PHYSICAL REVIEW B 1 NOVEMBER 1999-IIVOLUME 60, NUMBER 18

Adsorption-desorption isotherms and x-ray diffraction of Ar condensedinto a porous glass matrix

P. Huber and K. KnorrTechnische Physik, Universita¨t des Saarlandes, D-66041 Saarbru¨cken, Germany

~Received 3 August 1998; revised manuscript received 16 April 1999!

Ar condensed into a porous glass matrix has been investigated by simultaneous measurements of adsorption-desorption isotherms and x-ray diffraction patterns as function of the pore filling above and below the meltingpoint. The chemical-potential–temperature phase diagram has been established. It is consistent with a first-order phase transition between the adsorbate state and the capillary condensed state, above and below themelting temperature. The adsorbate and the capillary condensed state can also be distinguished in the diffrac-tion patterns. A consistent picture of the structure and the thermodynamics is obtained.@S0163-1829~99!09341-8#

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I. INTRODUCTION

Porous glasses such as Vycor are permeated by an iconnected network of tubular channels. The diameters ofpores are in the nm range. A large variety of materialsbeen condensed into the pores, ranging from simple vanWaals systems, such as the rare gases,1–3 to organic chainmolecules,4 semiconductors,5 and metals.6 Liquid metalshave to be forced into the pores, as is standard for Hg psimetry. Simple liquids are soaked into the ‘‘thirsty’’ glasand are obviously stable in the pore condensed state, atas long as they do not solidify at lower temperatures. Tadsorption isotherms of such liquids at low vapor pressuare similar to those obtained on planar substrates. At higpressures a transition to complete pore filling is oftenserved which is due to capillary condensation. This situathas been called complete wetting in analogy to the terminogy used for adsorbates on plane substrates.@For the case ofincomplete wetting, the final thickness of the adsorbed laon the walls when approaching the saturated vapor presPsat(T) is less than the pore radiusR.# The transition fromthe adsorbate state with a thin adsorbate layer coveringinner walls of the pores and vapor in the central part ofpores to the capillary condensed state is of first orderhence shows an intrinsic hysteresis on adsorptiondesorption.7 Figure 1 shows a schematic adsorptiodesorption isotherm which is based on the idea of a mefield first-order phase transition between the two statescylindrical pore of uniform diameter. At the pressurepcoexthe two states are in equilibrium~see inset of Fig. 1!, theupper and lower spinodal pressurespu and pl give the endpoints of the metastable sections of the adsorption andsorption branches. The variations of the pore diameter ofirregular pore network of porous glasses are expected toto a distribution of transition points which smear out thysteresis and to pore blocking.8 Note, however, that thefirst-order character of the transition and not pore blockingthe primary reason for the hysteresis. This is demonstraby measurements on the matrix material MCM-41 which hpractically uniform pores.9

The fundamentals of this transition have been worked

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by Saam and Cole.10 In contrast to adsorption on a planasubstrate, two essential points have to be considered forsorption on the inner walls of cylindrical tubes:~i! The areaof the adsorbate-vapor interface decreases with increathickness of the adsorbed layer; thereby, the adsorbate-vsurface tension enters into the problem.~ii ! The adsorbate-substrate interaction potential is different from the plancase. In particular, the gradient of the potential vanishesthe center of the pore. The transition from the film statethe capillary condensed state occurs when the cost of adstion energy related to the fact that molecules move fartaway from the substrate equals the gain of interfacial enerelated to the reduction of the vapor-condensate interfaceto the formation of a concave meniscus. Thus, wheneveradsorbate layer grows to a thickness close to the pore rada phase transition will necessarily take place betweencondensed film state and the filled pore state. The transiis of first order since it involves a fundamental change ofshape of the fluid-gas interface.

This model, complemented by assumptions on the disbution of the pore diameter, has been successfully fittedadsorption isotherms of liquids in porous glasses. The mois confirmed by calculations based on the density functioapproach.7

One wonders whether the same concept applies to spore condensates, whether solid condensates are at all sin the pores. Dewetting transitions observed on coolingsome pore condensates suggest that this is not necesso.11 In Sec. III A we will discuss this point on the basis ochemical potentialm vs temperatureT phase diagrams whichhave been derived from adsorption isotherms above andlow the melting point.

The confinement changes the properties of the materiathe pores with respect to the bulk. Thus the melting asolidification temperaturesTm andTs are both lower than inthe bulk. The reductionDT is proportional to the inversepore radius 1/R.12 The transition appears to evolve in a cotinuous way, but nevertheless shows considerable therhysteresis.1 Several models have been proposed for thesefects. Referring, for instance, to the standard nucleatmodel, solidification can only occur whenT is lowered tosuch a point that the radius of the critical nucleus is sma

12 657 ©1999 The American Physical Society

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12 658 PRB 60P. HUBER AND K. KNORR

than the pore radius. Models of this type have been criticifor the reason that they refer exclusively to the geometrconfinement, but ignore the role played by the interactwith the substrate.13 In fact, for strong adsorbate-substrainteractions one expects a layered structure of the poredensate and a layerwise propagation of the boundarytween the wall coating and the vapor. The multilayer phdiagrams of adsorbates on graphite~001! substrates give examples of this type.14 The pore walls of the porous glassform a heterogeneous substrate. The adsorption isothefail to show layering transitions. Hence the distinctionindividual layers or shells is obscured; nevertheless, loprobes can discriminate particles of the first and seconders from particles in the pore center.15–17 The m,T phasediagram resulting from our measurements will give alsoinsight into the melting of pore condensed Ar.

Diffraction studies on simple molecules such as H2,18

D2,19 Kr, Xe,20 O2,

21 and CO2 ~Ref. 22! in Vycor have re-vealed that the solidified capillary condensates are crystain the sense that some Bragg peaks, which are, of coubroadened due to finite-size effects, do exist in the diffrtion patterns. On the other hand, the patterns are not idento those of the bulk system. Solid H2 was said to be crystalline, but neither hcp nor fcc. Recently, we have shown t

FIG. 1. Adsorption-desorption isotherms in the liquid state atK and in the solid state at 65 K. A schematic isotherm as expefor a first-order phase transition between the adsorbate state ancapillary condensed state is shown in the top frame. The insesualizes the coexistence of these two states in a capillary.

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the pattern of pore condensed N2 can be quantitatively described as that of a hcp crystal with a tremendous amounstacking faults.23

In Sec. III B we report diffraction results which elucidathe structure of the pore condensate, both in the adsorstate at low fillings and in the capillary condensed statehigher fillings. We will also explain why we disagree witthe interpretation of a recent x-ray diffraction studyBrown et al.24 on Ar ~and Kr! in Vycor according to whichthe solid pore condensates show a completely random sting of spherical close-packed net planes just below thelidification temperature and transform at about 40 K into fcrystallites which are still embedded in the porous matrbut have sizes of at least 1000 Å.

II. EXPERIMENT

The porous glass used in this study is a controlled pglass, ‘‘Gelsil,’’ a xerogel of almost pure SiO2 with a nomi-nal pore diameter of 75 Å as specified by the manufactu~Geltech Inc., Orlando, FL!. Our routine characterization othe matrix by N2 adsorption at 77 K based on the BrunaueEmmett-Teller~B.E.T.! analysis for the initial part of theisotherm and on the Kelvin equation for the vapor pressnecessary for emptying the pores shows that this matrisimilar to the standard Vycor glass~No. 7930!, but has asomewhat larger porosity.17 For comparison, we will alsoshow some diffraction data obtained on standard Vyglass, on another type of Vycor glass with a pore diamete130 Å, and on a xerogel with 25 Å.

The experimental setup allows the simultaneous measments of volumetric adsorption isotherms and x-ray diffration patterns. A tablet of the porous glass substrate~0.3 g! ismounted in a copper sample cell. The cell is equipped wtwo Be windows for the passage of the incoming and outing x-ray beams. The cell is attached to the cold plate oclosed-cycle refrigerator. Temperatures above and belowbulk triple point (T3

bulk583.8 K) have been investigated. Thtemperatures are stabilized to better than 0.05 K. A capillconnects the cell to an all-metal gas handling system at rotemperature. The vapor pressureP is measured with capacitive membrane gauges of 10 and 1000 mbar full scale.thermodynamic state of the pore condensed material wilgiven in terms of the variables temperatureT, reduced pres-surep5P/Psat, and filling fractionf 5N/N0 . Psat(T) is thesaturated vapor pressure of bulk Ar,N the number of Arparticles in the cell, andN0 the number necessary for thcomplete filling of the pores.N0 is about 12 mmol. AtT3

bulk

the vapor in the dead volume of the cell and the gas handsystem is of the order of 1 mmol. The use of reduced qutities for the pressure and the amount of material minimithe effect of systematic errors. Thus one can safely ignthe effect of the temperature difference between the samand the gauge on the reduced pressurep.

For the x-ray powder diffraction experiment the BragBrentano configuration has been chosen. The radiacomes from a rotating Cu anode operating at 10 kW andmonochromatized by reflection from a graphite~002! crystal(l51.542 Å). The diffraction patterns cover scatterinangles 2u from 15° to 65°. All features observed in th

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PRB 60 12 659ADSORPTION-DESORPTION ISOTHERMS AND X-RAY . . .

patterns of the pore condensate exceed the instrumentauresolution by far. Hence resolution effects can be ignore

III. RESULTS AND DISCUSSION

A. Thermodynamics

In total seven adsorption-desorption isotherms have bmeasured. Figure 1 shows one isotherm of the liquid s(T586 K) and one of the solid state (T565 K) of the porecondensate. For the measurement of an isotherm smalltions of gas are consecutively introduced into the cell andpressure is allowed to relax towards equilibrium in each sThe equilibration time increases with decreasingT and withp approaching 1. Figure 2 shows the time dependence ovapor pressure for several data points of the adsorpbranch of the 65 K isotherm. As can be seen in the figure,relaxation can be extremely slow with intermediate lonlived instable states. The most prominent intermediate sis p51; here, the extra material offered to the substrateviously forms bulk crystallites outside the pores first aenters into the pores via distillation only afterwards. Tintermediate states withp,1 remain unexplained; perhapthey are related to temporary pore blocking. At 60 K equbration times of several days have been observed. It is cthat the intermediate instable state atp51 occurring in thefinal part of the adsorption branch can be easily misinpreted as incomplete wetting.

The initial part of the isotherms is similar to what is otained for physisorption on planar heterogeneous substrThere are no steps due to layering transitions. At higherp, aclear hysteresis between adsorption and desorption shup. This type of behavior is well known from experimentsa large number of liquids in porous substrates. We usepoint of steepest ascent on adsorption and of steepest deon desorption to define the characteristic pressurespa andpd . The final change of the isotherm from horizontal to vetical definesp51, f 51. For f .1, bulk material condenseoutside the pores atp51.

FIG. 2. The approach of the reduced pressure to equilibriumsome data points of the adsorption branch of the 65 K isothermFig. 1. Thef values from bottom to top are 0.23, 0.32, 0.45, 00.85, and 0.93. The saturated vapor pressure~corresponding top51! is 26.3 mbar.

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The characteristic pressures of the isothermspd , pa , p51 and additionally the pressures atf 50.15 andf 50.30~ofthe adsorption branch!, which according to the B.E.T. analysis correspond to wall coatings of, respectively, one andmonolayers, have been converted into the chemical poteDm5m2mbulk5kBT ln(p). Figure 3 shows them-T phasediagram. Here the chemical potential of the bulk liquid fT.T3

bulk and its linear extrapolation to temperatures belT3

bulk is used as the origin of them scale. Data points deriveddirectly from the isotherms are indicated in Fig. 3 by biggsymbols. The other data points, onp51 andpd , have beenobtained from heating scans on fixed amounts of gasclosed in the cell. In such heating scans—and in an angous way also in cooling scans—the amount of condenmaterialf and the reduced pressurep are not strictly constanbecause of evaporation~5desorption! of the condensate intothe dead volume or condensation~5 adsorption! of the vaporout of the dead volume into the pores. The corrections canestimated and turn out to be negligible in all instances whwe refer to such quasi-isosteric data.

Although this is not of direct importance for the interprtation, we suggest to relate the filling pressurepa of theexperiment to the upper spinodal pressurepu of the mean-field model described in the Introduction. The emptyipressurepd is presumably closer topcoex than to the lowerspinodal pressurepl . Note that even in the completely fillestate, the condensate in every capillary terminates at theternal surface of the porous glass sample with a menisOn desorption the pores simply empty by a retreat ofmeniscus, which just meanspd5pcoex. pl would be relevantin the unrealistic case of a closed pore where the menishad to be created as the necessary first step toward empt

The variations of the characteristic pressures with teperature are best discussed in terms of them-T diagram~Fig.3!. The T variation of thembulk is in agreement with theliterature. The kink at about 84 K definesT3

bulk . The enthalpyof fusionHbulk, Hbulk5T3

bulkDS, as obtained from the differ-

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FIG. 3. The difference of chemical potential for various statesthe adsorption-desorption isotherms with respect to the bulk liqsquares,p51, the bulk vapor pressure; triangles,ma , capillary con-densation on adsorption; circles,md , pore emptying on desorptionasterisks,m for f 50.3 ~equivalent to two monolayers! on adsorp-tion; stars,m for f 50.15 ~equivalent to one monolayer!.

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12 660 PRB 60P. HUBER AND K. KNORR

ence of the slopes (]m/]T52S) above and belowT3bulk is

(13006100) J mol21, in agreement with the literaturemd(T) andma(T) are both smaller thanmbulk(T) throughoutthe T range investigated. Thus the pore condensate is stwith respect to the formation of bulk droplets or crystallitoutside the pores. Note, however, that the difference withbulk is small, of the order of a few K with respect to bothTandm.

The kink of md(T) at about 78 K defines the transitiofrom the solid to the liquid state of the capillary condensmaterial, which is relevant if the system is held on the dsorption branch, as is the case for quasi-isosteric heatingwill therefore refer to this temperature as the melting teperatureTm . Calculating the point of intersection of thlow-T section with the high-T section ofmd(T), one arrivesat the following relation forTm :

T3bulk2Tm5~Dmd

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in the solid and liquid states atT5T3bulk , respectively.~Here

and in the followingDm’s are always defined as positivquantities.! The values ofDmd

liq andDmdsolid are directly ac-

cessible from Fig. 3 by linear interpolation and extrapolatirespectively, of the high-T and low-T sections ofmd(T). Onearrives at a value of (78.060.3) K for Tm . Note that theentropy of the capillary condensate, as obtained fromslope ofmd(T), is slightly lower than for the bulk, for boththe liquid and solid states. The enthalpy of melting of tpore condensate,Hm , as obtained from the difference of thslopes ofmd(T) above and belowTm , is, however, withinexperimental error identical toHbulk.

The same reasoning can be applied to the adsorpbranch, represented byma(T). We define the temperatureTSof the intersection of the low-T and high-T linear sections ofma(T): TS5(76.260.5) K. TS is related to the quantitieDma

liq and Dmasolid by an expression analogous to that giv

above forTm . TS is the temperature of the liquid-solid transition when the system is held on the adsorption branch. Tis the case for solidification in a quasi-isosteric cooling scThe slopes ofma(T) above and belowTS are within experi-mental error identical to their bulk counterparts.

Combining the expressions forTm andTS gives

Tm2TS5~Dma-dsolid2Dma-d

liq !/DS.

Dma-dsolid and Dma-d

liq are the widths of the hysteresis of thadsorption-desorption isotherms, expressed as differencchemical potentials, of the solid and liquid states of the pcondensate. An inspection of Fig. 3 and of the two isotheof Fig. 1 clearly shows thatDma-d

solid.Dma-dliq and that accord-

ingly there should be a finite thermal hysteresis of solidcation and melting. Further, below, the temperaturesTm andTS of the m-T diagram will be compared with the meltinand solidification temperatures of diffraction measuremeobtained in the course of a quasi-isosteric heating and cing scan.

We summarize: The depression ofTm andTS relative toT3

bulk is due to the fact that the difference of the chemipotential with respect to the bulk system is smaller insolid than in the liquid state. The solid and liquid pore co

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densates are both stabilized by the interaction with the gmatrix, but the liquid state profits more from the attractipotential of the substrate than the solid.TS and Tm are di-rectly related toma(T) and md(T), as shown by them-Tphase diagram and by the expressions forTS and Tm . Thethermal hysteresis between solidification and meltingquasi-isosteric runs and the adsorption-desorption hysteof the isotherms are also directly related. Both types of hteretic behavior refer to the same reservoir of metastastates, namely, those lying betweenma(T) andmd(T) in them-T plane.

As just pointed out, the reduction ofTS andTm relative toT3

bulk is due to the fact thatDm iliq.Dm i

solid,i 5a,d. Can themodel of Saam and Cole account for the changes of theDm’supon solidification? The parameters of the model arestrength of the molecule-substrate interactionams and theproductsv of the condensate-vapor interfacial energy athe molar volume of the condensate. We have calculatedcharacteristic pressurespm (5pcoex; see Fig. 1! and pc(5pu) of the model for a homogeneous pore withD575 Å. We identify the pressurespm andpc with the pres-surespd and pa of our experiment. In the liquid state atT586 K ~Fig. 1! one arrives at a reasonable agreement wthe experiment (pd50.70,pa50.78,pm50.70,pc50.79) byusing literature values25 for s and v, s511.863103 N/m,v528.3631026 m3, and by settingams to 0.5 eV Å3. Thereis, however, no way to reconcile the model and experimin the solid state. We refer, for instance, to the value ofpd at65 K, which is again 0.70. The model can reproduce tvalue by reducing the parametersv by about 28% whileleavingams unchanged. The value resulting forpc is 0.80, incontradiction to the experimental value ofpa of 0.91. Thus itis in particular the increased width of the adsorptiodesorption hysteresis of the solid state where the model fHence the model needs to be refined, preferentially bmore microscopic view of the adsorbate-adsorbate intetion. Further, below, we will propose that the reduced staity of the pore solid with respect to the pore liquid is notmuch due to a reduced surface tensions, but to a largenumber of lattice faults.

Nevertheless, one should keep in mind that the solidifipore condensate is still stable with respect to the bulk stem. Obviously, it profits from the attractive adsorbasubstrate potential. In order to do so, the pore solid mnestle against the walls and form a concave meniscus, slar to the liquid. A smooth meniscus can be realized abothe roughening temperature, only. Estimates based onlayer critical points of Ar multilayers adsorbed on graphsuggest a roughening temperature of the~111! surface of fccAr of the order of 68 K.29 Our study extends to temperaturewell below this value without showing any evidence forchange of the thermodynamic behavior, let alone an indtion of a dewetting transition. Clearly, the concept of a mniscus becomes debatable in pores with diameters of stens of Å and a microscopic or at least layerwise mowould be more appropriate.

Them-T phase diagram of Fig. 3 also includes data onchemical potential for lower fractional fillings, namely,f50.15 andf 50.30, which roughly correspond to one antwo monolayers, respectively. The relatively large scatterthese data results from the fact that the adsorption isothe

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PRB 60 12 661ADSORPTION-DESORPTION ISOTHERMS AND X-RAY . . .

are close to horizontal at these low fillings. The results shthat the entropy of these layers is larger than for bulk soand even for the bulk liquid. This is quite common for asorbates on planar substrates and leads to a delayeringsition at the temperature where the layer line intersectsmbulk(T) curve. In the present case the intersections withcapillary condensation linesma(T) and md(T) are of moreinterest.m(T)u f 50.30 approachesma(T) at about 60 K, whichmeans that below 60 K capillary condensation is induceda wall coating of less than two monolayers. The correspoing extrapolated temperature for the first monolayer is oforder of 50 K.

Do the first and second layers undergo a melting tration? The gradual change of slope ofm(T)u f 50.30 around 70K may suggest this for the second layer. There is, howeno clear indication for a melting transition of the first monlayer. This would be consistent with the idea that the mecules of the first layer are immobile because of the hetgeneous nature of the substrate which is responsible fsubstantial corrugation of the adsorption potential. Onother hand, we cannot exclude that the melting of the fimonolayer takes place below 60 K.

At the end of this section we want to discuss a pitfallexperiments on pore condensates, namely, the formatiothe bulk solid under nonequilibrium conditions. Note thaccording to Fig. 3 the differenceDma5mbulk2ma for thesolid regime is of the order of 5 K, only. Due to its lothermal conductivity and possibly a bad thermal contacthe metal walls of the cell, the temperature of the glass mtrix lags upon temperature changes. For fast cooling or hing, distillation occurs. Rapid cooling transfers the mateout of the pores onto the colder walls, subsequent heareverses the process, and a layer of bulk solid is formedthe external surface of the glass tablet. This is in fact howprepared the bulk layer, the diffraction pattern of whichshown in the top frame of Fig. 4. Practically any desirpartition of capillary condensate and bulk solid can be pduced with this technique.

The effects are more subtle in case the cooling rateslow enough that the temperatures of the glass and celequilibrated. Nevertheless, there is a net transfer of partifrom the vapor to the condensate on cooling. The relaxatime of this process can be estimated from the equilibratime of isothermal adsorption~Fig. 2! and Dma . For 65 Kone arrives at a maximum tolerable cooling rate of the orof 5 K per 30 h. Faster cooling inevitably leads to a noneqlibrium state in which the residual vapor in the dead volucondenses~at p51! into parasitic bulk crystallites outsidthe glass matrix. In principle, quasi-isosteric heating acooling are a legitimate way of changing the thermodynamstate of the pore condensate, but it requires a careful conof the vapor pressure and of the amount of material reming in the pores. The formation of bulk crystallites atp51may simply be a nonequilibrium effect rather than a dewting transition.

We think that the sharp fcc diffraction peaks observedBrown et al.24 below 40 K are due to such bulk crystalliteOf course, we do not know the details of this experiment,obviously the vapor pressure has not been recorded. Furmore, we cannot imagine that the limited beam time asynchrotron source allows cooling rates as slow as those

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FIG. 4. Diffraction patterns of a bulk Ar film and Ar pore condensates in porous glasses with various pore diameters. Foxerogel Gelsil 75 Å, pattern~2! is obtained from capillary condensation in the solid state, pattern~1! from capillary condensation inthe liquid state and subsequent solidification by cooling. All pterns are for practically completely filled pores. The temperatur60 K, except for the 50 K pattern of the condensate in the 25pores. Also shown is a calculated pattern considering finite sizestacking faults.

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12 662 PRB 60P. HUBER AND K. KNORR

B. Structure

Diffraction patterns have been recorded in parallel wthe adsorption isotherms, that is, at constantT as function off and occasionally also as function ofT in quasi-isostericruns. As far as the position of the first maximum of the liqustructure factor is concerned, there are no detectable diences to liquid bulk Ar. We concentrate on the solidifipore condensate and start at a temperature well belowsolidification. Figure 4 shows diffraction patterns of the somaterial in several types of porous glass for almost compfilling. In all diffraction patterns the contribution of the substrate has been subtracted, properly accounting forf-dependent absorption of the x rays. The figure also inclua pattern on an evaporated bulk Ar film for comparison. Tbulk film shows a minority hcp component, with an idealc/aratio and a next-neighbor distance identical to that of themajority component.

Pore condensed solid Ar shows the reflections expefor a fcc solid. The width of the peaks increases with dcreasing pore diameter. For the matrix with the smallest pdiameter, namely, 25 Å, the~200! and ~222! reflections arepractically absent. The patterns of the pore solid in the 7xerogel and in the standard Vycor glass suggest that thtwo matrices are very similar, apart from the higher porosof the xerogel. This leads to a superior quality of the patteobtained with the xerogel. The patterns depend on the wapreparation of the pore condensate. For pattern~2! the con-densate has been adsorbed and the capillary condensedsolid state, whereas for pattern~1! the material has beecondensed in the liquid state and solidified by cooling. Oviously, the former method warrants a better sample quawith sharper diffraction peaks.

Figure 5 shows diffraction patterns for 0.17, f ,0.93 of

FIG. 5. The evolution of the diffraction pattern in the solid staat 65 K with fractional filling.

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the adsorption branch of the 65 K isotherm. At lowf, f,0.3, that is, for a coating of the walls with less than twmonolayers, the diffraction patterns show two bromaxima, the first one centered at 2u527° and the secondone extending from about 45° to 58°. Altogether, the pattesuggest that these thin layers are highly disordered, amphous. Further insight comes from a comparison to monoers on planar homogeneous substrates such as graphite.Ar monolayers form a close-packed triangular lattice, simito the ~111! plane of the bulk fcc crystal.27 We have calcu-lated the powder diffraction patterns of such a twdimensional planar array26 and arrive at a satisfactory fit othe f 50.17 pattern by choosing a coherence length oforder of 16 Å. The first maximum is identified as the~10!peak, the second as strongly broadened, overlapping~11! and~20! reflections. The next-neighbor distancednn, as calcu-lated from the~10! position, is close to the value obtainefrom the positions of the fcc Bragg peaks at highf. For thelowest filling fraction,dnn is slightly enlarged~Fig. 6!. Thiseffect is presumably due to the heterogeneous charactethe substrate, which does not allow a perfectly close trianlar packing.

The diffraction patterns change smoothly from the amphous type at lowf to the quasi-fcc type at highf. This isconsistent with the smooth, rather than stepped, shape oadsorption isotherm. Nevertheless, Figs. 5 and 6 shows sconspicuous changes of the patterns with increasingf. We tryto localize these changes by inspecting the fcc reflecti~111!, ~200!, and~220!. These reflections respond quite diferently to the number and to the stacking sequence of clopacked triangular planes. Examples have been calculatecontext with diffraction measurements on Ar multilayers a

FIG. 6. The position and width of the first maximum of thdiffraction pattern, which develops into the~111! peak, as functionof f at 65 K. The upper dashed line marks the~111! position of thebulk solid, the lower dashed line the~10! position of a triangularlattice having the same next-neighbor distance.

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PRB 60 12 663ADSORPTION-DESORPTION ISOTHERMS AND X-RAY . . .

sorbed on a homogeneous graphite substrate.27 The ~111!reflection develops out of the triangular~10! reflection, aprocess which involves a shift of the peak position. The fiposition is reached atf 50.45, which corresponds to abothree monolayers~Fig. 6!. The reflection is common to alclose-packed stacking sequences, such as fccABC or hcpAB. The width of the~111! reflection decreases with an increasing number of stacked planes. The drop of the waroundf 50.60~Fig. 6! is therefore due to an increase of thnumber of stacked planes. The~200! reflection is specific tofcc ABC stacking and does not have a counterpart in tdimensions. The~200! reflection becomes visible atf50.47. The~220! reflection is the successor of the~11! re-flection of the triangular lattice. The~220! reflection can beidentified as a sharp feature atf 50.40 and above. We suggest that the changes of the diffraction pattern are due toonset of capillary condensation. The most reliable estimfor the critical valuef C is presumably 0.40. This value compares favorably with the adsorption branch of the 65 K itherm of Fig. 1. The somewhat larger value extracted frthe ~111! width results from the fact that here the broad peof the adsorbate and the sharper peak of the capillary cdensate superimpose, the intensity of the latter increasinproportion to f 2 f C . As suggested by thef dependence othe ~111! position, the principle of close packing appearsbe obeyed already in the adsorbate state with a film thicknas low as three monolayers. The somewhat delayed appance of the~200! reflection may arise from the fact that thcapillary condensation starts in the narrow sections ofpores. Here deviations from the fcc stacking sequenceABCare particularly likely.

In the following we concentrate on the quasi-fcc-tycomponent of the patterns representing the capillary cdensed pore solid, which is obtained by subtracting thesorbate component, represented by the pattern atf 50.40,from the f 50.93 pattern. Analogous to our interpretationthe diffraction pattern of pore condensed hcp N2, we con-sider two line broadening effects: the finite sized of thecrystallites and stacking faults.28 a is the probability that anAB stack of ~111! planes is followed by a plane in theAposition rather than in theC position. Figure 4 includes asimulated pattern withd51000 Å and a50.055. Thischoice of parameters describes the difference pattern qwell, in spite of the simplicity of the model. The value ofdexceeds the pore diameter by far; this points to a rather lacoherence length along the pore. One is inclined to identidwith average chord length of a curved pore. There isadditional feature in the patterns at higherf ~Figs. 4 and 5!,namely, a shoulder on the low 2u wing of the ~111! peakwhich is not accounted for by the model, but which is neertheless related to stacking faults. The position of the shder coincides with the~100! peak of the metastable hcphase of Ar. For a random stacking of the net planes,diffracted intensity appears on a triangular array of Brarods perpendicular to the~111! plane. The minimum 2uvalue of the ~10! rod coincides with the Bragg angle o(100)hcp. Thus the shoulder is another piece of evidencea large density of stacking faults.

It appears that the solid condensate adapts to the restrgeometry in the pores by the formation of stacking faults apresumably other lattice faults which are not considered

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our model. The energy of these lattice defects causes ancrease of the chemical potential of the solid pore condensthereby reducing the difference of the chemical potenwith respect to the bulk solid~which in turn leads to a low-ering of the melting point!. The same idea has been metioned in connection with the pressure-induced solidificatof He in fine powders.30

Figure 7 shows the integrated intensity of the~220! peakfor a quasi-isosteric cooling and heating scan. The appance of the peak on cooling signals crystallization, the dappearance on heating melting. One notes a thermal hyesis of the order of 1–2 K, and solid-liquid coexistenregions both for cooling and heating in which the peaktensity varies withT in a roughly linear fashion. Both observations are in qualitative agreement with heat capacityultrasonic measurements on Ar in Vycor.1 We define thesolidification and melting temperatures of the quasi-isostescans,TS

qi andTmqi , as the high-T end points or alternatively

as the midpoints of the coexistence regions. The valuestained for the former case, 76 and 78 K, agree quite well wthe temperaturesTS5(76.260.5) K and Tm5(7860.3) Kextracted from them-T phase diagram. The values for thlatter case are 75.5 and 76.5 K. Here the agreement issatisfactory.

For temperaturesT, TS,T,Tm , the m-T the phase dia-gram suggests that the pore condensate of the adsorbranch with the chemical potentialma is still liquid, but thatthe condensate of the desorption branch with the chempotentialmd is already solid. Thus it should be possiblecontrol the structural state of the pore condensate by adstion and desorption. Pertinent diffraction results exist forK ~Fig. 8!. Note that this temperature is just outside tthermal solidification-melting hysteresis when referringthe temperaturesTS andTm of them-T diagram, but inside interms ofTS

qi andTmqi of Fig. 7. Indeed, the diffraction pattern

at 76 K ~Fig. 8! show that upon adsorption the diffractiopatterns are of the amorphous or liquid type. The fcc refltions appear only close to complete filling, abovef 50.9.~Ashas been discussed in Sec. III A, the critical valuef C of

FIG. 7. TheT dependence of the intensity of the~220! peak inthe melting region on quasi-isosteric cooling and heating fof50.8. The melting and solidification temperatures of them-T dia-gram,Tm andTS , are indicated for comparison.

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12 664 PRB 60P. HUBER AND K. KNORR

capillary condensation increases withT.! On desorption,however, remainders of the fcc reflections are still presenf 50.9 and extrapolate to zero at aboutf 50.8.

One wonders whether the capillary condensate formstop of the adsorbed layers or whether the capillary condsation leads to a reorganization of the material on the pwalls. The patterns of the adsorption branch of Fig. 8 shthat the growth of the fcc peaks betweenf 50.9 andf 51 isconnected with a decrease of the amorphous componentthe 2u range left of the~111! peak. The analysis of theschanges suggests that half of the adsorbate, two out ofmonolayers, takes part in the crystallization process ofcapillary condensate. It is natural to think of the third afourth layers, which means that the first and second layremain in the original state. The recrystallization of thigher layers of the film state can be only observed at higtemperatures since at lower temperatures, of the order oK, the capillary condensation starts already when the wcoating has reached two monolayers. As far as the hypothcal melting of the first and second monolayers is concernthe diffraction pattern cannot really decide whether theseers are liquids or amorphous solids. Altogether, the diffrtion results are in agreement with the conclusions drafrom the thermodynamic data and given further insight inthe structural organization of the pore condensate.

Brown et al.24 interpret the diffraction patterns of porcondensed Ar at 55 K and of Kr at 94 K in Vycor (f '0.9) interms of a disordered hexagonal close-packed structure

FIG. 8. Diffraction patterns atT576 K, i.e., at a temperatureintermediate to melting and solidification. Note that forf 50.9, thepeaks of quasi-fcc material are visible on desorption, but notadsorption.

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random stacking of net planes. However, the patternsBrown et al. clearly show a residual~200! intensity havingthe form of a shoulder. Analyzing these patterns, we arriveestimates of stacking fault probabilitya of around 0.2. Thisis larger than what we observe, but is by no means consiswith random stacking (a51). Considering the limitingcases of ideal fcc and random stacking, only, the pore cdensates of Brownet al. are closer to the former than to thlatter case, in qualitative agreement with the present res

As far as the evolution of the diffraction pattern with dcreasing temperature is concerned, we have often produdiffraction patterns similar to that shown in Ref. 24 for ArVycor at 37 K, but in our experiments the sharp componof the fcc peaks could be always removed by sufficienlong waiting or by pumping off small fractions of the totaamount of gas in the cell. The patterns obtained after streatments have no peculiarT dependence except shifts othe peaks due to thermal expansion and a slight increasthe intensities toward lowT. The peak widths which aresensitive to stacking faults and size effects are indepenof T down to 20 K.

IV. SUMMARY

The concept of adsorption and capillary condensationwell established by numerous experimental studies onder Waals liquids in porous SiO2 glasses. Our thermodynamic and structural study shows that the solid Ar condsate is also stable in the pores and undergoes a first-otransition from the adsorbate state into the capillary cdensed state very much like the liquid. An analogous behior can be expected for other van der Waals systems incing the other rare gases, and also for H2, CH4, N2, or CCl4.Even though we have no indications for dewetting or aother phase transition in pore condensed Ar, dewetting moccur in other systems at lower temperatures, in particulaconnection with subsequent structural phase transitionspractice, it may be difficult to produce convincing evidenfor a dewetting transition and to discriminate such a trantion from the formation of the bulk solid in a nonequilibriumstate because of the problem of slow kinetics at lowT.

The adsorbate-substrate interaction calls for a layestructure of the pore material, and diffractometry has pduced evidence that this is so. The first monolayer adsorcrystallizes in a highly disordered triangular lattice. Furthlayers stack on the ones underneath, obeying the principlspherical close packing. On capillary condensation, the ctals grow along the pore axis to a coherent linear extensof the order of several hundred Å. Altogether these resare not consistent with the naive view of the solidified mterial in the pores as fcc nanocrystals with an overall sphcal shape, the size being given by the diameter of the po

The depression of the melting temperature is due tofact that the reduction of the chemical potential of the pocondensate with respect to the bulk is larger in the liqthan in the solid state. We propose to relate the redustability of the solid capillary condensate to lattice defectsparticular stacking faults, which the solid has to form in oder to adapt to the pore geometry. The solidification-melthysteresis is due to the fact that the width of the adsorptidesorption hysteresis is larger in the solid than in the liq

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PRB 60 12 665ADSORPTION-DESORPTION ISOTHERMS AND X-RAY . . .

state. Presumably, the general features of the chempotential–temperature phase diagram established formelting transition of Ar also apply for other phase transitiofor which the pore condensate is in equilibrium with its vpor.

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ACKNOWLEDGMENTS

This work has been supported by the Sonderforschunbereich 277 ‘‘Grenzfla¨chenbestimmte Materialien.’’ Wethank D. Wallacher for helpful discussions.

, J.

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