VISCOSITY
OF VITREOUS
KN03-Ca(NO& MIXTURE
MHz for MeNC aligned in the nematic phase. Inasmuch as the independent T2* measurements of Loewenstein and Margalits also support our results, a remeasurement of the N14 TI of MeNC is de~irab1e.l~We
conclude, therefore, that there is a real difference between the e2qQ/hof MeNC in the gas phase (0.483 -I:
0.017 R.IHz)~and that in the liquid phase (0.27 MHz).
I n the case of EtNC, an upper limit to e2qQ/hof 0.5
MHz has been determined from the gas phase microwave spectrum.15 It would be premature to conclude
that the liquid phase value of 0.30 MHz is significantly
smaller than the gas phase value until higher resolution microwave data can be obtained.
As noted above, e2qQ/h for nitrogen compounds
generally is 10-15% less in the solid than in the gas.
I n the compounds in which this decrease is observed,
the nitrogen atom has been subject to large electric
field gradients, resulting in quadrupole coupling constants of the order of 4 MHz. It is entirely possible
that the effect is not a relative one, but an absolute one.
That is, e2qQ/hmay be reduced in magnitude by about
0.4-0.6 MHz in going from the gas phase to the
condensed phase. I n any case, the observed difference
in the values of e2qQ/hfor MeNC in the liquid and gas
phases may reflect alterations in the field gradient
caused by interactions with neighboring molecules.
4147
The dipole moment ( p ) of MeCN in the gas phase
(3.92 D)16 is substantially larger than that of the
liquid in benzene solution (3.47 D).17 The observed
decrease in the p of MeCN in going from the gas to
the condensed phase parallels the decrease in e2qQ/hin
going from the gas to the solid. It appears that a
change in p of comparable magnitude occurs in the case
of MeNC. MeNC in the gas phase has a p of 3.83 D.’6
While the p of MeNC in the condensed phase has not
been reported, that of EtNC in benzene is 3.47 D.18
Thus, it seems reasonable that the change in electron
distribution associated with the alteration of p would
be reflected in different nitrogen quadrupole coupling
constants for gaseous and liquid MeNC. The lack of
an observable N14 nmr resonance in solid MeNC3 suggests that further changes in electron distribution occur
upon solidification.
(14) H.S. Gutowsky, private communication.
(15) R. J. Anderson and W. D. Gwinn, J . Chem. Phys., 49, 3988
(1968).
(16) S. N. Ghosh, R. Trambarulo, and W. Gordy, ibid., 21, 308
(1953).
(17) J. W.Smith and L. B. Witten, Trans. Faraday SOC.,47, 1304
(1951).
(18) R. G. A. New and L. E. Sutton, J. Chem. floc., 1415 (1932).
Viscosity of a Vitreous Potassium Nitrate-Calcium Nitrate Mixture
by R. Weiler, S. Blaser, and P. B. Macedo
Vitreous State Laboratories, Catholic Universitg of America, Washington, D. C.
(Received April 14, 1969)
Viscosity measurements were made between 10-l and lo8P over the temperature range from 79 to 200’ in 60%
KN03-40% Ca(NO& (mole per cent). The data departed from the Fulcher equation as one lowered the temperature below 120’. The departure was similar to that observed in Bz03 and n-propyl alcohol, indicating that
the viscosity tended to a low-temperature Arrhenius region. The existence of such an Arrhenius region in so
many different types of liquids shows a major deficiency in the existing viscosity theories.
Introduction
Many of the viscosity equations can be approximated
by the Fulcher’ equation
In7
=
A
+ B/(T - To)
(1)
in which 7 is the viscosity and A , B, and T are material
constants. This expression predicts the temperature
dependence of any structural relaxation time, be it
involved in shear, volume, dielectric, or conductivity
processes. Thus the original equation now has a much
more general application in determining the structural
kinetics of liquids.
Of the many viscosity models which predict this relationship there are three most prominent ones which will
be reviewed here. Cohen and Turnbul12 assumed that
the limiting mechanism for an irreversible diffusional
motion was the ability that the ‘llattice cell” would
expand to permit molecules to “jump” over each other.
(1) G. S. Fulcher, J . Amer. Ceram. Soc., 8, 339 (1925).
(2) M.H.Cohen and D. Turnbull, J. Chem. Phus., 31, 1164 (1969).
Volume 78, Number 18 December 1969
4148
R. WEILER,S. BLASER,AND P. B. MACEDO
They calculated the distribution of excess or free volume
by maximizing the entropy of mixing. By a further
assumption that there was a minimum volume below
which no motion could occur, they calculated the
temperature dependence of the viscosity (or any other
structural relaxation time) to be
In q = A
+ B'/(V - V O )
N
A
+ B / ( T - To)
(2)
where Ti is the actual volume and Vo is the close-packed
volume. The approximation used here is that the
expansion coefficient, CY, is temperature independent
(B = B'/a).
A second approach is due to Adam and G i b b ~ . They
~
proposed that in order to have viscous flow, a flow unit
had to overcome a potential barrier. In addition, depending upon the configurational entropy, more than
one flow unit might have to perform the "jump" at a
time. This led to an equation of the form
lnq = A
+ B"/[T(S - So)]
(3)
in which (S - So) is the configurational entropy, 8,.
This equation is equivalent to the Fulcher equation if
one assumes that
S , _N Ac,(T
- To)/T
(4)
where Ac, is the relaxational part of the specific heat
When Gibbs4proposed the approxiand B = B"/Ac,.
mation in equation (4), he implicitly expected a second
order transition at To, and an infinite relaxation time at
T = Towas identified with the presence of this second
order transition6
A third approach grew from the hybrid concept of
Macedo and Litovitz.E There it was proposed that
the "jump" probability was the product of the probability of obtaining sufficient energy to break the bonds
and the probability of having the appropriate structural
configuration which permitted such rearrangements.
The ability to obtain such a structural rearrangement
could be quantitatively expressed in terms of the excess entropy or free volume, giving an equation of the
form
lnq = A
+ H / R T + V / ( V - Vo)
(5)
Angel15 had also considered the relative roles of structural configuration and bond energy in terms of the
Adam-Gibbs equation. He associated the parameter
B" (of eq 3) to relative bond strength and ( T - To)
with the excess entropy.
The conclusion of all these approaches can best be
seen from the shape of the curve on an Arrhenius plot
(log q vs. l / T ) . It is expected that at equivalent viscosities, the weaker the intermolecular bonding, the
higher will be the curvature. Also, as the temperature
is made to approach the vicinity of T , (Le., ( T - To)
gets smaller), the curvature increases. This latter
feature had not been observed in Bz03, where a second
Arrhenius region was found at low temperatures, but
The Journal of Physical Chemistry
since KN03-Ca(NO& mixture has minimum bonding,
it is expected to follow the Fulcher equation. In fact,
AngelP has reported that the electrical conductivity
of this mixture does follow the Fulcher equat'ion over a
limited temperature range. We therefore propose to
test the validity of the Fulcher equation for this mixture over nine decades of viscosity.
Experimental Method
( I ) Sample Preparation. The composition of the
sample was 60% KN08 and 40% Ca(NOa)z (mole per
cent) and was prepared by a method similar to that of
Ubbelohde.8 The Ca(NO& (certified reagent grade)
was dried a minimum of 12 hr at approximately 230"
before being placed in weighing bottles. The bottles
were replaced in the oven for 2-4 hr to drive out the
moisture contaminated during filling, then cooled in a
desiccator under vacuum. The salt was weighed and
the amount of K N 0 3 necessary for the given composition was calculated. The KN08 (primary standard)
was dried for 3-5 hr at 130" and cooled in a desiccator,
under vacuum, before weighing. The sample was prepared by melting the KN03 over a flame and allowing
the Ca(N03)2to dissolve the K N 0 3melt.
(2) Measurements. Viscosity of the sample was
measured over the temperature range 79-200". A
Pyrex test tube holding the sample was placed in a
bath, consisting of a glass dewar fitted with an electric
stirrer and filled with glycerol. The temperature of
the bath was controlled by a Fisher proportional temperature control unit, with an accuracy of =tO.Ol" and
0.02". Temperature measurereproducibility of
ments were done by a mercury thermometer, to the
nearest tenth of a degree.
Three methods were used to measure the viscosity
over various ranges. The Cannon-Fenske (capillary)
method produced data, reproducible to within 1%,
for viscosities up to about 750 P. The small volume of
sample used, as well as the short duration of a complete
run, eliminated most of the problem of crystallization.
A Brookfield rotating cylinder viscometer was used
to measure higher viscosities. The outer cylinder remained stationary in the temperature bath while the
inner cylinder was rotated at a known angular velocity.
The general equation of motion for the-viscometer is
I
d28
dt2
+ q ~ de
l - + Kze = o
dt
where I is the moment of inertia of the inner cylinder,
(3) G.Adam and J. H. Gibbs, J . Chem. Phys., 43,139 (1965).
(4) J. H.Gibbs, "Modern Aspects of the Vitreous State," Butterworth and Co., Ltd., London, 1960,Chapter 7.
(5) C.A. Angell, J . Amer. Ceranz. Soc., 51, 117 (1968);C. A. Angell,
ibid., 51, 125 (1968).
(6) P.B.Macedo and T. A. Litovitz, J . Chem. Phys., 42,245(1965).
(7) C. A. Angell, J . Phys. Chem., 65, 1917 (1964).
(8) E.Rhodes, W. E. Smith, and A. R. Ubbelohde, Proc. Roy. Soc..
A285, 263 (1965).
VISCOSITYOF VITREOUSKNOa-Ca(NO& MIXTURE
4 149
0 is its angular displacement, 7 is the dynamic viscosity,
and K 1 and Kz are constants of the apparatus. The
first term represents the resistance to acceleration
which is negligible, the system being overdamped.
The second term is the viscous drag, and the third term
represents as elastic torque produced by the angular
displacement of the suspension.
For viscosities up to 7 X lo4 P the Brookfield was
used in the conventional rotation mode. The spindle
was rotated at a constant angular velocity so that it
experienced a torque (proportional to the angular displacement) which was produced by the viscous drag
of the sample. The equation of motion of this mode
reduced to
cle/($)
(7)
where CI = Kz/K1 and was obtained in units of P-rev/
min per scale division by calibrating the viscometer.
dO/dt was obtained from the rpm rotation of a synchronous motor, with an error of 0.1% or less.
The range of the Brookfield was extended by using
the decay mode. In this method the spindle was displaced and allowed to return to its equilibrium position
by the torque of the suspension system. The equation
of motion for this mode is
T
d0
dt
~ -r
+ ~~0 = o
or
T =
C2(t2- tl)/ln(el/e2>
(9)
where Cz = K2/K1 with a conversion to units of P
rev/sec per radian. It was necessary to extrapolate
the scale to read 575 divisions per 360" in order to obtain the proper conversion factor for 0. This constant
was calculated by direct calibration with a standard oil.
For this mode (tz - tl)/ln (O1/O2) was calculated by
finding the slope of a plot of the logarithm of the displacement vs. time (Figure 1). The slope could be
calculated to within 1% uncertainty. Agreement between measurements of 7 made a t the same temperature
by the two modes was within 0.8%. Using a spindle
with a diameter of 0.118 cm, the range of the viscometer
was extended to approximately lo8P.
Crystallization was a major problem with the Brookfield viscometer. The large volumes of sample needed
and the presence of the rotating spindle tended to speed
up the crystallization process so that it was generally
impossible to leave the sample in the apparatus overnight. This problem particularly increased for temperatures around and above 100".
Results
The measured values and methods of measurement
are given in Table I. These results represent several
melts but since sample reproducibility was within the
I
I
1
1
I
1
I
\o
9'0 120
I
Id0 IJO 210 240 2o
;
TIME (sec)
Figure 1. Logarithm of the angular displacement of the
spindle of the Brookfield viscometer (used in the decay
mode) as a function of time.
loco
9 =
1
I
30
60
Table I : Viscosity Data as a Function of Temperature
T,OK
Log
'I P
Cannon-Fenske
T,OK
Log
'I P
Rotation method
473.4
466.1
459.7
454.0
449.0
441.6
435.3
430.7
425.4
-0.240
-0.129
-0.022
-0.094
0,193
0.351
0.513
0.642
0.814
384.8
383.0
380.0
377.3
374.6
372.4
371.0
419.8
414.8
413.0
411.1
406.9
404.0
400.5
399.2
397.8
397.6
395 2
392.9
390 6
387.8
1.039
1,226
1.309
1.552
1.610
1.792
1.976
2.046
2.140
2.154
2.346
2.468
2.636
2.868
369.0
366.8
364.6
362.5
361.2
360.0
359.0
357.6
356.7
355.9
354.1
354.8
352.5
3.199
3.350
3.662
3.981
4.309
4.599
4.763
Decay method
I
I
5.085
5.377
5.748
6.115
6.378
6,634
6.781
7.029
7.202
7.371
7.826
7.608
8.134
accuracy of the viscometer, sample identification was
dropped.
The results from the Cannon-Fenske method give
viscosity in Stokes which was converted to poises by
multiplying each value of 7 by the density of the sample
a t that temperature. The densities, D,were calculated
from the graph of Dietzelgusing the equation
D
=
2.2344
- 0.79364 X
10-3T
(10)
in which T is the temperature in degrees Centigrade.
(9) A. Dietzel and H. J. IJoegel,Proceedings of the Third International
Glass Congress, Venice, 1963,p 319.
Volume 73, Number 12 December 1969
4150
R. WEILER,S. BLASER,AND I?. €3. MACEDO
I
I
I
I
I
I A
k
7-
:
A
A
A
G
A
5-
.IO
2
.06
Q
Q
Q
0
0
0
@
Q Q0
QQQQ
0
"
Y
8 4-
L
6)
i"'i
A
6-
2
l3
3l
l3
E?
0
8
2 .
0O
8
7
6
5
4
3
2
1
0
-
1
-
2
L O O BOBS
Figure 3. Deviation of the viscosity data from the
Fulcher equation.
I
00 210
I
I
I
I
I
I
I
220 230 240 250 260 2/0 280
IOVT ( O K )
0
Figure 2. Arrhenius plot of the temperature dependence of
viscosity: V, Ubbelohde's data; 0 , Cannon-Fenske viscometer;
0 , Brookfield (rotation mode); A, Brookfield (decay mode).
In Figure 2, the logarithm of the viscosity is plotted
1/T. The values obtained by the Cannon-Fenske
method are in fair agreement with Ubbelohde'ss data,
the latter plotted as inverted triangles in Figure 2.
Unfortunately, ref S does not include any "raw" data.
It seems that the worst deviations between their curve
and the presently reported values are about 10%.
The results from the Brookfield viscometer, operated
in both modes, join smoothly with each other as well
as with the low viscosity values obtained from the
Cannon-Fenske viscometer.
In order to fit the Fulcher equation to the data by
a least-squares fit method, eq 1 has to be 1inearized'O
t o the form
os.
log q = A
+ --T1 ( B - ATo) + rTo log q
(11)
Obtaining an initial set of values A', B', To' for the
adjustable coefficients in eq 11, an improved set A , B,
and Tocan be calculated by a simple iterative procedure.
Differentiating eq 11 and rearranging terms, one has
A log q = AA
+ AB/(T - To) +
AToB/(T
- To12
(12)
Performing successive least-squares fits on eq 12 to
obtain a best set of values for AA, AB, and ATo, one
finally has
A
=
A'
+ AA
B = B'
+ AB
To
=
To'
+ AT0
The standard deviations of A, B, and To are those of
AA, AB, and ATo,respectively. If the initial values of
the coefficients A', B', and T'o are taken from a leastThe Journal of Physical Chemistry
squares fit to eq 11, then the second set of coefficients
AA, AB, and AT0 in eq 12 are comparable to their
standard deviations and only a €ew further iterations
are necessary.
If the viscosity data are limited t o the same temperature range as that of Angell's conductivity measurements (ie., roughly up to 76" above To),it follows the
Fulcher equation to within the experimental standard
deviation. The best fit is obtained with the following
values of the unknown parameters; A = -2.234 f
0.062, B = 635 f ll°K-l, To = 334.8 f l.S"K, where
f denotes the standard deviation in each case, and the
overall standard deviation of the data is O.,j%. I n
fact, the conductivity as well as the viscosity data can
be fitted to the same To indicating that both transport
processes display the same curvature in an Arrhenius
plot (Figure 2). Thus taking To = 324.4"K from ref 7,
the limited viscosity data can be fitted t o the Fulcher
0.024 and B = 798 A
equation with A = -2.602
5°K-1. However, this shows that the value of the
constant B from viscosity measurements is 1.30 f 0.02
times larger than the value of B obtained from Angell's
conductivity data, which was processed in the same
manner. This ratio is somewhat larger than that obtained by Ubbelohde, who found it to be 1.20 at higher
temperatures, and over a much smaller range of viscosity and conductivity.
If instead of stopping at 76" above To, we use all
our data (which extends up to 35" above To), the
Fulcher equation overestimates the data by a factor of
500. The data over the entire temperature range can
be fitted to the Fulcher equation using the described
least-squares technique, where now A = -3.245 j=
0.088, B = 1066 f ll"K-l, and To = 312.6 0.9"K.
*
(10) P. B. Macedo, Mechanical and Thermal Properties of Ceramics:
Proceedings of a Symposium, (U. S. Department of Commerce,
National Bureau of Standards Special Publication 303, 1969) p 169.
VISCOSITY
OF VITREOUS
KN03-Ca(N03)2MIXTURE
The new fit requires the lowering of Toby 22' and an increase of 68% in B. Even so, this new fit has the
characteristic "S" shape" (Figure 3) indicating that the
discrepancy between the data and the Fulcher equation
is well outside the experimental uncertainty.
Conclusion
The failure of the Fulcher equation to fit the data
as one approaches the glass transition has many implications. First, the relaxation time will not extrapolate to
infinity at a finite To. This raises the question as to
whether or not a second order transition exists below
T,,as has been proposed by several
The low-temperature Arrhenius behavior seems to be
a general liquid feature, since it has been observed in
molten oxides (B203) and organic liquids such as n-propyl alcohol12as well.
Secondly, Macedo and Napolitanoll showed that in
B203 (a melt with strong covalent bonds) the kinematics of viscous flow was controlled by the activation
energy, rather than the structural effects. This also
4151
seems to be the case in the KN03-Ca(NO& melts, a
rather unexpected c o n c l u ~ i o n . ~ ~ ~ ~ ~
Thirdly, even though the Fulcher equation fails
markedly to fit the data near the glass transition, it
still represents a good fit to the high-temperature data.
Even more significantly, Angell'4 has shown that the
parameters A and B are concentration independent in
this system and To varies linearly with concentration.
Thus the Fulcher equation may have a deeper physical
significance, beyond that of a purely empirical expression. However, the significances proposed so
f a r 2 ~ 3are
~ 6inconsistent
~6
with the data.
Acknowledgment. This work was sponsored by Air
Force Office of Scientific Research-Grant No. AFOSR68-1376.
(11) P. B. Macedo and A. Napolitano, J. Chem. Phys., 49, 1887
(1968).
(12) A. C.Ling and J. E. Willard, J. Phys. Chem., 72, 1918 (1968).
(13) C.A. Angell, L. J. Pollard, and W. Strauss. J. Chem. Phys., 50,
2694 (1969).
(14) C.A. Angell, ibid., 46,4673 (1967),
Volume 75 Numbw 18 December 1069