Calculation of Adsorption Energy Distributions of Silica Samples Using

Langmuir 1995,11, 1735-1743

1735

Calculation of Adsorption Energy Distributions of Silica Samples Using Nonlinear Chromatography Brett J. Stanley? and Georges Guiochon” Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1501, and Chemical and Analytical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -6120 Received August 11, 1994. In Final Form: February 2, 1995@ The estimation of adsorption energy distributions from nonlinear chromatographic data is considered in detail from both experimentaland theoretical viewpoints. The experimentaldata is obtained on DAVISIL, IMPAQ, and VYDAC silica samples. The adsorbates studied include diethyl ether, methanol, ethanol, tetrahydrofuran, pyridine, and heptane. The theoretical models of adsorption studied are the Langmuir, Jovanovic, Fowler-Guggenheim (both random and patchwise), and Brunnauer-Emmett-Teller local isotherms. The chromatographicdata is obtained with the use ofhigh-efficiencyporous layer open tubular columns. The experimental variables of temperature and maximum solute partial pressure are studied for their effect on the data and estimation. The validity of the technique is assessed. It is concluded that adsorption energy distributions may only be calculated accurately and without bias for systems in which the majority of the adsorption energy is 10 kJ/mol greater than the heat of vaporization of the solute. For adsorption energies less than this, intermolecular interactions decrease the accuracy and confidence of the results. Other experimental parameters also limit the scope of studies of this kind. These include the detector range and resolution, as well as the injection profile. The major obstacle in these studies is the difficultyin samplingthe entire range of adsorptionenergies,especiallythe low energieswhich correspond t o the required measurement of the retention times corresponding t o high solute partial pressures.

Introduction Adsorption energy distributions of solute probes on heterogeneous surfaces have been determined from their nonlinear chromatographic band profiles for the last 20 years as a method ofinverse chromatography. The general problem of obtaining distribution functions from adsorption isotherm data has proceeded for a substantially longer period of time, with many of the nagging fundamental obstacles being delineated and characterized along the way.lS2 The basic fundamental hurdle is centered around the solution of the linear Fredholm integral of the first kind:

In eq 1, q ( p ) is the experimentally measured global adsorption isotherm a s a function of the solute partial pressure,p; B(c,p)is the local model of adsorption for each adsorption site of energy, E ; and R E )is the distribution function of interest. The solution of this equation for the function RE)is mathematically ill-posed, which is to say that any error in the data, q(p), is unbounded upon inversion to the solution, RE). In order to accelerate research in this area, chromatographic retention volumes may be used to obtain the adsorption i ~ o t h e r m : ~ Current address: Department of Chemistry, California State University, 5500 University Parkway, San Bernardino, CA 924072397. *Author t o whom correspondence should be sent at the University of Tennessee. Abstract published in Advance A C S Abstracts, May 1, 1995. (1) Rudzinski, W.; Everett,D. H.Adsorption ofGase8 on Heterogeneous Surfaces; Academic Press: New York, 1992. (2)Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, The Netherlands, 1988. (3) Conder, J. R.; Young, C. L. Physicochemical Measurement by Gas Chromatography; Wiley: New York, 1979; Chapter 9.

where M, is the mass of stationary phase or adsorbent under study, y is the mole fraction of solute a t partial pressure p , j is a James-Martin correction term, q~is a correction term for nonideal gas behavior and the sorption effect, and c is the concentration of solute a t partial pressure p . In this way an adsorption isotherm can be obtained in one chromatographic run and potentially possess hundreds of data points. Use of a highly linear detector such as the flame ionization detector (FID)allows a n isotherm segment covering 5 orders of magnitude in partial pressure. In order to offer a n unbiased solution to eq 1,numerical estimation becomes necessary, as any assumption of a n analytical function for qk)dictates the properties of the solution RE). Any assumption of the analytical form of R E )obviously biases the solution, and there is no a priori reason to assume any type of distribution for studies of this type. Also, there is no way to validate the experimental estimation of distribution functions since no standard surface distribution function exists. Numerical estimation of fee) assuming only the measured q(p)and a local model B(E,P) necessarily results in artifactual information since perfect noise-free data may never be obtained. The nature of solution inaccuracy may be dictated, however, by the algorithmic approach of the solution estimation. A tradeoff typically exists: resolution vs smoothness. For this class of problems one may increase the resolution by approaching the minimum squared variance of the data, q ( p ) , from the estimated data, qest@).Unfortunately, this condition is not equivalent to that of the estimated solution, &(E), being minimally close to the true distribution, RE). In order to minimize the latter deviation, it is common practice to incorporate some form of smoothing into the solution with the estimation algorithm. This practice increases the estimation variance, and the concomitant loss of inherent

0743-746319512411-1735$09.00/0 0 1995 American Chemical Society

Stanley and Guiochon

1736 Langmuir, Vol. 11, No. 5, 1995

resolution is justified by assuming that a smooth solution is closer t o the true distribution in all probability than a possibly artifactual, highly curved one. The above numerical problem has been observed in simulations with well-defined functions, correct assumed models, 8(c,p),and well-behaved noise (truly random or Gaussian).l The problem ofill-definednoise or systematic error present in the data is less well-known and presents a further challenge in this research area. We have recently shown how the finite efficiency of chromatographic columns affects the solution when nonlinear chromatography is used to obtain the isotherm data.4 Finite efficiency places a systematic error on the isotherm because eq 2 is solely founded on the ideal model of chromatographywhich assumes an infinite efficiency.The effect on the isotherm is that it is biased high. The effect on the estimated distribution is a divergent function a t the low-energy side of the distribution. This is apparently because the kinetic or diffusional information imprinted on the isotherm due to the finite eficiency resembles adsorption on a lower-energy site not yet sufficiently sampled by the data. We have also shown that in order to sufficiently sample an adsorption isotherm, approximately 70% of the monolayer must be p r e ~ e n t .Thus ~ a significant amount of isotherm curvature at the lowest energy site that is present on the surface must be sampled in order for that site to be confirmed with respect to the model. Diffusional processes appear to be adsorption processes still in the linear range of coverage with respect to the model (note that the highest energy sites saturate first and the lowest energy sites saturate last). Another obstacle in solving eq 1 for the distribution function is the problem of assuming the correct local model of adsorption. The affect of an incorrect local model has been investigated and shown to influence the accuracy of the obtained distribution.2 One of the primary decisions that must be made in choosing a local model is the presence and nature of lateral adsorbate-adsorbate interactions on the surface. If these interactions can be neglected, either the Langmuir or the Jovanovic model may be chosen, with the Langmuir model being the most popular. The Langmuir model may be written as6

(3) where K is the preexponential factor, R is the ideal gas constant, and Tis the column temperature. The Jovanovic model7 may be written as (4)

These models also assume static adsorption and no multilayer effects. If lateral interactions are to be modeled, the FowlerGuggenheim model is the most popular choice. This model can be written as a simple modification of the Langmuir model+

where

5 is the interaction term:

(4)Stanley, B.J.;Guiochon, G. Langmuir, in press. (5) Stanley, B. J.; Guiochon, G. J . Phys. Chem. 1993, 97, 8098. (6) Roles, J.;Guiochon, G. J. Phys. Chem. 1991, 95, 4098. (7) Sircar, S.;Myers, A. L. Surface Sci. 1988,205, 353. (8) Jagiello, J.; Ligner, G.; Papirer, E. J. ColloidInterface Sci. 1990, 137, 128.

and c is the interaction parameter quantitating the total interaction energy of an adsorbate molecule with all its neighbors. 8' represents the relative coverage of the surface as it affects the density of adsorbates on sites of the same energy. In order to model the lateral interactions properly, a further assumption about the surface topography must be chosen in order to depict the relevant 8' correctly. The extreme cases are that of a patchwise surface and a random surface. The patchwise surface contains adsorption sites of the same energy in discrete patches with complete correlation of the lateral interactions to the neighbors within that patch. The random surface assumes a random distribution of adsorption energy across the entire space of adsorption sites with no correlation of neighbors to the same adsorption energy. As with many cases in physical theory, a situation in between most likely exists. However, an extreme must be chosen for practical reasons of model simplicity and tractability. For the patchwise surface, 8' is taken as the local coverage within the energy patch under consideration. For the random surface, 8' is taken as the global coverage across the entire surface. The interaction parameter, c, is estimated from theory. For a rigorous estimation, a statistical mechanical estimate should be developed. At the very least, the number of nearest interacting neighbors should be known, multiplied by the interaction energy between two adsorbates, which might be estimated from a known distance which separates the adsorbates, etc. Considering the problems and assumptions associated with AED estimation alluded to so far, it seems overzealous to pursue the estimation of c in any complicated sort of manner. In a homogeneous surface study, where only one adsorption energy is present, the entire 5 term may be fit to the data as one parameter, thus circumventing the patchwise vs random choice, or the topography issue may be investigated while fitting the c parameter. However when considering an adsorption energy distribution, the 5 term must be completely specified in order to keep the linear Fredholm integral, eq 1,linear. The effect of lateral adsorbate-adsorbate interactions is to increase the adsorption as the partial pressure and surface coverage increases. Thus as the higher energy sites become saturated, lower energy adsorption becomes influenced by these lateral interactions. The deconvolution of the lateral interactions from the low-energy adsorption can only succeed if the accuracy of the model and the quality of the data (signal-to-noise ratio) are very high. If the model is not accurate, or if the modeled response of lateral interactions is similar to that of lowenergy adsorption, it is impossible to estimate the lower energy adsorption. The intermolecular interactions sustain continued adsorption and the sites never appear to saturate. Compounding this impasse is the onset of multilayer effects. In cannot be assumed that a full monolayer (100%)is adsorbed before another one begins, nor can it be assumed that 70%of a monolayer is present before another one begins a t the high-energy sites. Thereforefrom a theoretical standpoint, it is a challenging task to accurately estimate an adsorption energy distribution due to the error incurred in the low-energy region of the distribution,resulting from an adsorption isotherm that does not appear sufficiently sampled. In this paper we show the ramifications of the above considerations for the case of organic adsorbates and silica adsorbents using nonlinear chromatography (eq 2) to determine the adsorption isotherms. Additional considerations more directed toward the chromatographyaspects of the problem will also be discussed. In particular, it is

Langmuir, Vol. 11, No. 5, 1995 1737

Adsorption Energy Distributions of Silica Table 1. Characteristics of the Silica Samples Used in This Study silica type

surface area (mZ/g)

particle size +m)

pore size (A)

Davisil grade 633 Impaq RGlOlOSi* Vydac 101 TP

480 366 106.5

35-74 8.8 6.583

60 101 380

a Data obtained from respective manufacturers. Note that the surface area of the Impaq in a previous study14possessed a different surface area (246 m2/g)

observed that in only certain cases may an adsorption energy distribution be adequately estimated. For many of the systems studied, the majority of the adsorption energy is too low, or too close to the adsorbate's heat of vaporization, to be adequately estimated by this technique. Accurate energy distributions are reported for diethyl ether on two silicas, and alcohol on one silica, revealing the heterogeneous nature of the pertinent interactions.

system is capable of a 10 Hz sampling rate, and possesses 19 bits of resolution for 5 orders of magnitude of signal output. Lower sampling rates were often used to reduce the memory requirements, as 230 min were often required for the complete elution of many strongly retained probes. The collected data is then uploaded to the VAX 6000 computer at the University of Tennessee Computing Center for the further necessary processing. Data Analysis. The isotherm is calculated via eq 2; further details can be obtained from the references. The adsorption energy distribution is calculated with the expectation-maximization method, which also has been described in detail.5 This latter calculation has been programmed in parallel on a MasPar MP-2 supercomputer, maintained by the Computer Science Department at the University of Tennessee. This allows largesized problems to be computed in a few minutes (up to 1000 data points and 500 energy points). Twenty thousand iterations were performed in all calculations. The energy axis end points were determined from the minimum and maximum pressure points measured by use of the condensation approximation:

rim)

Emin= -RT In -

Experimental Section Materials. The characteristics of the silica samples used in this study are given in Table 1. The Davisil was obtained from Aldrich (Milwaukee, WI), the Impaq from BTR Separations (Wilmington, DE), and the Vydac as a gift. The solute probes were all used as received: diethyl ether (99.9%, Aldrich), methanol (99.9%,Baxter), ethanol (200 proof, Warner-Graham Co.), tetrahydrofuran (99.9%,Baxter), pyridine (99.9%,Baxter), and heptane (99.9%, Baxter). Methane (99%)was used for the measurement of the holdup time. Helium and hydrogen were both used as a carrier gas; no difference in results were observed. Instrumentation and Methods. The chromatographic columns used in this work are 546ym i.d. fused silica capillaries (PolymicroTechnologies,Inc., Phoenix, AZ)of approximately 15 m in length. The silica samples are coated on the inside wall of these capillaries according to a published pro~edure.~ Briefly, a suspension (ca. 1-3% w/v) of the silica is placed within the capillary, and then the column is drawn slowly into an oven maintained above the boiling point of the coating solvent with the use of two hubs and a small motor. The coating solvents are chosen for their viscosity in order to maintain a stable suspension throughout the coating procedure (e.g. dimethyl sulfoxide, ethylene glycol, or pentanol). After this procedure the columns are conditioned in the GC oven (HP5840, Hewlett-Packard, Kennett-Square, PA) at an elevated temperature for an extended period oftime (typically280 "C for 2-3 days) under approximately 1psi of inlet pressure. The weight of silica coated on the column is obtained from a weight difference of the column before and after this procedure with a Sartorius analytical balance. The isotherms are obtained by the elution-by-characteristicpoints method of nonlinear chromatography (eq 2h3 Neat liquids are injected by rapidly plunging the needle of a Hamilton syringe (either 0-1, 0-5, or 0-25 yL) by hand both into and out of the septum in the split injection mode. The column was reconditioned after every run via a 20 "C/min ramp to the FID temperature and held there for 5-10 min. Inlet temperatures were kept at least 110 "C higher than the boiling point ofthe solute. The split ratio is ca. 2.5%, which corresponds to a flow rate through the inlet to the split vent of 200-250 mumin. This procedure provides a very rapid and sharp injection profile. Column flow rates are maintained at 3-4 mumin (ca. 25-30 c d s ) , and the column efficiency, as measured with the methane peak, is 20 00025 000 theoretical plates (-1200 plates/m). Detection is performed with a flame ionization detector (FID) held at least 160 "C higher than the boiling point of the solute with a lean flame. The FID is calibrated independently with a 320 ym i.d. DB-5 analytical column (J&W Scientific, Folsom, CAI. This is necessary since the peak shapes on the test column of interest are so highly skewed from the heterogeneous elution that integration ofthe entire amount injected is exceedinglydifficult or impossible. The data is acquired from the HP5840 via a D/A board installed in the GC (ARS Services, Minneapolis, MN) and an A/D board installed on an IBM AT personal computer. The Peaksimple I1 software was used to collect the data (SRI, Torrance, CA). This (9) Roles, J.;Guiochon, G. J . Chromatogr. 1992,591, 233.

(7)

and

E,,

= -RT In

-

Goodness of fit is estimated by the root-mean-squared fractional deviations of the estimated data from the raw isotherm data. The distribution function is analyzed by the method of moments to yield the total amount adsorbed (zeroth moment), the mean adsorption energy (first moment) and the variance (second moment) for each peak as well as for the total solution. Local Models. The Langmuir model (eq 3), the Jovanovic model (eq 4), and the Fowler-Guggenheim models (eqs 5 and 6) were tested, as was the Brunnaer-Emmett-Teller multilayer correction to the Langmuir model.lo Trapezoidal quadrature was used in the integration of eq 1. The preexponential factor was calculated asll

wherePs is the vapor pressure and E,, is the heat ofvaporization of the solute. For the Fowler-Guggenheim (FG) patchwise model, an estimate of the fractional coverage, W , at each energy site as a function of the experimentally determined pressure is needed. This value is not known or available experimentally and must be approximated. The Langmuir coverage at each energy (eq 3) was used for this estimate in eq 6, and as such is biased low. More elaborate schemes may be used to improve the accuracy of this estimate, such as successive approximations starting with the Langmuir estimate; or approximation of .!Keg) using a Wcalculated from @(q) at a lower energy andor pressure with the FG model (starting with the Langmuir estimate for the lowest pressure and energy). However, considering the reservations outlined in the Introduction and the results outlined below, these procedures were considered unnecessary to support the aim of this paper. For the FG random model, the globally measured coverage was used, normalized to a predicted monolayer coverage based on the density of silanol sites.12 The interaction parameter, c, was estimated as Evad4.8

Results and Discussion I. Diethyl Ether Adsorption. Diethyl ether is a slightly basic, relatively nonpolar molecule with a low boiling point (bp = 34.6 "C) and heat of vaporization (Evap FZ 25 kJ/mol). Since the silanol sites on the surface of silica are weakly acidic in nature, this molecule should be a good probe ofthese sites. It is generally recognized that (10)House, W. A.; Born, G.; Brauer, P.; Franke, S.; Henneberg, K. H.; Hofer, P.; Jaroniec, M. J. Colloid Interface Sci. 1984,99, 493.

(11)Jaroniec, M. Surface Sci. 1975,50, 553. (12) Scott, R. P. W.; Kucera, P. J . Chromatogr. Sci. 1975,13, 337.


1738 Langmuir, Vol. 11, No. 5, 1995

-g

Table 2. Characteristics of the EM Solution for the Adsorption Energy Distribution of Methanol on Davisil Silica

I

0.201

3

0.16

- 5.0pL -- 10. pL

m

0.14

20. p L

a

0.12

model

Langmuir Jovanovic BET

FG-random FG-patchwise

goodness of fita (rms) apparent capacity (moVg)

0.01328 0.01061 0.01303 0.01338 0.01057

4.7 x 4.6 4.3 4.5 4.7

10-4 10-4 10-4 10-4 10-4

rn-18 = {X[(qexp - qeat)/qexp12/n)1'2 where qest is the estimated data from the estimated distribution,qexpis the experimental data, and n is the number of data points. (I

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Time (min) \

_----

2 5E-4-

-50pL 10. pL 20 p L

2 2 OE-40 1 5E-4-

m

U

1 OE-4t S 5.OE-5Y

0,000

0.002

0.004

0.006

0.008

0.010

0.01:

Relative Solute Pressure, P/Ps

'6

< -3

1.OE-4

I

I

cU_ 9.OE-50

E W

C

5.0pL _-

8.OE-5.-

10. p L 20. p L

7.OE-5-

.g 6.OE-5-+J

2 2

5.OE-54.OE-5--

t 3.OE-50

2.OE-5-7

3

n .L -w .-U -m

1.OE-5 0.0 35

,J

40

7 45

50

55

60

65

Adsorption Energy (kJ/mol)

Figure 1. Diethyl ether adsorption on Davisil silica at 60 "C: (a)nonlinear chromatographicband profiles,(b)ECP isotherm, and (c) adsorption energy distribution. it is the silanol sites which are responsible for much of the high-energy adsorption observed with silicas. Therefore, in large part, the silanol activity is responsible for the surface energy exhibited by most silicas. Three nonlinear band profiles of diethyl ether on a Davisil silica column are shown in Figure l a . These chromatograms were obtained at 60 "C a t a flow rate of 32.0 cm/s and a n inlet pressure, Ap, of2.1 psi, with helium as carrier. The elution was monitored for 50 min. The profiles obtained with different injection sizes possess a high degree of overlap of the diffuse profile rears. A small disagreement is observed near the peak maxima. If finite efficiency, longitudinal broadening, or slow kinetic mass transfer may be neglected, the diffuse rears should overlap perfectly. This condition may be checked more visibly by inspecting the isotherms vs injection size, as the integrative nature of eq 2 accumulates the differences in the diffuse rears as a function of partial pressure. The isotherms corresponding to the band profiles in Figure l a are shownin Figure lb. The 5 and lOpL injections overlap perfectly, and the 20 pL injection is biased high. The

column appears to be close to saturation for the higher amounts injected, since the isotherm slope is low a t the high pressure end and the retention time of the shock layer is close to the holdup time (the adjusted retention time is t ~= ' 3.5 s for the 2 0 p L injection). The adsorption energy distributions (AEDs)corresponding to these injections, calculated with the Langmuir local model, are given in Figure IC. The larger, lower energy adsorption peak in Figure ICis centered a t 46.1 f 0.1 kJ/mol; the smaller, higher energy peak is centered a t 57.0 f 0.1 kJ/mol (first moments from minima to minima; uncertainties refer to the standard deviations of three sequential injections). The respective peak widths, as defined by the square roots ofthe second moments, are 1.4 f0.1 kJ/mol and 2.1 f0.2 kJ/mol. The amount adsorbed within each peak (zeroth moment) is 2.75 f 0.07 x and 3.94 f 0.07 x mol/g, respectively, with the total apparent adsorption capacity being 3.14 f 0.07 x moVg. The accuracy of the zeroth moment is dominated by the accuracy of the mass of stationary phase measured, which unfortunately is low ( f lmg). This can correspond to column-to-column uncertainties from f 1 0 %to f50%in the amount adsorbed per unit stationary phase, depending on the amount coated. The accuracy of the first moment is dominated by the accuracy of the detector sensitivity calibration. This accuracy is good for probes with relatively high boiling points (>50 "C).However, for diethyl ether, the accuracy is reduced because the high volatility causes an uncertainty in the amount actually injected into the inlet. Thus the column-to-column (or day-to-day) uncertainty can be as high as 5-10%. The accuracy of the second moment is dependent on the inherent resolution of the numerical method, which is dependent on the noise characteristics of the data and is generally not tractable. The goodness of fit of the estimated isotherm to the experimental isotherm is generally around 1%error (see Table 2). Overlaying plots are therefore indistinguishable. The reproducibility of the AEDs in Figure ICis seen to be very good. However, a n error is observed for the main low-energy peak for the 5.0 pL injection. The cause of this error is generally postulated as incomplete coverage of all the low-energy sites o r an incomplete sampling of the entire isotherm. This hypothesis was documented and explained in earlier work: as this problem is very common. Even though the isotherms overlap perfectly for the 5 and 10 p L injections, the error occurs with the 5 p L injection because the isotherm was not measured to a high enough partial pressure; i.e., the isotherm is still increasing too rapidly a t its end point in Figure l b (solid line). However, the complete understanding of the results for this series lies in the particular details of the experimental data. The results are perplexing because all the well-estimated energies for the 10 pL and 20 p L injections appear almost completely contained within the domain of the 5 pL result. Moreover, the discrepancy cannot be alleviated via recalculation with a lower energy end point, Emin.On the other hand, the discrepancy between the isotherms obtained with the 10 and 20 pL injections does not influence the estimation of the m D s

Adsorption Energy Distributions of Silica

Langmuir, Vol. 11, No. 5, 1995 1739

-

A

0, 3

% 1.4E-4 E

v

- 0.5UL -- 1.0

1.2E-4

.-.w5 1 OE-4

l

) &,

2.0

i

UL UL

l

8.OE-5 3

LL 6.OE-5

U

Adsorption Energy (kJ/mol) Figure 2. Diethyl ether adsorption energy distribution on Impaq silica; data were taken at 30 “C. for these experiments. This latter discrepancy can be explained because the error is not grave enough to appear as adsorption a t lower energy sites. The slight shift in the main energy peak (to the dotted line from the dashed line in Figure IC) is enough to account for the corresponding high bias in Figure l b (with the fit being worse). Both discrepancies can be further understood by inspection of the chromatograms. The 20 p L injection tail is eluted a t slightly longer times than the 1OpL tail. This causes the corresponding high bias in Figure lb. Conversely, the 10 pL tail elutes slightly before the top of the 5 pL peak (an error), and this causes the low-energy peak at the corresponding high pressures (low energies) to not be contained for the 5 p L as for the 10 pL peak. Thus these effects are precipitated by the important details of the data and do not necessarily reflect a problem with the inversion algorithm, but instead with the ill-posed nature of the general problem. The details are much more important a t the borders (the so-called “border effect” in signal treatment.13 Correction for this effect requires further assumptions and bias in the data analysis (e.g. extrapolation), which may be the only way to achieve true stability in these types of problems, for all experimental data. Diethyl ether adsorption on the Impaq column was similar in nature, except a smaller capacity is observed due to the lower surface area. The AEDs for three separate injections are given in Figure 2. The experiment was carried out a t 30 “Cto increase the retention time because of the reduced capacity. The distribution tails to higher energies for the Davisil experiment versus that of the Impaq experiment, but this could be due to the higher experimental temperature (discussed later). The reproducibility was again good, as was the band profile overlap. The AEDs in Figures ICand 2 are described here as “convergent” in nature. This is to be contrasted with “divergent” behavior, which exhibits a diverging solution a t one or both of the end points. A divergent solution must be regarded suspiciously, as it is not known if adsorption energy beyond the end point is present. More notably, as alluded to above, a divergent solution can correspond to a n inadequately sampled isotherm, and cause an error in the distribution function within the window of energies that was probed. Systematic error such as finite efficiency was also shown to yield divergent b e h a v i ~ r .Therefore, ~ the convergent solutions in Figures ICand 2 may be regarded a s accurate representations of the form and shape of the distributions from a maximum likelihood point of view. Further solution structure may (13)LumWan, 3. A,;White, L.R. J.Chem. SOC.Farad. Trans. 1991, 87, 3051.

3 4.OE-4-

2

E

3.5E-4. . ’

3.OE-4--

, . . ’

,...’,...

- 2.0 p L

2.5E-4-

:,/’

.....

f 0 2.OE-4.ffl

2

1.5E-4.-

,...’

5.0pL IO.pL

//


* -VI

n

40

45

50

55

60

65

70

Adsorption Energy (kJ/mol) Figure 3. Methanol adsorption on Davisil silica at 50 “C: (a) Nonlinear chromatographic band profiles, (b) ECP isotherm, and (c) adsorption energy distribution. be present in the true distributions, but only that structure presented here is needed to adequately fit the data. 11. Alcohol Adsorption. Methanol adsorption on Davisil a t 50 “C is depicted in Figure 3. The band profile overlap is good except for the highest amount injected, 10 p L . This is reflected in the isotherm for this injection lying above the others, a trend similar to that observed for the diethyl ether case (cf. Figures 2b and 3b). However, the AEDs are divergent except for this highest amount injected, which approached the amount needed to adequately sample the surface. The low-Eregion is not as well-described as the diethyl ether experiments because of the overlap problem (note the small divergence a t the end point, dotted line, Figure 3c). For the highest amount injected, the energy distribution was sampled to 42.8 kJ/ mol, which is 6.7 kJ/mol greater than methanol’s heat of vaporization (Evap= 36.1 kJ/mol), and 8.4% solute vapor was detected at the outlet. These conditions approach the limit a t which the experiment may be performed in terms of sampling the low-energy region. Near this limit, intermolecular in-


1740 Langmuir, Vol. 11, No. 5, 1995

teractions may become important, especially near the column inlet where the concentrations are substantially higher. The intermolecular interactions are postulated here to cause a degree of band broadening and profile nonoverlap, thus negating any opportunity to study lower energy adsorption. For methanol adsorption on Davisil, this effect does not prevent a n estimate of the adsorption energy distribution. The main peak shown for the highest amount injected in Figure 3 is centered a t 47.2 kJ/mol, which is 11.1 kJ/mol greater than Evap.In a previous publication, we reported this peak as centered a t 52.3 kJ/mol.14 The discrepancy is most likely not only due to possible errors in detector calibration. The reported distributions in ref 14 possessed a significant divergence a t the low-energy end point, and this can cause a significant shift in the reported energy to higher value^.^ The true peak center is probably closer to the divergence point in that publication. Advances in peak convergence were obtained in this work by altering the injection conditions (less injected faster with a higher split ratio). This resulted in sharper injection profiles, which improved the accuracy of the retention tails a t short retention times, which are observed for high partial pressures on capillary columns. In Figure 4, ethanol adsorption on Davisil is investigated. The band profile overlap is slightly better than that observed for methanol. This is consistent with the fact that the intermolecular interactions of ethanol are lower than for methanol (polarity lower). The calculated AED for the 1OpLinjection is convergent and reproducible. The main peak is 14.2 kJ/mol higher than ethanol's heat of vaporization, displaying a higher relative adsorption energy than that exhibited by methanol. 111. Alkane Adsorption. Heptane adsorption on Davisil is shown in Figure 5. The experiment was performed a t 40 "C, a t which temperature the vapor pressure of heptane is only 12.3 kPa. The detector sensitivity is high for heptane, however, and only 0.235 kPa of maximum partial pressure, or 1.9%of the vapor pressure, was obtained. Examination of the band profiles shows good overlap, but a cursory look a t the ECP isotherm suggests that the surface is not adequately covered or sampled by heptane a t 1.9%. The slope is too high, and the maximum amount adsorbed is 1 order of magnitude less than that observed for diethyl ether, methanol, or ethanol. The result of this undersampling is a strongly divergent AED a t the low-energy end point, with artifactual information appearing a t higher energies. The magnitude of these artifactual peaks is quite small, as can be seen by comparing the ordinate scale of Figure 5c with the other results. The 54 kJ/mol peak obtained with the 1.0p L injection cannot be reproduced with the 0.5 p L injection, and the 62 kJ/mol peak not reproduced well. These results are positive indications of artifactual peaks arising from undersampling or that the majority of the adsorption energy is lower than that estimated with the methodU4 W . Adsorption on Vydac Silica. Adsorption of all probes on Vydac exhibit a significantly lower energy. This is evidenced by irreproducible, divergent AED results. Figures 6 and 7 show the results for diethyl ether and the weakly basic pyridine. Good overlap is achieved a t high concentrations, but the distribution is still not adequately sampled and higher concentrations are needed. The data indicate that the majority of the adsorption energy is less than 45 kJ/mol. The pyridine results are better than those of the other probes, as the higher energy peak is reproducible when the experiment is performed a t 60 "C. The low-energy peak remains irreproducible. ,n

(14)Pyda, M.; Stanley, B. J.;Xie, M.; Guiochon, G. Langmuir 1994, .IC-"

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Figure 4. Ethanol adsorption on Davisil silica at 50 "C: (a) Nonlinear chromatographic band profiles, (b) ECP isotherm, and (c) adsorption energy distribution. The alcohol experiments proved to be interesting. Band profile overlap for methanol was significantly poorer on the Vydac silica than for the Davisil. At low concentrations the effect is remarkably clear (see Figure 8). I t appears that slow mass transfer kinetics may be influencing the elution, as longitudinal diffusional broadening is insignificant. The effect holds over 32-50 "C and 25-35 cm/s. The effect is diminished with ethanol a t 50 "C, but apparent at 40 "C. AEDs were not attempted for methanol or ethanol a t 40 "C, and a divergent AED was observed for ethanol a t 50 "C similar to that observed for diethyl ether. Tetrahydrofuran adsorption on Vydac was also investigated in order to obtain higher relative partial pressures while maintaining a slightly basic interaction with the silica. However, this tactic did not work because the heat of vaporization is increased relative to the heat of adsorption in comparison with the diethyl ether experiment. Results were comparable to those obtained with ethanol in terms of the artifactual AED and the band overlap occurring a t 40 "C, but not a t 32 "C (Figure 9).


Langmuir, Vol. 11, No. 5, 1995 1741 n

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These observations highlight the fact that the adsorption heat is too low and not adequately sampled by the

experiment. The observation of band profile overlap at higher temperatures but not a t lower temperatures supports the notion that it is the intermolecular forces that dictate the limiting mass transfer kinetics. At the higher temperatures, these kinetics become fast enough so that the ideal model of chromatography is an adequate representation of the band. It is interesting, however, that the effect takes place at such low concentrations. Apparently the reduced adsorption energy allows for increased intermolecular interactions relative to that observed with the Davisil and Impaq silicas. The reason for the substantially decreased adsorption energy of the Vydac silica may be due to a much larger pore size for the Vydac (see Table 1). This may also be evidence that the Vydac silica is less acidic than the other silicas studied. V. Chemically Bonded Silicas. Chemically modifying the silica surface with various chemistries only reduces the adsorption energy or the acidity of the surface. The heterogeneity is generally believed to be reduced as well. Judging from the distributions observed in this study, the high-energy tails should be significantly reduced in magnitude. The main adsorption peak should move closer to the heat ofvaporization and be reduced in width. These characteristics preclude its accurate measurement by the inverse GC/AED method studied here. Experiments were presented on C-18 Impaq silica that illustrated this p r ~ b l e m . Experiments ~ on other chemically modified silicas (e.g. C-1 and diol) have supported this evidence. The results for all the probes studied are similar to the results given for heptane adsorption on bare silica (Figure 5). They are not presented further here. It was not possible to study the surface heterogeneity of deactivated silica with this method.


1742 Langmuir, Vol. 11, No. 5, 1995 0

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VI. Infinite Dilution and Temperature Studies.A paradox is apparent when very low concentrations are investigated on the capillary columns. The first several points of the ECP isotherm are closely linear (Figure 10). However a n injection within the below the concentrations depicted in Figure 10 will always yield a highly skewed peak (data not shown). We are able to study elution profiles visually at concentrations approximately 1order of magnitude less than is able to be detected with the A/D system described in the Experimental Sectionby obsenring the filtered response a t the GC terminal. It is always skewed. The explanation could be that the initial concentration a t the beginning of the column is out of the linear range, which initiates the nonlinear response. Another possible explanation is that the decay of the elution profile into the noise looks linear upon ECP integration because of the lack of A/Dresolution bits. In other words, more structure or curvature in the isotherm exists but is below the detection limit and precision of the

Figure 11. Adsorption energy distribution functions of diethyl ether on Impaq silica as a function of temperature.

A/D board (i.e. 1OpVis the lowest response and the lowest step response possible and noise is prevalent at these values, but the tail theoretically decays to lower values in a nonlinear fashion). The effect ofnoise on the estimated AED was published earlier.5 It was shown that a hump becomes apparent a t high energy, corresponding to partial pressures that approach the detection limit (20, where a i s the root mean square of baseline noise), followed by a decay of the distribution function to zero at higher energies. This decay of the AED occurs even though a finite amount of true adsorption occurs a t the higher energies. The hump preserves the amount adsorbed that is observed, and thus the fit. This effect is shown experimentally in Figure 11. Note the progression of the hump to lower energies as the temperature is decreased. This is because the relative partial pressure is increased as the temperature is decreased; therefore the detection limit is decreased as well. The existence of the hump at a high energy suggests that high-energy adsorption may extend out to even higher energies for all the AEDs given. The high-energy adsorption may be simply a skewed tail of the main peak. In order for these high-energy tails to be adequately estimated, better A/D bit resolution must be provided along with high-frequency-noise filtering. VII. Applicability of the AED Method. Considering all the evidence presented thus far, it appears that certain conditions must be met in order for an accurate adsorption energy distribution to be obtained from nonlinear chromatography data. Firstly, the majority of the adsorption energy must be 10 kJ/mol greater than the heat of vaporization. If the adsorption energy is less than this, the partial pressures needed to adequately sample the corresponding sites become too large. At these pressures the retention time becomes very small with capillary columns, and the band profiles become less well overlapped because the effect of diffusion dominates the curvature of the ECP integration more so than the thermodynamic elution. As the maximum pressure is increased, eventually the assumptions of the local model become strained. In order to obtain large partial pressures a t the column outlet, extremely large concentrations must be present a t the column inlet. This furthers the nonideal effects of band profile nonoverlap. Secondly, intermolecular interactions must be relatively low in comparison to the adsorption heat. If polar molecules such as methanol are adsorbed on surfaces with relatively weak energies, the chromatographic band profiles will not overlap. Lastly, standard chromatographic equipment typically only provides around 5 orders of magnitude of signal, and the results here suggest that the heterogeneity is greater than this in pressure units.


Langmuir, Vol. 11, No. 5, 1995 1743 Better weighing methods or balances, and higher sampling frequencies are needed to improve these situations. The method of calculation used in this study is a minimum bias method. No analytical functions, except for the local model, or smoothing routines are used. No extrapolation of the isotherm to higher pressures o r of the AED to lower energies was practiced. This results in a n increased sensitivity of the estimated AED function to all the variables considered in this study, causing many types of adsorption isotherms to be unamenable to AED analysis. Such a n unbiased approach to the data analysis, coupled with a superior approach to the experimental data acquisition, results in a method that is a prime indicator of the reliability of the overall method of estimatingAEDs with nonlinear chromatography. Biasing the analysis of inferiorly acquired data may make the results look better and more reproducible; however, the accuracy and the sensitivity of the resulting method may be compromised.

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45

50

55

60

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Adsorption Energy (kJ/rnol) Figure 12. Adsorption energy distributionfunctions of methanol on Davisil silica as a function of local model.

The effect of the local model on the AED was investigated in order to ascertain if low energy was only apparent for the relevant cases above, when in fact lateral-lateral or multilayer effects provided this appearance. The results are given in Figure 12 for the methanol-Davisil experiment reported in Figure 3. Characteristics ofthe solutions are reported in Table 2. The FG model with patchwise topography provides the best fit to the data. The FG model with random topography provides the worst fit to the data. The BET and FG-random models estimate slightly lower apparent monolayer capacities. This is expected as some of the isotherm magnitude is modeled as intermolecular interactions. However, the solutions do not change drastically with the model chosen. The Jovanovic model splits the peaks estimated with the Langmuir model to provide a better fit. The FG or BET models cannot be expected to alter the solution too much, as the maximum relative pressure and coverage do not reach 10%. The chromatographic ECP technique is inherently a low concentration method and high concentrations may not be achieved. At the lower concentrations the results for different models become similar; it is the higher concentration data which provides the handle necessary to discriminate between different model results. Their applicability must be held in question regardless, since the magnitude of adsorbate-adsorbate interactions is initially higher a t the beginning of the chromatographic column, and the amount adsorbed is uncertain from the measurement of the mass of adsorbent coated on the column. Thus it is not very practical to study model validity with the chromatographic technique discussed here. Capillary columns were used in this study in order to decrease the effect of finite efficiency to a negligible amount. The price paid for this effort was a severely decreased column capacity, often resulting in retention times a t sufficient column saturation of 1 5 s. The nature of such a short retention time is to decrease the margin for error with respect to injection and detection. The profiles are rapidly decreasing a t the high partial pressures corresponding to these retention times. Furthermore, the use of capillary columns yields a highly uncertain mass of stationary phase. This uncertainty is propagated to the measured amount adsorbed per unit adsorbent, causing an uncertainty in the magnitude of lateral-lateral adsorbate interactions and the use of the FGlocal models.

Conclusions Adsorption energy distributions may only be calculated from nonlinear chromatographic data under certain conditions, namely, (1)the majority of the adsorption energy must be approximately 10 kJ/mol greater than the solute's heat of vaporization and (2) the solute's intermolecular interactions must not be significant in comparison to it's adsorption energy. The accuracy of the distribution function is a function of the local model of adsorption chosen; however, such a choice may have no meaning in this method. The use of capillary PLOT columns increases the quality of the chromatographic data; however, the benefits are offset by a n inability to measure the amount coated in the capillary accurately. Accurate adsorption energy distributions have been calculated for diethyl ether on Davisil and Impaq silicas, as well as methanol and ethanol on Davisil silica. The degree of heterogeneity is readily apparent, but may extend to higher energies; i.e. the heterogeneity may be greater than the range and resolution of the data acquisition system. These results have been acquired with the expectationmaximization method of parameter estimation. Correlation with results from other methods may not be appropriate. However, the method allows a good critique of the overall finite concentration/ECP/AED methodology. In particular, standard chromatographic equipment may not be adequate for studies of this type, which repeals the major advantage of the chromatographic technique. Theoretically, fundamental problems exist with the adsorption a t high relative pressures and surface coverages. These correspond to the measurement (overlap problem, maximum possible concentration) and the illposed nature of the data analysis, which amplifies the shortcomings of the measurements as a n error in the AED (divergent border effect). Other highly heterogeneous adsorption systems most likely would reveal the same problems. Other data analysis algorithms cannot alleviate them unless further assumptions and data are incorporated than allowed here. Further work in this field must be dedicated to addressing and solving these problems, or a n entirely different technique for assessing adsorption heterogeneity should be proposed. Acknowledgment. This work has been supported in part by Grant DE-FG05-88ER13859 of the Department of Energy and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. We acknowledge support of our computational effort by the University of Tennessee Computing Center. LA940632X

Calculation of Adsorption Energy Distributions of Silica Samples Using

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