Behavior of Viscosity and Thermal Conductivity of Fluids Near the Critical Point J. V. Sengers Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 1. Introduction irreversible fluxes associated with thethermody- namicforces by thelinear laws. This is the most This paper reviews the experimental information serious difficulty encountered in measuringthe on the behavior of the viscosity and thermal con- thermal conductivity and will be further discussed ductivity of fluids nearthe critical point. Section insection 4. As is well known,equilibrium or a 2 presents some general remarks on experimental steady state is approached veryslowly by a physical complicationsassociated with the measurement systemnear a critical point. Consequently, some of theseproperties in thecritical region. Ex- authorshave observed hysteresis effects in their perimental work on the viscosity of one component experimental observations of the viscosity [2, 6, 501. fluids near the liquid-vapor critical point is reviewed A prerequisite for thestudy of thebehavior of in section 3. Thethermal conductivity of one anyphysical property in thecritical region is a component fluids isconsidered in section 4. The knowledge of the values of the appropriate thermo- behavior of the viscosity and thermal conductivity dynamicvariables at which the property is meas- inbinary liquid mixtures near the critical mixing ured. Near the liquid-vapor critical point the tem- point is discussed in sections 5 and 6, respectively. peratureand density and near the critical mixing Section 7 is asummary of conclusionsbased on point of a binary liquid mixture the temperature and these experimental results. concentrationshould be known. If a P-V-T re- lationship is known, one can calculate the density 2. Some General Remarks from themeasured pressure and temperature. Closeto the critical point, where the isotherms Inaddition to the usual difficulties associated becomevery flat, thisprocedure becomes inac- with experiments in thecritical region, measure- curate.This inaccuracy is greatly increased by ments of thetransport coefficients are hampered uncertainties in the absolute value of the tempera- by the fact that they are nonequilibrium quantities. turesat which either the transport coefficients or Most accuratemeasurements have been carried theisotherm data were measured. Uncertainties out by introducingmacroscopic gradients into the in the temperature also hamper a comparison with system.In order to obtain the transport coeffi- otherproperties near the critical point. Such a cients the hydrodynamic equations must be solved comparisonfor abinary liquid mixture is com- undercertain imposed boundary conditions. plicated by thefact that the critical parameters Exact solutions of these equations are known only are very sensitive to impurities. forfluids with constantproperties in idealized If onemeasures the transport coefficients as a geometricalsituations. However, in thecritical function of pressure and temperature, care should regiona number of physicalproperties like local betaken to work atsmall enough pressure and densityand specific heat vary appreciably with temperature intervals as otherwise the whole critical position as a result of thegradients. This com- region is easilyoverlooked. In experimental work plicatesthe analysis considerably. Close to the on transport coefficients this has led to considerable critical point even the validity of the Navier-Stokes confusion,which will be discussed further in sec- .equations [41] and of thelinear laws has been tion4 in connectionwith thermal conductivity questioned [38, 42, 591. measurements.It is equallyimportant for the If in a fluid nearthe liquid-vaporcritical point study of anyother property near the liquid-vapor the temperature is not uniform,gravity can easily critical point. generateconvection currents andconvective Anotherprocedure consists infollowing an iso- heat flow which mustbe distinguished from the choricmethod where the temperature is varied 165 while the totalamount of gasis kept constant. it allows themeasurement of the viscosity at a thecritical temperature). Although theabsolute However,this method also hasits pitfalls. First, certain horizontallevel in the fluid and conse- if one lowers the temperature at constant density, quently, in principle,at acertain localdensity. accuracy of these old measurements may be limited, the relative precision is sufficiently good to warrant phasetransition occurs at a certain temperature In the capillary flow method the density varies along this conclusion. which is equal to thecritical temperature at the the capillary. After some inconclusive work on the subject by criticaldensity. Below thistransition tempera- An example of the "normal" behavior of the vis- PHILLIPS [ll], CLARK [2], and SCHROER and tureone measures either in the vapor or in the cosity outside the critical region is shown in figure BECKER [13], an experimental investigation of the liquid, depending on the experimental arrangement. 1; the viscosity coefficient of argon [4, 71 at three viscosity near the critical point of CO, was under- As a consequence the density, at which the proper- temperatures, high relative to the critical tempera- taken by NALDRETT and MAASS [lo] using an tiesunder consideration actually aremeasured, ture (t,=- 122.3 "C), as a function of density. One oscillatingdisk [6]. Thecritical region was ap- does change rapidly as a function of temperature, amagat unit of density is thedensity at 0 "C and proached by theisochoric method. One viscosity although the density averaged over vapor and liquid 1 atm.' At a given temperaturethe viscosityco- isothermat 31.14 "C wasalso determined.The phase is kept constant. The temperature at which efficient increases monotonically with thedensity: authors conclude that the viscositycoefficient shows thistransition occurs shouldbe determinedac- (dq/dp)r and (#7/dp2)r are bothpositive. The asmall anomalous increase close to thecritical curately.Moreover, even above this temperature temperature derivative (d~/aT),~is positive up to a point. At thecritical density this anomalous in- a largedensity gradient is developed close to the density well beyond the critical. At very high den- crease is about 1 percent at 32.0 "C and about 10 criticaltemperature so that the localdensity at a sities (d~/dT),~becomes negative [3, 81, but this is percent at 31.1 "C. The results confirm the conclu- certain position in the vessel is still not independent not relevant for ourpresent discussion. Some- sion of Warburg and von Babo that at 32.6 "C any of the temperature. Ultimately. close to the liquid- timesthe excess viscosity q-qo, where qo is the anomalyis smaller than 0.5 percent. The authors vapor critical point a knowledgeof the local value of viscosity coefficient in the dilute state at the given did not consider the influence of the variation of the the density is required. temperature, is considered to be only a function of density as a function of the height in the fluid. A The influence of criticalphenomena is not re- thedensity. This is, however, not true in general. cell with a height of about 13 cm was used,the stricted tothe immediate vicinity of thecritical Most measurements of the viscosity and thermal oscillatingdisk being located in the lower end of point.For instance,an anomaly in thespecific conductivity in the critical region have been carried the vessel, and an estimate from the density versus heat can be detected at temperatures considerably out for carbon dioxide because of its easily acces- height relationship publishedin the literature[65,67] higher than the critical temperature. In this paper siblecritical temperature., The discussion will shows that this effect could have caused an apparent the qualitative properties of viscosity and thermal therefore be mainly based on thephenomena 01)- increase of the same order as theanomaly observed. conductivityregarding the existence of similar served for CO2. The few investigations on other anomalies in the critical region will be discussed. substances sustain the conclusions obtained for CO2 Therefore,the conclusion thatthe viscosity ex- hibits an anomalous increase near the critical point However, theprecise mathematical character of The experimental investigation of the viscosity of is not justified. such anomalies can only be obtained from measure- CO2 in the supercritical region started with the work A pronounced anomalous increasein a wider tem- ments very close to thecritical point. In viewof of WARBURG and VON BABO in 1882 using a cap- perature range andwith the viscosity increased by a the difficulties mentioned above an analysis of the illary flow method [15]. Eventhat early these factor 2 at 31.1 "C was reported by MICHELS, BOT- latter kind is not yetpossible for thetransport authors were aware that knowledge of the density is ZEN,and SCHUURMAN [8]. Theirdata were coefficients. essential and were careful to determine the density obtained with a capillary flow method which was as well as pressure and temperature. They realized known to be accurateoutside the critical region. that thepressure difference along the capillary 3. The Viscosity of a One Component As pointed out by the authors, the method is not shouldbe kept small and applied corrections for Fluid Near the Liquid-Vapor Criti- reliable in the critical region due to the large density cal Point the variation of the density along the capillary. In gradient in the capillary resulting from the use of an illuminating