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Behavior of 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 thebehavior 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 presentssome 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 -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 and near the critical mixing Section 7 is asummary of conclusionsbased on point of a binary liquid mixture the 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 [4, 71 at three viscosity near the critical point of CO, was under- As a consequence the density, at which the proper- , 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 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 capillaryresulting from the use of an illuminating discussion of this experimental work apressure difference of about0.5 atm.Never- Accurate measurements of the viscosity of com- M. BRILLOUIN [l] remarkedthat these correc- theless,they conjectured that the anomalous pressed gases have been mainly carried out either tions are practically canceled by the variation of the behaviorobserved was essentiallyreal [8, 91. by a transpiration method or by the oscillating disk viscosity associated with thevariation of theden- In order to resolve the discrepancy between these method.In the first methodthe gas is caused to sity. The data of Warburg and von Babo at 32.6 "C results, new measurements of the viscosity of CO, flow through acapillary and the viscosity is ob- areshown in figure 2. Theyalso measured vis- werecarried out by KESTIN, WHITELAW, and tained from the flow rate. In the latter method the * One amagat unit of density lor argon corresponds to 0.0017834 p/cm3. cosity isotherms at 3.5.0 and 40 "C which have not ZIEN using an oscillating disk method [5]. They ap- 2In this review the values given by Michels et al. [60]are used as critical parameters damping of an oscillating disk due to the viscosity of CO2: beenincluded in figure 2 sincethe temperature proachedthe critical region alongdifferent iso- of thesurrounding mediumis determined. Both I, =31 04 Y:, I,, =iHi atm. p, =O.Mli g/cn1''=?:3h amagat. effect turned out to be small. Sincethe viscosity choresas didNaldrett and Maass. Their results methodshave been used to measurethe viscosity An alternative possible set of critical values was given by Schmidt and Traube [65] isotherms at 3.5 and 40 "C are nearly parallel to the show convincingly thatany anomalous increase is in thecritical region.Close to thecritical point /'=.11.00 Y:, I', -7'.Hh atm, p, =O.(hH p/mP=nf, amagat. one at 32.6 "C, theauthors conclude that the vis- much smaller than that reported by Michels et al. the oscillating disk method has the advantage that One amagat unit of density for CO2 corresponds to 0.00107h4 g/cm3 cosity does not show any anomaly at supercritical Close to the critical point they find ananomalous temperatures down to 32.6 "C (about 1.6 "(: above increase of thesame order of magnitudeas that 166 167 reported by Naldrettand Maass. Thedata ob- Close to thecritical temperature they find anin- the viscosity does not show a pronounced anomaly fromthe critical temperaturethe density is only tained by Kestin etal., at 31.1"C andat 34.1 "C crease of 1 to 2percent on oneisotherm, which in thecritical region.However, nearthe critical 185 amagatat the critical pressure, whereas the are shown in figure 3a, b as a function of density. is of theorder of themeasuring accuracy in this point neitherthe density at which the viscosity criticaldensity is 236 amagat.Consequently Forcomparison the behavior of the viscosity iso- difficult region. was determined nor the precision of the values for extrapolation of data at constant pressure 2s a func- therm at 50 "C is indicatedand data obtained by From theexperimental work described, we can the viscosity coefficient is sufficiently well known tion of temperature easilyoverlooks thecritical Naldrett and Maass and a few data of Michels et al., draw the following two conclusions. todecide whether a small increase, as observed region entirelyand therefore does not revealany outside the critical region are included in the figure. 1. Measurements of the viscosity of CO2 with by Kestin et al., exists. anomaly in thecritical region. Along anisotherm Recently the viscosity of CO, was also measured the oscillating disk method [5, 10] as well as meas- Recently DILLER reported that the viscosity of the density is similarly sensitive to a small change by BARUA and ROSS with a capillary flow method urements with the capillary flow method[12, 151 parahydrogen does not show any anomaly near the in pressure. [12]. By using a horizontal capillary and extrapolat- have shown thatthe viscosity coefficient does not critical point [3]. Measurements at temperatures Anotherprocedure sometimes used to obtaina ing to zero flow rate [4], they were able to avoid the show anyanomalous behavior supercriticalat for which T-Tc is tentimes smaller than in the thermalconductivity value associated with the difficulties inherent in theviscometer of Michels temperatures down to atemperature (7'- Tc)/Tc case of CO2 arerequired, because the critical critical point is based on theassumption that the et al. Theanalysis of theirexperimental data has = 1percent from thecritical. temperature of parahydrogen is only 32.976 OK. excessthermal conductivity X- Ao, where All is notyet beencompleted, yet it seemsthat their 2. Thesignificance of thesmall anomalous in- Consequently, only themeasurements at oneiso- the thermal conductivity at 1 atm at the tempera- results are in agreement with those of Kestin et al. crease at temperatures closer to the critical isnot therm (T=33.000 OK) are relevant for our present tureconsidered, is only dependent on thedensity clear at present. A correctionshould be sub- discussion.Indeed, the data on thisisotherm do and not on thetemperature. However,this rela- tracted from thedata of Naldrettand Maass in not show any appreciable anomaly near the critical tionactually has been verified outsidethe critical this region which in magnitude might becompar- density.However, this conclusion is based only region only and therefore cannot be used legitimately able to the anomalyreported. Using thesame on a few experimental points for which calcula- in thecritical region. “Critical thermalconductiv- apparatus Mason and Maass did not find an anom- tions of the density from the P-V-T data are diffi- ity values," obtained in this way, are often used as alous increase of the viscosity of ethylene near the cult. An analysis of the viscosity data on adjacent reduction parametrrs for correlation of the thermal criticalpoint, although the dataare somewhat isothermsshows that the precision is only a few conductivity of different substances [66]. It must preliminary. It is interestingthat the difference percent.Therefore, no decisiveconclusion can bekept in mindthat thesereduction parameters betweenthe data of Kestinet al., and the uncor- be drawn from this work regarding the existence of a do not necessarily represent the actual thermal con- recteddata of Naldrettand Maass is also of the small anomaly. ductivitynear the critical point.Similar pro- sameorder as the anomaly. Kestin etal., have cedures havebeen used to obtaincritical values discussed the possible influence on their measure- 4. The Thermal Conductivityof a One of the viscosity. The reductionparameters for ments of thevariation of thedensity with height. Component Fluid Near the Liquid- the viscosity obtained in this way are closer to the Because their oscillating disk is located at a posi- Vapor Critical Point actual viscosity in the critical region, because any tion somewhathigher than the center, the varia- anomaly in the viscosity nearthe critical point is tion of density to a correction in the opposite. Outsidethe critical region the behavior of the small.With few exceptions,values for thetrans- thermalconductivity is very similar to that of the direction from that to beapplied to the data of' portcoefficients in thecritical regionwhich are Naldrett and Maass and therefore tends to enlarge viscosity described in the previoussection. How- based on dataoutside the critical region are not theanomalous effect. Theauthors point outthat ever,there exists seriousa controversy in the discussed in this review. theirdata become less reliable in this difficult literaturewhether the thermal conductivity, con- Two methods for the measurement of t the ther- region andthey aretaking the attitude that their trary to the viscosity, exhibits a pronounced anomaly mal conductivity of compressed gases have had data prove only theabsence of thepronounced in thecritical region. An introductorysurvey wide acceptancethe concentric. cylinder and the anomaly suggested by Michels et al., and that any on the subject has been presented by Ziebland [37]. parallelplate method. In the tirst the fluidis en- anomaly is smallerthan 20 percent.The reality Thethermal conductivity is sometimesstudied closedbetween two coaxial cylinders, either in of thesmall anomalycan be decided definitely either as a functionof pressure at constant tempera- horizontal or in vertical position, and the heat flow only if the local value of the density is determined tureor as a function of temperatureat constant through the gaslayer is determinedas a function simultaneously and the effect of a variable density pressure, without consideringthe dependence on of the temperature differencebetween the cylinders. on the interpretation of the viscosity measurements the density. The region around the critical density In the latter method the fluid is contained between is carefully analyzed. is restricted to very narrow pressure and tempera- two horizontal plates anti the heat flowing from the This picture of the viscosity in the criticalregion is ture intervals in this procedure,because both upper plate to the lower plate is determined asa sustained by the work on other substances. STAR- ;(3p/i)/J)Tand (ijp/dT),, become infinite. at the critical function of the temperature difference. A special LING,EAKIN, DOLAN, and ELLINGTON have .,point.Measurements at the critical pressure are case of the concentric cylindermethod is the hot measuredthe viscosity of ethane,propane and n- .notequivalent to measurements near the critical wire methodThis method isnot suitable for the butane with acapillary flow method [14]. Although densityunless the temperature is equal to the critical region becausethe analysis the authors did not analyze their measurements as a 'criticaltemperature within atenth of adegree. is complicated too large temperature differences function of density,they have measured at suffi- 'For instance,considering the density dependence are usually involved and, most importantly,con- cientlysmall pressureintervals to conclude that dong anisobar, in CO, at atemperature 0.15 "C vection cannot be avoided. 169 Papersreporting ananomalous increase of the AT=O from the experimental data obtained for A' layer with athickness of 0.68 mm. 26, 301. Contrary to theconcentric cylinder thermal conductivity in the critical region are gen- usingvarious values of AT. In thecritical region could not be avoided in thisapparatus so that A method the horizontal flat layermethod has the erallyconsidered with much skepticism due to the contribution of convection becomes very large was determined from measurements at various advantagethat the anglebetween temperature the high probability of convection. An exact the- and such an extrapolation can diminish the accuracy temperature differences by extrapolation to AT gradient and the vertical can be made very small to oretical analysis of convection is very complicated considerablyeven when care is taken that only =O at constantaverage temperatures. He con- suppressconvection. The experimental results and has been carried out only in very special cases. values for A' corresponding to thesame average cluded that the thermal conductivity itself exhibits obtained for the thermal conductivity as a function We do not go into the details here, but remark that densityand average temperature arc used. In apronounced anomaly. This anomalousincrease of densityare presented in figure 4. Thethermal thepresence of convection ran be detected by some cases, such as a horizontal fluid layer heated is already 30 percent at 40 "C at the critical density. conductivityshows indeed apronounced anomaly varying the thickness of the gas layer or the tem- from below, convection only seems to start if the Heapproached the criticaltemperature to within at thecritical point. An anomalousincrease can peraturedifference across this layer. The heat Rayleigh numberbecomes larger than acertain 1 "(:. Although at 75 and 40 "C the influence of be detectedas much as 20 "C above thecritical transfer coefficient or so-called apparentthermal critical value [64]. Thishas alsobeen reported convection was small, closer to the critical tempera- temperature which becomes 250 percent at 1 "C conductivity coefficient, which includes a contribu- forconcentrica cylindrical layer in horizontal ture the influence of convection became large dimin- above thecritical temperature. Convection was tion from convection,depends on the geometry of position [23. 581. When thisinstability occurs, ishing the accuracy of the data. avoided by the use of asmall distance between theapparatus and of the boundary conditions extrapolationto AT=O is not legitimateand can On the other hand, VARGAFTIK stated that the theplates (d=0.4 mm) andsmall temperature involved. to overcorrections. thermal conductivity did not exhibit ananomaly differences (AT varied from 0.03 to 0.25 "C). The Heat transferred by convection is usually related The controversy over the existence of a thermal in the critical region 134.1. This assertion was based absence of convection was verified experimentally; to the R defined as conductivityanomaly started in 1934. KARDOS on experimental work of SHINGAREV, which unfor- in the whole critical region thethermal conduc- attempted to measurethe thermal conductivity tunately, has not been made available and therefore tivity values measured were independent of the of C02 using a hot wire method 1211. Hefound a cannot be criticallyexamined. According to the temperature difference used. Of course,one tremendous increase of heat transfer in the critical discussion given hy Vargaftik, theseconclusions should compare measurements with different AT, where g is the gravitation constant, a the expansion region, which was laterrecognized as an effect of weredrawn after applying corrections for convec- but at thesame average density and temperature. coefficient, p thedensity, c,, thespecific heat at convection [28]. SELLSCHOPP measuredthe tion by extrapolation to zero temperature difference. For thedetails we referto the previouspublica- constant pressure, d the thickness of the gas layer, thermalconductivity of CO, with two concentric It is clear that measurements in the critical region tion. The reality ofthe anomaly was independently AT the temperature difference, A the thermal con- cylinders in horizontal position [29]. Heobserved free of convection are highly desirable. For this pur- confirmed by a few measurements with alarger ductivity and 7 the viscositycoefficient. Inthe anincrease in theapparent thermal conductivity pose a detailed study of the thermal conductivity of plate distance. An attempt was made to include critical region theRayleigh numberbecomes nearthe critical density at supercritical tempera- CO2 was carried out by MICHELS, SENGERS, and ameasurement of thethermal conductivityiso- extremelylarge [26, 301. By theuse of many tures, but his measurements were also influenced by VAN DER GULIK using a parallel plate method[25, thermat 31.2 "C usingeven smaller temperature approximationsand assuming that the convection convection. The whole effect was attributed to differences.These results are shown in figure 5. is small,the error introduced by convectioncan convection and itwas assumed, withoutexperi- Obviously the values at this temperatureare less be derived explicitly: mentaljustification however, that in thecritical region the thermal conductivity was only dependent on the density and not on the temperature 118, 291. TIMROT and OSKOLKOVA [33] reported that the thermal conductivity of C02 did not show any where A' is theapparent thermal conductivity co- anomaly in the supercritical region, but their efficient as measured, l the length of the gas layer, measurementsobtained with a hot wire method E the angle between the and were actuallycarried out too far awayf'rom the thevertical, and C a proportionality constant.In critical point. order to obtain (2) oneneglects the variation with COMINGS et al., measured the thermal conduc- temperature of the fluid properties other than the tivity of several gases using two concentric cylinders density,linearizes the hydrodynamic equations, in horizontal position. Anomalies in the super- assumesa spatially uniform temperaturegradient critical region were reported for ethane, propane, d andconsiders the limitingsolution for - < 1. and butane (221 but the. reality of' the effect could 1 not be established again due to the possibility of Without theseapproximations the error may well convection An anomalous increase ofthe thermal depend on d and AT in a more complicated way. conductivity of' argon at 0 "C, originally reported The absence of convection is verified, nevertheless, by the present authoramong others, later turned if themeasured thermal conductivity coefficient out to be entirely caused by convection 1241. at a given density and temperature does not change The first critical study of the existence of a under sufficient variation of AT or d. thermal conductivityanomaly was carried out by If convection is present,one sometimes tries GUILDNER [19]. Hemeasured the thermal con- to obtain the correct value of A by extrapolating to ductivity of CO2 using avertical cylindrical gas 170 accuratethan those of figure 4, since the experi- ments using a hot wire apparatus at various tempera- conductivityanomaly can be noticed from the A similar result is obtained [31] if oneanalyzes mental verification of the absence of convection tures, butat constantpressure. As mentioned occurrence of intersections of thethermal con- thethermal conductivity dataobtained by IKEN- becomes more difficult due to theinaccuracy in beforedata at constantpressure as a function of ductivityisotherms asa function of density at BERRY and RICE forargon [20]. However, since the density. temperature do not approach smoothly the critical temperatures considerably higherthan thecritical theseauthors did not adequatelyexamine the ab- Due to the variation of temperature, the density density.Consequently the measurements were temperature. It is interestingtherefore to study sence of convection, we cannot rely on their results variation in the pressure vessel isvery compli- not carried out nearthe critical density. The in- as a function of density data given in the literature as decisive. cated. It is therefore difficult to approachthe terpretation of Abas-Zade that he followed the somewhatremote from thecritical point. As an In view of these results NEEDHAM and ZIEB- critical region by the isochoric method. The coexistence line up to the critical temperature example, we considered the experimental data ob- LAND studied the thermalconductivity of ammonia densitymust be calculated from the experimental is erroneous. tained by ZIEBLAND and BURTON [35] for the A in more detail [27, 371 than before [36]. The data pressureand temperature, which may introduce Very careful thermal conductivity measurements of . Thesedata were originally plotted wereobtained using an apparatus consisting of systematic errors at each isotherm.Thus the were carried out by AMIRKHANOV and ADAMOV along isobars and consequently did not reveal any two concentriccylinders with agap of 0.2 mm. detailedbehavior along the critical isochore par- using both a horizontal parallel plate method and a anomaly.However, if thedata are plotted asa The resultsobtained at two supercritical tempera- ticularly the character of the divergence of the vertical concentric cylinder method 1171. Using the function of densitythe thermal conductivity iso- tures are shown as a function of density4 in figure 7. thermalconductivity cannot accurately be de- parallel plate apparatus, a series of measurements therms shown in figure 6 are obtained,3 A pro- An anomalousincrease can be noticednear the duced from the data. was carried out with plate distances of 0.3 mm and nounced increase in the thermalconductivity can criticaldensity even at 157.1 "C. Theabsence It is notable that at 75 and 40 "(:. where the of 0.6 mm,and thetemperature difference was be noticed at - 140 "C (7 "C above the critical tem- of convection was verified by measurements with amount of convection in the apparatus of Guildner varied from 0.07 to 1 "C. With thesmaller plate perature).This observation is based 011 only two differentvalues of AT. At 138.8 "C thethermal was reasonably small. there is a satisfactory agree- distance the data turned out to be independent of experimentalpoints, but thecrucial experimental conductivity goes through a maximum. The den- ment with thedata of Guildner confirming the AT, which guaranteesabsence of convection. value at - 140 "C, showing an anomalous increase sities were estimated by Needhamand Ziebland anomaly of 30 percent at 40 'C. It turned out. Dataobtained with thelarger plate distance and of 35 percent, was obtained with two different values from P-V-T data published in theliterature. alsothose obtained using the concentric cylinder however. thatcloser to the critical temperature of AT, which excludes the possibility of convection. However,at 138.8 "C these P-V-T dataare vir- theanomalous increase is considerably smaller apparatus were dependent on AT showing the This is a clear illustration of how one can he misled tually unknown, so thatthe cannot be than that reported by Guildner presence of convection. The resultsobtained with if thedensity dependence of A is not considered calculated with sufficient accuracy to conclude The fact that the thermal conductivity measured the smallest value of AT agreed for all 3 series of explicitly. that the maximumdoes not occurat the critical was independent of thetemperature difference measurements.The valuesobtained for the density. used not only guarantees the absence of con- thermalconductivity therefore seem to be very 3 One amagat unit of N, correponds to 0.0012502 g/cm3 We conclude that the existence of a pronounced vection. but also verifies the validity of the law of reliable. As hasbeen pointed out repeatedly anomaly in thethermal conductivityhas been Fourier in thetemperature range considered. The before, it is equallyimportant to verify whether verified by convection free measurements obtained validity of the lawof Fourier in the critical region themeasurements were carried out sufficiently both with a parallelplate method and with a con- was questioned by SIMON and ECKERT [32]. From closeto the critical density. Unfortunately, the centric cylinder method. interferometricstudies of laminarfree convection authors do not describetheir experimental pro- 4 One. amagat unit of NH3 corresponds to 0.0007714 g/cm3. near a vertical wall a quantity k was derived, which cedure. They do not report whether and how either they identified with thethermal conductivity co- pressure or density was determined. In addition, efficient of the fluid. Thisquantity X. turned out althoughthey measurethe temperature difference to increase with increase of heat rateand it was AT carefully, no device was presented for meas- conjecturedthat the anomaly in thethermal con- uring the absolute average temperature of the gas ductivity would disappear in the limit of zero heat layer, which obviously differs from the temperature rate. However. an analysis of theirdata shows of thesurrounding thermostat. It isnot clear at thatthis quantity k remained dependent on the which temperatures and densities the thermal con- heat rate outside the critical region as well. where ductivity actually was determined. If the measure- the validity of the law of Fourier cannot reasonably ments were carried out at the coexistence line, one be questioned. Therefore, this quantity X. should would expect a large variation of the density when not be identified with the thermal conductivity thetemperature difference was variedfrom 0.1 coefficient. to 1.0 "C, at the liquid side even leading to evapora- The existence of an anomaly in the thermal con- tion of the liquid. On thecontrary, their results ductivity near the critical point was denied by indicatethat even up to a few hundredths of a Abas-Zade 1161 and by Amirkhanov and Adamov degree below thecritical temperature the density 1171. Theseauthors do not base this assertion on did not change when AT was variedfrom 0.1 to measurements in thesupercritical region. but on 1 "C. Thus it seems thatthe measurements were dataobtained at temperatures below the critical not carried out near the critical density. which theyclaim areassociated with the coexist- From the data given in figure 4 it is evident that ence line. ABAS-ZADE performedhis measure- the first indication of the existence of the thermal 1 72 5. The Viscosity of a Binary Liquid Mixture Near the Critical Mixing Point Many binary liquid mixturesexhibit acritical mixing point near room temperature.The vis- cosity of a number of differentmixtures of this typehas been studied. There are not enough data obtained by different authors for thesame substance. Moreover. thecritical parameters are very sensitive to the purity of the components Althoughsome attention has been given to the influence of a thirdcomponent on the viscosity in thecritical mixing region 143, 47, 50, 51, 53], theinfluence of any impurity on thisanomaly is not clear. It seemsthat the effect depends on the binary system as well as on thecharacter of the impurity. Most viscosity measurements for binary liquid mixtures have been carried out with some version of the capillary flow method. We have not made a critical analysis of theexperimental procedures Figure 9. The viscosity of the mixture isooctane-perfluro- used by thevarious authors. However, there is heptane as a function ofconcentration according to Reed and An interesting question is whetherthe anomaly Taylor [46] in thethermal conductivity is related to that in sufficient agreement to establish the existence the specific heat.For this reason the variation of of an anomaly in the viscosity nearthe critical SEMENCHENKO and ZORINA investigated the A of C02as a function of temperature at the critical mixing point for systems with either an upper viscosity of thesystems triethylamine- and isochore is compared with the variation of c,. and or lower critical solution point 139. 40. 4,3-53). nitrophenol-hexane [49] The first mixture has c,, in table 1. For c,. theexperimental data [62] In the beginning of thecentury. ananomalous a lower critical mixing temperature. For both are supplemented withvalues calculated from the I increase of' the viscosity near the critical mixing systems an anomalous increase of the viscosity P-V-T data at temperatures whereno experimental point was reported by various authors. among others up to 20 percent was found. The anomalous data for cl. areavailable [61]. Thevalues for c,, by FRIEDLANDER [43] for the systems isobutyric phenomenon only occurred at temperatures within areobtained by addingthe calculated difference acid-water and phenol-water, by SCARPA [48] for 1.0 or 1.5 "(1 from the critical and in a concentration c,,- cI. to the values for cI.obtained as mentioned phenol-water, by DRAPIER [40] fornitrobenzene- range of10 percent, above.It is evidentthat c,, increasesmuch faster hexaneand cyclohexane-aniline, and by TSAKA- Recent measurements of the viscosity of the in the critical region than A, so that the ratio A/c,, LOTOS 152) for isobutyric acid-water,trimethyl- nitrobenzene-isooctane mixture obtained by PINGS decreases by two orders of magnitudewhen the ethylene-aniline,triethylamine-water and nicotine- et al.. confirmed that for this system also a viscosity temperature is variedfrom 75 to 31.2 "C. Onthe water. anomaly occurs in a very small temperature range other hand the variation of A/c,. is an order of mag- For simple binary liquid mixtures the anomalous [45] No anomaly could be detected at 4.5 "C above nitudesmaller than that of A itself. Very close to increase of the viscosity is of theorder of 15 to the critical temperature but an anomalous in- thecritical point c,. as well as A is not accurately 25 percent at the critical mixing point. crease of 5 to 7 percent was noticed at 0.45 "(: known. A relationshipbetween the anomaly in REED and TAYLOR observed an anomalous in- above the critical temperature. A and cI.is well possible.In viewof therecent crease in the viscosity ofthe mixtures isooctane-per- We conclude thatthe viscosity does show an work on the cl. anomaly, it is natural to ask whether flurocylic oxide. n-hexane-perfluorocyclic oxide. anomalousincrease near the critical mixing point. A divergeslogarithmically. Suchquestions about isooctane-perfluoroheptane 146). The effect was but for a number of' simple mixtures the anomalous the detailed behavior cannot be decided at present most pronounced in themixture isooctane-per- behavior occurs only at temperatures within 2 "(1 and further experimental research is required. fluoroheptane: theirresults for thissystem are of' the critical temperature. According to the classical theory the line width of shown in figure. 9. The viscosity exhibits an thecentral component of thespectrum of thecorresponds to light from a He-Ne laserscattered anomalous increase up to 25 percent According 6. Thermal Conductivity of a Binary scattered light is determined by thethermal dif- through an angle of 60". Because c, divergesmuch to thesedata the anomalous behavior can be de- Liquid Mixture Near the Critical fusivity Alpc,, [57, 631. The linewidth to beex- faster than A, thethermal diffusivity [SI] andcon- tected as far away as IO "C from the critical tem- Mixing Point pected for COr from thestatic measurements of A sequentlythe linewidth of thecentral component perature.However. other authors found that for ispresented in figure 8 for k= IO5 cm, which decreases rapidly in thecritical region. some systems the anomaly occurs only at tempera- Experimentalinvestigations of thethermal con- tures much closer to the critical. ductivity of binary liquid mixtures near the critical 1 74 175 I mixingpoint are veryscarce. This is surprising The mostextensive study was carried out by methodhave shown that the viscosity does not 18) A. Michels, A. Botzen and W. Schuurrnan, Physica 23 (1957). 95. since the experimental complications are somewhat GERTS and FILIPPOV, who measured the thermal showany anomaly near the liquid-vapor critical 191 A. Michels, Proc.Intern. Symposium TransportProcesses less difficult thanthose encountered in themeas- conductivity of a number of binary using a point if thetemperature isnot within (T- TC)TC in Stat. Mech., Brussels, 1956, p. 365. [lo] S. N. Naldrett and 0.Maass, Can. J. Research 18B (1940). urement of thethermal conductivity near the hotwire apparatus [54]. Measurements of the = 1 percent of thecritical temperature. Whether 322. liquid-vapor critical point. thermal conductivity of the system nitrobenzene-n- ananomalous behavior of theviscosity occurs at 11 11 1’. Phillips, Proc. Roy. Soc. (London) A87 (1912). 48. [I21 J. Ross, private communication hexane were carried out with four different tempera- temperatures closer to the critical point is not clear [I31 E. Schröer and G. Becker, Z. physik. Chem. A173 (1935), turedifferences: 0.01, 0.05, 0.15, and 0.3 “C and atpresent. If suchan anomaly does occur it is 178. 1141 K. E. Starling, B. E. Eakin, J. P. Dolan and R. T. Ellington, foundto be independent of thistemperature dif- probablysmaller than 20 percent. Progress Intern. ResearchThermodynamic Transport ference.In addition the systems nitrobenzene-n- 2. Measurement withthe parallel plate method Properties, A.S.M.E., Princeton, 1962, p. 530. K. E. Starling, Critical region viscosity behavior of ethane, heptane,methanol-n-hexane andtriethylamine- as well as withthe concentric cylinder method propane and n-butane, Masterthesis, Institute of Gas waterwere investigated. The results obtained by have shown that the thermal conductivity exhibits Technology, Chicago, 1960. (151 E. Warburg and L. von Babo, Ann. Phys. Chem. 253 (1882), Gertsand Filippov tor the nitrobenzene-n-hexane a pronounced anomaly near the liquid-vapor critical 390. mixtureare shown in figure 10 andthose for the point.Further experimental work is required to nitrobenzene-n-heptanesystem in figure 11. It is study the precise character of the anomaly and to Thermal Conductivity of evidentthat, contrary to thebehavior near the investigate whether the anomaly of A is related to One-Component Fluids liquid-vaporcritical point, the thermal conduc- the anomaly of the specific heat c,.. 1161 A. K. Abas-Zade. Doklady Akad. Nauk SSSR 68 (1949) tivity of these binary liquid mixtures does not show 665: 99 (1054)227. A. K. Abas-Zade and A.M. Amuras- I” 3. The viscosity of binaryliquid mixtures shows lanov. Zh. Fix. Khim. 31 (1057) 1450. anypronounced anomaly near the critical mixing ananomalous increase near the critical mixing 1171 Kh. I. Amirkhanov and A. P. Adamov. Teploenergetika point.Similar results were obtained for theother 10, No. 7 (1963) 77: Primenenie Ultraakustiki k Issle- point.For simple systems this increase is between dovan Veschestva 18 (1963) 65. Kh. I. Amirkhanov, two systemsmentioned. An anomalyeven as 15 percentand 25 percent at thecritical point. A. P. Adamov and L. N. Levina, Teplo i Massoperenos. small as that found for the viscosity of binary mix- Pervoe Vses. Svesch., Minsk 1 (1961) 105. Fornumbera of liquidmixtures, however, this 1181 E. Borovik. Zh. eksp. teor. Fiz. U.S.S.R. 19 (1949) 561. turesseems to beabsent. The data were not phenomenon isonly observed attemperatures [19] L. A. Guildner, Proc. Nat. Acad. Sci. 44 (1958) 1140; J. Res. NBS 66A (Phys. and Chem.) No. 4. ilY62) 333. 341. obtainedatenough different concentrations to i ”(: within 2 of the critical temperature. 1201 L. D. Ikenberry and S. A. Rice, J. Chem. Phys. 39 (1963) study the behavior in detail and the results cannot 4. It seemsevident that the thermal conduc- 1561. beplotted as a function of concentrationas was [2IJ A. Kardos. Z. gesamte Kälte-Ind. 41 (1934) 1. 29; Forsch. tivity of anumber of binaryliquid systems does Gebiete Ingenieurwesens 5 (1934) 14. done for theviscosity in figure 9. The absence of notshow an appreciable anomaly near the critical 1221 I;. K. Kramer and E. W. Comings, J. Chem. Eng. Data 5 a pronounceda anomaly near the criticalmixing (1960) 462. D. E. Leng and E. W. Comings, Ind. Eng. mixingpoint. Additional experimental information Chem. 49 (1957) 2042. J. M. Lenoir, W. A. Junk and point seems to beestablished. This difference ishighly desirable, since this assertion can only E. W. Comings. Chem. Eng. Progress 49 (1953) 539. betweenthe behavior of thethermal conductivity be based on theresults of one experimental (231 J. M. Lenoir and E. W. Comings. Chem. Eng. Progress 47 (1951) 223. nearthe liquid-vapor critical point and that near investigation. 1241 A. Michels, .A. Botzen A. S. Friedman and J. V. Sengers. thecritical mixing point was noticed before by Physica 22 (1956) 121. A. Michels, J. V, Sengers and I., J. M. van de Klundert, SKRIPOV [56]. Additionalexperimental data are Physica 29 i106:3) 140. highlydesirable to verify thisconclusion and to 1251 2. Michels, J. V. Sengers and P. S. van der Gulik. Physica Theauthor is indebted to Dr. H. Zieblandwho 28 (1962) 1201. 1216. study smaller details. presentedthe thermal conductivity data for am- 1261 A. Michels and I. V. Sengers. Physica 28 (1962) 1238. On the other hand an anomalous increase of the i27j D. P. Needham and H. Ziebland,Int. J.Heat Mass Transfer moniashown in figure 7. Theauthor also ac- 8 (1965) 1387. thermal conductivity was reported for by OSIPOVA knowledgesthestimulating interest of several 1281 R. Plank and 0. Walger, Forsch. Gebiete Ingenieurwesens thephenol-water system [55]. Thedata were ob- 5 (1934) 289. colleagues at theNational Bureau of Standards 129) W. Sellschopp,Forsch. Gebiete Ingenieurwesens 5 tainedwith aparallel plate method. The uncer- and in particular that of Dr. M. S. G.Green. ( 1934) 162. tainty in the temperature difference and the scatter 1301 .J. V. Sengers,Thermal conductivity measurements at elevated gas densities including the critical region, thesis, of the experimental data suggest that these results References van der Waals Laboratory, Amsterdam, 1962. shouldbe considered with some reservations. No J. V. Sengers and A. Michels, Progress Intern. Research Thermodynamic and TransportProperites A.S.M.E., experimental verification forthe absence of con- Viscosity of One-Component Fluids Princeton, 1962. p. 434. vection was presented. [31] J. V. Sengers,Int. J. Heat Mass Transfer 8 (1965) 1103. [I] M. Brillouin, Leçons sur la viscosité des liquides et des 1321 H. S. Simon and E. R. G. Eckert, Int. J. Heat Mass Transfer gaz. I., pp. 184-195 (Gauthier-Villars, Paris,1907). 6 (1963) 681. [2] A. L. Clark,Trans. Roy. Soc. Canada III 9 (1915) 43; 18 1331 D. L. Timrot and V. G. Oskolkova, Izvestia V.T.I. NIB.4 7. Summary (1924)329; Chem. Revs. 23 (1938) 1. (1949). [3] D. E. Diller, J. Chem. Phys. 42 (1965). 2089. [34]N. H. Vargaftik,Proc. joint Conference Thermodynamic ’ [4] G. P. Flynn, K. V. Hanks, N. A. Lemaire and J. Ross, J. TransportProperties Fluids, London 1957 p. 211. Theexperimental information on the viscosity (:hem. Phys. 38 (1063). 154. 1351 H. Ziebland and J. T. A. Burton,Brit. J. Appl. Phys. 9 [5] J. Kestin, J. H. Whitelawand T. I;. Zien, Physica 30 (1964). (1958) 52. and thermal conductivity in the critical region can 161. [.36] H. Ziebland and D. P. Needham, Progress Intern. Research be summarized as follows: [6] S. G. Mason and 0.Maass, Can. J. Research 18B (1940), Thermodynamic and Transport Properties, A.S.M.E., 128. Princeton, 1962, p. 441. 1. Measurements of the viscosity with the oscil- [7] A. Michels, A. Botzen and W. Schuurman, Physica 20 [37] H. Ziebland N.P.L. ThermalConductivity Conference, lating disk method as well as with the capillary flow (1954). 1141. Teddington, England 1964. 177 is applied to the nuclear spin system such that the tional degrees of freedom of the system so that one Viscosity of Binary Liquid Mixtures [55] V. A. Osipova,Doklady Akad. Nauk Azerbaidzhan S.S.R. 13 (1957) 3. gradient G is small so that can usually write [38] W. Botchand M. Fixman, J. Chem.Phys. 36 (1962)310. [56] V. P. Skripov, Conference Critical Phenomena and Fluctu- [391 P. Debye, B. Chu and D. Woermann, J. Polymer Science ationsin Solutions, Akad. Nauk SSSR, Moscow, 1960, -=1 R,, + Rf, + Rr,. A1 (1963) 249. p. 117. (5) [40] P. Drapier, Bull. Acad. Belg. C1. Sci. 1911, p. 621. TI 1411 M. Fixman, J. Chem. Phys. 36 (1962)310. [42] M. Fixman, Adv. Chem. Phys. 6 (1964) 175. overthe entire volume of thesample. i and k R.1 is the relaxation rate due to intramolecular di- 1431 J. Friedlander Z. physik. Chem. 38 (1901) 385. Miscellaneous areunit vectors in the x- andz-directions respec- pole-dipole and/or electric quadrupolar interactions (441 W. Ostwald and H. Malss, Koll. Z. 63 (1933) 61. [6]; i.e.,those interactions which transform as [45] C. J. Pings, private communication. [57] N.C. Ford and G. B. Benedek,Conference Phenomena tively. The spin system is assumed to be in equilib- 1461 T. M. Reed 111 and T. E. Taylor, J. Phys. Chem. 63 (1959) Neighborhood Critical Points, N.B.S., Washington, D.C., riumbefore the first of the two r-fpulses applied YZtt,(CLi), where ai is the orientation of ,a vector 58. 1965. t t =T fixed in themolecule i. R, is therelaxation rate 1471 V. Rothmund, Z. physik. Chem. 63 (1908) 54. [58] H. Kraussold,Forsch. Gebiete Ingenieurwesens 5 (1934) at times = 0 and respectively. Assume that 1481 0. Scarpa, Nuovo Cimento (5) 6 (1903) 277; J. Chim. Phys. 186. the r-f pulses are sufficiently intense and that their due to dipole-dipoleinteractions between nuclear 2 (1904) [59] I. R. Krichevskii, N. E. Khazanova and L. S. Lesnevskaya, 447. frequency is sufficientlyclose to theLarmor spins on different molecules, i.e., those interactions [49] V. K. Semenchenko and E. L. Zorina, Doklady Akad. Nauk Inzh.-Fiz. Zh., Akad. Nauk SSR 5 (1962) 101. SSSR 73 (1950) 331; 80 (1951) 903: Zh.Fiz. Khim. 26 [60] A. Michels, B. Blaisseand C. Michels, Proc. Roy. Soc. frequency wo= yH0, where y isthe nuclear gyro- whichvary as YZttt(flij)/r:j, where rij = (rij, 0,) is (1952) 520. (London) A160 (1937) 358. magneticratio, that the lengths of thepulses can the vector joining a pair of spins on different mole- [50] V. K. Semenchenko and E. L. Zorina, Doklady Akad. Nauk [61] A. Michels and S. R. de Groot, Appl. Scient. Research A1 SSSR 84 (1952) 1191. (1048)04. be chosen such that the z-component of magnetiza- cules [71. ti!(. is the relaxation rate due to the inter- [SI] V. P. Solomkoand S. M. Smiyun,Dopovidi Akad. Nauk [62] A. Michels and J. C., Strijland,Physica 18 (1052) 61.3. Tr action between the nuclear spins and the rotational Ukrains’koi RSR 1961, 649. [63] R. D. Mountain, Rev. Mod. Phys. 38 (1966)205. tion is rotated by an angle by each of the pulses. 1521 D. E. Tsakalatos Bull. Soc. Chim. France 141 5 (1909) 307: [64] A. Pellewand R. V. Southwell,Proc. Roy. Soc. (London) 2 angularmomentum Ji of thesame molecule [8] Z. Physik. Chem. 68 (1910) 32. A176 (1940) 312. Then, if the z-component of the nuclear magnetiza- (the spin-rotation interaction). [53] E. L. Zorinaand V. K. Semenchenko, Zh.Fiz. Khim. 33 [65] E. H. W. Schmidt and K. Traube, Progress Intern. Research tion relaxestowards itsequilibrium value ex- (1959) 523, 961. Thermodynamic Transport Properties, A.S.M.E., Prince- Of the three terms contributing to Tr’ in eq (5), ton, 1962; p. 193. ponentially with a time constant TI, the maximum amolecular theory is availableonly for RA [7], Thermal Conductivity of Binary [66] L. I. Stieland G. Thodos,Progress Intern. Research amplitudes of the detected nuclear induction signals Thermodynamic Transport Properties, A.S.M.E., Prince- which is expressed in terms of Fourier transforms Liquid Mixtures ton, 1962, p. 352. A(0) and A(T)following the pulses are related by [l] of the correlation functions [67] S. A. Ulybin and S. P.Malyshenko, Advances Thermo- [54] L. G. Gerts and L. P. Filippov,Zh. Fiz. Khim. 30 (1956) physicalProperties at Extreme Temperatures and Pres- 2424. sures, A.S.M.E., Purdue Univ., 1965, p. 68.

where g(r, r’, T) isa time dependent pair distri- If the x and y components of thenuclear mag- bution function for the system; &r, r’, T) drdr‘ is Nuclear Magnetic Resonance Measurements Near the Critical Point netizationrelax towards their equilibrium value of the probabilitythat the vector joining a pair of of Ethane zeroexponentially with a common time constant moleculesis between r and rtdr initially and T2 and if the spatial motion ofthe spins in the sample between r’ and r’+ dr’ at a time T. For a system is adequatelydescribed by thesolution to the dif- M. Bloom 1 ,2 c)f identical nuclear spins RB is expressed in terms fusionequation, the maximum amplitude of the of J(w0) and ./(2w0). Notethat typical values of “ Harvard University, Cambridge, Mass. spin-echo)” at time 27 is given by [2] wo are 00 c lox sec-lwhich is anextremely low frequencyinsofar as the dynamical motions of fluids areconcerned. The otherimportant point Inthis talk I shalldiscuss the application of tonotice about eq (6) is that J(w) getsits major nuclearmagnetic resonance (NMR) techniquesto Equation (4) correctly gives relative values of A(%) contributionfrom pairs of moleculeswhich are themeasurement of two transportproperties of in terms of T?, G, 11 and T for any given angles of very close together, because of the r3dependence fluids nearthe critical point. The two transport rotation by thr. r-f pulses[3]. For systems with of the dipole-dipole interaction. properties we shalldiscuss are the nuclear spin- strong NMR signals, it iseasy to measure TI and Physically, one can say that RB gives a measure latticerelaxation time TI andthe self- D to an accuracy of k 5 percent and with care an of a specific type of inelastic scattering cross sec- constant D of the fluid. Both of these observables accuracy of 2 1 percent can be attained. tioncorresponding to collisions between pairs of areeasy to measureusing the NMR pulsetech- Notethat the self-diffusion constant measured molecules in which the total nuclear Zeeman energy nique of E. L. Hahn [l]. here is the“spin self-diffusion constant.” The is changed.The low frequency 00 correspondsto TheHahn pulse technique: Asimple version of interpretation of this type of self-diffusion constant thefact that the energy exchanged between the the NMR pulsetechnique isillustrated sche- , has been givenby Emery [4] and by Fukuda and spinsystem and the translational degrees of free- maticallyin figure 1. Supposethat a time inde- Kubo [SI. dom is very small(< lO-’eV). Interpretation of TI measurements: Formost Thereare no detailed molecular theories for simple systems there are three main types of inter- RA and R(. in dense fluids, except for liquid Hz [91, ‘ Alfred 1’. Sloan Fellow. 2 Permanent address: Department of Physics, University of British Columbia, actionswhich enable the nuclear spin system to and this has inhibited the interpretationof the many Vancouver, BritishColumbia. Canada. On sabbatical leave 1964-65 while holding exchangeenergy with the translational and rota- TI measurementsalready performed on liquids a John Simon Guggenheim Memorial Fellowship. 1 79 178