Znd. Eng. Chem. Res. 1991, 30, 865-873
865
Poly(furfuryl alcohol)-Derived Carbon Molecular Sieves: Dependence of Adsorptive Properties on Carbonization Temperature, Time, and Poly(ethylene glycol) Additives David S. Lafyatis, Jeannie Tung, and Henry C. Foley* Center for Catalytic Science and Technology, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
A systematic study of the effect of final soak temperature (650-900 "C) and final soak time (1.5-4.5 h) on the microporosity of poly(furfury1 alcohol)- (PFA) derived carbon molecular sieves (CMS) was performed. T h e surface area of the CMS samples displays a maximum in the 800-900 "C range for the samples with a 1.5-h final soak time. The micropore diameter decreases with higher heating temperatures, resulting in decreasing effective diffusivities for COz and n-butane. Longer final soak times also decrease the pore-mouth diameter, indicating that the ultramicroporous carbons are intermediate, nonequilibrium structures. The addition of high molecular weight (MW > 2000 amu) poly(ethy1ene glycol) (PEG) to the PFA prior to pyrolysis was shown to develop a system of transport pores in the carbon. Lower molecular weight (MW < 600 amu) PEG had much less effect on the carbon structure. Introduction Adsorptive separation processes have become increasingly important during the past decade. Design of these separative stages requires useful macroscopic models for mass-transfer diffusion and adsorption in packed beds. Robert L. Pigford was a leader in the development of these models for adsorptive processes, and he set the direction for a whole generation of engineering practitioners, as well as scholars. In this work we have focused upon the issues of microscopically engineering the structure of carbon adsorbents to control their molecular sieving properties, and we are pleased to dedicate this contribution to Professor Pigford. Ultramicroporous carbons, or carbon molecular sieves (CMS), are amorphous materials with narrow pores similar in size to small molecules. They have recently received considerable attention because of their commercial application in the pressure swing adsorptive separation process for the production of high-purity nitrogen from air (Yang, 1987). The ultramicropores of CMS materials are 3-7 A in size depending on preparation method, and they provide the sieving structure necessary for small molecule separations. Diffusivities of small molecules through a well-characterizedCMS material (Takeda 5A Carbon) have been measured, and they show a strong dependence on molecular size (Chihara et al., 1978). The opportunity for using metal-containing carbon molecular sieves for metal-catalyzed reactions had been noted early on (Tri" and Cooper, 1970, 1973; Schmitt and Walker, 1971, 1972). These reports indicated that a composite of molecular sieving carbon with platinum could provide reactant shape selectivity for catalytic olefin hydrogenation reactions. Other investigators have briefly pursued the catalytic and sieving properties of metal-CMS materials (Conner, 1980; Moreno-Castilla et al., 1980; Dessau, 1983; Bragin et al., 1980). More recent investigations have been carried out aimed at combining inorganic oxides and supported metals with molecular sieving carbon, referred to as inorganic oxide modified carbon molecular sieves or IOM-CMS (Foley, 1988). In this vein, Lafyatis and Foley (1990) have proposed a reaction with diffusion model for an IOM-CMS catalyst that suggests the design for a catalyst with improved selectivities for synthesis gas conversion in the
* Author to whom correspondence should be addressed.
Fischel-Tropsch reaction. In all of these applications the physical properties of the carbon, such as pore size and surface area, are crucial to the control of the molecular mass transfer within the carbon structure. Previous investigators have laid a solid foundation for the understanding of the chemistry that leads to the conversion of poly(furfury1alchohol) to ultramicroporous carbon. Work by Fitzer et al. (1969) and Fitzer and Schaefer (1970) is particularly noteworthy. By analysis of the gases evolved during the pyrolysis of PFA, a mechanism for the conversion of the polymer to carbon was proposed. In conjunction with this, gas uptake measurements were used to predict surface area and pore volumes in the resultant CMS material. Interesting work has also been performed by Lamond and Marsh (1963a,b), who concentrated on the properties of activated carbons. The difficulties in determining correct surface areas for carbons with large internal volumes are well documented in these reports. Similar work has also been done for poly(viny1chloride)- and poly(viny1idenechloride)-derived CMS samples. Lamond et al. (1965) performed a detailed study of the effects of temperature on the adsorptive properties of Saran-type carbons. As part of our ongoing research into the synthesis and characterization of carbon molecular sieves for separation and catalysis, in this work we sought to study the effects of final soak temperatures and time on the microporosity of poly(furfury1 alcohol)- (PFA) derived CMS materials. The addition of poly(ethy1ene glycol) (PEG) to the PFA prior to pyrolysis also was investigated as a technique to create mesoporosity for higher mass transport into the samples. The adsorptive and diffusional charactistics of these samples were studied by gravimetric uptake measurements of COz, n-butane, and isobutane, as well as nitrogen adsorption experiments, in order to gain a better understanding of the effects of the synthesis variables on the microporosity and mesoporosity of these samples. Thii effort was motivated by the need to control the porosity of CMS materials, a fador that is crucial to the design and preparation of catalysts from ultramicroporous carbons.
Experimental Procedures All of the carbon molecular sieve materials were formed by the pyrolysis of poly(furfury1alcohol) (Durez Resin No. 16470, Occidental Chemical Corporation). The PFA had a specific gravity of 1.21 and a viscosity of 200 CPat 25
0888-5885/91/ 2630-0865$02.50/0 0 1991 American Chemical Society
866 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 Table I. Pyrolysis Reactor Conditions carrier gas nitrogen flow rate 260 cm3 (STP)m i d 6 cm tube diameter sample size 12 g of PFA (neat) 1 atm pressure temperature variable
Table 111. Molecular Probes system press., probe molecule Torr carbon dioxide 700 n-butane 492 isobutane 705
re1 press., PIP 0.015 0.27 0.27
crit diam,"A 3.3 4.3 5.0
"Ruthven (1984). Table 11. Pyrolysis Temperature Programs: Time (in hours) at Temperature Range for PFA-1.5 Samples sample identification temp, "C PFA650-1.5 PFA750-1.5 PFA830-1.5 PFA900-1.5 25-500 1.0 1.0 1.0 1.0 500-650 1.5 1.5 1.5 1.5 650 1.5 0.5 0.5 0.5 650-750 0.5 0.5 0.5 750 1.5 0.5 0.5 750-830 0.5 0.5 830 1.5 0.5 830-900 0.5 900 1.5
k
kZ-+ 40 c r
Mineral 011
4
Figure 1. Pyrolysis reactor used for CMS synthesis.
"C. Table I displays the reactor conditions for the pyrolysis. The samples were made by placing the Durez resin into a quartz boat (approximately 12 cm long and 2 cm high). The boat was then placed in the heating zone of a quartz tubular reactor fitted with a high-temperature furnace. Nitrogen gas flowed through the pyrolysis tube and then bubbled through mineral oil to maintain a nearly inert atmosphere in the reactor. Figure 1 is a schematic diagram of the pyrolysis reactor. CMS materials were synthesized with final temperatures of 650,750,830, and 900 "C and final soak times of 1.5,2.5,4.5 and 5.5 h. Table I1 displays the temperature programs used for the PFA-1.5 samples investigated in this study. The initial heating rate of the samples was about 9.5 "C/min for the first hour. After the first hour the heating rates varied between 2 and 3 "C/min. The temperature was measured in the center and on both ends of the heating zone, and the reported temperature is an average of these three temperatures. The temperature in the center of the heating zone was always the highest, but all three temperatures were always within 12 "C of the average temperature. The PFA650-4.5, PFA750-4.5, PFA830-4.5, and PFA900-4.5 samples were prepared in precisely the same manner, except that the soak time at the final temperature was 4.5 h instead of 1.5 hours. PFA900-2.5 and PFA900-5.5 samples were also synthesized. As shown by Table 11, the temperature program during pyrolysis was identical for each sample; only the final soak temperatures and times were varied. This procedure was used to minimize possible differences induced by the heating rate on the properties of the CMS. The carbon yield in each case was approximately 25% based on the original mass of PFA regardless of the final soak time and temperature. A duplicate sample of PFA650-1.5 was prepared, and n-butane diffusivities for these materials were satisfactorily reproducible (2.4 X lo-" versus 3.2 X cm2/s). The poly(ethy1ene glycol) (PEG) in this work was obtained from Aldrich and was used without further purification. The PEG/PFA samples were made with the use
of a starting mix of 25% PEG and 75% PFA by weight. The 300- and 600-amu PEG samples were liquids at room temperature, and were simply mixed with the PFA. The PEG samples of 1000,2000,3400,and 8OOO amu were waxy solids at room temperature and were ground to a powder before dispersion in the PFA. All of the PEG/PFA samples were prepared using the following temperature program: PFA/PEG-MW: 25-400 "C, 1 h; 400-650 "C, 2.5 h; 650 "C, 1.5 h. The PFA/PEG samples are identified by the molecular weight of the polyethylene glycol that was dispersed in the PFA (e.g., PFA/PEG-300, PFA/PEG2000). The temperature program used to prepare all of the PFA/PEG samples was similar to that of the PFA650-1.5 sample. A separate experiment was performed in which neat 8OOO-amu PEG was subjected to the heating cycle used for the PFA/PEG samples. After the completion of the heating cycle, there was no residue, indicating that all of the PEG vaporized or decomposed as gaseous products during heating. Thus, it is probable that the final product of the PFA/PEG samples contained no residual PEG. Molecular probe adsorption studies were performed on a 10-port McBain Balance, using quartz springs with 0.01 g/cm sensitivity. All samples were outgassed at 350 "C and Torr prior to exposure to the molecular probe. All experiments were done at 20-21 "C. Table 111 shows the physical data and conditions for the probe molecules and uptake experiments. Nitrogen adsorption was done at 77 K with an Omnisorp 100 from Omicron Technologies. Again, all samples were pretreated at 350 "C and lo5 Torr. The versatility of the Omnisorp 100 allowed the experiments to be run in either a static mode (in which nitrogen was pulsed into the system, after which the sample and nitrogen were dowed to equilibrate for a set amount of time) or a continuous flow mode; both types of experiments were used in this study. Sample sizes of approximately 0.25 g were used, and the volume of the system was approximately 50 cm3. Experimental Results The usefulness of uptake data to evaluate intraparticle mass-transfer resistances is demonstrated in Figure 2. This figure shows COPand n-butane uptake on different sized samples of PFA650-1.5. C02sorption occurs rapidly on all of the samples, and after 1h much of the adsorption has taken place on even the largest particle size. With the n-butane probe molecule, the sorption rate is much slower. Indeed, even after a period of 250 h measurable sorption continues to take place on all of the samples. The combination of molecular probe and nitrogen adsorption data allowed the evaluation of the effects of soak time, final temperature, and addition of poly(ethy1ene glycol) on the pore structure and diffusional characteristics on the carbon molecular sieve materials. The effect of soak temperature on the adsorption of carbon dioxide is illustrated in Figure 3. Figure 3a shows the adsorption of C 0 2 versus final carbonization temperature for the PFA-1.5 samples at a series of adsorption times on 140/200 mesh ( f ; = 0.045 mm) particles. The PFA650-1.5-PFA830-1.5 samples show very little diffusional resistance to COz.
Ind. Eng. Chem. Res., Vol. 30,No. 5, 1991 867 SO U
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Figure 2. (a, top) COz adsorption ( P / P = 0.015). 600 "C carbonization temperature, 1.5-h soak time. (b, bottom) n-Butane ( P / P = 0.27). 600 O C carbonization temperatwe, 1.5-h soak time.
After 1 h these samples are all near their apparent equilibrium adsorption, with negligible additional sorption taking place between 1and 72 h. The PFA900-1.5 sample, however, shows significant continued adsorption after the 1-h measurement. A series of measurements made on 40/80 mesh (ii = 0.15 mm) CMS is shown in Figure 3b. On these larger particles, the rate of uptake is slower because of a greater diffusional path length. Thus, there are significant differences between the 1- and 24-h adsorption values on all particles. The resistance is particularly pronounced for the 900 "C sample. Figure 4 shows the adsorption data for n-butane and isobutane on the PFA-1.5 140/200-mesh particles. The diffusional resistance to adsorption is evident on all of the samples. The higher temperature treated particles now show almost no hydrocarbon adsorption, even after extended times. For n-butane adsorption, the PFA650-1.5 sample appears to reach equilibrium uptake after 72 h. Parts a-c of Figure 5 display the adsorption data for the three probe molecules on PFA-4.5 140/200-mesh CMS
Pyrolysis Temperature
(C)
Adsorption times 0
0.3 hr
*
48 hr
a
1.0 hr
x
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Figure 3. (a, top) COz adsorption ( P / P = 0.015). 1.5-h soak time; 140/200-mesh samples. (b, bottom) COzadsorption ( P / P = 0.015). 1.5-h soak time; 40/80-mesh samples.
samples. Figure 5a shows the results of C02 adsorption on these samples. The PFA650-4.5 an dPFA750-4.5 samples both show high levels of C02adsorption with relatively rapid uptake. The final adsorption levels for these two samples are higher than the corresponding PFA650-1.5 and PFA750-1.5 samples presented in Figure 3a. The PFA900-4.5 sample shows a very low value of C02 adsorption with very slow uptake due to significant diffusional resistance. Even after 72 h the sample does not appear to have reached its equilibrium C02 adsorption level. Figure 5b and Figure 5c display the adsorption of n-butane and isobutane on the same 140/200-mesh PFA4.5 samples. The rate of adsorption for both of these gases has been greatly reduced on the 650 and 750 OC samples compared to Figure 4. The PFA9OO samples show insignificant adsorption levels for both the 1.5- and 4.5-h soak times. Figure 5d shows C02 adsorption on a series of
868 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 Table IV. Nitrogen Adsorption Results on 40/80-Mesh Samples
sample ID PFA/PEG-8000 PFA/PEG-3400 PFA/PEG-2000 PFA/PEG-lOOOb PFA/PEG-GOOb PFA/PEG-300b PFA600-1.5b
rmewl
A
100
30 30
dmicror A 5.5 5.5 5.5
BET surf. area, m2/g 305 285 310 80
60 35 30
t-plot surf. area, m2/g 91 86 76 60 65 45 41
micropore vol., cm3/g 9.2 X 8.6 X lo-* 9.9 x 10-2 0 0
molec probes,’ mg/g COz n-butane 78 82 72 84 77 86 81 34 83 30 82 21 71 16
mesopore vol., cm3/g 0.45 0.24 0.28
0
0
24-h adsorption; 40/80 mesh. Nonequilibrium adsorption of nitrogen; thus reported values are lower than correct values.
PFA9OO samples with different final soak times. The increased diffusional resistance to adsorption with soak time is clearly illustrated. Nitrogen adsorption on PFA-derived CMS materials is difficult to measure, due to the high resistance to nitrogen diffusion or adsorption at 77 K. One experiment was carried out on the PFA650-1.5,40/80-mesh size. In this experiment, 2.9 X lo4 mol of nitrogen was pulsed into the sample holder. The sample was then allowed to equilibrate with the gas phase for 5 min, after which the pressure was measured. This sequence was repeated until the experimental pressure reached 10 Torr. After the pressure rose above 10 Torr, nitrogen flowed continuously at a rate of 0.07 cm3(STP) min-’ into the sample holder, and the pressure was monitored continuously. Nitrogen adsorption in this experiment clearly did not reach equilibrium. The results are shown in Table IV. The BET surface area of 30 m2/g determined in this experiment is clearly much too low to be accurate, and the value reflects the severe transport limitations imposed by the pore structure at 77 K. A 200/325-mesh sample was also investigated, and although the isotherm obtained was again clearly not a reflection of the true equilibrium isotherm, a surface area of 360 m2/g was calculated from it using the BET equation. Therefore, although reducing the particle size improved the results of the adsorption isotherm, it did not completely compensate for the problem imposed by the slow diffusion of nitrogen into the pores. The PFA/PEG samples were examined by molecularprobe experiments. Figure 6a shows the results of COz adsorption on 40/80-mesh particles. It is evident from the data that there was little diffusional resistance to the adsorption of COz on these samples, and the total COz adsorption level is similar on all the samples. Figure 6b presents the adsorption of n-butane on the 40/80-mesh samples, A dramatic increase in the adsorption of n-butane is apparent between the PFA/PEG-1000 and the PFA/PEG-2000 sample. Diffusional resistance to mass transfer is certainly apparent for the lower molecular weight samples. The high molecular weight samples also appear to have some continued adsorption after 1 h. Figure 7 displays the adsorption of n-butane and isobutane on 80/140 (r = 0.072 mm) mesh samples. These results also show more rapid hydrocarbon adsorption on the samples prepared with higher molecular weight PEG. Nitrogen adsorption was performed on the 40/80-mesh PFA/PEG samples. Nitrogen adsorption experiments were performed in both the static and the continuous mode. The results are summarized in Table IV. The experimental procedure for the data reported in Table IV was identical with the nitrogen adsorption on the PFA650-1.5 sample. For the samples that were prepared with the high molecular weight PEG (MW > 2000 amu), the experimental method made virtually no difference in the adsorption isotherm except at very low relative pressures. Thus, the calculation of BET surface areas, t-plot
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Figure 4. (a, top) n-Butane adsorption ( P / P = 0.27). 1.5-hsoak time; 140/200-mesh samples. (b, bottom) Isobutane adsorption ( P / P = 0.27). 1.5-hsoak time; 140/200-mesh samples.
surface areas, and t-plot micropore volumes were unaffected by the rate at which the experiment was conducted. The adsorption at low relative pressures is indicative of the micropore adsorption in the samples. Because the diffusion into these narrow pores is activated, the diffusion
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 869
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Figure 5. (a) COz adsorption ( P / P = 0.015). 4.5-h soak time; 140/200-mesh samples. (b) n-Butane adsorption ( P / P = 0.27). 4.5-h soak time; 140/200-mesh samples. (c) Isobutane adsorption ( P / P = 0.27). 4.5-h soak time; 140/200-mesh samples. (d) COz adsorption ( P / P = 0.015). 900 "C final soak Temperature; 140/200-mesh samples.
time at 77 K is significant. However, unlike the CMS samples prepared from Durez resin alone, on which virtually no adsorption could be measured, the high molecular weight PEG/PFA samples apparently contain larger pores for the transport of nitrogen. These transport pores significantly reduce the overall diffusion time necessary for adsorption in the micropores. This permits the analysis and characterization of these materials in experiments within a reasonable time frame. Thus, in Table IV an approximate value for the micropore width of 5.5 A is reported based on the Horvath-Kawazoe method for calculating pore sizes in microporous carbon (Horvath and Kawazoe, 1983). Figure 8 displays the calculated pore size distribution, where the distribution has been scaled by the total adsorption in the micropores. This distribution was nearly identical for the PFA/PEG-8000, PFA/PEG-3400, and PFA/2000 samples. The reported pore width is the maximum of a fairly broad pore size distribution. Because the nitrogen adsorption in the high molecular weight PFA/PEG samples reached equilibrium, desorption experiments could also be carried out to estimate mesopore and macropore size distributions and volumes by use of the Kelvin equation. The desorption results on these three
samples are again summarized in Table IV. Median mesopore sizes are reported from rather broad distributions. The 30-A pore size reported for the PFA/PEG-3400 and PFA/PEG-2000 samples is the mean of a broad pore size distribution calculated for these samples on the basis of the Kelvin equation, which accounts for adsorption in pores down to 20 A in diameter. As has been documented in the literature, the Kelvin equation is not valid at pore sizes of 20 A and below, due to the abrupt closure of the hysteresis loops between adsorption and desorption in the nitrogen isotherms at P/P" = 0.42 (Gregg and Sing,1982). Thus, an apparent peak in the pore size distributions at 20 A of the PFA/PEG-3400 and -2000 samples is more likely due to the occurrence of this artifact typically present in nitrogen adsorption experiments than to any real structural feature. However, since the nitrogen adsorption isotherm can be measured efficiently at 77 K and n-butane adsorption at 21 OC is apparently unhindered, these results indicate that there is a system of transport pores in these samples prepared with the higher molecular weight PEG which is not present in the other samples. The nitrogen adsorption experiments on the lower molecular weight PEG/PFA samples (MW = 300-1000 amu)
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991
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Figure 6. (a, top) C02 adsorption ( P / P = 0.015). PFA/PEG samples; 40/80 mesh. (b, bottom) n-butane adsorption ( P / P = 0.27). PFA/PEG samples; 40/80 mesh.
could not be completed in a reasonable time frame, again due to the slow diffusion of nitrogen into the pore structure at 77 K. This is not surprising, based on the diffusional resistance to n-butane and isobutane sorption at 21 "C exhibited by these samples (see Figures 6b, 7a, and 7b). The nitrogen adsorption experiments carried out did not reach equilibrium, even at the static adsorption conditions discussed previously. An interesting example of the diffusional resistances in these samples is presented in Figure 9. The variation in the apparent adsorption isotherm for the MW 1000 sample as a function of experimental conditions is a clear indication that the sample did not equilibrate in the time allowed. Thus, it is apparent that the transport pore system in these samples is inadequate to reduce the diffusion time sufficiently to allow nitrogen adsorption to occur in a useful experimental time. Nitrogen adsorption was also performed on a smaller mesh size (80/140) of the PFA/PEG-1000 material. It was clear from the isotherm shape that the sample had not equilibrated, but the surface area predicted by the BET equation rose to 210 m2/g. This isotherm is also shown in Figure 9. The apparent isotherms for the 40/80-mesh PFA/PEG-300 and PFA/PEG-600 samples are very sim-
00
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Figure 7. (a, top) n-Butane adsorption ( P / P = 0.27). PFA/PEG samples; 80/140mesh. (b, bottom) Isobutane adsorption ( P / P = 0.27). PFA/PEG samples; 80/140 mesh.
ilar to the apparent isotherm for the 40/80-mesh PFA650-1.5 sample. Thus, it appears that there is very little difference in the pore structures of these samples. Desorption experiments were not attempted on these samples since they did not approach equilibrium.
Discussion Interpretation of the experimental results from uptake measurements made on the McBain balance allows many conclusions to be drawn about the surface area and pore structures of the various samples. The 72-h adsorption times for C02,displayed in Figure 3a, appear to represent near-equilibrium uptake values. Although it is dangerous to calculate absolute surface areas from these data, it is reasonable to use uptake data as a measure of the relative adsorptive capacities and surface areas of the samples. Thus Figure 3a indicates that the surface area goes through a maxima in the 800-900 "C range. Fitzer et al. (1969) showed a surface area maximum on their PFA-derived carbons in the region of 700-800 "C. This is reasonably good agreement, considering the differences in preparation of the carbons. Lamond et al. (1965) showed a maxima
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 871
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0
Figure 10. Diffusivity of COBand n-butane in PFA-1.5 materials. 0.5
0.7
0.6
(
P o r e D i a m e t e r (nm)
Figure 8. Micropore size distribution estimated from HorvathKawazoe equation. 60
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the work of Lamond et al. (1965) on Saran-derived CMS materials. Lamond showed that, at temperatures above 1200 "C,Saran-derived carbons begin to sieve neopentane (based on 1-h adsorption times, with -100-mesh particle size). Although the micropores in their Saran-type carbons are clearly larger than in these PFA-derived carbons, the trend of decreasing pore size with increasing soak temperature is consistent. The McBain balance also provides uptake versus time data which can be used to calculate effective diffusivities of gas molecules through solids. Ruthven (1984) has pointed out some of the inherent problems in these calculations, the main danger being the possibility of nonisothermal conditions during the experiment due to exothermic adsorption. However, as a first approximation if isothermicity is assumed, the data from the McBain balance experiment can be fit to the equation
Figure 9. Apparent adsorption isotherms: PFA/PEG-1000 sample.
in surface area in the range of 1000-1200 "C for Saranderived carbons. The difference is probably due to the different pyrolysis chemistries of the polymer precursors. In Figure 4a, the adsorption of n-butane on the PFA650-1.5 sample appears to have reached equilibrium after 72 h. Estimating the surface area for this sample based on an area of 47 A2 (Gregg and Sing, 1982) for the n-butane molecule and assuming monolayer coverage at PIP. = 0.27 gives a value of 460 m2/g. This is in the range of expected values for microporous carbon. Much of the experimental data provides convincing evidence that higher temperature treatments induce greater diffusional barriers to gaseous sorption in the CMS structures, thus indicating a reduction in the size of the micropores. Figure 3b shows particularly dramatic evidence of diffusional resistance for COzsorption in a relatively large particle. Figure 4 shows the increased diffusional resistance for n-butane and isobutane for higher temperature treatments. This is in good agreement with
exp(-n2n2Dt/R2)
(1)
to estimate and compare effective diffusion coefficients for the materials. In this paper, the first 20 terms of the infinite summation were included in the calculation, and the result was shown to be insensitive to additional terms beyond these 20. Figure 10 shows the calculated effective diffusion coefficients of COz at 21 "C on the PFA-1.5 and the PFA900-2.5 samples. The equilibrium uptakes were determined from the 72-h adsorption times from Figures 3a and 5d. Figure 10 shows the expected downward trend in diffusivities as the final carbonization temperature is increased, thus supporting the hypothesis that the additional heating decreases the pore size. Figure 10 also shows the diffusivities calculated for n-butane in the PFA6501.5-PFA900-1.5 samples. The equilibrium adsorption level of n-butane on the 650 "C sample was determined by the 72-h reading on Figure 4a. The 750 "C sample did not reach equilibrium, but the equilibrium capacity was estimated by
where Q is the uptake at equilibrium. This same procedure was also used for the higher temperature samples. Because of the extremely low levels of adsorption measured for the PFA830-1.5 and PFA900-1.5 samples, the diffusivities shown in Figure 10 are highly uncertain and are meant to provide an order of magnitude estimation for the true value. As expected, the diffusivity of the n-butane in the
872 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991
CMS is several orders of magnitude lower than the COB diffusivity. The samples that were subjected to longer soak times showed somewhat surprising results. A comparison of Figures 5a and 3a shows that the additional treatment time increases the C 0 2 capacity for the PFA650 and PFA750 samples. This is consistent, because for the low temperhigher ature PFA-1.5 samples (PFA650-1.5-PFA830-1.5), final carbonization temperatures increased surface area. This establishes the similarity in effect on pore structure between raising the Carbonization temperature and additional heating at a constant temperature. For the PFASOO samples, the additional soak time has reduced the pore size so that even for 140/200-mesh size particles significant diffusional resistance now exist for COP adsorption. It is expected that the equilibrium surface areas for these samples would also be reduced, and this is consistent with the results shown in Figure 5d. In Figure 5b,c it is apparent that the diffusion of n-butane and isobutane was severely hindered in the samples that were prepared with the longer soak time. Comparison of the results from the PFA-1.5 and PFA4.5 samples indicates that the physical properties of the carbon are still evolving after 1.5 h of soak time at final temperature. The yield of carbon was the same (25%) regardless of the final soak time or temperature, indicating that the chemical conversion of PFA to carbon is complete. This is in agreement with Fitzer and Schaefer’s (1969) study of PFA pyrolysis chemistry. However, the allotropy of the carbon product is changing. Thus, it appears that the carbon molecular sieve structures formed from PFA are generally far from equilibrium. Indeed, it has been known for some time that additional heating of porous Saran-type carbons at elevated temperatures does eventually lead to the thermodynamically favored, graphitic, nonporous structure (Franklin, 1949a,b). Hence with time at elevated temperatures, the porous structures of these CMS materials also move closer to the nonporous, graphitic structure in the thermodynamic well. Thus, differences in porosity that arise with different treatment times and temperatures are attributable to the kinetics of this conversion of amorphous porous carbon to more crystalline graphitic carbon. The addition of the high molecular weight poly(ethy1ene glycol) in the PFA-derived CMS samples clearly affects the pore structure of the final product. Although using the nitrogen isotherm to predict absolute surface areas and pore volumes in carbon is problematic (Lamond and Marsh, 1963a,b), in this case it is clearly a useful tool for comparing samples on a relative basis. The t-plot analysis of these nitrogen isotherms indicate that the PEG8000PEG2000 samples have about 70 m2/g mesoporosity. The BET equation predicts total surface areas of about 300 m2/g for each sample, so that we can estimate a micropore area of about 230 m2/g for each sample. Thus, the ratio of areas between microporosity is very similar in all three samples. The micropore volumes are also very similar for all three samples, but the mesopore volume for the PFA/PEG-8000 sample is significantly larger. Analysis of the low-pressure portion of the isotherms for the PFA/PEG samples indicates that the micropore mouths have a diameter in the range of 5.5 A. This is a reasonable value for the PFA650-1.5 sample, which showed hindered adsorption of n-butane and isobutane. This is a slightly larger pore mouth than estimated by Chihara et al. (1978) for their Takeda 5A Carbon sample. Thus, it is possible that the PEG does not affect the pore size in the microporous regions of the PFA. This is further supported by
the fact that the micropore distributions determined for the PFA/PEG-2000, -3400,and -8000 samples were identical. It is likely that the pore mouths for the PFA830-1.5 and PFA900-1.5 samples are less than 5.0 A because nbutane and isobutane are almost completely restricted from sorption even after 72 h. The nitrogen adsorption and McBain balance results indicate that the lower molecular weight PEG has much less of an effect on the pore structure of the carbons. One possible reason for this is that the low molecular weight PEG may be flashing off during the pyrolysis before the pore structure of the PFA carbon begins to develop. In the higher molecular weight sample PFA/PEG samples, the PEG may not vaporize (or decompose to vaporizable products) until the carbonization of the PFA is further along toward complete conversion to carbon. Thus when the PEG vaporizes or decomposes, it must leave through the partially solidified carbon structure, creating relatively large transport pores in the CMS structure. The large mesopores found in the PFA/PEG-8000 sample may be the extreme case of this, in which the PFA has completely carbonized before the PEG begins to exert its effect.
Conclusions In the application of carbon molecular sieves, the diameter of the pore mouth is a critical parameter. This work provides certain proof that, for PFA-derived CMS materials, carbonization at higher temperatures decreases the micropore size. This narrowing of the pore size affects the diffusivities of carbon dioxide and n-butane in the CMS materials by several orders of magnitude. Also, CMS samples that are left for a longer time at their final carbonization temperature exhibit a decrease in micropore size. The surface area of the CMS materials, as estimated by carbon dioxide adsorption, goes through a maximum at intermediate carbonization temperatures. Transport pores may be necessary to provide efficient sorption of gases into the narrow micropores of CMS materials. The PFA/PEG materials prepared here had both microporous and mesoporous regions. The addition of poly(ethy1ene glycol) with a molecular weight above 2000 amu to the PFA prior to pyrolysis has a pronounced effect on the mesopore and macropore structure of the final product. Poly(ethylene glycol) with molecular weights below 600 had little effect on the final pore structure. Acknowledgment We gratefully acknowledge the contributions of Mr. Akira Saito in assembling the nitrogen adsorption data and Mr. Ravindra Mariwala for valuable discussions about carbon molecular sieve materials. The National Science Foundation is thanked for its generous support through the Presidential Young Investigator Award (NSF Grant No. CBT-965714), as is the Mobil Foundation. Registry No. PFA, 25212-86-6;PEG,25322-68-3;carbon, 7440-44-0.
Literature Cited Bragin, 0. V.; Olfereva, T. G.; Ludwig, J.; Fiebig, W.; Heise, K.; Schnabel, K. H. Effect of the Pore Structure of the Support on the Catalytic Properties of Platinum-Carbon Molecular Sieves. Z. Chem. 1980,20,387-388. Chihara, K.;Suzuki, M.; Kawazoe, K. Interpretation for the Micropore Diffusivities of Gasas in Molecular Sieving Carbon. J . Colloid Interface Sci. 1978,64, 584-587. Conner, H. DE Patent 3006105,1980. Dessau, R. M.US Patent 441354,1983. Fitzer, E.; Schaefer, W.; Yamada, S. The Formation of Glasslike Carbon by Pyrolysis of Polyfurfuryl Alcohol and Phenolic Resin. Carbon 1969,7,643-648.
Ind. Eng. Chem. Res. 1991, 30, 873-881 Fitzer, E.; Schaefer, W. The Effect of Crosslinking on the Formation of Glasslike Carbons from Thermosetting Resins. Carbon 1970, 8, 353-364. Foley, H. C. Carbon Molecular Sieves: Properties and Application in Perspective; Perspectives in Molecular Sieve Science; Flank, W. H., Whyte, T. E., Jr., Eds.; American Chemical Society: Washington, DC, 1988; pp 335-360. Franklin, R. E. Study of the Fine Structure of Carbonaceous Solids by Measurements of True and Apparent Densities. I. Coals. Trans. Faraday SOC.1949a, 45, 274. Franklin, R. E. Fine Structure of Carbonaceous Solids by Measurements of True and Apparent Densities. 11. Carbonized Coals. Trans. Faraday SOC.1949b,45,668. Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982; pp 68, 154-160. Horvath, G.; Kawazoe, K. Method for the Calculation of Effective Pore Size Distribution in Molecular Sieve Carbon. J . Chem. Eng. Jpn. 1983, 16, 470-475. Lafyatis, D. S.; Foley, H. C. Molecular Modelling of the Shape Selectivity for the Fischer-Tropsch Reaction Using a Tri-Functional Catalyst. Chem. Eng. Sci. 1990, 45, 2567-2574. Lamond, T. G.;Marsh, H. The Surface Properties of Carbon41 The Effect of Capillary Condensation at Low Relative Pressures Upon the Determination of Surface Area. Carbon 1963a, 1, 281-292. Lamond, T. G.; Marsh, H. The Surface Properties of Carbon-111: The Process of Activation of Carbons. Carbon 1963b, 1,293-307.
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Lamond, T. G.; Metcalfe, J. E., III; Walker, P. L., Jr. 6A Molecular Sieve Properties of Saran-Type Carbons. Carbon 1965,3,59-63. Moreno-Castilla, C.; Mahajan, 0. P.; Walker, P. L., Jr.; Jung, H. J.; Vannice, M. A. Carbon as a Support for Catalysta-111. Carbon 1980, 18, 271-276. Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984; p 189. Schmitt, J. L., Jr.; Walker, P. L., Jr. Carbon Molecular Sieve Supports for Metal Catalysta-I. Preparation of the System Platinum Supported on Polyfurfuryl Alcohol Carbon. Carbon 1971, 9, 791-796. Schmitt, J. L., Jr.; Walker, P. L., Jr. Carbon Molecular Sieve Supports for Metal Catalysts-11. Selective Hydrogenation of Hydrocarbons over Platinum Supported on Polyfurfuryl Alcohol Carbon. Carbon 1972, 10,87-92. Trimm, D. L.; Cooper, B. J. The Preparation of Selective Carbon Molecular Sieve Catalysts. Chem. Commun. 1970,477-478. Trimm, D. L.; Cooper, B. J. Propylene Hydrogenation over Platinum/Carbon Molecular Sieve Catalysts. J . Catal. 1973, 31, 287-292. Yang, R. T. Gas Separations by Adsorption Processes; Butterworths: Boston, 1987; pp 14-17.
Received for review June 5, 1990 Revised manuscript received August 30, 1990 Accepted December 11, 1990
Unified View of Transport Phenomena Based on the Generalized Bracket Formulation+ Brian J. Edwards and Antony N. Beris* Department of Chemical Engineering and Center for Composite Materials, University of Delaware, Newark, Delaware 19716
The Hamiltonian formulation of equations in continuum mechanics through generalized brackets is presented here in order to demonstrate the inherent structure and similarity between a variety of transport phenomena. The bracket formulation presented in this paper is based upon the Poisson bracket description of continuous systems and the entropy dissipation postulated in irreversible thermodynamics. This general formulation is presented for both single-component and multicomponent systems, as well as for systems with internal structure, for example, viscoelastic fluids. Thus, in addition to providing an alternative formulation for transport phenomena (of value for possible new numerical schemes), this paper represents the initial stages of a generalization of nonequilibrium thermodynamics to complex media (i.e,, materials with internal structure), which has never been accomplished t o date in a fully satisfactory manner using traditional approaches. 1. Introduction
For many years, intuition and experience have told us that the physics of transport phenomena is governed by an underlying structure or symmetry that is the same regardless of the type of transport involved (Bird et al., 1960; Sherwocd et al., 1975). Only recently, however, have the methods become apparent through which this underlying physical symmetry can be demonstrated. The major device in this task is the generalized bracket formulation, which has been only recently proposed (in a number of different variations) for dissipative continua (Kaufman, 1984; Grmela, 1985,1986,1989; Beris and Edwards, 1990a,b). The generalized bracket formulation defines the time evolution of an arbitrary functional (integral function) in terms of the Hamiltonian (totalenergy) and the dissipation present in the system. Dedicated to the memory of the late R. L. Pigford.
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The generalized bracket formulation reduces to the more familiar Poisson bracket description when dissipation is absent. The Poisson bracket formulation of nondissipative continua has been developed only relatively recently (in the past 30 years) despite the fact that ita counterpart for discrete systems, developed at the beginning of the last century by Poisson (1809), had astonishing successes in the development of quantum mechanics at the beginning of this century (Lanczos, 1972). The Poisson bracket formulation has recently stirred considerable interest (Dzyaloshinskii and Volovick, 1980; Morrison and Greene, 1980; Holm et al., 1985; Grmela, 1986,1988; Salmon, 19881, mainly due to the advantages gained through its application to nondissipative systems with respect to nonlinear stability analyses (Holm et al., 1985). The objective of this paper is to reveal the underlying Hamiltonian structure of a variety of transport processes through the generalized bracket formulation. This can serve many purposes, such as the development of better 1991 American Chemical Society
