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Langmuir 1997, 13, 1016-1019
A Simple General Relationship between the Dielectric Losses Measured on Divided Solids and Adsorption Thermodynamic† J. C. Giuntini,‡ V. Mouton,‡ J. V. Zanchetta,‡ J. M. Douillard,*,§ J. Niezette,⊥ and J. Vanderschueren⊥,| URA CNRS 407, CC 003, Universite´ des Sciences, 34095 Montpellier, France, URA CNRS 79, CC 015, Universite´ des Sciences, 34095 Montpellier, France, and Chimie Macromole´ culaire et Chimie Physique, Universite´ de Lie` ge, Institut de Chimie au Sart Tilman, B4000, Lie` ge, Belgium Received November 13, 1995X Dielectric losses measured on a zeolite sample have been investigated to correlate these measurements with surface thermodynamic. The polarization conductivity measurement allows the determination of variables which are linked to the adsorption enthalpies and entropies but also to the system dynamic. To quantitatively interpret the results, it is necessary to know the energetic heterogeneity of the solid sample. The thermally stimulated current measurement (TSDC) appears as a possible tool. Therefore this new method of studying adsorption by combining two techniques seems promising. This paper is an introduction to the work under development.
I. Introduction The interactions between vapors and solids can be considered as a physical problem clearly defined at the present time.1,2 Volumetric and calorimetric studies have allowed the processes to be clearly pictured and accurate values of the thermodynamic variables to be given. But the usual techniques cannot give values necessary to describe the microscopic level. New theoretical developments need reliable measurements of the microscopic interactions in the interface. For example new theories treating reaction dynamics need to determine excited energy levels of the molecule before reaction. Therefore such a modeling of the heterogeneous catalysis needs to know the energy transferred from the solid to the molecule during the adsorption process. In this paper, we show that it is possible to relate a microscopic treatment of the polarization conductivity applied to divided solids to adsorption thermodynamic variables. Measurement of dielectric losses is a technique often used in the field of solid state research,3-7 but until now the explanation remains qualitative. Most models explain the behavior of the polarization conductivity in * To whom correspondence should be addressed (fax, 33 67 14 33 04; e-mail,
[email protected]). † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. ‡ URA CNRS 407. § URA CNRS 79. ⊥ Universite ´ de Lie`ge. | Research Associate of the National Fund for Scientific Research (Belgium). X Abstract published in Advance ACS Abstracts, September 15, 1996. (1) Adamson, A. W. Physical Chemistry of Surfaces; Wiley: New York, 1976. (2) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (3) Fripiat, J. J.; Jelli, A.; Poncelet, G.; Andre´, J. J. Phys. Chem. 1965, 69, 2186. (4) McIntosh, R. Dielectric behavior of adsorbed species; Marcel Dekker: New York, 1966. (5) Imai, J.; Kaneko, K. Langmuir 1992, 8, 1695. (6) Pissis, P.; Daoukaki-Diamanti, D. J. Phys. Chem. Solids 1993, 54, 701. (7) Gun’ko, V. M.; Zarko, V. I.; Voronin, E. F.; Tischenko, V. A.; Chuiko, A. A. Langmuir 1995, 11, 2115.
S0743-7463(95)01031-6 CCC: $14.00
protonic conductors by a simple mechanism of conduction by charge carrier hops between localized states.8 One of these models has been extended to study the influence of water adsorption upon dielectric properties of powders.9 It has even been possible to apply this model to the adsorption onto clays.10 We develop here a model of this type, allowing explanation of the dielectric losses on the surface of a solid covered with vapor, in the range of low partial pressures, and the linking of these losses to classical adsorption theories. As a example, we have studied a zeolite, a wellknown material.11,12 The main interest of this work is the possibility of relating microscopic movements of the molecule to macroscopic variables, which appears to us as progress in adsorption research. II. Experimental Section Material. The experiments have been performed at 25 °C on hydrated synthetic zeolite Na-13X (supplied by Rhoˆne-Poulenc, France) in the form of pellets, 1 mm thick and 10 mm in diameter, compacted under a pressure of 2.5 × 108 Pa. Polarization Conductivity Measurements. Contacts were deposited by cathodic pulverization of gold on each face of the pellet. The samples were held in a conductivity cell, thermally controlled. The variation of the vapor pressure is controlled by a second thermostat in which a vapor reservoir is located. The value of the relative vapor pressure P/P0 depends on the difference of temperatures between the two cells.13 The pellets were left in contact with the vapor for 24 h. Ohmic resistance and capacity of the pellets were measured with an impedance meter General Radio, in a frequency range of 1-200 kHz, with a voltage of 1 V. Thermally Stimulated Depolarization Current Techniques. The measurements have been performed with a thermally relaxation map analysis (TSC/RMA) spectrometer, model 9180+, from Solomat Instruments. The method consists of the determination, following a strictly controlled temperature (8) Anderson, O. L.; Stuart, D. A. J. Am. Ceram. Soc. 1954, 37, 573. (9) Bensimon, Y.; Belougne, P.; Giuntini, J. C.; Zanchetta J. V. J. Phys. Chem. 1984, 88, 2754. (10) Giuntini, J. C.; Jabobker, A.; Zanchetta, J. V. Clay Miner. 1985, 20, 347. (11) Meier, W. M.; Olson, D. H. Atlas of Zeolithe Structure Types; Butterworth: London, 1988. (12) Dubinin, M. M.; Isirikian, A. A.; Rakhmatkariev, G. U.; Serpinskii, V. V. Izv. Akad. Nauk SSSR, Ser. Khim. 1973, 4, 934. (13) Partyka, S.; Rouquerol, F.; Rouquerol, J. J. Colloid Interface Sci. 1979, 68, 30.
© 1997 American Chemical Society
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Langmuir, Vol. 13, No. 5, 1997 1017
program, of either the current created by the return to an equilibrium state of a dielectric which has been previously polarized (TSDC) or the charge current resulting from a thermally activated transition, obtained by passing from an equilibrium state to a polarization state (TSPC). The details of the method have been previously described.14 It can be noted that this method is useful in the case of high resistivity samples, because their polarization is high enough during the return to the thermal equilibrium. The complementarity of this method with the dielectric losses measurements has been discussed recently.15 Comparing the results allows the determination of the energies involved during the processes and of the distribution law of the charge carriers.
III. Dielectric Losses Observed with Divided Solids III.1. Introduction. It is generally admitted that the solid-vapor adsorption can be considered as an equilibrium between molecules of the vapor phase and molecules interacting with some sites of the solid surfaces. Of course this equilibrium has a dynamic character. Each local arrangement of the solid site/adsorbed molecule has a lifetime, and therefore the actual adsorption process is related to a distribution of these lifetimes. Each site can be characterized by the energy given to fix the vapor molecules. The probability of desorption of a molecule from a site varies with the temperature, but decreases when the energy of the site increases. For this reason the equilibrium state corresponds to the coverage of the more energetic sites. If the adsorption site is considered as a potential well, in low coverage cases the vapor molecules are strongly linked to the surface and strongly polarized. Each pair site/molecule corresponds to a dipole. Without an external electric field, all these dipoles are randomly oriented, implying that the total resulting dipolar moment vanishes. However, when an electric field is applied, a dipolar moment appears, trying to follow the external field, against the viscosity forces of the system. The polarization current density Jp is linked to the polarization vector P by the following relation
JP ) ∂P/∂t
(1)
where t is the time and P is the measure of the dipolar moment by volume unit. When the external field is periodic, with a frequency ω, the polarization conductivity σ′(ω) has the following form, called universal.16 2
2 2
σ′(ω) ) ω′′(ω) ) A(T)(ω τ/(1 + ω τ ))
(2)
In this equation ′′(ω) is the imaginary part of the dielectric permittivity and A(T) is an experimental factor depending of the temperature T. Moreover all the dipoles have the same relaxation time τ. In an actual system, each dipole has a relaxation time. We will admit there is a distribution of the relaxation times G(τ), such as G(τ) dτ is the number of jumps corresponding to a time between τ and τ + dτ. Hence it is possible to write
σ′(ω) )
2
∫ττ G(τ) 1 +ωωτ2τ2 dτ ∞
0
(3)
Calling Pj the value of G(τ) corresponding to the value τj and putting all the terms independent of τ outside of (14) Vandershueren, J.; Gasiot, J. In Field Induced Thermally Stimulated Currents; Bra¨unlich, P., Ed.; Topics in Applied Physics, Vol. 37; Springer Verlag: Berlin, 1979. (15) Giuntini, J. C.; Vanderschueren, J.; Zanchetta, J. V.; Henn, F. Phys. Rev. 1994, 50, 12489. (16) Jonscher, A. K. Nature 1977, 267, 673.
the sum in a term B(T) one gets
∑j
σ′(ω) ) B(T)
Pj
ω2τj
(4)
1 + ω2τj2
The choice of a distribution function Pj allowing a description of dielectric losses due to adsorption on divided solids is discussed below. III.2. The Model. The model derived below is strictly equivalent to the Fowler-Guggenheim model, as recently reviewed by Everett and Rudzinsky.2 This simple model assumes there are not lateral interactions between the adsorbed molecules. Moreover the sites are equivalent. From this model, it is possible to derive the BET equation. It is then rigorously valuable for partial pressure values lower than 0.2. We will consider that the charges handled by the polarization phenomenon are due to the molecules adsorbed, which can freely move on the surface, from site to site. This is handled on the basis of a random distribution of molecules among the adsorption sites. It is the energetic distribution of molecules which determines the electric response. We consider a surface phase composed of a certain number “i” of homogeneous “subphases”, of which molecules are characterized by the same energetic state (i.e. the same chemical potential µiσ). One possible way of writing the partition function of the Nσ adsorbed molecules is as follows σ
Z ) [Ns!/Ni!(Ns - Ni)][Jiσ(T) exp(U0/kT)]Ni
(5)
where U0 is the energy of the surface layer, NS the total number of sites, Niσ the number of adsorbed molecules in the i sites, and Jiσ(T) the internal partition function corresponding to
Jiσ(T) ) exp(∆µiσ/kT)
(6)
∆µiσ is the variation of the chemical potential during the adsorption process. The number of molecules in the subphase i is defined by
Niσ ) Ns(ξi/1 + ξi) with ξi ) exp(µiσ + U0/kT)
(7)
and then the distribution in the surface layer is
Niσ/Nσ ) [exp(-µiσ + U0/kT)](1 - exp(-U0/kT))
(8)
In this model, the energy origin is taken as the molecule non-adsorbed and at rest. Therefore, the relaxation time corresponding to the change from the energetic state i (i.e., in the subphase i) to j is written, assuming a simple Maxwell-Boltzmann distribution of sites
τij ) τ0 exp(j - i/kT)
(9)
where τ0 is the phonon frequency (typical values are 10-13 to 10-14 s). The adsorption of a molecule at rest in a thermodynamic state, µj is then related to an energy, WG, as follows
(
τj ) τ0 exp
)
WG - j kT
(10)
Furthermore we assumed that a pair site/molecule is also characterized by a local organization, with some symmetry properties. This symmetry is connected with the movement of the molecule, and then with its degree of freedom. Calling γ a number linked to this degree of freedom, it can be shown that the energy of a molecule
1018 Langmuir, Vol. 13, No. 5, 1997
Giuntini et al.
trapped in a site can be written
(11)
j ) γ1,j
In this equation, 1,j is the energy value corresponding to an equivalent movement following a single degree of liberty. γ can be considered as a scale factor. A very simple relation between the distribution of energy levels (nj) and the energies can be written
nj )
N -j/kT e f(T)
(12)
(In fact, it is clear that we have j ) U0 - µiσ). In this equation, both j and f(T), the partition function, depend on the temperature; N is the total number of molecules. Carrying the definition of 1,j in (12) and assuming a weak value for 1,j, we obtain
nj )
N 1,j/kT -γ ] [e f(T)
(13)
By the same way, eq 10 gives
1,j )
()
τj WG kT ln γ γ τ0
(14)
Consequently, one obtains
[
nj ) b(T) 1 +
( )]
τj WG 1 - ln γkT γ τ0
-γ
(15)
In this equation, b(T) contains all the terms independent of the relaxation times. The polarization Pj due to the dipoles is proportional to the number of adsorbed molecules, then
[
Pj ) a(T) 1 +
( )]
τj WG 1 - ln γkT γ τ0
-γ
(16)
a(T) is a factor independent on the relaxation time. This expression of Pj leads to an expression of the polarization conductivity depending on two parameters, characteristics of the pair solid/vapor. This relation has the general form
∫τ
σ′(ω) ) Φ(T)
[
τ∞ 0
( )]
WH 1 τ - ln γkT γ τ0
ω2τ dτ 1 + ω2τ2
Figure 1. Variation of the polarization conductivity versus the frequency (log-log scale) obtained for zeolite NaX at 25 °C.
Tp, called the polarization temperature. (ii) The whole system is cooled at a temperature T0 (77 K). (iii) Keeping the temperature constant, one modifies the polarization tension. Vp becomes Vd, the depolarization tension. In our case, we have taken Vd ) 0. (iv) With the same depolarization tension Vd, the system is heated, and one records the current created by the depolarization of the solid. When the tension Vd is zero, the current is the result of the return of the sample to an equilibrium state. This method is useful in the case of samples of high resistivity, because their polarization is sufficiently high during the return to the thermal equilibrium. The complementary of this method with the dielectric losses measurement method has been discussed recently.15 Comparing the results allows determination of the energies evolved during the processes and the distribution law of the charge carriers. V. Experimental Results
-γ
(17)
Φ(T) is a function grouping all the terms independent on τ and WH is the following sum:
WH ) γkT + WG
(18)
A numerical integration of (17) can be performed. We obtain a general law of the following form:
σ′(ω) ) A(T)ωS(T,ω)
(19)
Then, the determination of S(T,ω) allows determination of the parameters WG and γ. In this model, WH can be obviously related to the enthalpy of adsorption. On the other hand γ has clearly an entropic character.
V.1. Polarization Conductivity. Figure 1 shows an example of the polarization conductivity as a function of frequency (in a log-log scale). The conductivity is an increasing function of frequency. Measurements are performed in the case of this work as a function of the relative pressure P/P0. The polarization conductivity always increases with P/P0.17 In Figure 2 is pictured a typical variation of the polarization conductivity versus the partial vapor pressure of water. The temperature of the experiment is 300 K and the frequency has been fixed at 5 and 20 kHz. V.2. TSDC Measurements. In Figure 3 are reported the depolarization current density, measured as a function of the heating temperature. One observes a double peak. This implies the existence of two different types of dipoles in these experimental conditions.
IV. Thermally Stimulated Currents
VI. Analysis of the Results
The measurement is as follows: during a programmed heating process, the current is produced in an electric circuit by a solid previously polarized and placed between the armatures of a condensator. There are four steps in the process: (i) One applies a continuous polarization tension, Vp, to the sample, maintained at the temperature
The similar behavior between the experimental curves reporting the conductivity σ′(ω) versus P/P0 and typical adsorption isotherms as measured by Dubinin et al. is a confirmation of eq 17 relating the current to the number (17) Mouton-Chazel, V. Thesis, Montpellier Sciences, 1994.
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Langmuir, Vol. 13, No. 5, 1997 1019
Figure 4. Variation of WG (black triangles) and of the adsorption free enthalpy measured by Dubinin et al. (open circles) versus the partial pressure of water.
Figure 2. Variation of the logarithm of the polarization conductivity of zeolite NaX at 25 °C versus the partial pressure of water for two frequencies (square, 5 kHz; circles, 20 kHz).
Figure 3. Variation of the thermally stimulated depolarization current versus the temperature for a zeolite NaX at 25 °C and a water partial pressure of 0.8.
of charge carriers, i.e., in our model the adsorbed molecules. Of course the type of charge carriers depends on the experimental frequency. For the high frequencies the atoms of the solid act like dipoles; this explains the difference between the curves obtained at different frequencies. Hence for high frequencies, there is a difference between the quantity of dipole observed and the number of adsorbed molecules. A second point must be emphasized: experimentally it will be difficult to estimate the surface area involved by a pellet during the experiment. These two points are a difficulty if one wants to use the dielectric method as a quantitative method of measurement of adsorption isotherms. On the other hand, classical adsorption methods cannot give indications about modification of the energetic of the solid surface atoms during the adsorption process. Figure 4 shows the results obtained with our analysis for given values of γ and WG. The experimental curves have been fitted by varying WG and taking for γ a constant value: γ ) 2. The term γ corresponds to a surface heterogeneity, implying
that the retained value corresponds to the simplest model of the surface topology (i.e., no roughness) and energy. We obtained for every value of the partial pressure a value of WG. Comparisons have been made between these values and the surface Gibbs free energy of adsorption as measured by Dubinin et al.12 Differences are observed. At this stage of the work it seems difficult to state if these differences are due to an experimental error or to the influence of the roughness of the interface or to the simplifications of our model. In the quoted paper are also reported the values of the differential heat of adsorption. One observes two plateaus corresponding to two types of adsorption sites. This result is in good agreement with the results obtained by TSDC. Obviously this agreement suggests a value of γ more complex that the one we used. But at the present step of the analysis, it is too difficult18 to quantitatively interpret these results of TSDC, to obtain values of WH which are useful in our model. At this stage of the work, we want only to show that coupling TSDC and polarization measurement could be an efficient tool to study adsorption mechanisms. VII. Conclusion This work tries to demonstrate that it is possible to use electric measurements to analyze adsorption phenomena. By using a very simple polarization model, based on the hopping process of charge carriers between localized states, it is possible to link the dielectric losses due to adsorption to classical adsorption thermodynamic equations. Moreover the combination of dielectric losses experiments and thermally stimulated current experiments (TSDC) appears complementary and very fruitful as an alternative way to study solid surfaces. Two parameters appear in our treatment, WH and WG, which are clearly related to the adsorption enthalpy and the adsorption free enthalpy, respectively. This work is under development. The present approach combined with classical adsorption methods could be an efficient tool to determine microscopic arrangements on the surface of divided solids. LA951031O (18) Ibar, J. P. Fundamentals of Thermal Stimulated Current and Relaxation Map Analysis; SLP: New Canaan, 1993.
