Langmuir 1994,10, 235-240
235
On Thermodynamics of Permanent Hysteresis in Capillary Lyophobic Systems and Interface Characterization? Vladimir Yu. GusevJ Kiev Polytechnic Institute, pr. Pobedy 37,Kiev 252056, Ukraine Received April 23,1993. I n Final Form: November 8,199P Permanent hysteresis, meaning the hysteresis which is stable with regard to the change in the process rate, is known to occur in capillary condensationtevaporationof wetting fluid and intrusiontextrusion of nonwetting liquid in porous solids. No satisfactorythermodynamic description of the permanent hysteresis in these processes exists at present, despite their use in thermodynamic interface characterization of dispersed systems. Also, in the case of lyophobic systems, no direct determination of total energetic exchange of the system exhibitinghysteresis has been reported. Considering quasi-isothermalirreversible processes in capillary lyophobic systems, we obtained expressions of energy and entropy balances which relate equilibriumphysical characteristics of the system, energeticexchange,and internalentropyproduction. This provides a strict base for a modelless thermodynamicmethod of evaluation of the cumulative interface areas and entropy of the change of interface areas and then the pore volume distribution and surface fractal dimension of porous solid. For the first time, along with p-V measurements, direct determinations of heats of high-pressureintrusion and extrusion of water in pores of hydrophobized silica gel have been undertaken. Analysis of these data shows validity of the developed approach.
Introduction It is well-known that spontaneous percolation of nonwetting liquid in a porous media of solid at elevated pressures in capillary lyophobic systems (CLS) may result in either complete irreversibility of intrusion of the liquid or hysteresis on the intrusion-extrusion (IE)curves. The hysteresis of this kind does not vanish when slowing down the IE process and is therefore called true or permanent. Among conventional uses of the CLS are liquid reversedphase chromatography and mercury porosimetry.' Hysteretic processes in CLS can also be employed in promising alternative technologies of developing new materials2and energy accumulators.3 The pressure required to force a nonwetting fluid into a cylindrical pore of radius r is given by the Laplace equation p=--
u3 cos 0
r where u3 is the interfacial tension and 0 is the three-phase contact angle. Thermodynamic consideration of Rootaire and Prenslow" of the processes af IE in mercury porosimetry led them to the equation relating change in interface solidliquid areas to the work done by external means in this process:
wll
bQ2= -u3 cos 0
Here 02 is the solid-liquid interface area and W,= -Sip dV+ Sip dV1 is the net external work (the work excluding that of reversible compression of liquid -Sip dV1) done on the system in the process i-j of intrusion or extrusion. Based on paper A.23 presented at the Second International Symposiumon SurfaceChemistry, Adsorption and Chromatography, MOSCOW, June 29-July 3,1992. t Present address: Department of Chemical Engineering, Yale University, P.O. Box 2159 Yale Station, New Haven, CT 06620. published in Advance ACS Abstracts, January 1,1994. (1) Ritter, H. L.; Drake, L. C. Znd. Eng. Chem., A d . Ed. 1946,17,782. (2) Bogomolov, V. N. Usp. Fiz. Nauk 1978,124, 113 (in Russian). (3) Eroahenko, V. A. USSR Author's Certificate 1333870. (4) Rootaire, H. M.; Prenelow, C. F. J. Phys. Chem. 1967, 71, 2733.
9 Abstract
Equations 1and 2 are used for estimation of pore size distributions and surface areas of the materials from the data of mercury porosimetry and analyzing of CLS in the other appli~ations.~*3 Attempts at improving the mercury porosimetry method, such as extending it for characterizing fractal properties of dispersed solids (see, e.g., refs 5 and 6), are mainly based on these equations. A certain progress in understanding of the CLS behavior has been achieved due to their statistic simulation (see, e.g. ref 7). This direction is mainly restricted by studies of dependencies of the IE processes on different model pores shapes and their interconnectivities. In phenomenological thermodynamics of the CLS, external influences on the system (expressed as p vs V dependencies) are related with internal equilibrium parameters without considering any models of porous media and possible causes of h y s t e r e s i ~ However, . ~ ~ ~ ~ ~beginning ~~ with the work of Rootaire and Prenslow, overwhelmingly irreversibleprocesses are traditionally treated as reversible ones. This strong assumption leads to an inconsistency of characteristics obtained from direct (intrusion) and reverse (extrusion) processes. In the present paper the problem of hysteresis in CLS is treated within the phenomenological approach without assumptions of reversibility of the IE processes. Relationships between the total energy exchange of the CLS (which can be characterized by p, V data and heats of IE process)occurred in a quasi-isothermal irreversible process and internal equilibrium parameters of the CLS such as interface area, contact angle, and entropy of the change of interface area have been obtained. This has been done by means of a method of Shottky, Ulich, and Wagner10 ~
~~
(6) Friesen, W. I.; Laidlaw, W. G. J. Colloid Interface Sci. 1993,160,
226-236. (6) Neimark, A. V. Adsorpt. Sci. Technol. 1991, 7, 210. (7) Characterization of porous solids (COPS-11);Proceedings of the IUPAC Symposium. Elsevier: Amsterdam, 1991; pp 67-74,76-84,106113,169-178. (8) Huisman, H. F. J. Colloid Interface Sci. 1983,94, 26. (9) Eroahenko, V. A. Kolloidn. Zh. 1987,49, 876 (inRussian). (10)Shottky, W.; Ulich, H.; Wagner, C. Thermodynamik, reprint of the edition 1929 Springer: Berlin, 1973.
0743-7463/94/2410-0235$04.50~0 0 1994 American Chemical Society
Gusev
236 Langmuir, Vol. 10, No. 1, 1994 (see also ref 11) considering irreversible IE processes starting in some equilibrium state and converging to some other equilibrium state of the CLS. An emphasis on modelless (in the sense of not assuming any geometric structure model of dispersed solid) methods of interface characterization is made. Usage of the developed method assumes availability of both experimental heats and p , V data of IE processes. A new and conceptually simple high-pressure calorimetric technique for getting such data is described in the experimental part, where the proposed method is applied to the CLS water-hydrophobized mesoporous silica gel.
Thermodynamics of the Hysteresis in CLS Let a closed externally reversible12 thermodynamic system contain an inert porous solid, a nonwetting liquid of mass m and volume VI, and its vapor. A certain part of the CLS containing the solid and part of the liquid is surrounded by a calorimetric cell. The whole system is subjected to an action of a thermostat of temperature T and external source of pressure p via a movable wall (bellows, see Figure 1). NonequilibriumTransition Between Equilibrium States. Consider now a real, Le., nonequilibrium, transition between equilibrium states i and j of equal termperature T. Apply the first law of thermodynamics to this irreversible transition, and the second law of thermodynamics to the i and j equilibrium states to obtain in both cases the difference in the internal energy corresponding to these states. This allows equatingan exchange of CLS by work Wand heat Q with its surroundings caused by the movement of the piston in the real process, and the combination of the respective differences in the CLS Helmholtz free energiesAF,entropies AS, and temperature T
W + Q = hF+ TAS
(3) where W = -Sip dV. For the entropy change being due to an external heat flux Jpand an internal entropy production occurring at the rate y = djS/dt (t, time), a balance equation holds dt
AS= f[&+y]
where Q, is a heat evolved to the calorimeter, m, is a mass of liquid in the calorimetric ampoule, and s denotes a specific entropy of the bulk liquid. Imaginary Reversible Transition. Following refs 10 and 11,the differences in the free Helmholtz energies and entropies corresponding to the equilibrium i and j states of the CLS can be determined as the changes in these values along an imaginary quasi-static isothermal transition through equilibrium states laying between the i and j states. In the assumption of sufficiently large pores, inert solid, and negligibility of the vapor pressure, quasistatic reversible changes in parameters of the closed CLS are connected in the fundamental Gibbs' equation 3
dF = c
u i dQi - p
dVl - S dT
r=l
Here ai and Qi are respectively the surface tension and area of the ith interface (i = 1 , 2 , and 3 being assigned to the solid-vapor, solid-liquid, and liquid-vapor interfaces correspondingly). An elementary quasi-static change of the entropy as a function of independent (ai),VI, T is
On assuming the interfaces to be homogeneous, by integrating eqs 9 and 10 at T = const (notice, that AQ1= -An2, and AQ3 may be neglected at the sufficiently big transition) one can get
AF = -(a2 - al) AQ, - f p dVl
(11)
and, in view of Maxwell's expressionsfor partial derivatives of eq 10
(4)
Tb
In the case of infinitesimal difference between the CLS boundary temperature T b and that of the thermostat T, it can be reduced to AS =
3+
Ais
(5)
where Ais = SJy dt > 0. Combining eqs 3 and 5 leads to Ais =
-M+W
Hence w>A'F>O and O > W > A % (7) Here and below the subscripts + and - correspond to the direct (intrusion) and opposite (extrusion) processes. Provided the compression and expansion of liquid in the CLS outside the calorimeter are reversible, one can get ~~
Here, the value of the specificentropy of change in interface areas Asn d(a1- az)/dT is introduced. Invoking the Gibbs notion of dividing surface between phases, it is possible to show13 that Asn is the difference between excess entropies of the interfaces liquid/solid and vapor/solid defined with respect to equimolecular positions of dividing surfaces. Below, eq 12 will be used in form where the term responsible for the change of entropy of liquid at ( Q i j = const is written as J j m ds A S = As, An2
+fm
ds
Main Equations. Now, from eqs 3 and 5 using eqs 8, 11, and 13 one can get for the nonequilibrium process between equilibrium states, the energy balance in the form of the extended version of Gibbs-Helmholtz equation and the entropy balance
~~
(11)Prigogine, I.; Defay, R. Chemical Thermodynamics; translated by Everett. London. 1954. (12) Haywood, R. W. Equilibrium Thermodynamics For Engineers and Scientists: Wiley and Songs: Chichester, New York, Brisbane, Toronto, 1980.
(13)
(13)Melrorre, J. C . J. Colloid Sci. 1965, 20, 801.
Thermodynamics of Permanent Hysteresis
Langmuir, Vol. 10, No.1, 1994 237
Q, = [As, AQ2 - Ais] T where the value of net calorimetric IE heat Qn Qc- T.J {mc ds is introduced. Equation 14 relates the heat and work exchange of the CLS and its isothermal surroundings in the irreversible IE processes with equilibrium physical characteristics-parameters of state of the CLS. Equation 15 shows that at the assumptions adopted the net calorimetric heat is equal to the heat of equilibrium isothermal change of CLS interface areas less the Clausius’ noncompensated heat of the process. Among the physical causes of entropy production in the IE processes in CLS are evidently the pressure micropulsations of liquid in the pores of solid due to unstable liquid configurations-Haines type14 jumps. Young’s expression for the equilibrium contact angle 0 (0 > 90° in the CLS) on the three-phase contact line: f
63
cos e = c1- u2
(16)
may be used in order to transform eqs 14 and 15
Equations 17 and 18 can be used for determination of CLS interface areas and evaluation of the entropy of interface changes by means of both measurements of p , V properties and heats of IE quasi-isothermal processes. The corresponding technique is described in the experimental part of the present article. By analogy with ref 15, a degree of irreversibility of the IE process in the CLS may be characterized by the ratio of the loss of work in the hysteresis cycle to the work of intrusion
where the superscript “h” means that the respective value is determined in the hysteresis cycle. If e 0.5 usually (see, e.g. ref 16),and therefore, an application of reversible approach to the CLS, as a rule, is not justified.
Interface Characteristics of the CLS A known group of modelless methods of assessment of interface areas (Kiselev” and Derjagin,18lyophilicsystems, Rootaire-Prenslow,2 an analog of Kiselev’s method for CLS), pore volume distribution (Brunauer, Mikhail, and Bodorlg), and fractal surface dimension (Neimark6) of dispersed solids have been developed on equating the work done in the process of physical interaction of a solid with wetting or nonwetting fluid and respective change in the free energy of the system. However, this approach, in the case of capillary hysteresis, ignores (see eq 6) the considerable irreversibility of the processes which accompany (14) Haines, W. B. J.Agric. Sci. 1930, 20, 97. (15) Neimark, A. V. Dokl. Acad. Nauk SSSR 1985,281, 383. (16) Gregg, 5.J.; Sing,K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (17) Kiselev, A. V. Usp. Khim. 1945, 14, 367 (in Russian). (18) Derjagin, B. V. Zh.Fiz. Khim. 1957, 31, 516 (in Russian). (19) Brunauer, S.;Mikhail, R. Sh.; Bodor, E. E. J. Colloid Interface Sci. 1967, 24, 451.
the fluid-solid interaction. Therefore, in reality these methods may give only upper and lower estimates of interface characteristics.16 The expressions of energy and entropy balances of the real process taking place between equilibrium states-eqs 14 and 15 for CLS-may provide a stronger base for the experimental interface characterization. An expression of the interface area solid-liquid appeared or disappeared along the IE process is due to eq 17
The denominator of the last equation represents the specific interface energy corresponding to the unit in interface area solid-liquid change. If the IE process is supposed to be a reversible one, eq 20 may be readily transformed to the Rootaire-Prenslow equation (2) used in the mercury porosimetry. As the IE processes are always substantially irreversible, eq 2 should not give accurate values of AQp (in fact, the error of determination of An2 by means of the RootairePrenslow method is usually not less than 50%). Attempts to adjust values of equilibrium contact angles 0 to bring Rootaire-Prenslow interface areas into coincidence with these obtained by other methods (BET)8 in view of the above cannot satisfy a strict thermodynamic consideration. Inequalities (7) in view of eq 9 and 13 lead to the following inequalities instead of eq 2
which can be used in mercury (or “lyophobic”)porosimetry for rough estimates of the areas mentioned when the heat of the IE processes is not available. Pore Volume Distribution. The dependence of Q2 on p being obtainable from eq 17 can be used for calculation of pore volume distribution V, = f(rh) in terms of cumulative pore volume Vc and hydraulic radius b@) = AVJAQp.18 If the value of AVc (and AQ2) is sufficiently large, and the porous media is isotropic, the hydraulic radius rh is one-fourth of the mean linear pore size of the participating domain of porous media.20 Surface Fractal Dimension. An idea that the solidliquid cumulative interface area could be measured by means of spheres of Laplace’s radius rl@) used as a yardstick for monolayer covering of the dispersed solid provides the basis of Neimark’s thermodynamic methodg for calculating surface fractal dimension d,P1 according to:
In order to use this method one needs to know only one Vc@) curve obtainable routinely in the mercury porosimetry. However, strong (as has been shown earlier) assumption of reversibility of the IE process and implicit assumption of ideal self-similarity (not a self-affinity) of the solid surface structure narrow the possibilities of this method. The first restriction eliminates in view of the proposed here method of determination of interface areas due to eq (20) Heifeta, L. I.; Neimark, A. V. Multyphase processes in porous media; Khmia: Moscow, 1982 (in Russian). (21) Mandelbrot, B. Fractal geometry of nature; Freeman: SanFrancisco, CA, 1982.
Gusev
238 Langmuir, Vol. 10, No. 1, 1994
20 d 1n(Wn@)+ (23) d In@) The second one should presumably remedy if the notion of hydraulic radius is used d,, = 2 -
Equations 23 and 24 express the fact that the fractality of a homogeneous lyophobic surface should lead to the respective fractal dependence of the energy needed for its forced wetting on geometric scale. The differentiation in eqs 23 and 24 implies using very accurate experimental data. Entropy of the Change of Interface Areas. The value of Asa can be estimated using the feature Ais > 0 in view of eq 15
Assuming, as a first approximation, that the changes in the areas Ql and 8 2 are proportional to the change in the mass of pore water, mp, and introducing the value of net differential heat of IE q=- Qn
Amp assessment of the ASQvalue can be made using inequality 24
Here, k = mo/(AT), where mo is the mass of liquid in pores at the end of intrusion and A is an area of the surface of solid accessible for the liquid. The value of ASQfor the CLS presumably cannot be evaluated by other methods at present.
Experimental Example Capillary Lyophobic System. For experimental investigation, the CLS has been chosen to consist of water and hydrophobized silica gel, partly because of relevance to our experimental installation p - V parameters. The main reason, however, was that the surface properties of hydrophobized silica gel-one of the most versatile adsorbents in t h e high performance liquid chromatography-has been the subject of an intensive characterizationby virtuallyall availableanalyticaltechniques. Naked silica gel KSK-G (SILBET, Estonia) possessed specificBET surface area, SBET = 270 m2/g,and the mean Kelvin pore diameter of 13nm. Surface of the naked silica gel was grafted with n-hexadecyldimethylchlorsilaneand end-capped with trimethylchlorsilane following ref 22. By use of a carbon content of 11%in the modified sample, a bonded layer $16 chains) density has been determined following the procedure in ref 23 as p = 2.26 groups/nm2,which presents the upper limit for alkyldimethylchlorosilanes. The mean thickness of the bonded layer db = 0.95 nm has been determined from the change in the pore volume as described elsewhere.24 Outgassed bidistillate of water was used in all experiments. (22) Buzsewski, et al. J. Liq. Chromatogr. 1987,10, 2335. (23) Lisichkin,G. V., Ed. Modified silicas insorption, chromatography and catalysis; Chimia: MOSCOW, 1986, 248 p (in Russian). (24) Fadeev, A. Yu.;Staroverov, S. M. J. Chromatogr. 1988,447,103.
L + = > nonwettingliquid
I
I I I
I I I I I
I I I I I I I I I I I
I
I I I I
Loadina - .Diston manometed
I
0.1 -30MPa
I I I
I I I I I I I I I I
-----------1
Figure 1. Principle of the high-pressure calorimetricexperiment.
Experimental Setup and Procedure. Experiments were done using a high-pressurecalorimetricsetup (Figure 1)previously used for high-pressure gas adsorption measurements.% Quasi-isothermalcalorimeter DAC1-1A (NTO of the USSR Academy of Sciences) of Tian-Calvet type has been used to measure the IE heats with an accuracy ca. 3 % . A calorimetric ampule contained a part of the system water (m,) and the sample of porous solid. The remaining part of the water was held inside of the bellows of an inductive volumeter connected with the ampule by a 1 mm external diameter capillary. Special blank experiments showed no non-detectable heat losses from the ampule in the calorimeter to the outside environment. Pressure in the range of 0.1-30 MPa has been generated and measured with a loading piston manometer (accuracy 0.05% of reading). The volume of the system has been read by means of an inductive transformer and digital alternator and voltmeter with precision of 0.3 % of reading. Heats of the IE processes and simultaneous measurements in the CLS were measured at the temperature T = 308 K over the pressure range of 0.1-25 MPa, changing the pressure by steps of 0.4-1 MPa. Results and Discussion. The behavior of the CLS is characterized by dependencies p(mp),q(mp),and Qn(mp) presented in Figures 2-4. At the pressures not exceeding the minimum intrusion (break-through) pressure in the CLS (section AB) an ordinary compression of the bulk CLS parts takes place, and values of net differential calorimetricheat q did not exceed experimental error. The intrusion of water (sectionBC) is essentially an exothermic process (Q < 0 and Q n < 0). The bulk expansion of the CLS (section CD), as might be expected, is characterized by the negligible values of q. The extrusion of water (sectionDE) as well as intrusion is an exothermic process. The remarkable fact was that the expansion of the CLS at the decreasing pressure was an endothermic process until the beginning of water extrusion (point D) and exothermic process after beginning of extrusion (section DE). (25) Gusev, V.
Yu.;Fomkin, A. A. J. Colloid Interface Sci., in press.
Thermodynamics of Permanent Hysteresis 25
Langmuir, Vol. 10, No. 1, 1994 239 C-
I
120
0 0 0 0 8 ~
0
8
a 0
a
80 i 8 0
0
0
0.05
0.1
,
0.2
0.15
8 8
ob
0
I
0.25
0.3
0.35
Figure 2. Compression (B) and expansion (0)of the capillary lyophobic system water-hydrophobized silica gel KSK-G/ClG: BC, intrusion; DE, extrusion.
5 0.05
0.2
0.15
0.25
0.3
$ -10 &
0.36 8
-15
40 -35
8
a
8
m
.
8
Figure 3. Differential net heats q of compression (M) and expansion (0)in the capillary lyophobic system water-hydrophobized silica gel KSK-G/ClG: BC, intrusion; DE, extrusion.
-2 -4 3
-7
8
a a
-8
Figure 4. Net heats Q, of compression (M)and expansion (0) of the capillary lyophobic system water-hydrophobized silica gel KSK-G/ClG: BC, intrusion; DE, extrusion. The following condition for the IE cycles within the limits of experimental errors was met (Q,
5
10
15
20
25
p. MPa
mp , gig
0 -6
0
+ W,)' - (Q, + W J
(28)
This indicates that the CLS always returned to the initial state after ahysteresis cycle, i.e. irreversible changes in the properties of the CLS (due to, e.g., possible hydrolysis of the grafted layer) were negligible. The analysis of these data in principle allows obtaining the dependencies of ai@)(i = 1,2) and V, = f(rh),provided the value of specific energy of interface areas change-the denominator of eq 20-is known. This implies knowing the value of the equilibrium contact angle 0 and its temperature derivative for the system in question. As a first approximation, we will try to evaluate these twovalues using our own data at simplifying assumptions about the surface area of the solid. An estimated value of the area of accessible surface of grafted silica gel A = 130 m2/g (taking into account the thickness of grafted layer) in view of eq 20 leads to the assessment of the value of denominator as 0.016 J/m2(the
Figure 5. Dependence of the interface solid-liquid area Wz on pressure at compression (B)and expansion (0)of the capillary lyophobic system water-hydrophobized silica gel KSK-G/ClG.
+
correspondent sum of Q,, Wn was equal to 2.17 J/g). The dependence Q2@) presented on Figure 5 has been obtained by using this value of the denominator in eq 19. The inequalities (27) lead to the estimation -9.2 X 106 < A ~ 0 for hydrophobic surfaces (see,e.g., ref 26, it is ascribed to disordering of water at the transition from bulk volume to the boundary layer near the hydrophobic surface). The possibility of realization of condition As,-,> 0 in the CLS under question has been circumstantially evidenced by the decreased breakthrough pressure Pb(Vp) = c(V,) r ~ 3cos 0 (c(V,) is a value characterizing the form of pores at agiven filling of the pores V,) at increased temperature. The value of (Ap/AT)v was ca. 0.06 MPa/K; however, it cannot be used for tke calculation of As,, since it characterizes the nonequilibrium situation of the breakthrough process. Taking the value As0 = 0-2.3 X 106 J/(m2 K) in view of eq 17, the equilibrium contact angle can be assessed as 0 = 100.4 f 2.5O. The contact angles in the system of water sessile dropsmooth hydrophobized silica gel support have been directly studied by Kessaissia et The support was prepared by compacting at sufficiently high pressure modified silica gel, and the reported angles corresponded to the plateau on the curve 0 vs surface roughness, where 0 apparently did not depend on the roughness. For the silica gel grafted with C16 alkyl chains (surface layer density p = 2.0 groups/nm2) the value of advancing contact angle Oa = 100' was observed. This could be considered as an indication on validity of the proposed method of determination of equilibrium contact angle 0. Using the value As, = 0-2.3 X lo3 J/(m2 K) in view of eq 27 (see also Figure 3) leads to the following conclusions: (1) Ai+S(m)> Ai-S(m), Le., adopted in the thermodynamic of the CLS assumption that the intrusion is less irreversible than extrusion2 is not justified for the CLS in question; (2) a t the intrusion in the CLS studied, the entropy production exceeded the corresponding increase in the entropy of interfaces, due to which the condition q < 0 has been observed. In principle, the opposite situation may occur, therefore for a CLS exhibiting smaller hysteresis (e), higher monoporosity of the solid, and larger values of As,, one can expect an endothermic intrusion. The pore volume distribution calculated by means of the above method and that obtained from the nitrogen desorption isotherm using an equation of Kelvin are in agreement if one takes into account the width of the grafted layer (db = 0.95 nm, see Figure 6). (26) Tarasevich, Yu. 1. Kolloidn. Zh. 1991,53, 1111 (in Russian). (27) Kessaissia, Z.; Papirer, E.; Donnet, J. B. J . Colloid Interface Sci. 1981, 82, 526.
Guaeu
240 Langmuir, Vol. 10, No. 1, 1994 0.8 0.7
0.6
1
*
average pore size of 10 nm derivatized with octadecylchlorosylane to various degrees of coverage. It has been shown that derivatizing of silica gel surface affecta its structure only at the geometric scale smaller than 5 nm. Schmidt et al. also concluded that derivatized silica gels under question did not exhibit surface fractal properties.
*
a 0.5
??
0.4
>v 0.3
0.2
A
t
:4 A
A
Conclusion
A
0.1
4 A.
0
?
0
5
10
15
20
d. 4 * r g m
+
Figure 6. Integral pore volume distribution: ,naked silica gel KSK-G, the dependence on Kelvin’s pore diameter; A, grafted KSK-G/ClG, the dependence on 4*n,.
Analysis of the same data using eqs 23 and 24 reveals no surface fractal properties of the modified silicagel KSKG/C16 at the considerable changes of the geometric scale at 100 nm > 4rh > 7 nm (the corresponding log-log representation of experimental data could not be linearized with reasonable tolerance). This conclusion does not contradict with the results of the small angle X-ray scattering (SAXS)investigation28of a series of naked silica gels characterized by a mean pore size of 6.8-57.2 nm and BET surface area in the 48-409 m2/grange, if one assumes that the surface modification of the silica gel does not affect its fractal properties within the considered geometrical scale. The former is confirmed by work of Schmidt et al.29, who studied by SAXS silica gel with (28) Drake, J. M.; et nl. Chem. Phys. 1988,128, 199. (29)Schmidt, P. W.; Avnir, D.; Ley, D.; H6hr, A.; Steiner, M.; Rdl, A. J. Chem. Phys.1991, 94 (2), 1474.
Phenomenologicalthermodynamicsof quasi-isothermal hysteresis IE processes in the CLS is developed. A modelless method for calculation of interface characteristics of the CLS from the calorimetric and p V data of quasi-isothermal irreversible processes is proposed. Analysis of the available experimental data supports the thermodynamic description. The developed approach can be also applied for description of other hysteresis systems, e.g. for interface characterization of lyophilic systems using the capillary condensation hysteresis data. A corresponding paper is in the process of preparation.
Acknowledgment. The author is indebted to Professor V. A. Eroshenko (KievPolytechnicInstitute) who initiated the present work and to Dr. S. G. Tkachenko (Kiev Polytechnic Institute), Dr. A. A. Fomkin, and Dr.N. I. Regent (Institute of Physical Chemistry of the Russian Academy of Sciences, Moscow) for their help in the experiment. Dr. A. Yu. Fadeev (MoscowState University) is acknowledgedfor modification of silica gel and providing characteristics of grafted layer. Drs.A. V. Neimark and V. A. Bakaev (Institute of Physical Chemistry, Moscow) are gratefully acknowledged for useful remarks.
