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
