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NMR spin echo studies on mobility and diffusion of benzene adsorbed on silica
NMR Spin Echo Studies on Mobility and Diffusion of Benzene Adsorbed on Silica B. B O D D E N B E R G , R. HAUL, ~ AND G. O P P E R M A N N 2 Institut fi~r Physikalische Chemie und Elektrochemie, Technische Universitdt Hannover,s Federal Republic Germany Received December 30, 1970; accepted February 24, 1971 Diffusion coefficients of benzene in porous plugs compressed from Aerosil were measured by the spin echo technique with the use of constant and pulsed field gradients. In order to obtain surface self-diffusion coefficients D"* the contribution from molecular mobility in the pore volume has been evaluated. D'* shows a maximum near 0 = 1 in agreement with earlier results obtained from rates of adsorption. Differences between the limiting experimental diffusion coefficients in the region of capillary condensation and the bulk liquid are attributed to the tortuosity of the pore system. Activation energies for surface self-diffusion are larger than for T2.
INTRODUCTION
of liquids. Only recently this method has been applied to adsorbed phases 4 (2-5). I n previous publications (4, 5) no corrections had beeh made for the contribution from mobility of molecules in the pore volume. Thus only overall diffusion coefficients were obtained. Furthermore the effect of the tortuosity of the pore system has to be considered.
Nuclear magnetic resonance (nmr) relaxation studies of adsorbed molecules have been repeatedly carried out in recent years. Efforts have been made to separate the different contributions to the relaxation rates, viz., intra- and intermolecular proton-proton interactions, interactions of the adsorbate protons with paramag~aetie impurities and surface OH groups. Although a detailed relaxation analysis has been carried out for benzene adsorbed on silica gel (i) it is not possible to evaluate diffusion coefficients from such data. Even if a particular diffusion model is assumed calculations would require the knowledge of jump times as well as jump distances on the energetically heterogeneous surface. Such detailed information is not as yet obtainable. Diffusion coefficients of adsorbed molecules can, however, be directly determined from nmr spin echo measurements, which are frequently used in self-diffusion studies
EXPERIMENTAL Materials. Samples of high surface area silica consisting of spherical particles (Aerosll 200 and 380, Degussa, Frankfurt) were compressed in Pyrex glass tubes. After removal of the cracked tubes the plugs were transferred to the nmr sample holder. The properties of the porous plugs are listed in Table I. The adsorbents are well suited for nmr studies since the concentrations of paramagnetic impurities are sufficiently low and the texture of the porous media, although complex, is 4 The following studies have been recently carried out in the NMI~ Laboratory, Section Physics of the Karl-Marx-University, Leipzig, D.D.R. D. Freude has studied diffusion in the systems n-hexane, diethylether/silica gel and J. Kaerger in the system It20/zeolites. Personal communication.
i Dedicates this paper to Stephen Brunauer. 2 Part of thesis C-. Oppermann TU IIannover
(1971). s 3 ttannover, Callinstr. 46.
Journal of Colloid and Interface Science, Vol. 38, No. 1, J a n u a r y 1972
210
Copyright @ 1972 b y Academic Press, Inc.
211
STUDIES ON MOBILITY AND DIFFUSION OF BENZENE TABLE I POl~OUS S I L I C A
P R O P E R T I E S OF BET ~,~. Mean Plug Type of ~z~ surface ~'particle No. m Aerosil - (m~/gm) area ' diameter (~)
1 2 3
200 !'380 :200
170 360 170
160 80 160
OH groups per° 100 A2
Mass of(gin) plug
Diem/length of plug (me)
Porosity e
3 to 4 (17) 2 to 3 (17) --
0.20 0.12 0.21
6.5/7.0 6.0/6.0 5.4/17
0.50 0.68 0.78
better defined than for precipitated silica gels. Sample 1 and 2 were compressed at room temperature; sample 3 was compressed at about 550°C. The plugs were degassed at 10-5 torr and at the temperatures listed in Table I for about 60 hours. Benzene, p.a. (Merck, Darmstadt), was thoroughly degassed and dried with molecular sieve Linde 5 A. Nuclear
Magnetic
Resonance--Technique.
N m r relaxation times T, and T2 as well as diffusion coefficients were measured with a 60 MHa spin echo spectrometer (BrukerPhysik, Karlsruhe). For measurements at low surface coverages a time average computer (Fabri-Tek, 1052 LSH, Madison, Wisconsin) was used to improve the signal-tonoise ratio. For TI, pulse sequences rr/2 -- rr/2 -- rr, for T~, rr/2 -- rr, were used. In cases where T2 was larger than about 50 msec the Car> Purcell technique was applied since the H a h n method was no longer reliable owing to the influence of diffusion. Diffusion coefficients for surface coverages 0 > 0. 5 have been measured according to the constant field gradient method (6). The spin echo attenuation due to Tp-relaxation and diffusion is given by ~./(-%)
=
• exp
M(0)
T2
12
PLUGS
2GPD.(Pr) a ,
[1]
where r i~ the distance between the re/2 -- rrpulses, M ( 2 r ) and M ( 0 ) are the spin echo amplitude at 2r and zero time, respectively, is the gTromagnetic ratio, G the linear field gradient, and D* the self-diffusion coefficient.
Mean hydraulo. Degass. radius (A) temp. (°C)
27 27 85
170 200 450
Conveniently, the spin echo amplitude is measured as a function of G at a fixed pulse distance r. D* is obtained from a plot of log M (Pr) against G2. Linear field gradients were used up to 4 gauss/cm. In order to measure diffusion coefficients at low surface coverages 0 < 0.5 when relaxation tfines T~ are relatively small, pulsed field gradients were used up to 64 gauss/cm and pulse widths up to 5 msec. With respect to the more complicated evaluation of diffusion coefficients from these experiments we refer to the literature (7). Measurements were carried out at several fixed distances of the rr/2 - rr-pulses, resulting in the same D* values within experimental error. Furthermore, with both methods the same diffusion coefficients were obtained in the range of coverage 0.5 < 0 < 1.5. The field gradients were calibrated with a 1/1000n aqueous CuSO4 solution using the diffusion coefficient for pure water 2.3 X 10-~ cm 2sec at 25°C (8). The porous silica plugs were contained in a Pyrex glass tube which could be separated from the vacuum and gas handling system by means of a valve with a Teflon seat. The dead space was kept small by insertion of a glass rod which could be removed while the sample was degassed or loaded with the adsorptive. After adsorption of benzene at a known temperature and equilibrium pressure the valve was closed and measurements were made at various temperatures. Because of the small dead space the surface coverage remained practically constant. Only above q-50°C was desorption noticeable as could be seen from a decrease in signal intensity larger than to be expected from Curie's law.
Journal of Colloid and Interface Science, Vol. 38, No. 1, J a n u a r y 1972
212
BODDENBERG, HAUL, AND OPPERMANN RESULTS T2
Adsorption and Deso~\ption Isotherms. T h e a m o u n t of benzene adsorbed was determined from the spin echo heights extrapolated to zero time using calibrated attenuators. T h e isotherms obtained with plug 1 and plug 2 at 23°C are shown in Fig. 1, in which the signal intensity is given in arbitrary units. F r o m a B E T plot the monolayer capacity n~ ~ and thus 0 values are calculated. The absolute values for n~ ~ (plug 1: 0.7, plug 2: 1.6 m m o l e s / g m ) were determined from adsorption of known amounts of benzene. For plug 1 b o t h adsorption and desorption branches were measured which join at about 0 = 1 exhibiting the usual hysteresis loop. T h e isotherm thus determined was practically identical with the isotherm previously measured for a plug with the same properties and p r e t r e a t m e n t (5). Within experimental error no difference was found for the adsorption and desorption branches with respect to n m r relaxation times and diffusion coefficients. Relaxation Times. Figure 2 shows the re-
0
-
t
I
-60
t
I,
i
z -~0
l plug
1, /
J
/
-20
0.2
O.t,
I 200
0.6 =, p/Po
0.8
Fie. 1. Adsorption and desorption isotherms for benzene adsorbed on silica plugs at 23°C. Saturation pressure of liquid benzene, po = 78.1 torr. Plug 1: ©--adsorption; O--desorption. Plug 2 : [~--adsorption. Journal of Colloid and Interface Science, ¥ o l . 38, N o . J, J a n u a r y
~
plug3
Esec]
plufl
1
l (
150
100
5O
FIG. 2. Relaxation times T1 and T2 for benzene adsorbed on silica plugs at 23°C. Plug 1:A--T1 ; ©--T~ . Plug 2: D--T~ • Plug 3:~)--T2 •
tlS i
TI
Cmsecl
laxation times TI and T2 as a function of surface coverage at 23°C, For plug 1 T2 increases from 5 msec at 0 = 0.04 to 250 msec at about 0 = 9. For plug 2 the initial sharp rise of T~ from 2 msee at 0 = 0.02 to 10 msec at 0 = 0.1 is followed b y a smaller linear increase to about 45 msec at 0 = 3. For plug 3 the dependence of T~ on surface coverage was not studied in detail and only two values are shown in Fig. 2. T1 increases monotonously from 1.5 sec at 0 = 0.1 to 6 sec at about 0 = 9 for plug 1. T h e T1 values for plug 2 (not shown in Fig. 1) are almost the same as for plug 1 in the range 0 --- 0.02 to 1. The t e m p e r a t u r e dependences of T~ and T2 for the three plugs were studied in the t e m p e r a t u r e ranges and at the surface coverages listed in Table I I . Only activation energies for T2 are given since the corresponding values for Tt (about 1.6 kcal/mole for plug 1) are uncertain and too low because the temperatures were in the vicinity of a T~ minimum. For plug 3 with a m i n i m u m at 56°C an activation energy of 2.3 kcal/mole for T1 was estimated. Diffusion Coe~cients. The diffusion coefficients at 23°C as a function of surface coverage are shown in Fig. 3. The data for plug 1 and 2 were obtained from measurements with constant field gradients (4, 5). For plug 1 these measurements were confirmed and supplemented b y data obtained
1972
213
STUDIES ON MOBILITY AND DIFFUSION OF BENZENE TABLE II ACTIV.~TION ENERGIES
FOR t~ELAXATION TIME
T 2 ±kND D I F F U S I O N
~OEFFICIENTS
Activation energies (kcal/mole) Plug No.
Surface coverage
Temp. range (°C)
1 1
0.5 1
1 2
2
+10- +35 --5- +35 --15- +35 0- +62
1
T2
3.4 2.7 2.7 1.8
± 444-
D*(nmr)
0.2 0.2 0.2 0.1
with pulsed gradients. Above 0 = 0.5 agreem e n t was found between the two methods within the experimental error of about 4-10 % whereas at lower surface coverages only the pulse technique could be used with an error increasing to about 4-20% at the lowest surface coverages measured. For comparison diffusion coefficients of benzene in silica gel at 25°C published b y Freude (3) as well as the self-diffusion coefficient of liquid benzene (9) are included in Fig. 3. The activation energies derived front the experimental diffusion coefficients D* are listed in Table I I . D ISCUSS ION I n contrast to methods in which diffusion is studied under the influence of a gradient in chemical potential, measurements with the spin echo technique are carried out under equilibrium conditions thus resulting in selfdiffusion coefficients D*. The diffusion process in porous media is complicated b y the fact t h a t mobility of the molecules in the pore volume as well as in the adsorption layer has to be considered. As we are primarily interested in the mobility of adsorbed molecules the problem arises to estimate the contribution from displacements of molecules in the gas phase. K r u y e r (10) has treated the following model. On the average a molecule spends a time r in the adsorption layer and after desorption a time t' in the pore volume before being readsorbed. During these periods the corresponding mean-square displacements are Z, 2 and Zg 2, respectively. Applying the Einstein equation to the overall diffusion process in the pore system we obtain :
4.9 4.6 4.0 4.3
4444-
D s* (nmr)
0.5 0.5 0.5 0.5
3.5 3.5 3.5 3.5
DS*(kin)
4- 0.5 4- 0.5 4- 0.5 4-' 0.5
4.0 4- 0.2 3.2 4- 0.2 -
-
--
T
£¢m2sedt]|
2
t ,
plug 1
, I
[
~×
/
i
|
2
0
Z,
6
8
FIG. 3. Nmr spin echo diffusion coefficients (D*) and surface self-diffusion coefficients (D ,*) for benzene adsorbed on silica plugs at 23°C. Plug 1: © ~ * ; • D ~*. Plug 2: O--D*; []--D ~*. Silica gel (3); X--D*. -
D* -
-
-
-
m
Z2 -
Z~ ÷
2t
2(t' + r)
Z~2
[2]
With Zg2/2t ' = D ° and Z~2/2r = D ~*, where D ° and D ~* are the self-diffusion coefficient in the gas phase and in the adsorption layer, respectively, D* -
t' DO + r D~, t' + ~ t' + ~ "
[3]
Under the assumption t h a t the molecules behave statistically and t h a t the adsorbent is homogeneously packed the ratio t ' / r is equal to the ratio of the n u m b e r of molecules in the pore volume N ° and in the adsorption layer N ~. Thus from Eq. [3] D* -
Ng-- D °
No + N~
N~
+ / g o + N~
D ~*
• [4]
If the temperature is not too high and the
Journal of Colloid and Interface Science, Vol. 38, No. 1, J a n u a r y 1972
214
BODDENBERG, HAUL, AND OPPERMANN
diameter is sufficiently small, as is true in the present case, then we have N g << N 8 and consequently D* = Ng ~ D ~ + D ~*.
[5]
The ratio N g / N ~ can be taken directly from the adsorption isotherm. If the mean free path of the molecules is large compared with the mean pore d i a m e t e r - - a condition which is valid in the present experiments--the gas phase diffusion coefficient D ~ is identical with the Knudsen diffusion coefficient representing a self-diffusion coefficient. Once D ~ is known, the surface self-diffusion coefficient D "* can be evaluated from the spin echo diffusion coefficient D*. When diffusion coefficients are measured with the nmr spin echo method using constant or pulsed field gradients circumstances m a y arise where Eq. [4] is not valid. KBrger (11) has discussed quite generally the effect of two-phase diffusion on the diffusion-controlled damping of the spin echo amplitudes. I t follows from his analysis that Eq. [4] holds true if rapid exchange of molecules between the two phases occurs. This is the case for the system studied in the present investigation. 5 In earlier studies (12) flow experiments with helium and benzene were carried out with a porous silica plug practically identical with plug 1 in every respect. For this plug a Knudsen diffusion coefficient of benzene D ~ 5.2 X 10-8 cm2/sec at 30°C was determined which can be used to calculate surface self-diffusion coefficients. I t can be seen in Fig. 3 (dashed line) t h a t the correction for molecular mobility in the pore voluume results in a lowering of the experimental diffusion coefficient, e.g., of about 25% at 0=1. Before discussing the results obtained with plug 2 it is instructive to examine the diffusion coefficients at the highest amount adsorbed. I t can be seen in Fig. 3 t h a t D* approaches values which for plug 1 are disT h e v a l u e of ~/~a2g2Dir~ ( r e f e r e n c e 11) is of t h e o r d e r 10-4.
tinetly, for plug 2 only slightly lower than, for bull; liquid benzene. From the adsorption isotherms (Fig. 1 ) it is obvious t h a t the pore space is mostly filled with condensate under these conditions. On the assumption that the diffusion coefficients in the condensate are the same as in the bulk liquid the lowering of the experimental diffusion coefficient for plug 1 can be attributed to the tortuosity of the pore system. In the spin echo experiments the molecular displacement is measured in the direction of the field gradient and not along the actual paths in the pores. In this way a tortuosity factor K(eond) = 1.8 is obtained for plug 1 from the ratio of the limiting D* value and D* (liquid). On the other hand a Knudsen diffusion coefficient can be calculated for a model system consisting of a bundle of parallel cylindrical capillaries with a radius equal to twice the hydraulic radius of the actual porous system (13) D~(eyl) = g8 x~
/2~
[o]
where e is the porosity, A the surface area per unit volume of the plug, and M the molar mass of benzene. With the data for plug 1 (Table I ) D ~ (cyl) is 1.08 X 10-2 cm2/ sec, and thus with the above value for D ~ a tortuosity factor K(gas) = 2.1 is obtained in fair agreement with K(cond). Since plug 2 has almost the same hydraulic radius as plug 1 the difference in the limiting values for D* must be attributed to a different tortuosity factor. Although the diffusion experiments were not carried out up to 0 values as high as for plug 1 a value of about 1.2 can be estimated. Accordingly the correction for molecular mobility in the pore volume has been calculated with a D ~ value larger than for plug 1 by a factor 1.8/1.2. The resulting curve 6 for the surface self-diffusion coefficient While the correction for gas phase mobility has been calculated for the two plugs with the relevant K (cond) = K (gas) values, the tortuosity factor for surface mobility K (surf) is assumed to be the same for both plugs. For a system packed
Journal of Colloid and Inlet face Science, Vol. 38, No. 1, January 1972
STUDIES ON MOBILITY AND DIFFUSION OF BENZENE is shown in Fig. 3. I t can be seen that for both porous plugs the dependence of D* on surface coverage is rather similar showing maxima near 0 = 1. At low coverages the molecules are located preferentially at strong adsorption sites with long residence times resulting in small T~ values (Fig. 2) and diffusion coefficients. As 0 becomes larger more and more weaker sites come into play leading to an increase in D 8.. With rising coverage above 0 = 1 the diffusion process is determined more and more by molecular mobility in the second and higher layers and in the capillary condensate, the formation of which becomes noticeable in this region (Fig. 1). Since no differences between the diffusion coefficients have been detected on the adsorption and desorption branches of the adsorption isotherm, as mentioned previously, it follows that the diffusion coefficients in the multilayer region cannot be significantly different from those in the capillary condensate. Since the measured apparent diffusion coefficient in the capillary condensate is smaller than in the adsorption layer near 0 = 1 a maximum in the D ~* curve occurs. Since the molecular mobility in the liquid phase is smaller than in the adsorption layer near 0 = 1 taking into account the tortuosity factor for surface diffusion6 the maximum in the D ~* curve observed in the present case can be visualized. Whether or not a maximum in the D ~* -- 0 curve is exhibited depends not only on molecular mobility in the adsorption layer and the capillary condensate b u t also on the tortuosity factors for the transport processes in these phases (6). Comparing the D ~* values for plug 1 and 2 one has to consider that the density of surface OH-groups is smaller for plug 2 (Aerosil 380) than for plug 1 (Aerosil 200). I t is well known (15) that a specific interacfrom spherical particles K (surf) can expected to be in the order of about 2 (14). The D~* curve shown in Fig. 3 is not corrected for the tortuosity of surface diffusion; this would result in correspondingly larger diffusion coefficients.
215
tion occurs between OH-groups and the ~r-electron system of benzene leading to relatively large energies of adsorption on such sites. Therefore, the surface diffusion coefficient is greater for plug 2 than for plug 1. On the basis of these arguments one would expect D ~* to rise already at relatively small values in the case of plug 2 with low OHgroup density. However, one has to consider another t)~pe of adsorbed molecules located in the region of contact between the spherical silica particles where the benzene molecules are strongly adsorbed owing to overlapping potential fields. Since the mean particle diameter of Aerosil 380 is smaller (Table I ) these centers are more frequent and therefore of greater influence for plug 2 than for plug 1 explaining that the rise in D ~* does not occur already at lower coverages. The strong adsorption centers other than OHgroups discussed in reference 1 could correspond to this type of sites. For comparison the diffusion coefficients of benzene in silica gel recently published by Freude (3) are included in Fig. 3. The lower limiting D* value found for silica gel as compared to the plugs compressed from Aerosil is due to the tortuosity factor, which is known to be larger for silica gel type adsorbents (14). In an earlier paper (5) we have calculated surface self-diffusion coefficients of benzene in porous silica plugs from rates of adsorption. The shape of the D e* versus ~ curves is almost the same as those obtained by the nmr spin echo method. The D ~* values from kinetic measurements are, however, lower b y a factor of about 1.7. Obviously the influence of the complex pore texture on the diffusion coefficient obtained from kinetic and nmr measurements is more involved than can be allowed for by the above tortuosity treatment. On the whole the fair agreement between the two methods is encouraging. From the temperature dependence of the experimental diffusion coefficient D* apparent activation energies of about 4 to 5 kcal/mole can be evaluated for 8 values be-
Journal of Colloid and Interface Science, Vol. 38, No. 1, J a n u a r y 1972
216
BODDENBERG, HAUL, AND OPPERMANN
tween 0.5 and 2 (Table I I ) . A value of the same order can be estimated from the data given b y Freude (3) for 0 = 1.5. Correction for molecular mobility in the pore volume, however, leads to a distinctly smaller activation energy of about 3.5 k c a l / mole for surface self-diffusion, which is in fair agreement with the activation energies for D ~* obtained earlier from rates of adsorption (Table I I ) . This value amounts to approximately one-third of the corresponding isosteric heats of adsorption (12). I n the range 0.5 to 2 little change is observed for the heats of adsorption as well as the activation energy for D ~*. Below 0 = 0.5 the energies of activation for D ~* obtained from rates of adsorption increase from 4.0 to 6.4 k c a l / m o l e between 0 = 0.5 and 0.07. T h e heats of adsorption on the energetically heterogeneous adsorbent increase b y almost the same amount. I t is noteworthy t h a t the activation energies for surface self-diffusion are greater t h a n for the relaxation times T2. This finding is not clear in the light of the relaxation analysis carried out b y 5lichel (16) and the discussion given b y Freude 4 based on the assumption of two types of adsorption sites. This point requires further investigation. ACKNOWLEDGMENT The authors gratefully acknowledge the financial support of this investigation by the "Deutsche Forschungsgemeinschaft" and "Fends der Chemischen Industrie."
REFERENCES 1. WINKLER, H., NAGEL, M., MICHEL, D., AND PFEIFER, I~., in "Tagung ttochfrequenz-
spektroskopie der Physikalischen Gesellschaft der D.D.R. 1969." Leipzig, 1969. 2. BECKERT, D., Phys. Lett. A25,502 (1967). 3. FREUDE, D., Z. Phys. Chem. (Leipzig) 242,
57 (1969). 4. BODDENBERG, B.~ HAUL, R., AND OPPERMANN, G., Naturwissenschaften 56,635 (1969). 5. BODDENBERG, B., ]:IAUL, R., AND OPPERMANN, G., Surface Sci. 22, 29 (1970). 6. CARR, ~-L Y., AND PVRCELL, E. M., Phys. Rev.
94,630 (1954). 7. STEJSKAL,E. 0., AND TANNER, J. E., J. Chem. Phys. 42,288 (1965).
8. Mc CALL, D. W., AND DOVOLASS, D. C., J. Phys. Chem. 69, 2001 (1965). 9. DIETRICH, W., GROSS~ B., AND KOSFELD, R., Z. Naturforsch., A 25, 40 (1970). 10. KRUYER,S., Prec. Koninkl. Ned. Akad. Wetenschap. Prec. Set. B 56,274 (1953). 11. KXRGER,J., Ann. Physik (Leipzig) 24, 1 (1969). 12. HAu~, R., AND BODDENRER% B., Z. Phys. Chem. (Frankfurt) NF64, 78 (1969). 13. BARRER, R. M., AND GABOR, T., Prec. Roy. Soc. Set. A 251,353 (1959); BARRER, R. M., in E. A. Flood, Ed. "The Solid-Gas Inter-
face," Vol. 2, p 557. Dekker, New York, 1967. 14. CARMAN, P. C., "Flow of Gases through Porous Media." Butterworths, London, 1956. 15. KISELEV, A. V., Zh. Fiz. K h i m . 35,233 (1961). (Russ. Y. Phys. Chem. (English Transl. 35, 111 (1961)); BA~RER, R. M., Or . Colloid Interface Sci. 21,415 (1966). 16. MICHEL,D., Z. Naturforsch., A 23,339 (1968). 17. BOEH~, H. P., AND KAEMPE, G., Z. Phys. Chem. (Frankfurt) [NF] 23, 257 (1960); BODE, R., FERCH, H., AND FRATZSCHER,H., Kautschuk Gummi. Kunststoffs 20,578 (1967).
Journal of Colloidand Inter/ace Science, Vol. 38, No. I. January 1972
INTRODUCTION
of liquids. Only recently this method has been applied to adsorbed phases 4 (2-5). I n previous publications (4, 5) no corrections had beeh made for the contribution from mobility of molecules in the pore volume. Thus only overall diffusion coefficients were obtained. Furthermore the effect of the tortuosity of the pore system has to be considered.
Nuclear magnetic resonance (nmr) relaxation studies of adsorbed molecules have been repeatedly carried out in recent years. Efforts have been made to separate the different contributions to the relaxation rates, viz., intra- and intermolecular proton-proton interactions, interactions of the adsorbate protons with paramag~aetie impurities and surface OH groups. Although a detailed relaxation analysis has been carried out for benzene adsorbed on silica gel (i) it is not possible to evaluate diffusion coefficients from such data. Even if a particular diffusion model is assumed calculations would require the knowledge of jump times as well as jump distances on the energetically heterogeneous surface. Such detailed information is not as yet obtainable. Diffusion coefficients of adsorbed molecules can, however, be directly determined from nmr spin echo measurements, which are frequently used in self-diffusion studies
EXPERIMENTAL Materials. Samples of high surface area silica consisting of spherical particles (Aerosll 200 and 380, Degussa, Frankfurt) were compressed in Pyrex glass tubes. After removal of the cracked tubes the plugs were transferred to the nmr sample holder. The properties of the porous plugs are listed in Table I. The adsorbents are well suited for nmr studies since the concentrations of paramagnetic impurities are sufficiently low and the texture of the porous media, although complex, is 4 The following studies have been recently carried out in the NMI~ Laboratory, Section Physics of the Karl-Marx-University, Leipzig, D.D.R. D. Freude has studied diffusion in the systems n-hexane, diethylether/silica gel and J. Kaerger in the system It20/zeolites. Personal communication.
i Dedicates this paper to Stephen Brunauer. 2 Part of thesis C-. Oppermann TU IIannover
(1971). s 3 ttannover, Callinstr. 46.
Journal of Colloid and Interface Science, Vol. 38, No. 1, J a n u a r y 1972
210
Copyright @ 1972 b y Academic Press, Inc.
211
STUDIES ON MOBILITY AND DIFFUSION OF BENZENE TABLE I POl~OUS S I L I C A
P R O P E R T I E S OF BET ~,~. Mean Plug Type of ~z~ surface ~'particle No. m Aerosil - (m~/gm) area ' diameter (~)
1 2 3
200 !'380 :200
170 360 170
160 80 160
OH groups per° 100 A2
Mass of(gin) plug
Diem/length of plug (me)
Porosity e
3 to 4 (17) 2 to 3 (17) --
0.20 0.12 0.21
6.5/7.0 6.0/6.0 5.4/17
0.50 0.68 0.78
better defined than for precipitated silica gels. Sample 1 and 2 were compressed at room temperature; sample 3 was compressed at about 550°C. The plugs were degassed at 10-5 torr and at the temperatures listed in Table I for about 60 hours. Benzene, p.a. (Merck, Darmstadt), was thoroughly degassed and dried with molecular sieve Linde 5 A. Nuclear
Magnetic
Resonance--Technique.
N m r relaxation times T, and T2 as well as diffusion coefficients were measured with a 60 MHa spin echo spectrometer (BrukerPhysik, Karlsruhe). For measurements at low surface coverages a time average computer (Fabri-Tek, 1052 LSH, Madison, Wisconsin) was used to improve the signal-tonoise ratio. For TI, pulse sequences rr/2 -- rr/2 -- rr, for T~, rr/2 -- rr, were used. In cases where T2 was larger than about 50 msec the Car> Purcell technique was applied since the H a h n method was no longer reliable owing to the influence of diffusion. Diffusion coefficients for surface coverages 0 > 0. 5 have been measured according to the constant field gradient method (6). The spin echo attenuation due to Tp-relaxation and diffusion is given by ~./(-%)
=
• exp
M(0)
T2
12
PLUGS
2GPD.(Pr) a ,
[1]
where r i~ the distance between the re/2 -- rrpulses, M ( 2 r ) and M ( 0 ) are the spin echo amplitude at 2r and zero time, respectively, is the gTromagnetic ratio, G the linear field gradient, and D* the self-diffusion coefficient.
Mean hydraulo. Degass. radius (A) temp. (°C)
27 27 85
170 200 450
Conveniently, the spin echo amplitude is measured as a function of G at a fixed pulse distance r. D* is obtained from a plot of log M (Pr) against G2. Linear field gradients were used up to 4 gauss/cm. In order to measure diffusion coefficients at low surface coverages 0 < 0.5 when relaxation tfines T~ are relatively small, pulsed field gradients were used up to 64 gauss/cm and pulse widths up to 5 msec. With respect to the more complicated evaluation of diffusion coefficients from these experiments we refer to the literature (7). Measurements were carried out at several fixed distances of the rr/2 - rr-pulses, resulting in the same D* values within experimental error. Furthermore, with both methods the same diffusion coefficients were obtained in the range of coverage 0.5 < 0 < 1.5. The field gradients were calibrated with a 1/1000n aqueous CuSO4 solution using the diffusion coefficient for pure water 2.3 X 10-~ cm 2sec at 25°C (8). The porous silica plugs were contained in a Pyrex glass tube which could be separated from the vacuum and gas handling system by means of a valve with a Teflon seat. The dead space was kept small by insertion of a glass rod which could be removed while the sample was degassed or loaded with the adsorptive. After adsorption of benzene at a known temperature and equilibrium pressure the valve was closed and measurements were made at various temperatures. Because of the small dead space the surface coverage remained practically constant. Only above q-50°C was desorption noticeable as could be seen from a decrease in signal intensity larger than to be expected from Curie's law.
Journal of Colloid and Interface Science, Vol. 38, No. 1, J a n u a r y 1972
212
BODDENBERG, HAUL, AND OPPERMANN RESULTS T2
Adsorption and Deso~\ption Isotherms. T h e a m o u n t of benzene adsorbed was determined from the spin echo heights extrapolated to zero time using calibrated attenuators. T h e isotherms obtained with plug 1 and plug 2 at 23°C are shown in Fig. 1, in which the signal intensity is given in arbitrary units. F r o m a B E T plot the monolayer capacity n~ ~ and thus 0 values are calculated. The absolute values for n~ ~ (plug 1: 0.7, plug 2: 1.6 m m o l e s / g m ) were determined from adsorption of known amounts of benzene. For plug 1 b o t h adsorption and desorption branches were measured which join at about 0 = 1 exhibiting the usual hysteresis loop. T h e isotherm thus determined was practically identical with the isotherm previously measured for a plug with the same properties and p r e t r e a t m e n t (5). Within experimental error no difference was found for the adsorption and desorption branches with respect to n m r relaxation times and diffusion coefficients. Relaxation Times. Figure 2 shows the re-
0
-
t
I
-60
t
I,
i
z -~0
l plug
1, /
J
/
-20
0.2
O.t,
I 200
0.6 =, p/Po
0.8
Fie. 1. Adsorption and desorption isotherms for benzene adsorbed on silica plugs at 23°C. Saturation pressure of liquid benzene, po = 78.1 torr. Plug 1: ©--adsorption; O--desorption. Plug 2 : [~--adsorption. Journal of Colloid and Interface Science, ¥ o l . 38, N o . J, J a n u a r y
~
plug3
Esec]
plufl
1
l (
150
100
5O
FIG. 2. Relaxation times T1 and T2 for benzene adsorbed on silica plugs at 23°C. Plug 1:A--T1 ; ©--T~ . Plug 2: D--T~ • Plug 3:~)--T2 •
tlS i
TI
Cmsecl
laxation times TI and T2 as a function of surface coverage at 23°C, For plug 1 T2 increases from 5 msec at 0 = 0.04 to 250 msec at about 0 = 9. For plug 2 the initial sharp rise of T~ from 2 msee at 0 = 0.02 to 10 msec at 0 = 0.1 is followed b y a smaller linear increase to about 45 msec at 0 = 3. For plug 3 the dependence of T~ on surface coverage was not studied in detail and only two values are shown in Fig. 2. T1 increases monotonously from 1.5 sec at 0 = 0.1 to 6 sec at about 0 = 9 for plug 1. T h e T1 values for plug 2 (not shown in Fig. 1) are almost the same as for plug 1 in the range 0 --- 0.02 to 1. The t e m p e r a t u r e dependences of T~ and T2 for the three plugs were studied in the t e m p e r a t u r e ranges and at the surface coverages listed in Table I I . Only activation energies for T2 are given since the corresponding values for Tt (about 1.6 kcal/mole for plug 1) are uncertain and too low because the temperatures were in the vicinity of a T~ minimum. For plug 3 with a m i n i m u m at 56°C an activation energy of 2.3 kcal/mole for T1 was estimated. Diffusion Coe~cients. The diffusion coefficients at 23°C as a function of surface coverage are shown in Fig. 3. The data for plug 1 and 2 were obtained from measurements with constant field gradients (4, 5). For plug 1 these measurements were confirmed and supplemented b y data obtained
1972
213
STUDIES ON MOBILITY AND DIFFUSION OF BENZENE TABLE II ACTIV.~TION ENERGIES
FOR t~ELAXATION TIME
T 2 ±kND D I F F U S I O N
~OEFFICIENTS
Activation energies (kcal/mole) Plug No.
Surface coverage
Temp. range (°C)
1 1
0.5 1
1 2
2
+10- +35 --5- +35 --15- +35 0- +62
1
T2
3.4 2.7 2.7 1.8
± 444-
D*(nmr)
0.2 0.2 0.2 0.1
with pulsed gradients. Above 0 = 0.5 agreem e n t was found between the two methods within the experimental error of about 4-10 % whereas at lower surface coverages only the pulse technique could be used with an error increasing to about 4-20% at the lowest surface coverages measured. For comparison diffusion coefficients of benzene in silica gel at 25°C published b y Freude (3) as well as the self-diffusion coefficient of liquid benzene (9) are included in Fig. 3. The activation energies derived front the experimental diffusion coefficients D* are listed in Table I I . D ISCUSS ION I n contrast to methods in which diffusion is studied under the influence of a gradient in chemical potential, measurements with the spin echo technique are carried out under equilibrium conditions thus resulting in selfdiffusion coefficients D*. The diffusion process in porous media is complicated b y the fact t h a t mobility of the molecules in the pore volume as well as in the adsorption layer has to be considered. As we are primarily interested in the mobility of adsorbed molecules the problem arises to estimate the contribution from displacements of molecules in the gas phase. K r u y e r (10) has treated the following model. On the average a molecule spends a time r in the adsorption layer and after desorption a time t' in the pore volume before being readsorbed. During these periods the corresponding mean-square displacements are Z, 2 and Zg 2, respectively. Applying the Einstein equation to the overall diffusion process in the pore system we obtain :
4.9 4.6 4.0 4.3
4444-
D s* (nmr)
0.5 0.5 0.5 0.5
3.5 3.5 3.5 3.5
DS*(kin)
4- 0.5 4- 0.5 4- 0.5 4-' 0.5
4.0 4- 0.2 3.2 4- 0.2 -
-
--
T
£¢m2sedt]|
2
t ,
plug 1
, I
[
~×
/
i
|
2
0
Z,
6
8
FIG. 3. Nmr spin echo diffusion coefficients (D*) and surface self-diffusion coefficients (D ,*) for benzene adsorbed on silica plugs at 23°C. Plug 1: © ~ * ; • D ~*. Plug 2: O--D*; []--D ~*. Silica gel (3); X--D*. -
D* -
-
-
-
m
Z2 -
Z~ ÷
2t
2(t' + r)
Z~2
[2]
With Zg2/2t ' = D ° and Z~2/2r = D ~*, where D ° and D ~* are the self-diffusion coefficient in the gas phase and in the adsorption layer, respectively, D* -
t' DO + r D~, t' + ~ t' + ~ "
[3]
Under the assumption t h a t the molecules behave statistically and t h a t the adsorbent is homogeneously packed the ratio t ' / r is equal to the ratio of the n u m b e r of molecules in the pore volume N ° and in the adsorption layer N ~. Thus from Eq. [3] D* -
Ng-- D °
No + N~
N~
+ / g o + N~
D ~*
• [4]
If the temperature is not too high and the
Journal of Colloid and Interface Science, Vol. 38, No. 1, J a n u a r y 1972
214
BODDENBERG, HAUL, AND OPPERMANN
diameter is sufficiently small, as is true in the present case, then we have N g << N 8 and consequently D* = Ng ~ D ~ + D ~*.
[5]
The ratio N g / N ~ can be taken directly from the adsorption isotherm. If the mean free path of the molecules is large compared with the mean pore d i a m e t e r - - a condition which is valid in the present experiments--the gas phase diffusion coefficient D ~ is identical with the Knudsen diffusion coefficient representing a self-diffusion coefficient. Once D ~ is known, the surface self-diffusion coefficient D "* can be evaluated from the spin echo diffusion coefficient D*. When diffusion coefficients are measured with the nmr spin echo method using constant or pulsed field gradients circumstances m a y arise where Eq. [4] is not valid. KBrger (11) has discussed quite generally the effect of two-phase diffusion on the diffusion-controlled damping of the spin echo amplitudes. I t follows from his analysis that Eq. [4] holds true if rapid exchange of molecules between the two phases occurs. This is the case for the system studied in the present investigation. 5 In earlier studies (12) flow experiments with helium and benzene were carried out with a porous silica plug practically identical with plug 1 in every respect. For this plug a Knudsen diffusion coefficient of benzene D ~ 5.2 X 10-8 cm2/sec at 30°C was determined which can be used to calculate surface self-diffusion coefficients. I t can be seen in Fig. 3 (dashed line) t h a t the correction for molecular mobility in the pore voluume results in a lowering of the experimental diffusion coefficient, e.g., of about 25% at 0=1. Before discussing the results obtained with plug 2 it is instructive to examine the diffusion coefficients at the highest amount adsorbed. I t can be seen in Fig. 3 t h a t D* approaches values which for plug 1 are disT h e v a l u e of ~/~a2g2Dir~ ( r e f e r e n c e 11) is of t h e o r d e r 10-4.
tinetly, for plug 2 only slightly lower than, for bull; liquid benzene. From the adsorption isotherms (Fig. 1 ) it is obvious t h a t the pore space is mostly filled with condensate under these conditions. On the assumption that the diffusion coefficients in the condensate are the same as in the bulk liquid the lowering of the experimental diffusion coefficient for plug 1 can be attributed to the tortuosity of the pore system. In the spin echo experiments the molecular displacement is measured in the direction of the field gradient and not along the actual paths in the pores. In this way a tortuosity factor K(eond) = 1.8 is obtained for plug 1 from the ratio of the limiting D* value and D* (liquid). On the other hand a Knudsen diffusion coefficient can be calculated for a model system consisting of a bundle of parallel cylindrical capillaries with a radius equal to twice the hydraulic radius of the actual porous system (13) D~(eyl) = g8 x~
/2~
[o]
where e is the porosity, A the surface area per unit volume of the plug, and M the molar mass of benzene. With the data for plug 1 (Table I ) D ~ (cyl) is 1.08 X 10-2 cm2/ sec, and thus with the above value for D ~ a tortuosity factor K(gas) = 2.1 is obtained in fair agreement with K(cond). Since plug 2 has almost the same hydraulic radius as plug 1 the difference in the limiting values for D* must be attributed to a different tortuosity factor. Although the diffusion experiments were not carried out up to 0 values as high as for plug 1 a value of about 1.2 can be estimated. Accordingly the correction for molecular mobility in the pore volume has been calculated with a D ~ value larger than for plug 1 by a factor 1.8/1.2. The resulting curve 6 for the surface self-diffusion coefficient While the correction for gas phase mobility has been calculated for the two plugs with the relevant K (cond) = K (gas) values, the tortuosity factor for surface mobility K (surf) is assumed to be the same for both plugs. For a system packed
Journal of Colloid and Inlet face Science, Vol. 38, No. 1, January 1972
STUDIES ON MOBILITY AND DIFFUSION OF BENZENE is shown in Fig. 3. I t can be seen that for both porous plugs the dependence of D* on surface coverage is rather similar showing maxima near 0 = 1. At low coverages the molecules are located preferentially at strong adsorption sites with long residence times resulting in small T~ values (Fig. 2) and diffusion coefficients. As 0 becomes larger more and more weaker sites come into play leading to an increase in D 8.. With rising coverage above 0 = 1 the diffusion process is determined more and more by molecular mobility in the second and higher layers and in the capillary condensate, the formation of which becomes noticeable in this region (Fig. 1). Since no differences between the diffusion coefficients have been detected on the adsorption and desorption branches of the adsorption isotherm, as mentioned previously, it follows that the diffusion coefficients in the multilayer region cannot be significantly different from those in the capillary condensate. Since the measured apparent diffusion coefficient in the capillary condensate is smaller than in the adsorption layer near 0 = 1 a maximum in the D ~* curve occurs. Since the molecular mobility in the liquid phase is smaller than in the adsorption layer near 0 = 1 taking into account the tortuosity factor for surface diffusion6 the maximum in the D ~* curve observed in the present case can be visualized. Whether or not a maximum in the D ~* -- 0 curve is exhibited depends not only on molecular mobility in the adsorption layer and the capillary condensate b u t also on the tortuosity factors for the transport processes in these phases (6). Comparing the D ~* values for plug 1 and 2 one has to consider that the density of surface OH-groups is smaller for plug 2 (Aerosil 380) than for plug 1 (Aerosil 200). I t is well known (15) that a specific interacfrom spherical particles K (surf) can expected to be in the order of about 2 (14). The D~* curve shown in Fig. 3 is not corrected for the tortuosity of surface diffusion; this would result in correspondingly larger diffusion coefficients.
215
tion occurs between OH-groups and the ~r-electron system of benzene leading to relatively large energies of adsorption on such sites. Therefore, the surface diffusion coefficient is greater for plug 2 than for plug 1. On the basis of these arguments one would expect D ~* to rise already at relatively small values in the case of plug 2 with low OHgroup density. However, one has to consider another t)~pe of adsorbed molecules located in the region of contact between the spherical silica particles where the benzene molecules are strongly adsorbed owing to overlapping potential fields. Since the mean particle diameter of Aerosil 380 is smaller (Table I ) these centers are more frequent and therefore of greater influence for plug 2 than for plug 1 explaining that the rise in D ~* does not occur already at lower coverages. The strong adsorption centers other than OHgroups discussed in reference 1 could correspond to this type of sites. For comparison the diffusion coefficients of benzene in silica gel recently published by Freude (3) are included in Fig. 3. The lower limiting D* value found for silica gel as compared to the plugs compressed from Aerosil is due to the tortuosity factor, which is known to be larger for silica gel type adsorbents (14). In an earlier paper (5) we have calculated surface self-diffusion coefficients of benzene in porous silica plugs from rates of adsorption. The shape of the D e* versus ~ curves is almost the same as those obtained by the nmr spin echo method. The D ~* values from kinetic measurements are, however, lower b y a factor of about 1.7. Obviously the influence of the complex pore texture on the diffusion coefficient obtained from kinetic and nmr measurements is more involved than can be allowed for by the above tortuosity treatment. On the whole the fair agreement between the two methods is encouraging. From the temperature dependence of the experimental diffusion coefficient D* apparent activation energies of about 4 to 5 kcal/mole can be evaluated for 8 values be-
Journal of Colloid and Interface Science, Vol. 38, No. 1, J a n u a r y 1972
216
BODDENBERG, HAUL, AND OPPERMANN
tween 0.5 and 2 (Table I I ) . A value of the same order can be estimated from the data given b y Freude (3) for 0 = 1.5. Correction for molecular mobility in the pore volume, however, leads to a distinctly smaller activation energy of about 3.5 k c a l / mole for surface self-diffusion, which is in fair agreement with the activation energies for D ~* obtained earlier from rates of adsorption (Table I I ) . This value amounts to approximately one-third of the corresponding isosteric heats of adsorption (12). I n the range 0.5 to 2 little change is observed for the heats of adsorption as well as the activation energy for D ~*. Below 0 = 0.5 the energies of activation for D ~* obtained from rates of adsorption increase from 4.0 to 6.4 k c a l / m o l e between 0 = 0.5 and 0.07. T h e heats of adsorption on the energetically heterogeneous adsorbent increase b y almost the same amount. I t is noteworthy t h a t the activation energies for surface self-diffusion are greater t h a n for the relaxation times T2. This finding is not clear in the light of the relaxation analysis carried out b y 5lichel (16) and the discussion given b y Freude 4 based on the assumption of two types of adsorption sites. This point requires further investigation. ACKNOWLEDGMENT The authors gratefully acknowledge the financial support of this investigation by the "Deutsche Forschungsgemeinschaft" and "Fends der Chemischen Industrie."
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spektroskopie der Physikalischen Gesellschaft der D.D.R. 1969." Leipzig, 1969. 2. BECKERT, D., Phys. Lett. A25,502 (1967). 3. FREUDE, D., Z. Phys. Chem. (Leipzig) 242,
57 (1969). 4. BODDENBERG, B.~ HAUL, R., AND OPPERMANN, G., Naturwissenschaften 56,635 (1969). 5. BODDENBERG, B., ]:IAUL, R., AND OPPERMANN, G., Surface Sci. 22, 29 (1970). 6. CARR, ~-L Y., AND PVRCELL, E. M., Phys. Rev.
94,630 (1954). 7. STEJSKAL,E. 0., AND TANNER, J. E., J. Chem. Phys. 42,288 (1965).
8. Mc CALL, D. W., AND DOVOLASS, D. C., J. Phys. Chem. 69, 2001 (1965). 9. DIETRICH, W., GROSS~ B., AND KOSFELD, R., Z. Naturforsch., A 25, 40 (1970). 10. KRUYER,S., Prec. Koninkl. Ned. Akad. Wetenschap. Prec. Set. B 56,274 (1953). 11. KXRGER,J., Ann. Physik (Leipzig) 24, 1 (1969). 12. HAu~, R., AND BODDENRER% B., Z. Phys. Chem. (Frankfurt) NF64, 78 (1969). 13. BARRER, R. M., AND GABOR, T., Prec. Roy. Soc. Set. A 251,353 (1959); BARRER, R. M., in E. A. Flood, Ed. "The Solid-Gas Inter-
face," Vol. 2, p 557. Dekker, New York, 1967. 14. CARMAN, P. C., "Flow of Gases through Porous Media." Butterworths, London, 1956. 15. KISELEV, A. V., Zh. Fiz. K h i m . 35,233 (1961). (Russ. Y. Phys. Chem. (English Transl. 35, 111 (1961)); BA~RER, R. M., Or . Colloid Interface Sci. 21,415 (1966). 16. MICHEL,D., Z. Naturforsch., A 23,339 (1968). 17. BOEH~, H. P., AND KAEMPE, G., Z. Phys. Chem. (Frankfurt) [NF] 23, 257 (1960); BODE, R., FERCH, H., AND FRATZSCHER,H., Kautschuk Gummi. Kunststoffs 20,578 (1967).
Journal of Colloidand Inter/ace Science, Vol. 38, No. I. January 1972