The Viscosity of Water under Pressure

2182 R. A. HORNE AND D. S. JOHNSON

The Viscosity of Water under Pressure

by R. A. Horne and D. S . Johnson

Arthur D. Little, Inc., Cambridge, Massachusetts (Received December 16, 1966)

The effect of hydrostatic pressure on the viscosity of pure water has been measured with a rolling-ball type of viscometer over the pressure and temperature ranges of 1-2000 kg/cm2 and 2-20’. The application of pressure tends to destroy the structured regions in liquid water.

Introduction Over the past few years we have been concerned with

the electrical conductivity of aqueous electrolytic solutions, including sea water, under high hydrostatic pres-

A modified Walden’s rule has been proposed relating empirically the conductivity with the viscosity under p r e s s ~ r e . ~ * ~ The viscosity of water, relative to 1 atm, exhibits a minimum at about lo00 kg/cm2,’ and this anomaly has been attributed to the breakup of the rarified regions of structure in liquid waters8 At somewhat higher pressure (about 2000 kg/cm2), the relative conductances of aqueous solutions of strong 1: l salts exhibit maxima, the activation energies of such transport processes exhibit minima,a and the “mixture effect” in mixed electrolytic solutions exhibits inflection^.^ Presumably the same pressure-induced structural changes in liquid water are responsible for these phenomena even though the mechanism of electrical conductivity and viscous flow in water are not iden- ticalg

Reliable data on the viscosity of aqueous solutions under pressure are needed for purposes of interpreting conductivity data and for correlating the structural changes that have been indicated by the changes in the activation energies of conduction.

Unfortunately, although the earlier work of Bridg- man,lO Tammann and Rabe,” and Lederer12 for the viscosity of pure water under pressure appear to be in agreement at 30’, at lower temperatures significant discrepancies are apparent (see Figure 1). Recent new measurements by Bett and Cappi13 and Weber14 are in disagreement with the earlier work of Bridgman, Tammann and Rabe, and Lederer at 30” (Figure 1). Although the discrepancies in the relative viscosity (Figure 1) are appreciable, it should be remembered that the differences in absolute viscosity are quite small.

Experimental Section The details of the thermostat and the high-pressure

producing and measuring equipment are described elsewhere. The rolling-ball type of viscometer, while it may not be as accurate a t 1 atm as other types, lends itself so easily to high-pressure work that, although other types of high-pressure viscometers have been developed,16 it has been used by a majority of investi- gators of the viscosity of fluids under high hydrostatic pressures. 16-29

(1) R. A. Horne and G. R. Frysinger, J. Geophys. Res., 68, 1967 (1963). (2) R. A. Horne, W. J. Bannon, E. Sullivan, and G. R. Frysinger, J. Electrochem. SOC., 110, 1282 (1963). (3) R. A. Horne, B. R. Myers, and G. R. Frysinger, J. Chem. Phye., 39, 2666 (1963). (4) R. A. Horne and R. A. Courant, J . Chem. Soc., 3548 (1964). ( 5 ) R. A. Horne, Nature, 200, 418 (1963). (6) R. A. Horne and R. A. Courant, J. Geophya. Res., 69, 1971 (1964). (7) R. Cohen, Ann. Phys., 45, 666 (1892). (8) 8. D. Hamann, “Physicc-Chemical Effects of Pressure,” Butter- worth and Co. Ltd., London, 1957, p 82. (9) R. A. Horne, R. A. Courant, D. S. Johnson, and F. F. Margosian, J. Phy8. Chem., 69, 3988 (1965). (10) P. W. Bridgman, Proc. Am. Acad. Arts Sci., 61, 57 (1926); Proc. Natl. Acad. Sci., 11, 603 (1925). (11) G. Tammann and H. Rabe, Z . Anorg. Allgem. Chem., 168, 73 (1927). (12) E. L. Lederer, KoZlo&Biehejte, 34, 270 (1932). (13) K. E. Bett and J. B. Cappi, Nature, 207, 620 (1965). (14) W. Weber, Z . Angew. Phys., 15, 342 (1963). (15) See, for example, M. M. Kusakov, L. A. Konovalova, and A. A. Konstantinov, Chem. Abstr., 61, 6417d (1964). (16) A. E. Flowers, Proc. Am. SOC. Testing Materials, 14, 565 (1914). (17) M. D. Hersey, J . Wash. Acad. Sci., 6 , 525 (1916). (18) M. D. Hersey and H. Shore, Mech. Eng., 50, 221 (1928). (19) B. H. Sage, Ind. Eng. Chem., Anal. Ed., 5,261 (1933). (20) F. Hoeppler, Z . Tech. Physik, 14, 165 (1933).

The Journal of Physical C h k t r y

VISCOSITY OF WATER UNDER PRESSURE 2183

1.10 - C, Cohen1Zi0C

Q, Phuz: cl ol,20°C

HI Zhurc et oL,40°C

A, Coh:nlIoC 8, c P h q 1 5 ° C

El Zhuzc #I at, 5 o C

I, Brldgmon, O°C dl Brld#mon, 10°C

108 Fl Zhuz: tl oL1 10°C - 106 L, Tomaan # ot,OOC MI Tommann clal,lOaC -

Ol Ltdtrtr, 0% Ltdortr,1O0C

1.04 ----Bill ond CoppI ----RctI and Cappi - E - 0

N, Tom"? tt ol., 30%

4, Ltdtrtf, 3OoC

---Belt ond Cappl

1 I 1

i 1000 1500 500

kpf" Pressure

Comparison of literature values for the relative viscosity of pure water as a function of temperature and pressure. Figure 1.

Although the literature contains many descriptions of high-pressure, rolling-ball type of viscometers, 16-29 inasmuch as the discrepancies mentioned above probably arise from differences in apparatus and experimental technique, a detailed description of the present experiment is necessary.

The high-pressure, rolling-ball viscometer consists of a length of a/rin. o.d., '/*-in. i.d. stainless steel, nonmagnetic tubing, 30 in. in length and closed at one end with a dead-end plug. The viscometer is enclosed in a Lucite jacket through which transformer oil circulates. Thermocouple probes measure the temperature a t both of the pickup coils.

The viscometer design incorporates one important innovation. The ball rolls, not along the inside of the high-pressure tubing itself, but rather along a loose- fitting inner lining consisting of thin-wall stainless steel tubing, 0.0955-in. i.d., slightly crimped near the lower end to stop the ball's descent. Pressure forces the walls of the high-pressure tubing outward, thereby increasing the inside diameter, but there is no pressure differential across the walls of the inner liner; thus the ball moves in a tube of essentially constant diameter and, inasmuch as the compressibility of the ball is very small, there is no significant correction factor with

00

increasing pressure. We feel that the present measurements are more accurate than those previously reported largely because the uncertainties of a pressure- deformation correction have been thus avoided.

In order to avoid corrosion, initially the 0.0625-in. diameter balls were nickel striked and gold plated. Afterashort time, the fall time of these balls became erratic and tended to increase. Microscopic examina- tion revealed that the gold plate had blistered. Nickel- plated balls were substituted. These performed satis- factorily and were quite corrosion resistant. The balls were very uniform; no differences among the balls were detectable from viscometer fall times at 1 atm

~~ ~ ~

(21) B. H. Sage and W. N . Lacey, Am. Inst. Mining Met. Engis., Tech. Pub., 845 (1937). (22) B. H. Sage and W. N. Lacey, Ind. Eng. Chem., 30, S29 (1938). (23) R. B. Block, J . Appl . Phys., 11, 635 (1940). (24) B. H. Sage and W. N. Lacey, I d . Eng. Chem., 32, 587 (1940). (25) R. M. Hubbard and G. G. Brown, I d . Eng. Chem., Anal. Ed., 15, 212 (1943). (26) E. M. Griest, W. Webb, and R. W. Schiessler, J. Chem. Phys., 29, 711 (1958). (27) S. E. Babb, Jr., and G. J . Scott, ibid., 40, 3666 (1964). (28) G. E. McDuffie, Jr., and M. V. Kelly, ibid., 41, 2666 (1964). (29) D. E. Harrison and R. B. Grosser, Rev. Sci. Instr., 36, 1S40 (1965).

Volume 70, Number 7 J d g 1966


AUDIO OSCILLATOR

,--kid-, BOGEN AMPLIFIER

150Turns of No28Wira

VISCOMETER PICK-UP COILS -

TWO CHANNE~ AMPLIFIER

Channel Channel

BECKMAN/ BERKLEY EPUT METER

Start

Figure 2. Viscometer circuit diagram.

and a given temperature, weightings, or diameters as measured with a micrometer.

The angle used was 30°, this optimum value being determined by trial and error. A more steep inclina- tion sacrificed a significant figure in the fall time, while less steep inclinations tended to give erratic fall times. The latter was probably due to the greater likelihood of a very slowly moving ball getting retarded by dust motes or slight inhomogeneities on the inner surface of the liner.

The rolling-ball viscometer timing circuit is shown in Figure 2. Using a magnet, the steel baIl is brought to the top of the inclined viscometer. When the magnet is withdrawn, the ball rolls down the inclined section, and its fall time between the two pickup coils is measured.

Two sets of coils are arranged a fixed distance apart (approximately 26 cm) along the viscometer. Each set of coils contains two primary sections, wound in series opposition, and a centrally located pickup coil. The primaries are energized by an audio oscillator feeding 2000 c to a 100-w amplifier. The amplifier is adjusted so as to put out a 2-v signal. (The dc resistance of each coil is approximately 6 ohms.) In the absence of the steel ball, the effect of the two primary sections on the pickup coil cancels out, and the output voltage should be zero. Although each section of the primary coils has been designed to be exactly the same, a 10-mv background signal remains in each pair of coils owing to residual unbalance in the mutual inductance. This is caused by slight differences in geometry when the coils are wound.

The EPUT meter contains a precision quartz crysta controlled clock. As the ball passes through the first set of coils, it produces a gross imbalance in the coupling and thus induces a voltage in the pickup coil. This signal is amplified and fed into the start channel of the EPUT meter. It triggers the start channel and actuates the running-time readout. As the ball passes through the second set of coils, the process is repeated except that the amplified signal is fed into the stop channel of the EPUT meter and terminates the readout of the elapsed time. The interval of time between the start and stop signals is read directly off the digital display on the EPUT meter. Since the distance between the two pairs of coils is known and the time for the ball to roll between the two pairs of coils is measured, the velocity of the rolling ball can be calculated.

Immediately prior to a run the viscometer is dis- assembled, and the high-pressure tube and its stainless steel liner are rinsed successively with washes of (1) acetone, (2) distilled water, and (3) the solution to be studied. The liner is next inserted and the viscometer filled. The liquid in the viscometer is subjected to a mild vacuum to remove the gases and the ball carefully inserted with the tube enclosed in such a manner as to minimize gas entrainment. The viscometer is loosely connected to the high-pressure system and a slight pressure applied until the escape of hydraulic fluid through the loose coupling indicates that the air space between the fluid to be studied and the hydraulic fluid is removed. The proper angle of the viscometer is checked with a special level, the connection tightened, and the level checked again. The angle is also checked in the course of the experiment since it is very critical.

The hydraulic fluid used was Esso Univis P-38. The electrical conductivity of aqueous solutions is very sensitive to molecular and especially ionic contamina- tion. Conductivity measurements on aqueous electrolytic solutions in direct contact with Univis P-38 before, during, and after pressurization and over ex- tended periods of time (weeks) at 1 atm showed that, in the absence of strong acids or bases, water is not significantly contaminated by this hydraulic fluid. As a further check on water purity, following each pressurization series the sample was immediately returned to 1 atm and its viscosity remeasured.

Following each pressure increment the apparatus is allowed to stand from 10 to 15 min in order that the temperature increase resulting from adiabatic compres- sion be dissipated. This equilibration time was determined experimentally and is quite small as a result of the relatively large surface-to-volume ratio of the "sample" (about 32 in.-l). The ball is raised to the top of the viscometer by hand with a 2495-gauss mag-

The Journal of Physical Chemistry


Table I : Relative Viscosity of Compressed Water

Pressure, kg/cml

70 141 141 141 281 281 281 281 281 422 562 562 633 633 703 703 703 773 773

70 70

141 141 141 281 281 281 423 423 562 562 633 633

70 352 352 422 42 2 422

Re1 viscosity, Pressure, R P / R I atm kg/cm*

At 2.252 f 0.073' 0.9938 844 0,9875 844 0.9869 844 0.9838 984 0,9708 984 0.9713 984 0.9702 984 0.9689 1125 0.9688 1195 0.9539 1125 0.9457 1406 0.9434 1406 0.9420 1406 0,9417 1547 0,9403 1547 0.9392 1547 0.9396 1828 0.9318 1968 0,9322

At 4.009 k 0.015' 0.9985 70 3 0.9976 703 0.9938 703 0.9942 844 0.9933 844 0.9889 844 0.9875 844 0.9884 984 0.9839 1265 0.9814 1265 0.9775 1476 0.9786 1547 0.9777 1828 0.9749 1828

At 6.241 f 0.053' 0.9986 703 0.9930 844 0.9922 984 0.9902 984 0.9907 1195 0.9911 1265

Re1 viscosity, rlP/lll atm

0.9370 0.9358 0.9353 0.9358 0.9349 0.9343 0.9352 0.9326 0.9346 0.9350 0.9382 0.9373 0.9369 0.9383 0.9394 0.9388 0.9425 0.9455

0.9765 0.9750 0.9745 0.9726 0.9765 0.9748 0.9748 0.9745 0.9775 0.9781 0.9800 0,9818 0.9876 0.9861

0.9865 0.9850 0.9855 0.9850 0.9875 0.9852

Pressure, kg/cmr

492 562 562 633 633

211 281 281 562 562 633 703 703 703 703 844 844 844

70 211 422 703

1055 1055 1406

141 211 281 422 562 562 703 703 703

R el viscosity, Pressure,

OP/'Il etm kg/cm2

At 6.241 f 0.053' 0.9898 1828 0.9889 2109 0.9880 0.9865 0.9875

At10.090 f 0.035' 0.W60 984 0.9955 984 0.9952 984 0,9908 1265 0.9917 1265 0.9895 1265 0.9906 1547 0.9896 1898 0.9906 1898 0.9910 2109 0.9894 2109 0.9894 2109 0.9889

At 15.112 f 0.085' 0.9990 1406 0.9985 1406 0 * 9974 1758 0.9968 1758 0.9994 2109 0.0985 2109 1,0050

At 20.359 i 0.183" 0.9992 1055 0.9990 1055 0,9986 1055 0.9982 1406 0.9979 1406 0.9981 1406 0.9993 1406 0.9988 1758 0.9984 1758

Re1 viscosity, RP/R1 stm

0.9954 1,001

0.0901 0.9911 0.9901 0.9938 0.9938 0.9929 0.9977 1,0039 1 ,0060 1.0086 1.0091 1.0089

1.0036 1.0045 0.0125 1.0130 1.0216 1.0210

1.0024 1.0026 1.0034 1,0096 1.0092 1.0089 1,0081 1,0181 1.0176

net with a 3.5-cm gap. The ball is held a t the top about 4-5 cm above the first pickup coil for about 30 sec (longer for more viscous solutions). If this pre- caution is not taken, the roll times tend to be erratic and not reproducible. Apparently this pause allows eddies set up by raising the ball to die away. The ball is then released and its fall time between the two pickup coils measured. The procedure is then repeated.

The initial position of the ball at the top of its roll is crucial. If the ball is too close to the top coil, it may

still be accelerating while in the timed zone between the two coils. On the other hand, if it is too near the top of the viscometer, it may come in contact with the interface between the fluids. Fortunately, when the ball does accidently come in contact with the interface, erratic results obtain so the difficulty is easily recognized.

At a given temperature and pressure an experiment was repeated from four to as many as ten times. In the course of a given series of runs, the measured

Volume 70, Number 7 J d g 1966


average deviation of the temperatures of the viscometer ranged from +0.005 (for temperatures near room temperature) to as much as =?=0.02” (for the lowest temperatures). The larger temperature deviations listed in Table I are averaged over series of series.

The ball-roll time is read to four significant figures and the roll times range from 5 to 10 see. The average deviation in the measured roll time is about 0.10%.

Over the temperature interval 0-lo”, thermal expansion of the vitreous silica spacer between the pickup coils increases the distance between them and thus the roll time by approximately 0.008%. The diameter of the rolling ball increases by about 0.034%, but the effect of this increase is partially compensated by an approximately 0.018% increase in the diameter of the inner tube in which the ball rolls. These effects of thermal expansion are well within the measured average deviation of 0.10%; therefore no attempt is made to apply corrections for them. However, the temperature dependencies of the densities of the ball and of the fluid are taken into consideration (see below).

The pickup coil spacer is not exposed to pressure, and the application of pressure does not appreciably alter the inside diameter, DT, of the liner, and its effect on the diameter of the ball, DB, is slight. The expres- sions derived by Hubbard and Brown,25 contain the terms DB/DT and (DT + DB), and over the pressure range 1-10,000 kg/cm2 they change by only 0.02 and 0.06%, respectively. However, again, the pressure dependencies of the densities of the ball, PB,

and of the fluid, P F , are taken into consideration. In the present work the simplified relation developed

by Sagelg was used

7 = Ct(PB - P F ) (1) where 7 is the viscosity, C is a constant, and t is the roll time. Over the temperature and pressure ranges in- volved, the density of the steel ball is very nearly a constant, but the density of water and thus of (PB - PF) varies significantly. The density of steel, P F ~ , was calculated from the expression

PB = P F ~ = 7.8835/(1 + 3.5 X 10-’T) X

(1 - 6.1 x 10-7[p - 11) (2)

where T is the centigrade temperature, P is the hydrostatic pressure in atmospheres, and the constants are based on the densities, compressibilities, and coefficients of cubic thermal expansion from “The Inter- national Critical Tables.”

Up to about 2500 kg/cm2 there is good agreement in the density of water as reported by Amagat30 and B14dgman.~~ In view of this agreement, reliance was placed on Dorsey’s compilation of specific volumes, 32

and, for a given P and T, densities were calculated from specific volumes read or interpolated from Dorsey’s Table 95.

A calibration curve of the viscosity us. ~ ( P B - PF) for aqueous NaCl solutions (using data of she el^^^) and water-sucrose solutions (using data of Bingham and Jackson34 and more recent data of Swindells, Snyder, Hardy, and Golden3‘) shows that in the region of in- terest in the present experiments the curve is linear; thus eq 1 is applicable, and the following useful rela- tionship is valid.

? P / ? l a t m = tp(PB - PF)P/tlatm(PB - P F ) l a t m (3) Earlier viscosity measurement made on pure water

and sea water at 1 atmg with this same viscometer yielded values in good agreement with those compiled by D o r ~ e y . ~ ~ Deionized and degassed water was used.

Results The effect of pressure on the absolute viscosity a t

a given temperature is quite small (Figure 3) ; therefore a much more meaningful way of reporting the results is in terms of the viscosity at pressures relative to the viscosity at 1 atm and the same temperature. The observed temperature and pressure dependencies of the relative viscosity of pure water are summarized in Table I and Figure 4. Absolute viscosities may be obtained by multiplying the relative viscosity given in Table I by the 1-atm viscosity for the corresponding temperature obtained from Figure 1 or Figure 2 in Horne, Courant, Johnson, and Marg~sian,~ or the tables in D o r ~ e y . ~ ~

Discussion Qualitatively the present results show the same fea-

tures as revealed by earlier studies, namely an initial decrease in the relative viscosity as pressure is applied followed by an increase, the minima in the relative viscosity becoming more shallow and gradually disap- pearing as the temperature increases (Figure 4). Quantitatively, however, the agreement among the values obtained by different experimentalists is poor (cf. Figures 1 and 4). In particular, the present results are not in agreement with the older values of

(30) E. H. Amagat, Ann. Chim. Phya., 29, 68, 505 (1893). (31) P. W. Bridgmah, Proc. Am. Acad. Arts Sci . , 48, 307 (1913). (32) N. E. Dorsey, “Properties of Ordinary Water-Substance,” Reinhold Publishing Corp., New York, N. Y., 1940, pp 182-189. (33) M. L. Sheely, Ind. Eng. Chem., 24, 1060 (1932). (34) E. C. Bingham and R. F. Jackson, Bull. Natl. Bur. Std., 14, 59 (1918). (35) J. F. Swindells, C. F. Snyder, R. C. Hardy, and P. E. Golden, National Bureau of Standards Circular 440, Supplement, U. S. Government Printing O5ce, Washington, D. C., July 31, 1958.



17

16

15

8 - 14 s P # 13 s

E

12

I I

IO 0 2 4 6 8 IO 12 14 16 1 8 :

Temperature,'C

Figure 3. Absolute viscosity of pure water as a function of temperature and pressure.

Bridgman'O as quoted by D o r ~ e y . ~ ~ Bridgman's values of q r / q a t 30" appear much too high compared with the present 20" values, while his values at 10' appear much too low. Tammann and Rabe'l have criticized Bridgman's results, and the present values at 2 or 10" appear to be in fair agreement with their 0 and 10" values, respectively, as quoted by D o r ~ e y . ~ ~ Although the values of Zhuze, Sergeevich, and Chestnov a t 40", as quoted by Richardson, et u Z . , ~ ~ and Tammann and Rabell at 30" as quoted by Dorsey appear to be in agreement with Bridgman,'O the more recent work of Weber14 a t 32" and Bett and CappiI3 at 30" indicate that Bridgman's values are much too high and seem to be in better harmony with our values at 2Q0, yet Weber's values at 20" appear to be considerably less than our values and those of Bett and Cappi13 a t the same temperature. At temperatures close to 0" the present values and those of Cohen,' Bridgman,'O Tammann and Rabe,l' and Lederer12 are in fair agreement. Reading values from the small graphs pub- lished by Bett and Cappi13 is difficult, yet their values and the present ones appear to be in good agreement a t 2 and 20", but in poor agreement at 10".

In summary, the present results are in agreement with the recent results of Bett and Cappi;l8 however, by making many measurements over a rather narrow

1.02

1.0 I

1.00

0.99 E

P

L O -

,P 0.98

< .t 0.97 0 UI .- > w 0.96 > d w

.- - - a 0.95

0.94

0.93 0 500 1,000 1,500 2,000

kg /cm2 Pressure

Figure 4. as a function of temperature and pressure.

Relative viscosity of pure water

temperature region the present results give evidence of phenomena which escaped the notice of Bett and Cappi.I3

The structured regions in liquid water are less dense than the "free" water; thus cluster breakup results in a volume decrease and is therefore enhanced by the appIication of pressure. The thermal destruction of the clusters superimposed on the "normal" thermal expansion of the "free" water gives rise to the well- known maximum in the density of pure water near 4". If the data in Figure 4 are plotted isobarically us. temperature (Figure 5) , two significant features become clear: (1) the isobars exhibit inflections in the form of an abrupt change in slope near 4" and (2) isobars above and below 1000 kg/cm2 appear to form two distinct families of curves. As mentioned in the Introduction, this pressure appears to have consider- able structural significance. Above 1000 kg/cm2 and 4 O , Figure 5 suggests that the dominant effect is an increase in relative viscosity with increasing pres-

(36) J. L. Richardson, P. Bergsteinson, R. J. Gets, D. L. Peters, and R. W. Sprague, "Sea Water Mass Diffusion Coefficient Studies," Philco Corp. Publication No. U-3021 (Feb 26, 1961), Office of Naval Research Contract No. Nonr-4061(00).

Volume 70, Number 7 July 1966


I .03

I . 0 2

E' 1.01 - a

w !& 1.00

5 0.98

> .- c 0

p: 0, 0.97

0.96

0.95

0.94

0.93 0 2 4 6 8 IO 12 14 16 18 20

T e m p e r a ture, 'C

Figure 5. The effect of temperature on relative viscosity isobars.

sure as the water molecules are crowded closer together, the effect becoming more marked with increasing temperature. The pressure-induced destruction of the water clusters is greatest in the region below 4", where the clusters are largest, and, as a result, a given relative viscosity isobar drops off very abruptly as the temperature is lowered below 4". The data of Bett and Cappi13 when plotted isobarically (Figure 5 ) give no hint of these phenomena.

Pressure is a water structure-breaking influence; thus one expects the low-temperature structural transition in water and hence its temperature of maximum density to be displaced by the application of pressure.

Nemethy and Scheragaa7 have estimated that in going from 0 to 100" the average cluster size decreases from 90.6 to 21.0 molecules, while the fraction of unbroken H bonds decreases from 0.529 to 0.325. Al- though transport processes such as electrical conductivity and viscous flow correlate nicely with this esti- mate of cluster size, the phenomenological transitions in viscosity (Figure 5) and c o n d u c t i ~ i t y ~ ~ ~ ~ ~ ~ ~ at 4" are so sharply defined and abrupt that one is somewhat a t a loss to account for them in terms of the more or less gradual cluster breakup, and one is tempted to revert

5.2 5'4 7

5 3.4 W c W 3.2 c

3.0 t 2.0 > .- 4

2.6 2.4 2.2

5.0 4.8

p 4.6

:- 4.2

\

r 3 4.4

$ 4.0 3.8

5 - 3.6

0 2 4 6 8 10 12 14 16 18 20

T e m p e r a t u r e , 'C

Figure 6. energy of viscous flow of pure water under pressure.

The temperature dependence of the activation

to a more radical structural transition such as that postulated in the classical paper of Bernal and F~wler.~O The greater elegance of the Frank-Wen-Nemethy- Scheraga model for liquid water is very much to its credit; however, this model may be a more serious oversimplification than presently realized.

The pressure at which the relative viscosit,y is at its minimum is decreased with increasing temperature. The qualitative conclusions to be drawn from this are in harmony with experience: as the freezing temperature is approached, very high pressures are necessary to destroy the structure, while a t temperatures above about 40-50" there is so little structure that the effect of pressure is slight and the relative viscosity us. pressure isotherms cease to exhibit minima. The pressure at which the relative viscosity is again equal to unity also increases with decreasing temperature, the rate of increase becoming very great as the freezing temperature is approached.

(37) G. Nemethy and H. A. Scheraga, J. Chem. Phgs., 36, 3382 (1962). (38) R. A. Horne and R. A. Courant, J. Phys. Chem., 68, 1258 (1964). (39) R. A. Horne and R. A. Courant, J . Geophys. Res., 69, 1152 (1964). (40) J. D. Bernal and R. H. Fowler. J . Chem. Phys., 1, 615 (1933).



I: 4.6

g 4.4 E h 4.2 8 Js 4.0

g- 3.8

3.6 ii % 5: 3.4 F

3.2 F 01 3.0 15 g 2.8 .s e 2.6

=t 2.4

2.2 2.0

0- W

0

.- - 0

0 500 !,OOO 1,500 2,000 kg/cm2 Pressure

Figure 7. energy of viscous flow of pure water.

The pressure dependence of the activation

Viscous flow can be treated as a rate process and activation energies, E,, calculated from the integrated form of the Arrhenius equation. At 1 atm the Arrhenius activation energy of viscous flow of pure water, unlike the activation energy of electrical conductance in water, continues to increase with decreasing temperature.9 However, under hydrostatic pressure a maximum appears in the temperature dependence of E, (Figure 6), and the value of E, appears to be related to T,,, the temperature a t which the maximum occurs, in a very simple way

(Ea)max = 5.2 - 0.126Tmax (4) The rapid decrease in Ea with decreasing temperatures in the low-temperature region a t a given pressure was not expected, and its explanation is not clear a t this time. However, the pressure dependence of E a for a given temperature (Figure 7) is as anticipated: a t 3”, where water structure is extensive, increasing the pressure destroys this structure and E, decreases ac- cordingly, but at 16” where there is less water structure the effect of pressure is correspondingly less and E, decreases much more slowly with increasing pressure.

Finally, returning now to Figure 4, at some pressure below 1000 kg/cm2 the relative viscosity of water goes

3.0

2.8

Ln 2.6 E!

.G 2.4 f

0”

x e

0

4I

2.2 I I p. % 2.0 4I a - 3 o1 1.8 ’ 1.6

c s -

I .4

1.2 0 5 IO 15 20 25

Temperature O C

Figure 8. Pressure coefficients of specific volume and viscosity.

through a minimum and then begins to increase with increasing pressure as a “normal” unassociated liquid. Does this observation imply that by the time such pressures have been reached, all of the structure in liquid water has been destroyed and water has become a “simple,” unassociated fluid? If such is the case, then pressure ia indeed a powerful structure-breaking influence, for, as we have seen above, even a t 100” the fraction of unbroken H bonds in water is still as high as 0.325. Unfortunately, highly specific molecular interactions strongly dependent on molecular structure appear to be important in determining the viscous properties of fluids under high pressure;41 hence there is little point in trying to compare water’s behavior with that of other “simple” liquids. However, a t 20” above 1500 kg/cm2 the vp/v l os. P curve (Figure 4) is linear and its slope corresponds to a pressure coefficient A ( q p / q l a t m ) / A P of +2.7 X 10” cm2/kg. At 20” the pressure coefficient of the specific volume of water, A(VP/VI atm)/AP, calculated from Dorsey’s compilation of data of Amagatj30 is -2.75 X cm2/kg. Thus the viscosity at this temperature and these pressures appears to depend only on the reciprocal of the specific volume (Figure 8). Whether the water molecules

(41) See ref 8, pp 81-84.

Volume 70, Number 7 July 1966


would cease behaving as compact spheres upon further crowding a t even greater pressures remains to be seen; but, if the present finding is not fortuitous, then the state of affairs in liquid water above moderate temperatures and pressures may be incredibly simple.

Acknowledgment. The authors are indebted to Dr. I. Simon and Mr. F. F. Margosian for the design, con- struction, and testing of the high-pressure viscometer. This work was supported in part by the Office of Naval Research.

Ths Journal of Physical Chemistry

Documents

The Viscosity of Water under Pressure