
-3.196
- 2.568
- 2.567
- 2.575
-2.610
- 2.607
- 2.620
- 2.629
- 2.637
-2.614
- 2.636
- 2.634
- 2.622
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