J. Chem. SOC., Faraday Trans. 1, 1989, 85(7), 1545-1556 Study of Ultramicroporous Carbons by High-pressure Sorption Part 2.-Nitrogen Diffusion Kinetics Jacob E. Koresh", Tae Han Kim, David R. B. Walker and William J. Koros Department of Chemical Engineering, University of Texas at Austin, Austin, Texas 78712, U.S.A. Sorption-desorption kinetics for nitrogen in the as-received TCM 128 ultramicroporous carbon fibre are reported at 35 "C over a wide range of pressures. The nitrogen sorption kinetics at low pressures follow the Fickian model. As the pressure increases, the deviations from the model become more pronounced, and at sufficiently high pressure a sigmoid kinetic response shape is observed which is indicative of a non-Fickian diffusion process. An additional timescale must be active to account for the non- Fickian transport behaviour.This second timescale process may correspond to adsorbate surface rearrangements leading to locally time-dependent clearing of constrictions at a rate with a characteristic kinetic constant. The nitrogen desorption kinetics are found to be less affected by the non-Fickian transport, thus leading to higher apparent diffusion coefficients for the same average pressures. At high pressures and very long sorption times, slow protracted uptake becomes apparent into restricted regions. The kinetics for this process can be described by a barrier model and are sufficiently slow to allow treatment independent of the processes occurring in the more open pore system. In a previous study' with permanent gases under cryogenic temperatures for several activated samples of TCM 128 carbon fibres, direct measurements of the molecular diffusion coefficients were not made. In this case, in fact, severe non-Fickian sorption behaviour was observed. Moreover, the as-received carbon was not studied, because these gases are effectively non-sorbing at cryogenic temperatures for the as-received materia1.2~3 It was an objective in this study to determine if Fickian sorption kinetics could be observed at higher temperatures such as those of interest in actual separation applications.It was desired to measure actual molecular diffusion coefficients for nitrogen movement within the as-received TCM 128 under different degrees of saturation of the micropores. Also, because of the paucity of data on the as-received substrate material, we chose to focus our attention on it to establish a basepoint for later comparisons with more activated samples. We studied the ambient temperature range, since work with only slightly activated carbons had demonstrated22 the increased ability to penetrate restricted pore environments as temperature increases.The present study, therefore, extends our characterization of TCM 128 to include high-pressure nitrogen sorption-desorption kinetic data in the as-received material near ambient temperature. The data lead us to an improved understanding of the complex morphological nature and energetic heterogeneity of this ultramicroporous material. In this respect, the nitrogen penetrant serves as an ultrafine probe of the structure of the material.15451546 Nitrogen Difusion in Ultramicroporous Carbon Experimental The fibrous carbon cloth, TCM 128, and the high-pressure sorption apparatus and procedures used in this work were the same as described in Part 1 of this series dealing with equilibrium aspects of gas sorption4esorption. The nitrogen gas used was obtained from Linde at a purity of 99.999% and was used as-received. Theory Assuming a homogeneous continuum for the carbon fibres, Fick’s second diffusion equation in cylindrical coordinates becomes : - ac = --[iz(c)rg] 1 a at r a r where 9(C) is an effective local diffusion coefficient that may be concentration dependent owing to at least two factors. The most obvious of these factors is simple energetic site heterogeneity which causes the local diffusion coefficient to increase with increasing degree of site saturation. This effect occurs because of the higher activation energy for diffusion of the molecules occupying the more energetic sites.Obstruction of critical constrictions, on the other hand, causes the local diffusion coefficient to decrease, rather than increase with degree of site saturation. In this case, a literal ‘traffic jam ’ can occur at the constrictions where the molecular sieving process tends to occur. The concentration dependence of this obstruction function modifies the concentration dependence of the inherent diffusion coefficient which arises from simple energetic heterogeneity. Since a constant-volume sorption cell was used in the present study, the pressure drops slightly as the adsorption proceeds; however, the pressure in the gas phase is always uniform throughout the cell.For the case where 9 ( C ) NN D (a constant) the solution of eqn (1) with a changing boundary condition has been rep~rted.~ After integration with respect to r one obtains the expression for M,/M,, the normalized amount of material sorbed (or desorbed) in time t relative to the amount of material sorbed (or desorbed) at equilibrium ( t = 00) during a particular incremental run: where the q, are the positive, non-zero roots of and a = Kell/( Kdsorbent K ) the ratio of the volumes of adsorption cell and the adsorbent modified by the partition function, K = C(carbon)/C(gas). J , and J1 are the Bessel functions of the first kind of order zero and one, respectively.The qn values were obtained by solving eqn (3) with the experimental value of a for our particular system. The a value used in the above equation for each incremental sorption or desorption run was determined using the equilibrium IC values for each run as calculated from the sorption or desorption equilibrium isotherms reported in Part 1. The simple analytical expression given by eqn (2) in terms of a constant diffusion coefficient, D, has been shown to be quite effective for describing sorption kinetics over the entire sorption kinetic run if an average effective diffusion coefficient, D, is used for the concentration range of interest. The diffusion coefficients were obtained by curve- fitting of the experimental data up to M,/M, < 0.5 with eqn (2) using 32 qn values.J. E.Koresh, T. H. Kim, D. R. B. Walker and W. J . Koros 1547 1 .o 0.8 0.6 0.4 0.2 0.0 1 .o 0.8 0.6 0.4 0.2 0.0 0 1 0 2 0 3 0 4 0 0 1 0 2 0 3 0 4 0 5 0 6 0 J;/min: Fig. 1. Nitrogen adsorption kinetics at 35 "C (a model fit): (a) 1.3 atm, (b) 18 atm. Solid line without superimposed experimental data is the Fickian model. The average diffusion coefficient for use in eqn (2) is appropriately defined as: D = /I:9(C)dC/(C2-Cl) (4) where C, and C, refer to the concentrations in the carbon at equilibrium at the end and beginning of a sorption (or desorption) run, re~pectively.~ Clearly, the average diffusion coefficient in eqn (4) reflects the complex effects of site heterogeneity and obstruction factors discussed in terms of the local diffusion coefficient, 9(C).For cases in which even more complex effects cause additional time-dependent phenomena not linked simply to local concentration, the preceding Fickian analysis is insufficient. It is anticipated that relaxation of the rigid carbon matrix is highly unlikely. Nevertheless, non-Fickian effects which will be presented in the experimental results suggest that an additional process with a timescale longer than that of the local diffusive process is occurring. Results Nitrogen Kinetics at Low and Medium Pressures Sorption Fig. 1 (a) and 1 (b) show typical N, sorption kinetics at low (ca. 1 atmt) and medium pressure (ca. 18 atm). The lines through the points, corresponding to the Fickian model using the D values, fit the experimental data very well up to 0.65 of the normalized amount adsorbed at low pressure and only up to 0.45-0.5 at high pressure before starting to deviate.Fig. 2(a) shows the trend of the nitrogen kinetics as a function of pressure increase during the time interval where the Fickian model fits. The apparent diffusion coefficients as calculated using the curve-matching technique t 1 atm = 101 325 Pa.1548 Nitrogen Difusion in Ultramicroporous Carbon 0.6 1 0.4 0.2 0.0 1 .o 0.8 0.6 0.4 0.2 0.0 0 2 4 0 2 0 4 0 6 0 Fig. 2. Nitrogen adsorption kinetics at 35 "C for low and medium pressures: (a) short times, (6) long times; (0) 1.3 atm, (A) 8.5 atm, (0) 13.4 atm, ( x ) 17.9 atm. described above from eqn (2) for the data in fig. 2(a) are plotted as a function of the average pressures? in fig.3. The initial slope of the kinetic curves tends to increase with increasing pressure, as reflected by the D values in fig. 3. On the other hand, at intermediate and large normalized uptake ( M J M , > 0.4) a complex crisscrossing of the kinetic responses occurs with a general trend for the approach to final saturation to be more protracted as the average pressure increases [fig. 2(b)]. In order to explain the above behaviour, it is useful to consider an additional factor besides the simple energetic site heterogeneity that is generally believed to cause the increasing trend in D as a function of the extent of pore volume filling. Specifically, in addition to such penetrant-pore interactions, the mobility of a given molecule through the partially filled pores may decrease owing to steric blockage of free motion within the pore by the presence of other sorbed and diffusing penetrants as pressure increases.If the kinetic constant for establishment of such impediments is sufficiently fast compared to the local diffusion coefficient divided by the square of a characteristic length such as the fibre radius, the sorption kinetics will be Fickian. This factor will tend to cause the diffusion coefficient to decrease with increasing adsorbate concentration. On the other hand, when time-dependent adsorbate rearrangements occur over longer timescales than that of the local diffusion process, an even more complex non-Fickian case occurs owing to time dependence in the diffusion coefficient. If the observed negative deviations at long times from eqn (2) were due only to concentration dependence, the diffusion coefficient for sorption at high pressure should be a strongly decreasing function of pressure.In this case, however, at the end of run i, the same 'morphology ' should exist as at the very beginning of run i + 1. Clearly, since the long-time slopes decrease with increasing pressure, more resistance to motion exists at the end of the run than at the beginning. In this case the initial slope for the next run 7 The average pressure refers here to the average of the pressures at the beginning and end of the sorption or desorption run being considered.J . E. Koresh, T. H . Kim, D. R. B. Walker and W. J. Koros 0 10 2 0 30 4 0 4, /am Fig. 3. Adsorption diffusion coefficient as a function of average pressure.0.6 0.4 0.2 0.0 1549 0 2 4 6 Jtlrnin 5 Fig. 4. Nitrogen desorption kinetics for low and medium pressures: (0) 4.2 atm, (0) 6.7 atm, (a) 11.3 atm, (H) 19.6 atm, (0) 31.9 atm, (A) 42.7 atm. should be smaller than the preceding one, which is opposite to the observed results in fig. 2(a). These facts, therefore, suggest that the deviations are actually due to time- dependent rearrangements that relax out by the time the next sorption run starts. Presumably, energetic heterogeneity produces a concentration-dependent diffusion coefficient up to the point at which sufficient penetrant loading of the constrictions1550 Nitrogen Diffusion in Ultramicroporous Carbon 1 .o 0.8 0.6 0.4 0.2 1 .o 0.8 0.6 0.4 0.2 0.0 0 2 0 4 0 6 0 8 0 0 2 0 4 0 6 0 8 0 Jt/m;lf Fig.5. Nitrogen desorption kinetics at 35 "C (a model fit): (a) 4.2 atm, (b) 19.6 atm. Solid line without superimposed experimental data is the Fickian model. makes the time-dependent rearrangements strongly influence the further movement into the pores. Desorption Fig. 4 shows the normalized desorption kinetics for different pressures. Qualitatively these kinetics resemble the behaviour of the adsorption kinetics at short times, namely a rate increase with pressure reflecting the heterogeneous energetics of the sites. In the desorption data the deviations from the Fickian model all begin to occur at approximately the same value of M J M , (fig. 5 ) and therefore, the desorption kinetic response curves (fig. 4) do not cross as in the adsorption curves (fig.2). This indicates that the desorption kinetics are affected less by the time-dependent reduced mobility in the pores believed to be responsible for the deviations at long times in the sorption runs. The desorption diffusion coefficients calculated from eqn (2) (fig. 6) reveal much larger values at higher pressures for the desorption than the corresponding adsorption runs at the same average pressure. This is inconsistent with a Fickian response in which the local diffusion coefficient increases monotonically with concentration or pressure. This is a general conclusion based on the fact that in cases where diffusion coefficients increase monotonically with increasing local penetrant concentration the desorption rate is lower than the sorption rate.6 Clearly, at low sorption levels, steric obstruction effects like those indicated in the sorption results at long times and high pressures will be of minimum importance. Thus at low pressures the kinetics should approach the Fickian limit in which the sorption response occurs more rapidly than that of desorption.The data in fig. 6 are consistent with this expectation, indicating a higher effective diffusion coefficient for the sorption versus the desorption experiment in the limit of sufficiently low pressures. If the form of the local concentration-dependent diffusion coefficient displays a maximum at low concentration^,^ behaviour such as ours with a larger desorption diffusion coefficient as compared to the sorption coefficient for the same incremental concentration interval occurs.While the use of such average coefficients could obscure aJ . E. Koresh, T. H . Kim, D. R. B. Walker and W. J . Koros 1551 0 10 2 0 30 P., /am Fig. 6. Adsorption-desorption diffusion coefficient as a function of average pressure : (0) adsorption, (+) desorption. maximum occurring in local diffusion coefficients for a single run spanning the local concentration range over which the maximum occurs’ this cannot be the cause of the observed results, since the average diffusion coefficient data in fig. 6 increase monotonically with increasing concentration. Clearly, to maintain the trend of higher desorption us. sorption rates, even the average diffusion coefficient must decrease with increasing pressure above the concentration corresponding to the maximum in the local coefficient.No such maximum occurs in fig. 6, indicating that no form of concentration dependence alone can explain the observed data. Therefore, both the sorption and desorption observations indicate strongly that the deviations from eqn (2) are due to time-dependent rearrangements rather than concentration dependence of the diffusion coefficient. We believe that no relaxation of the rigid carbon matrix occurs at medium pressures; however, some additional timescale process must be active to account for the non- Fickian transport behaviour. Specifically, adsorbate present in the vicinity of critical constrictions may inhibit the random walk of other penetrants through the passages. If these obstructing adsorbates move from these easily accessible high-energy sites into less accessible (and less obstructing) low-energy sites, the random walking of other penetrants again becomes less obstructed.Therefore, we suggest that the two time scales controlling movement within the carbon correspond to : (i) concentration-dependent diffusion as discussed earlier in the context of time-independent obstruction and site heterogeneity ; (ii) obstruction-clearing rearrangements that have a characteristic kinetic constant not linked to the diffusion process, without which diffusive passage through local constrictions is obstructed to various degrees.1552 0.6 - 0.4 - \ s x 0.2 - 0.0 Nitrogen Difusion in Ultramicroporous Carbon . 0.0 - 0 1 2 3 4 5 0 1 2 3 4 5 Jrlminf Fig. 7. Nitrogen kinetics at 35 "C at high pressures.(a) adsorption: (A) 31.5 atm, (0) 45.3 atm, (0) 57.2 atm; (b) desorption: (A) 42.7 atm, (0) 52.5 atm, (0) 60.3 atm. If the kinetic constant for the adsorptive rearrangements is at least an order of magnitude smaller than the diffusion coefficient divided by the square root of the fibre radius, the sorption kinetics will tend to be non-Fickian and strongly affected by the adsorption rearrangement kinetics. In the limit where the rearrangements required to allow diffusion are very slow, they could totally control the uptake. As discussed above, time-dependent rearrangements affect the sorption process more than the desorption process as indicated by the earlier onset with increasing pressure of deviations from eqn (2) for sorption compared to desorption.This is consistent with the higher apparent diffusion coefficient for desorption as compared to sorption. Although the kinetic data can be fitted using these apparent diffusion coefficients in eqn (2) for a certain range of MJM,, we believe the response is non-Fickian over the entire range for pressures above 10 atm. This conclusion is based on the earlier argument that the sorption response must lie above that for desorption for Fickian kinetics with a monotonically increasing diffusion coefficient. Since this is clearly not observed in our data above 10 atm (fig. 6), the apparent diffusion coefficients are actually complicated measures of the overall diffusion and relaxation processes. Rigorous decoupling of these effects would require either steady-state permeation or transient sorption measurements using samples with different characteristic dimensions.Neither of these options was available to us with the current material. High-pressure Sorption and Desorption Kinetics The sorption and desorption responses for high pressures (> 30 atm) are shown in fig. 7 (a) and 7 (b). Clearly the qualitative shapes of these responses show a marked increase in sigmoid nature of the kinetic response and differ significantly from those in fig. 1 for the low and medium pressure ranges. This sigmoid shape starts to appear at about 30 atm for the sorption kinetics and at ca. 50 atm for the desorption. It is another indication that the desorption kinetic process is less affected than the sorption process by the factors responsible for the non-Fickian response at high pressure.J .E. Koresh, T. H. Kim, D. R. B. Walker and W . J. Koros 1553 High-pressure Sorption Kinetics into Restricted Regions At pressures above 35 atm a very slow adsorption into restricted regions occurs through tiny constrictions which are responsible for the unusual sorption-desorption isotherm hysteresis discussed in our earlier paper.* Three important points relevant to the current study were found in this earlier paper: (i) the kinetic sorption rates into these restricted regions are orders of magnitude lower, (ii) the small amount adsorbed in these regions is much less than that for the more open pores for the timescales studied and (iii) no desorption from the restricted regions was observed even at the lowest desorption pressure studied (4 atm).Quantitative estimation of the diffusion coefficient for uptake into these regions will be given below. Since the kinetics of adsorption into restricted regions is very slow, we can describe the high-pressure sorption as composed of two parallel uptake processes with two different average diffusion coefficients. The restricted-regions kinetic rate does not interfere with the uptake process for the more open pores which saturate much more quickly; however, these regions do affect the total amount adsorbed. To avoid incorrectly scaling the M J M , response for the faster process, the total apparent M , must be corrected to eliminate the contribution of the restricted regions. Correction of the M , could be done in two independent ways: (a) M , for adsorption could be evaluated from the desorption isotherm.Since there is no long-time desorption apparent from the restricted regions, taking the amount desorbed, from the isotherm, at the same pressure interval for the adsorption point, should give the corrected M,; or (b) extrapolation of the last points of M, us. square root of time to zero time for the sorption run. The second method could give an erroneous M , value for the following reasons. Towards the end of the high-pressure runs the order of magnitude of the adsorption rate into both the open and restricted regions become very similar. This phenomenon occurs because the driving gradient is largely dissipated for the uptake into the more open pores. The tremendous adsorption transport resistance of the restricted regions, on the other hand, forces the uptake rate in this region always to be small in spite of the significant driving force.The first method clearly distinguishes the amount adsorbed in the restricted regions. Corrections for the two highest pressure points which were found to be the only cases affected by the restricted-region penetration, were made. The corrections of other points were found to be negligible, consistent with our earlier conclusion that adsorption in restricted regions can occur to a significant extent only above 35 atm. The corrected kinetic responses corresponding to the results in fig. 7(a) are given in fig. 8. It is clearly seen that the curves in fig. 7 ( a ) are shifted upwards without changing the nature of the sigmoid shape of the high-pressure kinetic responses.A plot of the kinetics of uptake into restricted regions is shown both as a function of time and as a function of square root of time over the entire period of the high-pressure incremental sorption run at 52 atm [fig. 9 ( a ) and (b)]. The linearity of the plot versus time is striking and suggests that the additional uptake process is not simply a diffusion- controlled sorption into a network of pores with uniformly smaller diffusion coefficient. If this latter situation occurred, the response would be expected to be linear in square root of time with a zero intercept unlike the observed behaviour in fig. 9(b). The linear behaviour in fig. 9(a) is, in fact, consistent with a situation that has physical significance relative to the morphology of the as-received carbon’s restricted regions. Specifically, we believe that the bulk of the volume of these regions is similar to the more open regions in terms of typical pore diameters.These regions are, however, restricted due to a very small number of limiting constrictions that forbid ready access to the considerable volume bordered by them. In other words, a barrier exists to penetration into the restricted regions, and an effective permeability could be assigned to this barrier based on the number and size of the tiny limiting constrictions. An indication of the magnitude1554 Nitrogen Difusion in Ultramicroporous Carbon 0.8 0.6 0.4 \ s 0.2 0.0 0 1 2 3 4 5 6 J;lmin: Fig. 8. Nitrogen adsorption kinetics at 35 "C at high pressures after M , correction.0 10 2 0 30 4 0 5 0 6 0 7 0 filmin; 0 .o 0 1000 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 tlmin Fig. 9. Nitrogen adsorption kinetics at 35 "C at high pressures into restricted regions. ( a ) 2's. \ t. (b) vs. t .J. E. Koresh, T. H. Kim, D. R. B. Walker and W. J. Koros 1555 of these regions can be seen by results of earlier studies where incremental activation from 1 to 2% weight loss compared to the as-received TCM increased the volumetric uptake by 60% even at liquid nitrogen temperatures [compare fig. 2 ref. (3) with fig. 3 ref. (9)]. Since these tiny constrictions can be destroyed to a sufficient degree to allow access to the restricted regions that was previously precluded, the small amount of activation has a tremendous effect on the sorption level.Based on the above barrier interpretation, the slope of the plot in fig. 9(a) will be a function of the driving pressure across the limiting tiny constrictions as well as the number of such constrictions. As time progresses, the dissipation of the driving force between the restricted regions and the more open porosity will cause an eventual breakdown in the Mtl us. t relationship; however, this has clearly not occurred to a significant degree in fig. 9(a) even after 5000 min. For a barrier model under pseudo- steady-state conditions, an exponential time response would be observed as the pressure difference between the open and restricted regions seeks to equalize itself.' The exponential behaviour appears as the observed linear time response because only a small fraction of the percentage equilibration has been achieved.The slope of fig. 9(a) normalized by the capacity of the restricted regions? gives the effective time constant for the exponential function governing the equilibration of these regions. For example consideration of the 52 atm point indicates a time constant of 7 x lo4 min. The time to reach 90 % equilibration of the restricted regions would be 1.6 x lo5 min. This is much more than two orders of magnitude longer than the time to reach an equivalent degree of equilibration in the more open pores, therefore it is clear why this extremely long- timescale process does not interfere with the kinetics of the normal sorption process in the open pores.Conclusions Sorption4esorption kinetics for nitrogen in the open-pore system of the as-received TCM are complicated by concentration-dependent diffusion and additional slow kinetic rearrangements. No form of concentration dependence of the local diffusion coefficient can explain the observed data without also invoking time dependence of this coefficient. This time dependence may occur when a molecule adsorbs at an easily available but relatively high-energy site that can obstruct easy diffusion of other molecules. Rearrangement from such an unstable mode into a lower energy and unobstructing site presumably produces this additional time-dependent effect. The apparent diffusion coefficients at pressures above ca. 10 atm are complicated measures of the combined diffusion and relaxation processes which cannot be further decoupled with the presently available data. In addition to the above effects in the open porosity, protracted sorption into restricted regions occurs in a process moderated by tiny barrier constrictions. This supports the idea that the majority of these regions are composed of pores whose dimensions are similar to those in the accessible open porosity, but whose access is precluded by a limited number of tiny constrictions. t The restricted regions capacity is conservatively estimated to be 40 YO of the more open pore capacity at the same pressure.1556 Nitrogen Diflusion in Ultramicroporous Carbon References 1 J. Koresh and A. Soffer, J. Chem. SOC., Faraday Trans. 1, 1980, 76, 3005. 2 J. Koresh and A. Soffer, J . Chem. SOC., Faraday Trans. 1, 1980, 76, 2457. 3 J. Koresh and A. Soffer, J . Chem. SOC., Faraday Trans. 1, 1980, 76, 2472. 4 J. Crank, The Mathematics of Diffusion (Clarendon Press, Oxford, 2nd edn, 1975), p. 77. 5 V. T. Stannett, G. R. Ranade and W. J. Koros, J . Membrane Sci., 1981, 10, 219. 6 Ref. (4), p. 183. 7 Ref. (4), p. 186190. 8 J. Koresh, T. H. Kim and W. J. Koros, J . Chem. SOC., Faraday Trans. I , 1989, 85, 1537. 9 J. Koresh and A. Soffer, J . Colloid Interface Sci., 1983, 92, 517. Paper 8/02361A; Received 16th August, 1988