1801
Second Viscosity Coefficient of Water
Isotope Effect and the Molecular Mechanism of the Second Viscosity, Coefficient of Water E. McLaughlin Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803 (Received March 30, 1972)
The sound absorption data in water and heavy water are analyzed to yield the second viscosity coefficient which is discussed in terms of the basic molecular mechanistic processes involved compared with those operative in shear viscosity.
Introduction In the formulation of statistical theories of transport processes in liquids information is required on the molecular mechanisms of transport. One method of obtaining such information is the study of the transport properties of isotopic molecules where the masses and moments of inertia can be altered without significant changes occurring in the parameters of the intermolecular potential. This technique has already been usedl,2 in connection with shear viscosity and thermal conductivity. In the present paper it is used to examine the basic molecular mechanisms in. the viscous processes determining the second viscosity coefficients of water. Viscosity Coefficients In the absence of external forces the one-dimensional Navier-Stokes equation for an isotropic fluid can be written
where D I D t is the substantial time derivative, u the x component of the hydrodynamic or center of mass velocity u, p the thermodynamic pressure, and 7 and qf the first and second viscosity coefficients. The first coefficient of viscosity is more commonly known as the coefficient of shear viscosity. The two coefficients arise naturally in the equation of motion from the assumption of the linear Cauchy-Poisson or generalized Newton’s law between stress and rate of strain.3 To date the most important experimental source of information on v’ and K = ($ 2/37) comes from a study of the absorption of sound waves.4 For the case of plane sound waves where motion is in the x direction alone and the waves are assumed infinitesimal then eq 1reduces to
+
where PO is the density of the medium before passage of the sound wave. This equation combined with the linearized equations of continuity, energy balance, and equation of state forms a wave equation the solution of which is5 represented by a damped harmonic wave with an absorption coefficient angiven by - := u2
__I
POUO,3 2a2
[(7’ + 2q) + i 2 L 5 q CP
(3)
v is the sonic frequency, u,, the Laplacian speed of sound, X the coefficient of thermal conductivity, and y the specific heat ratio. Equation 4 shows that the absorption of sound arises from the various dissipative processes associated with the transport phenomena in the fluid. In practice the absorption contribution arising from thermal conduction is negligible compared with that arising from viscous processes and can be discarded. The resulting expression then provides a method of evaluating K or 7’ from experimental absorption measurements as
K = 4/3t((Ya/(Yc) - 117 where ac is the classical absorption calculated assuming K = 0. Figure 1 gives the experimental sound absorption data of water6 and heavy water7 (D2O) for temperatures in the range between 5 and 50” in the form of the ratio anlac with the spread on the water results given by Pinkerton included. For the case of heavy water the 15 and 50” points included are based on interpolated and extrapoiated values, respectively, of ac. Examination of this figure shows that although there is considerable scatter the results apparently decrease with increasing temperature, and in what follows values of aa/ac from the smooth curves are used in the calculations in order to detect trends as a function of temperature. Table I lists the various viscosity coefficients of water and heavy water. The shear coefficients are from Hardy and Cottinghams and the smoothed aa/a, values from which K and 7’ were obtained are also given. Before discussing these results based on continuum mechanics in terms of the molecular processes included, it is necessary to consider the isotope effect which has already been used as a diagnostic criterion in this context.132,S Isotope Effect To obtain some experimental information on the molecular mechanistic processes determining the viscosity coefficients the study of these properties for pairs of isotopic molecules can be used. If a pair of molecules interact with a potential characterized by two parameters and LT which are respectively a characteristic energy of interaction and a characteristic molecular diameter then the three viscosity coefficients may be written9 in the dimensionless form
9
= vo2/+
The Journal of Physical Chemistry, Vol. 77, No. 1 4 , 1973
1802
E.
McLaughlin
TABLE I: Absorption and Viscosity D a t a for Water and Heavy Water Water
11,10-3 kgm-' sec-I
T,oc
5
1.519 1.307 1.139 1.002 0.798 0.653 0.547
10
15 20 30 40 50
Heavy water
vi, 10-3
, K , 10-3
v',1 0 - 3
kgm-' sec-1
kgm-1 sec-1
cualac
kg r r - 1 sec-1
kg m-1 sec-1
10-3 kg m - l sec-'
4.638 3.813 3.231 2.793 2.169 1.750 1.458
3.625 2.942 2.472 2.125 1.637 1.315 1.093
3.290 3.188 3.128 3.090 3.040 3.010 3.000
1.983 1.675 1.437 1.247 0.973 0.785 0.650
5.102 4.234 3.587 3.079 2.277 1.902 1.567
3.781 3.117 2.629 2.248 1.728 1.379 1.134
I
332
5
I IO
1
I3
,
20
4
I
25
I
I
35
40
45
I
50
Figure 1. Ratios ( a a / a c )measured to classical absorption coefficients of water (0) and heavy water ( 0 )as functions of temperature.
K* where
v*,
7'
=
Ka2/&c
*, and K* are universal functions of k T / t and
V/Nu3 only. For a pair of isotopic molecules where e and identical it follows that
u are virtually
v 1 / v=
K i / K = (milm)"'
(5) where the subscript i refers to the isotopically substituted species. Equation 5 shows that all three viscosity ratios should be given by the square root of the mass ratio provideds the intermolecular potential is spherically symmetric and hence the equation of motion is of the form vl'/r]'
=
F = m(du/dt) For nonspherical molecules Pople has pointed out that eq 5 would not be expected to hold as conservation of angular momentum would also have to be considered. Discussion Examination of Table I shows for the shear viscosity, as has already been pointed out by Pople, that as the temThe Journal of Physical Chemistry, Voi. 77, No. 14, 1973
K ~10-3 ,
a a /.c
2.930 2.896 2.872 2.852 2.832 2.817 2.808
Ki/K
1.100 1.110 1.110 1.102 1.096 1.087 1.075
IIl'/Vf
1.043 1.059 1.063 1.058 1.056 1.049
1.055
91/17
-
1.305 1.282 1.262 1.245 1.219 1.202 1.188
perature is increased the ratio decreases from its value at the lowest temperature where it is close in magnitude to the square roots of the three principal moments of inertia (I*l/I*)'/' = 1.340; (IB~/IB)'/' = 1.414; (Icl/Ic)l~'= 1.390. This indicates that rotational motion is an important mechanism for momentum transport in water but that this contribution decreases in importance as the temperature increases. For the second viscosity, however, as in the case of thermal conductivity1 the ratio vl'/v, is, within the experimental accuracy, constant at 1.055 f 0.004 which corresponds closely to the square root of the mass ratio (rn,/rn)1/2 = 1.054. This suggests that rotational motion is not an important contribution to the mechanism of momentum transfer involved in the second viscosity coefficient. In the case of thermal conductivity it has been suggestedlO that the result there is consistent with the picture that to transfer heat the molecules need only oscillate about their mean equilibrium positions in the temperature gradient which contrasts with the case of shear viscosity where imposition of the velocity gradient b u / ~ y will necessarily require that intermolecular forces between adjacent layers are continually disrupted and this is unlikely to occur with the absence of rotational motion. The acceptance of these conclusions then leads us back to the physical process of the motion involved in div u . As agreement with the square root of the mass law for the second viscosity coefficient implies no rotational motion of the molecules as occurs in shear flow then it is suggested that 7' is the proportionality constant in the Case where the dilation of the fluid is a change of volume at constant shape where u11 = u22 = u33 = - p and the fluid is contracting uniformly with duldx = a u / ~ ) y= b w / d z such as a uniformly contracting sphere where the molecules do not rotate, When such an element contracts nonuniformly as, for example, in the more general case of a sphere dilating to an ellipse of smaller volume, the normal stresses are then not all equal and 7' and both enter the proportionality between @ - p ) and d u , / , p , in the form of the bulk viscosity. References and Notes (1) E. McLaughiin, Physica, 26, 650 (1960). (2)J. K. Horrocks, E. McLaughlin, and A. R . Ubbelohde, Trans. FaradaySoc., 59, 1110 (1963). (3) J. Serrin, "Handbuch der Physik," Vol. V l l l / l , J. Springer, Berlin. 1959,p 178. (4) L. Rosenhead, Proc. Roy. SOC.,Ser. A, 266,1 (1954). (5) P. Biquard, Rev. d'Acoustique, 1, 93 (1932). (6) J. Pinkerton, Nature (London), 160, 128 (1947). ( 7 ) R. Pancholv. J . Acoust. SOC.Amer., 25, 1003 (1953) (8) R. C. Hardv and R L Cottlnqham, J Res N a t Bur Stand U S 42, 572 (1949). (9) J. A. Pople, Physica, 19, 668 (1953), (10) E. McLaughlin, Quart. Rev. Chem. SOC.,14,236 (1960)
