Physical Adsorption on Patchwise Heterogeneous Surfaces
4 59
Physical Adsorption on Patchwise Heterogeneous Surfaces. 1. Heterogeneity, Two-Dimensional Phase Transitions, and Spreading Pressure of the Krypton-Graphitized Carbon Black System near 100 K Frederick A. Putnam and Tomlinson Fort, Jr.’ Deparfment of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 152 13
(Received July 18, 1974)
Adsorption isotherms, adsorption heats, and spreading pressures are presented for the krypton-graphitized carbon black system from 0.002 to 4.5 statistical adsorbed layers, in the 95-105 K temperature range. The existence of a second-order fluid-solid phase transition in the submonolayer film is confirmed, and evidence is presented for an in-registry to out-of-registry transition. The sample surface is shown to be composed of 0.5% high-energy sites and 99.5% lower energy sites with adsorption energy cgslk = 1466 f 6 K. From spreading pressure calculations, the dispersion force contribution to the surface free energy of (0001) graphite is estimated to be yd = 151 mN/m.
Introduction This is the first in a series of papers whose main subject is the problem of determining the adsorption site energy distribution and surface energy of heterogeneous solid surfaces from physical adsorption data. The investigation is restricted to a certain type of solid surface, herein called “patchwise heterogeneous.” This means a surface which has only a few types of adsorption sites and a surface in which these sites are present in large enough “patches” so that adsorption on each patch can be treated independently. The approach t o the problem of determining the adsorption site energy distribution is to model accurately the submonolayer adsorption function with a statistical mechanical model which contains as parameters the energies of the adsorption sites and the fraction of the surface which each type of site occupies. The parameters are determined by a fit of the model to the experimental data. This approach has been used by Pierottil and Ross.2 If the adsorption function on each patch is known, it is a simple matter to obtain the adsorption function for the entire ~ u r f a c e To .~ find a correct model for adsorption on a homogeneous surface is, thus, a subproblem of this study. Many different descriptions of submonolayer adsorption have appeared,4 and there is considerable theoretical work being done currentl~.~-~ In this paper, a “homogeneous” surface is one which has only one type of adsorption site (for example, any perfect low-index crystal plane). A “uniform” surface is one which presents negligibly small barriers to translation across the surface. No real surface is perfectly uniform, though some homogeneous surfaces are effectively uniform, a t high enough temperatures. “Heterogeneity” is used to denote the deviation of an adsorbent surface from being perfectly homogeneous (not uniform). The purpose of this first paper is to present and discuss experimental results for the adsorption of krypton on graphitized carbon black (gcb). Adsorption isotherms were obtained in a high accuracy ultraclean volumetric systern.lOJ1 The data were taken in the 95-105 K temperature range, from 0.002 to 4.5 statistical adsorbed layers. An inert gas was chosen as an adsorbate because of its lack of rotational degrees of freedom, which complicate the theory.4 Graphi-
tized carbon black was used because of its low heterogeneity, high uniforrnity,l2 inertness, lack of porosity, and its similarity to another patchwise heterogeneous material that has been investigated in the authors’ laboratory, graphitized carbon fibers.13 The temperature range of the experiments is above the two-dimensional critical point, which simplifies the model, yet low enough so that the Sshaped submonolayer isotherms have a well-defined step. Though many previous investigations of the krypton-gcb system have been made, this temperature range has not been investigated previously. Also, in the past, it has been difficult to test adsorption theories because of the often noted ability of a variety of different adsorption models to fit experimental data equally well, within experimental error.3 For this reason, extreme care has been taken in this work to obtain the highest accuracy data possible. One object of this work which is not directly related to the goals stated above is to study the two-dimensional fluid-solid phase transition. This phenomenon was first observed by Thomy14 and investigated further by Thomy and Duval,15 for rare gases and methane on exfoliated graphite and gcb. However, in a recent study, Newsome16J7 did not observe the transition for the krypton-gcb system. Consequently, it has been doubted that the effect observed by Thomy and Duval is real. In this study, then, considerable attention was paid to the phase transition portion of the isotherm. 11. Experimental Section A. Materials. Sterling FT graphitized carbon black, code no. FT-D5, was obtained from the Cabot Corp., Boston, Mass. This sample had been prepared by heating to 27003700’ in the absence of air. The BET N2 area of this sample is listed by the manufacturer as 11.5 m2/g. Gases were Research grade, from Air Products and Chemicals Co., Emmaus, Pa., of purity 99.99% (Kr) and 99.9995% (He). B. Apparatus. The volumetric adsorption apparatus, constructed by Drza1,ll is of a type described by Pierotti and Thomas.* I t is constructed from Varian minicomponents and valves (Varian Associates, Palo Alto, Calif.) with copper gaskets. The system was pumped by mechanical and mercury diffusion pumps. The pumps were isolated by double liquid nitrogen traps which were kept constantly The Journal of Physical Chemisfry, Vol. 79, No. 5, 1975
Frederick A. Putnam and Tomlinson Fort
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full. Capacitance manometers (MKS Instruments, Burlington, Mass.)18 were used for pressure measurements. Thus, the system is grease and mercury free. A cryostat similar to that of Morisson and Young,lg stable to f0.005 K, controls the temperature. A platinum resistance thermometerMueller bridge system was used for temperature measurement. The remainder of the system is air thermostated to f0.05 K. The apparatus is described in greater detail elsewhere.lOJ1 Before the adsorption experiments, the absolute accuracy of the pressure- and temperature-measuring systems were checked against each other in an experiment where the vapor pressure of krypton was measured in an empty sample tube. The measured temperature was compared to the saturation temperature calculated from the pressure by the equation of Freeman and Halsey.zOAt a measured temperature of 100.37 K, the calculated temperature was 100.25 K. This difference was found to depend on the helium pressure in the cryostat and became less as helium pressure increased. It was attributed to a lack of thermal equilibrium between the sample tube and the thermometer. When the adsorption runs were made, the sample tube contained adsorbent, so that better equilibrium was expected. Also, helium pressure was kept constant. Therefore, the above error is judged to be an upper bound. The effect of this error on the accuracy of the thermodynamic data will be discussed below. C. Technique. The sample was heated to 450’ for 12 hr in uucuo for the initial outgassing. Between runs, a 12-hr 250° sample tube bakeout was used.21 During this heat treatment, the remainder of the system, up to the liquid nitrogen traps, was heated to 150’. A steady pressure below 5 X Torr was achieved. During a run, the cryostat can be used to change the sample temperature, and more than one isotherm can be obtained in a single run. Due to a cancelation of accumulated error, this procedure yields more accurate thermodynamic quantities than are obtained from separate isotherms. This procedure is detailed elsewhere.1° The ability to change the sample temperature was also used to check for hysteresis, as follows. Suppose the system is Gat equilibrium, with a given pressure and amount adsorbed and the valve between the dosing chamber and sample tube is open. The sample temperature is increased (decreased), and desorption (adsorption) takes place. A new data point may be taken at the new equilibrium temperature and pressure. Then the sample temperature is decreased (increased) back to its original value, whereupon adsorption (desorption) takes place. In the absence of hysteresis, the pressure returns to its initial value. In this way, any experimental point may be approached via adsorption or desorption. Previous ~ o r k e r s ~have ~ J found ~ that in rare gas adsorption at temperatures below the triple point and at low pressures the approach to equilibrium can be very slow. There is a danger that when a new dose is added, most of it will condense on the top part of the sample and redistribute slowly, or even form metastable states. In the present experiments, to eliminate these effects, the dose was always leaked into the sample tube slowly. When this procedure was followed, equilibrium was attained in about 1 hr, even at the lowest pressures. The calculation scheme takes into account the change in volume of the system which occurs when the bellows valve between the dosing chamber and the sample tube is closed. The Journalof Physical Chemistry, Vol. 79, No. 5 , 1975
Miller’s equationz4 was used to calculate the thermal transpiration pressure difference, with the mean free path calculated from the viscosity data of Clarke and Smith,26according to the equation of McConville, et al. z 6 This equation was found to fit Rosenberg’s krypton data27,28very well, better than the Liang e q ~ a t i o n . ~Gas-phase g nonideality was accounted for by using a virial equation of state with the second virial coefficients of Beattie, et al.,30 and Fender and H a l ~ e y . ~The l second virial coefficients of Fender and Halsey were extrapolated from the temperature range of their experiments (105-140 K) to lower temperatures using their 6-12 potential parameters u = 3.591 A and elk = 182.9 K.32 A complete error analysis of the experimental data showed that accumulation of error throughout the isotherm in these runs was not a problem. While the absolute value of the expected accumulated error increased with each dose, the per cent expected accumulated error stayed roughly constant, and was always less than 0.2%. In a previous paper,1° these errors have been discussed in detail. 111. Results
A. Adsorption Isotherms. 1. Multilayer Region. Figure 1 shows the multilayer region of the isotherms. The saturation pressure PO was calculated from the equation given by Freeman and H a l ~ e y . ~The 3 surface excess amount adsorbed, nu,is given in mol of krypton/g of gcb. In this lowpressure work, nu is equal within the precision of the measurements to n *, the amount of adsorbed substance. In these experiments, after 4.5 statistical layers were reached, the desorption points (open circles) were taken. No evidence of hysteresis was seen. Hysteresis was also checked for by the temperature-change method previously described (flagged points) and by conventional desorption at the end of the run (solid circles). The pressure always returned to its initial value, to within the pressure measurement error of 0.08% (standard deviation), indicating absence of hysteresis. The isotherm’s second step appears at PIP0 = 0.43, and the third step, a t PIP0 = 0.78. 2. Monolayer Region. Figure 2 shows the isotherms for coverages up to 1 monolayer. These exhibit typical S shapes characteristic of adsorption on uniform surfaces. Each of the three isotherms exhibits an abrupt change in slope near 1 monolayer. This transition can be seen better on an expanded-scale plot, Figure 3. In this figure P T , the transition pressure, is the pressure at which the discontinuity in slope occurs. Plotting ns us. P I P , clearly shows that, although the isotherms exhibit a discontinuity in slope, the step is not completely vertical. A t these pressures it is quite difficult to carry out conventional desorption measurements with a volumetric apparatus. Thus, the temperaturechange method was used to check for hysteresis (flagged points). No hysteresis was found, even in the region of the phase transition. 3. Low-Coverage Region. Figure 4 shows the low-coverage region. Although these isotherms appear at first glance to be straight lines, they are slightly curved concave to the P axis even in the region 0.5 < n u < 4 pmol/g. If the best straight lines are drawn through the data in this region, however, they all intersect at about n u = 0.3 pmol/g. The actual experimental data are listed in the supplementary material section of this paper. B. Adsorption Thermodynamics. 1. Isosteric Heat. The isosteric heats, Figure 5, were calculated by plotting the five isotherms (two of which were replicates, see Figure 3)
461
Physical Adsorption on Patchwise Heterogeneous Surfaces
s Pa
d w P L
0
v)
P
4 I-
2
s
2 torr
PRESSURE,
Figure 1. Krypton-Sterling FT adsorption isotherms, multilayer region: open circles, adsorption points; flagged circles, hysteresis checks; solid circles, conventional desorption points taken at end of run; Dashed lines indicate Po,the saturation vapor pressure at the various temperatures.
2a 100
9
80 60
P
c
40
'li
I
00
I
I
0.4 0.8 PRESSURE, Torr
I.2
Figure 2. Krypton-Sterling FT adsorption isotherms, monolayer region: flagged points, hysteresis checks; circles, squares and triangles represent different experimental runs.
I 1
""; 110
.\" O
Figure 4. Krypton-Sterling FT adsorption isotherms, low-coverage region. Symbols are the same as Figure 2.
on large-scale graph paper, and drawing smooth curves through the data points. These curves were interpolated at constant n u )and linear regression was used to fit the data to the Clausius-Clapyron equation in the usual form34
3 120 m
PRESSURE, Torr
100.
I
I
I
I
The isotherms were spaced at equal increments of 1/T to effect maximum accuracy. The estimated standard errors of the heats given by regression analysis were less than 63 J/mol for n < 115 Mmol/g. The small error noted points up the precision of volumetric adsorption measurements obtainable with the apparatus and measurement technique previously described.1° These errors are mostly due to the interpolation process. In the region near the monolayer point, a significant temperature dependence was detected for the isosteric heat, as is expected when a phase change is occurring. To show this fact, the isosteric heat calculated using the 94.72 and 99.34 K isotherms only is shown as a dotted line in Figure 5. In the 0 < n * < 0.3 Mmol/g region of the isotherm it was not possible to obtain accurate isosteric heats because the corresponding pressures were below the range of accurate measurement with the capacitance manometer. Consequently, a run was performed with an ionization gauge included in the dosing chamber and operated at low emission current to reduce the pumping effect to a minimum. A value of q st at n u = 0.27 Mmol/g of 18,800 f 800 J/mol was obtained. This is the value which lies almost on the n = 0 axis in Figure 5. The systematic error in temperature measurement mentioned earlier may have caused all the temperature readings to be high by as much as 0.12 K. This would cause a maximum error of -0.25% in the isosteric heat, which is less than the errors quoted above. 2. Differential Entropy. The differential entropy of adsorption,34Figure 6, was calculated from
flK F
110 P/ P,
112
114
Figure 3. Krypton-Sterling FT adsorption isotherms, phase transition region. PT is the pressure at which the slope is discontinuous. Symbols are the same as Figure 2.
where AHs is the heat of sublimation, so that the reference state is the bulk solid-vapor equilibrium. Error in this quantity follows directly from the heat errors. C. Spreading Pressure. The spreading pressure was calculated from the adsorption isotherm by the Gibbs adsorption equation in the form The Journal of Physical Chemistry, Vol. 79, No.5, 1975
462
Frederick A. Putnam and Tomlinson Fort
....
I
I
I
I
G 20
60
?
a i
I
0
”
100
I
I
I
200
300
400
I
500 pnol/g
AMOUNT ADSORBED,
I
Figure 5. lsosteric heat of adsorption: solid line, obtained from isotherms at all three temperatures; dashed line, obtained from isotherms at 94.72 and 99.34 K only. Points “A,” “B,” and ‘ID” are discussed in section 1V.C.
6o i Y
IO
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a 0 n 70
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0
En 60
20
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40
0
2
LT
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20
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94.72 K
IO
.30
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OO
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.I 0 .20 AREA PER MOLECULE, nm2
Figure 7. Spreading pressures obtained from adsorption isotherms by eq 3. Different symbols represent separate experimental isotherms, which were integrated separately.
0
100
200
I
1
I
300
400
500
AMOUNT ADSORBED,
-
prnol/g
Flgure 6. Differential entropy of absorption: solid line, obtained from isotherms at all three temperatures; dashed line, obtained from isotherms at 94.72 and 99.34 K only. Points “A,” “B,” and “C” are discussed in section IV.B.2.
(3)
-
a 2.0 0
I-
where f is the gas fugacity and A is the surface area. To perform the integration, third-order polynomials were fit through the data, and integrated. For this calculation, the slope of the isotherm a t the phase transition point (Figures 2 and 3) was allowed to be discontinuous.35 A surface area of 11.4 m2/g (see section 1V.C) was used. Results of these calculations are shown in Figure 7, where each data point corresponds to an experimental point on the n u us. p isotherm.36 The two-dimensional compressibilities, Figure 8, were calculated from the T data, so that in this plot also each data point corresponds to a point on the experimental isotherm.
IV. Discussion Since the experiments were performed below the bulk triple point of krypton, the mobility of adsorbate molecules was relatively low. Thus, the danger of reaching nonequilibrium states was always present. Consequently, hysteresis checks were performed in all adsorption regions. As no hysteresis was detected, it is believed that the data represent states of true thermodynamic equilibrium. In the following discussion, the adsorbent is assumed to be inert and unperturbed by krypton adsorption. A . Adsorption Isotherms. 1. Multilayer Region. Stepwise-type multilayer isotherms have been obtained by many previous i n v e s t i g a t ~ r s . l Thomy ~ ~ ~ ~ -and ~ ~ Duva140,41 The Journal of Physical Chemistry, Val. 79, No. 5, 1975
o 1.5
2
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1.0
i
m
:2.0 W
a P
1.5
u I .o 0.5 0
5
io
SPREADING PRESSURE,
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15 mN/m
Figure 8. Two-dimensional compressibility plots obtained from adsorption isotherms: solid line, spreading pressures obtained via eq 3 from raw experimental data; dashed line, spreading pressures obtained after subtracting out adsorption on the high-energy sites. The phase transition points are marked “A,”
obtained five steps a t 77 K on exfoliated graphite and found that the step positions were well described by the Frenkel-Halsey-Hill (FHH) equation in the form given by Singleton and H a l ~ e y ~ ~ In (P,,/P,) = Ei/(n3RT) + W(l - g ) / R T (4) where P , is the pressure at the formation of the nth layer, El is a characteristic energy, and W and g are factors
Physical Adsorption on Patchwise Heterogeneous Surfaces which account for dimensional incompatibility between the adsorbate and ‘adsorbent lattices. The positions of their second and third steps were at PIP0 = 0.38 and 0.77, respectively, in contrast to 0.43 and 0.78 in this study. Equation 4 implies that the product RT In (PJPO) will be a constant independent of temperature. Using the above results, this product is -146 at 77 K and -159 a t 94.72 K for the second layer and -39 at 77 K and -47 a t 94.72 K for the third layer. Thus, the FHH equation does not describe the temperature dependence of the step positions well. Larher43 modified the FHH equation, adding an entropy term as in RT In (P,/P,) = (U” - Uo) + T ( S o - S”) (5) Here “ n” refers to the n t h layer, and “0” refers to the saturated vapor state. This equation accounts for the temperature dependence of RT In (PJPO). Fitting eq 5 to the above data, one obtains UO - U2 = 360 f 83 J mol-l and SO - S2 = 3.3 f 0.8 J mole1 K-l. These figures are quite approximate; they are included here mainly to show that Larher’s entropy term is sizable for this system. There are differing opinions in the adsorption literature as to whether the isotherms for this system either (1)approach the PO axis asymptotically or (2) intersect the Po axis at an angle, leading to the solid equivalent of a contact angle and a u t o p h o b i ~ i t y . ~The ~ , ~high-coverage ~*~~ region was investigated in this study, but the results are not presented here because the isotherm in this region is influenced by pendular ring c0ndensation.4~ 2. Monolayer Region. The striking discontinuity in the slope of the isotherm at n u = 100, 105, and 107 rmol/g a t 94.72, 99.34, and 104.49 K, respectively, was first observed for this system by Thomy and Duval,15 who worked with both gcb and exfoliated graphite. Recently, Larher45 obtained krypton isotherms on natural graphite which also show the transition. Figure 9 shows a comparison of Thomy and Duval’s data for the krypton-exfoliated graphite system with this work, where the n u scales were adjusted for the best fit. The excellent agreement, combined with the absence of any hysteresis, confirms that the “substep” phenomenon observed by Thomy and Duval is a true characteristic of rare gas adsorption on the graphite (0001) surface and not an artifact of exfoliated graphite. The agreement was obtained in spite of the fact that the adsorption apparatus used in this work is quite different from that used by Thomy and D ~ v a lTheir . ~ ~ apparatus is constructed of glass, has mercury cutoff valves, and has a cold trap between the dosing system and the sample. Also, McLeod gauges are used for pressure measurement, and vapor pressure thermometers are used for temperature measurement in their apparatus. Since there is no vertical discontinuity in the isotherm itself, but only a discontinuity in the slope, the transition is a second-order phase change according to conventional clas~ i f i c a t i o nIn . ~ Thomy ~ and Duval’s more extensive study of this phenomenon,15 the transition is shown as first order below about 90-95 K and second order a t higher temperatures. Thus, it appears that a tricritical point47*48for this phase change is located in the 90-95 K temperature range. The fluid-solid transition of two-dimensional films has been extensively investigated theoretically. The transition has been predicted by mobile and localized equations of ~ t a t e .Molecular ~ ~ , ~ ~dynamics studies on a two-dimensional gas of hard spheresb1 have provided convincing evidence of a first-order fluid-solid transition. More recently, Monte Carlo studies52 on a two-dimensional gas with Lennard-
4 63
0 PRESSURE, Torr
Flgure Q. Comparison of the krypton-graphite adsorption isotherms obtained in this study with those of Thomy and Duval: solid lines and left-hand scale, this work; dashed lines, and right-hand scale, Thomy and Duval. The amount adsorbed scales were adjusted for the best fit. Temperatures are as follows: “A,” 94.72 K; “B,” 96.3 K; “C,” 99.34K; “D,” 101.1K; “E,” 103.0K; “F,” 104.49K; “G,”105.1 K.
Jones interactions have also indicated the presence of a first-order transition. All of these models, however, ignore the effect of the potential energy wells on the graphite lattice, which tend to localize the krypton atoms on specific adsorption sites. The tendency to localize will be significant at high density since the lateral mobility of the adsorbate atoms is restricted due to adsorbate-adsorbate interactions. A model of the localized (or “in-registry”) film is the two-dimensional lattice gas. Due to the geometry of the krypton and graphite lattices, Figure loa, the occupancy of nearest-neighbor sites is forbidden. C0mputer5~and other theoretical on this model has found a continuous second-order transition to a state with long-range order. The order of the experimentally observed transition, then, suggests that the krypton monolayer film corresponds to this last model and is in registry with the graphite lattice in the transition region. The above theoretical lattice gas treatments do not take the attractive forces between adsorbate molecules into account. These forces would be capable of transforming the fluid-solid transition from second to first order a t lower temperatures, producing a tricritical p0int.5~ The “substep” and its phase transition cannot be attributed to second layer formation on the high-energy adsorption sites. The monolayer capacity of the high-energy sites (see section IV.A.3) is about 0.7 pmol/g. If second-layer adsorption on the high-energy sites occurred stepwise, a step of only 0.7 rmollg would result. However, the height of the “substep” a t 95 K (referring to Figure 3) is approximately 10 Fmol/g. The submonolayer isotherm below the substep is typical of a two-dimensional film with lateral interactions. Various two-dimensional equations of state may be fit to these data. 3. Low-Coverage Region: Heterogeneity. The initial gas adsorbed a t very low pressures, seen as a small “step” in the low-coverage isotherm, Figure 4, can be attributed to the presence of a small number of high-energy sites or “strong sites.” This effect was observed by Graham56 for nitrogen adsorption at 77 K on P33 (2700) and Graphon graphitized carbon blacks. Graham assumed that the linear portion of the isotherm above the step indicated complete saturation of the strong sites and extrapolated this linear portion to the n axis to obtain the number of strong sites. In this way, he found 0.1% and 1.25% strong sites on P33 (2700) and Graphon, respectively. The Journal of Physical Chemistry, Vol. 79, No. 5 , 1975
Frederick A. Putnam and Tomlinson Fort
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with decreasing nu,as has been found in numerous studies on nearly homogeneous graphites. Though the high-energy sites occupy only 0.5%of the surface, they affect qst even at a coverage of 3%. It is only in the limit of zero temperature that the isosteric heat curve for a patchwise heterogeneous surface has completely sharp stepsn3At higher temperatures, there is a smearing out which depends in the patchwise heterogeneous case on the value of kT relative to the difference in the adsorption energies on the patches. A least-squares fit to the approximately linear qst us. nu curve in the range 4 < nu < 20 pmol/g yields the limiting low-coverage isosteric heat for the (0001) part of the surface, qst = 12,608 f 50 J/mol. This may be used to obtain an approximation to cgs, the depth of the gas-solid potential energy well, using the relation
GRAPHITE LATTICE / KRYPTON
a.
b.
Figure 10. Monolayer structures for the krypton-graphite (0001)system: (a) in-registry structure, with krypton atoms localized at alternate lattice sites; (b) out-of-registry structure, where the krypton and
graphite lattices are unrelated and the krypton lattice is closepacked as in the fcc (111) plane.
If this procedure is applied to the present data (see section IILA.3), the present sample shows about 0.25% heterogeneity. This figure is approximate, however, as it is extremely difficult to tell from a simple examination of the isotherm when the strong sites are saturated. This is because over a certain range of pressures, adsorption on the strong sites can cause a curvature of the isotherm concave to the pressure axis, while lateral interactions on the (0001) plane cause a curvature in the opposite sense. When these cancel, the isotherm is deceptively linear. If this portion of the isotherm is used for extrapolation, the heterogeneity estimate will be too low. A much better estimate of the number of strong sites, n is obtained if one fits an accurate equation for the adsorption isotherm to data taken at high enough pressures so that the strong sites are truly saturated. The isosteric heat curve indicates that this is the case for n > 4 Nmollg. This fit has been done, using the virial adsorption isotherm with the two-dimensional virial coefficients given by Morrison and Ross.s The results of the virial coefficient theory analysis will appear in the second paper of this series. However, it is noted that the best-fit value of numsfrom this study was found to be 0.67 f 0.05 pmollg, the variation being due to various choices of the adsorbate-adsorbent and adsorbate-adsorbate potential energies. This coverage corresponds to a heterogeneity of 0.5%. The surface of Sterling FT-D5, then, is found to be composed of 99.5% (0001) face and 0.5% “strong sites.” The (0001) part of the surface is homogeneous. While Graham was of the opinion that the strong sites are of a single type, there is r.eally no definite information available about their energy distribution. Thus, the surface under study is best termed patchwise heterogeneous. Thomy and Duva140 stated that their exfoliated graphite has less heterogeneity than the highly graphitized blacks, since the steps in the multilayer region were sharper on exfoliated graphite. Actually, the multilayer step sharpness may be influenced both by heterogeneity and by pendular ring condensation, which is likely to differ strongly between the polyhedral-shaped gcb and the flakelike exfoliated graphite. Thus, the low-coverage intercept method seems to be a better way of assessing the heterogeneity of the nearly homogeneous graphites. B. Adsorption Thermodynamics. 1. Isosteric Heat. The isosteric heat, qst, at low coverages (Figure 5 ) rises rapidly (I
The Journal of Physical Chemistry, Vol. 79, No. 5, 1975
Equation 6 was first obtained by thermodynamic reas0ning.5~It can also be derived by differentiating a lowtemperature approximate form for the Henry’s law con~ t a n t The . ~ ~model to which eq 6 corresponds is that of a two-dimensional ideal gas on a structureless surface, where the “vertical” degree of freedom is that of a one-dimensiona1 harmonic oscillator. Using eq 6, one obtains cgs/k = 1466 f 6 K. This value is within 1%of that which is obtained from the same data with more accurate virial models.59 The isosteric heat us. nu curves in the monolayer region and in the region where the second layer is forming both show a definite S shape. This shape is not predicted by two-dimensional equations of state which employ the Bragg-Williams approximation (e.g., the two-dimensional van der Waals, Fowler-Guggenheim, and significant structures equations), all of which predict a linear isosteric heat curve.4 Theories which take association into account, however, predict this S shape in the heat curve.6 Below the critical point, the qst curve of course becomes horizontal near 8,,15 the critical coverage. The sharp peak in qst in the range 100 < nu < 125 pmol/g is due to the phase change previously discussed. A similar peak, but of smaller height, was reported by Thomy and Duval. l5 The value of qst in the fluid-solid transition region is also greater than that reported by Larher,45 qst = 18,627 J/mol, from experiments done a t lower temperatures. As was noted in section II.B.l, our data reveal an apparent temperature dependence of qst (as defined by eq 1) in this region, with qst increasing with temperature. This dependence may be an effect of the influence of the nonuniformity of the graphite surface on the krypton monolayer. The behavior of the heat curve is expected to be complex in the phase transition regions if the transitions are of the X type. As is well known, the heat capacity becomes singular at a X transition, and the differential heat capacity is the temperature derivative of the isosteric heat. Consequently, the observed temperature dependence of qst is not surprising. Clearly, this coverage region deserves more detailed study over a wider temperature range. The isosteric heat curve in the multilayer region oscillates around the heat of sublimation, as previous workers have found for homogeneous adsorbents.60*61 2. Differential Entropy. The general shape of the differential entropy curve, Figure 5 , during the formation of the first, second, and third layers is predicted by all theories of submonolayer adsorption on homogeneous surfaces which include lateral interactions. At low coverage the adsorbed molecules have high mobility, which is gradually reduced
Physical Adsorption on Patchwise Heterogeneous Surfaces as the coverage increases. Thus the differential entropy falls during the deposition of each successive monomolecular layer of krypton on the substrate. Now consider the very rapid variation of the differential entropy which occurs near the completion of the first layer. This rapid variation is due to the phase transitions occurring in the first layer, though a gradual rise in AS must take place as the first layer is completely filled and the second layer begins, even in the absence of phase transitions.60 If the experiments were done at low enough temperatures so that the phase transitions were first order, it would be found that over a finite range of n u, where both phases were coexisting, AS would be constant. At the upper and lower boundaries of this range, AS would be discontinuous. At the temperatures of the present experiments, however, the phase transitions are clearly of higher order. In this case there is a more gradual transition, and the horizontal segment with discontinuities in AS which applies to the first-order transition is effectively smoothed out. In its place, there is instead a peak (or valley) in the entropy. The first peak (actually valley) which appears in is the sharp drop in the range 100 < nu < 120 /rmol/g, labeled A in Figure 6. This valley A is interpreted to correspond to a transition of the partially completed monomolecular film from a two-dimensional (2d) fluid to a 2d solid which is in registry with the graphite lattice. The differential entropy attains a very low value, which is in accord with expectation since the “2d in-registry” solid is much more ordered than the 2d fluid and should have a lower entropy. The second peak which occurs in AS is the peak labeled C in Figure 6. The rise from valley A to peak C is extremely sharp. This peak C is interpreted to correspond to a transition from the 2d in-registry solid to a 2d out-of-registry solid film which is complete at point B. The differential entropy of this transition is larger than the differential entropy of the previous transition. This is to be expected since on theoretical grounds the entropy of the 2d out-of-registry solid is expected to be greater than the 2d in-registry solid. Ying62and Dash63 have theoretically studied the vibrational frequency spectra of the 2d in-registry and out-of-registry solids. The frequency spectra differ in that the 2d inregistry solid has an absence of low-frequency vibrational modes. It can thus be expected to have lower entropy than the 2d out-of-registry solid. Additional evidence for the above interpretation of the data is provided from the areas per molecule of the two solid films. This is presented in the following section. C. Surface Area. Several previous studies of gas adsorption on gcb have encountered anomalies when attempting to measure the surface areas. Ross and Winkler64 found that two Langmuir equations were needed to fit their Krgcb data, with a transition a t 6 = 0.89. Singleton and Halsey65found that the point B and surface titration methods gave monolayer values differing by 10%. These anomalies were probably due to the presence of the phase transition(s), which were not identified a t the time. The application of the BET theory has also led to some anomalous results. Pierce and Ewing@ found that the BET Nz monolayer point disagreed by a factor of 1.19 with the monolayer point computed from the universal nitrogen isotherm in the multilayer region. In addition, numerous studies have shown that (1) the range of linearity for the krypton BET plot is very limited67 and (2)in order to make krypton BET surface areas agree with nitrogen areas, an area per molecule of 0.19-0.22 nm2 must be used for kryp-
465
ton.68 This area per molecule value is inconsistent with the solid density of krypton at the temperature of the experiments, usually 77 or 90 K. This inconsistency of the BET theory results, a t least in part, from the failure of its assumption that the heat of adsorption for multilayers is constant and equal to the heat of liquefaction. This assumption breaks down for homogeneous adsorbents at temperatures so low that kT is of the same magnitude as the difference in adsorption energy between successive layers, and stepwise adsorption results. Thus, the BET equation in its usual form is not straightforwardly applicable to krypton adsorption below its triple point and has not been used in this study. The choice of the (statistical) monolayer point, nu,, depends on the area per molecule which is chosen to associate with 6 = 1. Various monolayer densities may be defined.69 Lander and Morrison70 established by LEED that xenon monolayers are in registry with (0001) graphite under certain conditions. More recently, electron microscope studies of Venables and B a l F have established that this is the case for krypton also. In addition, Larher45 concluded from his recent study of the phase properties of krypton on natural graphite that “there is evidence that this first solid layer which is formed is in registry with the substrate.” The registry structure is shown in Figure loa. The application of monolayer registry to surface area measurement has been discussed.70 The area per molecule for the registry structure is 0.1574 nmz. In order to assume the registry structure, the krypton lattice must be expanded 4.2% from its equilibrium configuration. Thus, another low-energy monolayer structure is one in which the krypton lattice is out of registry with the graphite lattice, but where the area per molecule is that of the (111) plane of the fcc crystal, Figure lob. At 99.34 K, this area per molecule is 0.1450 nm2.72In Thomy’s study of four rare gases on exfoliated graphite,73 data were scaled according to a monolayer point, designated V B ~This . point was defined similarly to the classical point B. It was found that this monolayer point was consistent with the (111) crystallographic density to within 1%for Ar, Kr, and Xe. This correspondence is evidence that the equilibrium fcc (111)out-of-registry structure was assumed at VB1. Thomy, et al., observed two steps in their krypton isotherms. The first step corresponds to the fluid-solid transition and the second step is completed at V B ~This . second step was thought to be a rearrangement of the first l a ~ e r . 7 ~ Recently, Price and V e n a b l e ~developed ~~ a model for this transition. This model postulates that at lower pressures, the localized registry structure applies and that there is a first-order phase change to the nonlocalized out-of-registry structure. The fit of this model to the data resulted in a value of the average gas-solid interaction energy tgs/k = 1456 K, which is very close to that obtained in the previous section. There is, then, experimental and theoretical evidence that both of the monolayer structures discussed above occur. Referring to our own isotherms, it is clear that there is no first-order transition present in the monolayer solid region. There is, however, a small but definite peak, labeled “B” in Figure 5, in the isosteric heat of adsorption in the range 127 < n u < 135 /rmol/g. This peak was observed in all of our experiments but is much more pronounced at the lower temperatures. A comparison of Thomy and Duval’s ismtherms with ours using the scale factor defined by Figure 9 shows that their V B ~which , coincides with the top of their The Journal of Physicai Chemistry, Voi. 79, No. 5, 1975
Frederick A. Putnam and Tomlinson Fort
466
highest step, corresponds to n u = 130 wmol/g. Consequently, the Thomy-Duval step and peak B occur a t nearly the same area per molecule. If the qst peak B corresponds to the out-of-registry structure, then the in-registry structure must occur a t a coverage which is 8% less, or n = 120 wmollg, which corresponds to the top of the step which has been identified with the fluid-solid transition, labeled “A” in Figure 5. Thus, a consistent explanation of the observed features of the present, data and that of Thomy and Duval is to associate point A with the structure shown in Figure 10a and point B with the structure shown in Figure lob. In order to determine the statistical monolayer point, the amount of second-layer adsorption must be considered. At n u n , the number of vacancies in the first layer equals the amount adsorbed in higher layers. According to the lattice vacancy theory of Pace4 and another more recent n occurs approximately a t the midpoint of a line joining points A and D in Figure 5. This midpoint coincides fairly closely with point B. The approximation that the number of vacancies is equal to the number of second-layer adsorbed atoms at point B is probably not in error by more than the precision to which point B can be located, about f 5 pmol/g. Thus, n = 130 f 5 bmol/g, which is to be associated with an area per molecule of 0.1450 nm2. From these data, the surface area of the sample is calculated to be A = 11.4 f 0.4 m2/g. Unfortunately, the pressures in the monolayer region are too high for LEED work, so that the presence of the monolayer structures cannot be directly verified a t the temperatures of the present experiments. However, recent work by Suzanne, et al.,‘’ in which isotherms were measured by AES and structures determined by LEED for the xenongraphite (0001) system, has identified a very sharp step and its structure. This work points the way to much more accurate graphitized carbon black area measurements in the near future. Very recent neutron-scattering experiments by Kjems, et a1.,78 have verified the in-registry to out-of-registry transition for the nitrogen-graphite system. D. Spreading Pressure. 1. Monolayer Region. The spreading pressures shown in Figure 7 below 45 mN/m are typical of what one observes in film balance studies. Of particular note is the striking similarity between the second-order phase transition in these data with the “liquid expanded”-“intermediate” transition of the same order which is commonly found in experiments with insoluble monolayer^.^^ Transitions at higher pressures also have been observed in film balance studies.80 There is considerable uncertainty about how to find the “monolayer point” for a film balance isotherm. The analogy to the present study suggests that the monolayer point would be better taken at B than as the “limiting area” obtained by extrapolation of the steep portion of the isotherm to zero pressure. 2. Multilayer Region. As Figure 7 shows, the second layer forms at 52.5 mN/m and the third at about 64 mN/m. The equilibrium spreading pressure, re, is the spreading pressure a t PO, equal to yo - ysat,where yo is the surface energy of the graphite (0001)-vacuum interface, and Ysat is the surface energy of the graphite (0001)-saturated krypton vapor interface. Since it has been found that the FHH equation describes step positions of the krypton-graphite system well in the multilayer region (section IV.A.l), this equation was used to extrapolate the isotherm to the P = P O axis to obtain an estimate of re.The simplest form of the equation was used. (This form can also be arrived a t (r
The Journal of Physical Chemistry, Voi. 79, No. 5, 1975
from the Larher equation, by making certain simplifying assumption^.^^)
The energy A6 was first evaluated from the step positions by treating n as an integral. The result was A€ = 5560 J/ mol. For the extrapolation, n was treated as a continuous variable and fit to the upper portion of the isotherms. This procedure was used by Melrose.81 The resulting extrapolation is shown as the dotted lines in Figure 7, the results being A , = 72.5 mN/m at 94.72 K and re = 74.4 mN/m a t 104.9 K. Spreading pressures on solid surfaces have been used to estimate their surface tensions by Fowkes,82 who obtained a value for the dispersion force contribution to the surface tension of the solid, ydgcb = 123 mN/m, for gcb at 77 K by analysis of nitrogen isotherms. Applying the Fowkes equation to the present system, one arrives a t Here YKr is the tension at solid krypton-krypton vapor interface and y d ~is, the dispersion force contribution to this tension. For krypton YKr = ydKr and has been estimated to be 52 mN/mE3 for the krypton (111) plane. Substituting this value into eq 8 and using 73.4 as an average value for r e , one arrives a t ydgcp= 151 mN/m for graphite in this temperature range. This calculation assumes a duplex film of krypton at PO. 3. Low-Coverage Region. The low-coverage spreading pressure is best observed in the two-dimensional compressibility plots, Figure 8. These curves have the general shape of a normal van der Waals dense gas, with two important differences: (a) the leveling out of the curve at the “substep,” labeled “A” and (b) the sharp rise in the curve at low pressures. Although it is difficult to see on Figure 8, all three curves have a A I R T = 1 at A = 0. As A is increased, they rise very sharply and then decrease. This behavior is due to the heterogeneity which affects the low-pressure part of the isotherm. At low pressures most of the adsorption is on highenergy patches comprising only 0.5% of the surface. The effect of the high-energy sites is to cause the initial part of the nu us. p isotherm (Figure 4) to be concave to the p axis, instead of convex to the p axis as would be the case for adsorption on a perfectly homogeneous surface with adsorbate lateral interactions. From the virial expressions7
and
it is apparent that the initial curvature of the n u us. p isotherm is the negative of the initial slope of the compressibility plot, and this explains why an initially concave isotherm causes the behavior seen in Figure 8. In order to calculate the true compressibility plot for the graphite (0001) surface only, it is necessary to integrate the isotherm obtained by subtracting out the adsorption on the strong sites. The results of this calculation are shown in Figure 8 in dotted lines. The very large effect which only a small amount of heterogeneity has on B 2 d and the fact that it causes an increase
Physical Adsorption on Patchwise Heterogeneous Surfaces
in the value of B 2d (in this case even changing its sign) have obvious implications for virial coefficient theory analysis of isotherms. An increase in B 2d caused by heterogeneity will cause the value of egg to be too low. This may be one reason for the discrepancy between the recent analysis of Rosss and that of previous treatments.
Acknowledgments. We are pleased to acknowledge helpful discussions with Dr. L. T. Drzal on experimental technique and with Dr. R. B. Griffiths and Dr. J. C. Wheeler on the physics of phase transitions. Supplementary Material Available. The isotherm data shown in Figures 1-4 will appear following these pages in the microfilm edition of this journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D.C. 20036. Remit check or money order for $3.00 for photocopy or $2.00 for microfiche, referring to code number JPC-75459 * References and Notes (1) H. E. Thomas, R. N. Ramsey, and R. A. Pierotti, J. Chem. Phys., 59, 6163 (1973). (2) S. Ross and J. P. Olivier, "On Physical Adsorption," Interscience, New York, N.Y., 1964. (3) A. W. Adamson, "Physical Chemistry of Surfaces," 2nd ed, Interscience, New York, N.Y.. 1967. (4) R. A. Pierotti and H. E. Thomas, "Surface and Colloid Science," Vol. IV, E. Matijevic. Ed., Wiley-Interscience, New York, N.Y., 1971, pp 93-259. (5) E. Bergmann, J. Phys. Chem., 78,405 (1974). (6) G. I. Berezin and A. V. Kiselev, J. Colloidlnterface ScL, 46, 203 (1974). (7) W. A. Steele, Surf. Sci., 39, 149 (1973). (8) I. D. Morrison and S. Ross, Surface Sci., 39, 21 (1973). (9) B. C. Kriemer, B. K. Oh, and S. K. Kim, Mol. Phys., 26, 297 (1973). (10) L. T. Drzal, F. A. Putnam, and T. Fort, Jr., Rev. Sci. hstrum., 45, 1331 (1974). Figure 5 of this paper shows the variation of error with amount
adsorbed for some of the same data presented here. (1 1) L. T. Drzal, Ph.D. Dissertation, Case Western Reserve University, Cleveland, Ohio, 1974. (12) Theoretical calculations (see, for example, Surface Sci., 36, 317 (1973)) indicate that the potential energy barriers to translation for the krypton-(0001) graphite system are about 167 Jlmol, which is less than '/..kT at the temperatures of the experiments. (13) P. J. Moller and T. Fort, Jr., ColloidPolym. Sci, in press. (14) A. Thomy, Dissertation, University of Nancy, 1968. (15) A. Thomy and X. Duval, J. Chim. Phys., 67, 1101 (1970). (16) D. S. Newsome, Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, Ga., 1972. (17) R. A. Pierotti and D. S. Newsome, Presented at the 167th National
Meeting of the American Chemical Society, Los Angeles, Calif., Abstract COLL No. 120. (18) "Baratron," Type 90, MKS Instruments, Burlington, Mass. (19) J. A. Morrison and D. M. Young, Rev. Sci. Instrum., 25, 518 (1954). (20) M. P. Freeman and G. D. Halsey, Jr., J. Phys. Chem., 60, 11 19 (1956). (21) The surface properties of the highly graphitized carbon blacks are relatively insensitive to outgassing conditions. (22) J. H. Singleton and G. D. Halsey, Jr., J. Phys. Chem., 58, 330 (1954). (23) A. Levy, M. S. Thesis, Georgia Institute of Technology, Atlanta, Ga., 1966. (24) G. A. Miller J. Phys. Chem., 67, 1359 (1963). (25) A. G. Clarke and E. B. Smith, J. Chem. Phys., 48, 3988 (1968). (26) G. T. McConville, W. L. Taylor, and R. A. Watkins, J. Chem. Phys., 53, 912 (1970), eq 7.
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