A Dynamic Volumetric Method for Measuring Adsorption of Water on

Article pubs.acs.org/JPCC

A Dynamic Volumetric Method for Measuring Adsorption of Water on Glass Fibers Victor A. Bakaev,* Lymaris Ortiz Rivera, and Carlo G. Pantano Materials Research Institute, The Pennsylvania State University, University Park, Pennsylvania 16802, United States ABSTRACT: A new dynamic volumetric method (DVM) is described for measuring adsorption isotherms on small surface area solids such as glass fiber. The experimental setup of the method is the same as that of frontal analysis (FA) chromatography. However, whereas FA analysis is based on the local mass balance equation (mass balance in each cross section of the column), this DVM is based on the global mass balance between the inlet and outlet of the system. It differs from the static volumetric method in that the latter measures an adsorption isotherm in the state of adsorption equilibrium, whereas the DVM does the same in the steady state. This difference creates a problem of determining equilibrium adsorption concentration in the DVM which does not exist in the static method. It is shown how to resolve that problem. The DVM is more general than FA and is applied here for determination of the BET specific surface areas (SSA) of glass fibers using water. The results are compared with those obtained by standard SSA determinations with Kr and those obtained by measuring glass fiber diameter and density. In some cases, the SSAs obtained by the three different methods reasonably agree with each other. In other cases, however, the SSA obtained by water proved to be considerably larger than that obtained by the two other methods. Because water is reactive with glass fibers, this could be an indication of nanoscale surface properties of glass fibers which cannot be detected by Kr adsorption. Surface area analysis is not typically performed in packed columns, so this development expands the applicability of chromatographic columns in general.

1. INTRODUCTION Glass fibers are important materials for insulation, filtration, and reinforcement of polymers. They are usually coated with a polymer or an adhesive for functionality, but the bonding can degrade due to interaction with water; silane additives or primers are often used to mitigate the loss of adhesion in aqueous environments.1 Thus, measuring the water adsorption characteristics of glass fiber is an important step for understanding of deleterious surface and interface reactions of glass fibers with water. Motivation for the development of a new method for measuring surface area by adsorption of water came from a phenomenon we observed while measuring the concentration of OH-groups on glass fiber surfaces. The dynamic method for measuring OH-group concentration by the hydrogen/deuterium (H/D) exchange on low area surfaces was published earlier.2 The experimental setup of the method was basically a gas chromatograph with a mass sensitive detector (MSD) working in the regime of frontal analysis (FA) chromatography. However, in contrast to the successful determination of OH surface concentration on silica nanoparticles (of high surface area), the low surface area of the glass fibers and limited sensitivity of FA and the MSD precluded reliable and reproducible measurements. Afterward, the FA and MSD were considerably improved and the concentration of OHgroups on glass fibers was reliably measured. However, of particular concern was the fact that the measured concentration of OH-groups on some glass fibers seemed unusually high. Of © 2015 American Chemical Society

course, what is actually measured in the H/D experiment is the total quantity of OH-groups per one gram of glass fiber. To determine OH per square nanometer, the specific surface area (SSA) of glass fiber must be used. Initially, we used the SSA provided by BET with Kr. Therefore, we considered the possibility that the SSA obtained by Kr BET might be too low and considered the possibility that better results could be obtained if SSA would be measured by H2O instead of by Kr. A dynamic gravimetric instrument for measuring water vapor adsorption, dynamic vapor sorption (DVS), is provided by Surface Measurement Systems.3 It is widely used in many areas (see “Scientific Bulletins” in ref 3). However, gravimetric adsorption techniques are more appropriately used for measuring relatively large SSA surfaces. Glass fibers present a challenge for surface adsorption studies because of their very small SSA. For example, a typical glass fiber with 10 μm diameter has a SSA of about 0.2 m2/g. Adsorption isotherms for such low SSA materials are usually measured by volumetric methods at low pressures. For example, the ASAP 2020 (Micromeritics) is an instrument for the static volumetric measurement of nitrogen or krypton adsorption at low temperatures. It is used mainly for measuring SSA and porosity. Unfortunately, the ASAP 2020 is not designed for measuring water adsorption. Received: July 13, 2015 Revised: September 8, 2015 Published: September 9, 2015 22504

DOI: 10.1021/acs.jpcc.5b06723 J. Phys. Chem. C 2015, 119, 22504−22513

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The Journal of Physical Chemistry C

gas adsorption. That problem was either ignored by assuming the carrier gas to be incompressible5−7 or the concentration scatter was considered as an intrinsic experimental error of the method.4 In section 3.1 a more systematic approach to that problem is considered. The validity of the DV method is confirmed by experiments with gaseous n-decane described in section 3. Decane is used mainly because it allows one to use a flame ionization detector (FID). Unfortunately, one cannot use FID for water. In the latter case we use MSD which is a quadrupole mass spectrometer. Employing a MSD is necessary for measuring OH-concentration on the surfaces of glass fibers by H/D exchange mentioned above. However, as a detector of single component concentration, MSD is far less robust than FID. This is seen by comparison of Figures 2 and 3 where the same concentration versus time plots of n-decane are measured by both FID and MSD detectors. In this case, n-decane makes it possible to understand which corrections one has to make when using a MSD for measuring water concentration. The last part of the paper is concerned with the comparison of SSA measured by water, krypton, and calculated from glass fiber diameter and density for different glass fibers. For krypton and water (in fact D2O), SSA is obtained by the BET method.10 The analysis of the BET plot for D2O (see Figure 8 in section 4) shows that the requirements of the BET method are observed. When the molecular area of a D2O molecule is chosen to be 0.09 nm2, it is found that the SSAs determined from Kr and water (D2O) adsorption are in agreement for three of the six fibers evaluated (cf. Table 2) but does not bring correspondence for the other three fibers. We do not have a firm explanation for this but make preliminary suggestions.

The new method of measuring water adsorption described below can be classified as a dynamic volumetric method. (It is dynamic because the adsorptive is delivered to the surface where it becomes the adsorbate via the carrier gas and not by self-diffusion as in the static method). In this respect the method differs from DVS in being volumetric and from ASAP 2020 in being dynamic. The basic idea of the method was put forth long ago.4 It was presented as a method of measuring adsorption isotherms by FA chromatography. The same approach, but less clearly substantiated, was employed later.5−7 FA chromatography is widely used for measuring adsorption isotherms from liquid solutions.8 The method is based on the local mass balance equation (mass balance at each cross section of the column) which for a one-component solution leads after several approximations (known as ED model) to a partial differential equation which can be solved numerically.9 The solution gives a breakthrough curve which is the concentration of solute at the outlet of the column versus time, when the concentration at the inlet is maintained constant. The area between the breakthrough curve and the concentration of solute at the inlet allows one to calculate the adsorption versus inlet concentration which is one point of the adsorption isotherm. Thus, in FA chromatography, one obtains an adsorption isotherm together with the profile of breakthrough curve. It is known that despite the approximate character of the ED model and correspondingly the breakthrough curve, the experimental adsorption isotherm obtained by FA chromatography is considerably more accurate. This was explained by the self-sharpening nature of FA.8 However, another explanation for that fact can be given. It was shown4 that if only the adsorption isotherm is desired, it can be obtained by the method based on the global mass balance rather than on the local mass balance. This means that the rate of total mass change in the column should be equal to the balance of in- and out-flow of the mass through the column ends. Using global mass balance dramatically simplifies the calculations, as shown in section 3. Such factors as adsorption kinetics or mass transfer in the column, which determine the shape of a breakthrough curve, do not influence the adsorption isotherm. The homogeneous column packing, which is implicitly a necessary condition of the basic chromatography equation, also does not influence the adsorption isotherm. Finally, as shown below, the global mass balance allows one to consider situations when chromatography cannot be applied at all. For example, in section 3.2 we consider filling of an empty ampule by a gas solute (adsorptive) when the solute is homogeneously distributed over the volume (no moving front of concentration as in FA). Thus, despite using the FA chromatography experimental setup, the method itself is more general than FA which is the reason why we call it a dynamic volumetric (DV) method. One principal difference between the static and DV methods is that the former measures adsorption in the state of adsorption equilibrium, whereas the latter does this in the steady state. In both states, adsorption and equilibrium concentration (equilibrium pressure) do not depend on time. However, whereas in the state of adsorption equilibrium the equilibrium concentration is unique and does not depend upon the position in the system, there may be a scatter of concentrations due to concentration gradients in the steady state when the carrier fluid is compressible. Therefore, a problem of determining the equilibrium concentration which should correspond to adsorption arises in the DV method of

2. EXPERIMENTAL SECTION The main part of the experimental setup is an Agilent 7890A GC with FID and 5975C MSD. This GC/MSD is supplied with a homemade injection system shown in Figure 1. The main part

Figure 1. Injection system: 1−6, six-port injection valve; T, thermostat with saturator; S, splitter; V1 and V2, ball valves; R, adjustable restrictor; coil between points A and B, guard column; cylinder between points B and C, packed column.

of the system is the saturator (T in Figure 1). The latter consists of a U-tube packed with fused quartz rods with 2 mm diameter and 2 mm length (Quartz Scientific, Inc.) or glass sphere 1 mm (Sigma-Aldrich). The tube is filled either with liquid D2O (99.9 atom % D deuterium oxide, Sigma-Aldrich) or with liquid n-decane (assay ≥99% Sigma-Aldrich). The saturator is submerged in a Dewar flask connected to the circulation bath (Thermo scientific, Haake A10). The temperature in the Dewar is controlled by a platinum thermometer (Omega Engineering probe PRP-1 and DP95-X-XX-41-41-X-X monitor) with the accuracy of about 0.01 K and resolution of 22505


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ports and adjustable restrictor, R, to the atmosphere. The restrictor R is adjusted to provide the same flow rate through the saturator as when it is in the INJECT position. In the INJECT position dashed lines in the six-port valve are open and dotted ones are closed. Now carrier gas flows through the sixth and fifth ports to the saturator T and then through the second and first ports to the columns. The other part of the He flow in this case is discharged in the atmosphere through the fourth and third ports. Thus, in the LOAD position pure He flows through the columns, whereas in the INJECT position the carrier gas is saturated with solute (D2O or n-decane in this paper) whose concentration corresponds to the saturated vapor of solute at the temperature of saturator. (In fact, as shown in section 2.1, the concentration of solute in the packed column is somewhat smaller than that in the saturator because of the pressure drop in the tubing.) Additional flow of the carrier gas through the valves V1 and V2 maintains about the same regime of the saturator in both LOAD and INJECT positions. This provides a sharp front of the pulse in Figure 2. Without that additional flow, the front would be considerably distorted because of the transition regime of the saturator.

0.001 K. The same thermometer is used to calibrate the GC oven. The geometric SSAs of fibers were calculated from the fiber diameters and densities. The fiber diameters were measured with a Hitachi S3500N SEM instrument. Their densities were measured with AccuPyc 1330 (Micromeritics) gas pycnometer. The BET surfaces of fibers were measured by an ASAP 2020 (Micromeritics) instrument with Kr or by the method described below with water. Six glass fibers (E-glass 1 through E-glass 6) were used in our experiments. The fibers E-glass 1 and E-glass 2 were drawn in our laboratory from commercial marbles. The other glass fibers were commercial ones. All of them were E-glass fibers with minor composition differences. Their densities and diameters are presented in Table 1. Only the diameters of the first two Table 1. Parameters of Glass Fibers fiber E-glass E-glass E-glass E-glass E-glass E-glass

1 2 3 4 5 6

density

diameter

2.588 2.595 2.561 2.533 2.582 2.556

4.25 4.10 12 12 12 12

fibers were measured as described above. The diameters of the other fibers were given by their producers. The fibers were packed in a fused quartz tube, a packed chromatographic column. Fibers were packed into the column by tying wax-free dental floss around the center of a 52 cm long monofilament bundle. The strand of floss was then used to pull the fibers through the column ensuring alignment of the fibers along the column axis. The column was packed as densely as possible to maximize the amount of surface area available for adsorption. On average, 3 g of fibers could be packed within the 23 cm long, 3.9 mm ID column. The cleaning of fibers was performed by treating the fiber in the flow of pure He (six-port valve in Figure 1 in position “LOAD”) at 200 °C for 30 min. Although this paper is concerned with adsorption of water, in fact, we use D2O. This is because the same experimental setup is used for the measurement of OH-group concentration on the surfaces of glass fibers by the H/D exchange. In the latter case, one needs to fill the saturator with D2O. As for the measuring of SSA by BET, it is not important what molecule we use (H2O or D2O). The carrier gas (He) goes through the GC pneumatic system which consists of a regulated proportional valve (PV in Figure 1) which maintains constant pressure, P2, displayed on the GC panel. Then it goes to the sixth port of the six port injection valve (Valco Instruments Co. Inc.). The valve can be either in the “LOAD” or “INJECT” positions. In the LOAD position, the connections between the ports of the valve designated by dotted lines in Figure 1 are open whereas those designated by the dashed lines are closed and vice versa in the INJECT position. Thus, in the LOAD position, one part of the carrier gas flows from the sixth port to the first port, then to the guard column AB, to the packed column BC, to splitter S, and finally to the MSD. In the case when the FID detector is used, the carrier gas flows directly to it from the outlet of the column (point C in Figure 1). When the ball valves V1 and V2 are open, another part of the He flows through the fourth and fifth ports into the saturator and finally through the second and third

Figure 2. FID response on injection of n-decane by saturator. No column AB in Figure 1. Saturator temperature 15.04 °C, solid line; 1.01 °C, dashed line.

There are two columns in the system: the guard column (AB in Figure 1, a fused quartz capillary 5 m length, ca. 0.45 mm I.D.; see section 2.1) and a packed column (CD in Figure 1, a glass tube 23 cm length, 3.9 mm I.D.). The guard column is necessary because the requirement of the dynamic volumetric method is the small pressure drop over the packed column, but the pneumatic system of GC cannot regulate such a small pressure drop. The column is packed with glass fibers whose characteristics are measured. It can be removed as in Figure 2 or substituted for a much larger empty glass volume as in Figure 5. The volumetric flow rate of the carrier gas is about 10 mL/min. It can be measured at the outlet of the column (point C in Figure 1) and designated below as F. To measure F, a bubble flow meter can be connected to the outlet of the FID detector or to the middle outlet of the splitter S in Figure 1. The direct measurements by the flow meter are performed only to evaluate the pressure drop over the column, the main part of the volumetric flow rate, F, is calculated (see section 2.1). The above-mentioned F exceeds the maximum input flow for 5975C MSD by about 5 times. The splitter S in Figure 1 22506


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The Journal of Physical Chemistry C resolves this problem. The splitter consists of a tee, the central outlet of which is open to the atmosphere. The 500 μm internal diameter (ID) capillary is inserted through a ferrule in the left end of the tee. Another end of the capillary is connected to the outlet of the packed column at the point C in Figure 1. The restrictor capillary 27 cm length, 75 μm ID is inserted in the right end of the tee (also through a ferrule). One end of this capillary is located inside the wider capillary as shown in Figure 1. Another end enters the ionization cell of MSD through the transfer line of MSD. Figure 2 shows a FID response to the injection by the saturator of n-decane at two temperatures of the saturator. Initially, the six-port valve in Figure 1 is in the position LOAD for 0.5 min. Then it is switched to the position INJECT and in 15 min back to LOAD. The detector signal I(t) (current for FID or abundance for MSD) at time t is proportional to the current of solute molecules carried by the carrier gas into the detector

I(t ) = kFdC(t )

Figure 3. MSD response to the same n-decane injection that gives the solid line in Figure 2. EM voltage = 2200 (dashed line) and EM voltage = 1812 (solid line).

use the SIM regime of MSD and the group of four lines (m/z = 17, 18, 19, 20). Before the run in Figure 3, the MSD was adjusted by the low mass auto-tune which narrows the calibrated scan to the low mass area that is used in our measurements. The electron multiplier (EM) voltage was established at 2500 V. Even with somewhat lower voltage, the broken line in Figure 3 decreases at a rate excluding MSD application in our experiments. We decreased the EM Voltage to about 1800 V, which makes the creep tolerable but distorts the front of the pulse in Figure 3. The latter value of EM voltage was accepted in our experiments with water. The distortion of the pulse shown in Figure 3 hampers using water for substantiation of the DV method. For that reason we use exclusively n-decane and FID for substantiation of the DV method. Isotope substituted water is used for H/D exchange where MSD is necessary and for measuring SSAs of glass fibers. It should be emphasized that the problem of using the 5975C MSD in our experiments with water arises not because of the defects of the MSD, but because we have to use it in a mode which this MSD is not designed for. Conventionally this MSD is designed to detect a series of very short chromatographic peaks so that it never works continuously for such a long time as in Figure 3. We can partially overcome the problem by decreasing the EM voltage. Because of that, although its gain drops with time, the drop is not as drastic as that in Figure 3. This decreases the MSD sensitivity; however, the sensitivity is still sufficient for our work. 2.1. Calculation of Volumetric Flow Rates. The volumetric flow rate at the outlet of the packed column (point C in Figure 1) is

(1)

Here k is the proportionality coefficient; Fd = F where F, as mentioned above, is the volumetric flow rate of the carrier gas at the outlet of the column for FID or Fd = FMSD where FMSD is the volumetric flow rate in the restrictor capillary of the splitter at its inlet; C(t) is the concentration of solute in the carrier gas. To determine k in eq 1, we choose the time interval where I(t) is almost constant. For example, the interval between 14 and 15 min in Figure 2. The interval contains 301 points, and the average fluctuation of I(t) over those points is 0.15%. The value of F is calculated by the method explained in section 2.1. The value of C(t) is as stable as I(t) and corresponds to the saturated vapor pressure of n-decane at the temperatures of the saturator indicated in Figure 2. For example, at 15 °C the saturated vapor pressure of n-decane11 is 87 Pa. At such a pressure the correction for nonideality of gas is far less than other experimental errors and C(t) can be calculated from the ideal gas equation of state. Another correction to C(t) comes from the pressure drop in the system and is considered in section 2.1. Finally, the values of k obtained from the run shown in Figure 2 are 1.28 × 104 A·s·mol−1 (solid line) and 1.25 × 104 A·s·mol−1 (broken line). Theoretically, it should be the same. The volume flow rate, F, and the pressure drop correction to C(t) theoretically depend on the temperature of the saturator. Those corrections come mainly from the dependence of the carrier gas viscosity on the concentration of the solute. The latter, however, is so small that those corrections can be neglected (see section 2.1). Thus, the discrepancy of about 2.5% in the values of k mentioned above should be ascribed either to the error in the calculation of the vapor pressure11 or to the work of saturator. We can decrease the error in k to 1.3% by taking the average between those two above-mentioned values. This is the accuracy of converting the FID current into the concentration of n-decane. Figure 3 shows exactly the same injection pulse of n-decane generated by the saturator as that shown in Figure 2 but measured not by FID but by MSD. The MSD works in the single-ion monitoring (SIM) regime. It monitors the four most abundant lines of the n-butane mass spectrum: m/z = 41, 43, 57, and 71. The purpose of Figure 3 is to determine the problem (and its correction) one runs across when using MSD instead of FID. That problem arises when we apply the dynamic volumetric method to water or isotope exchanged water where FID cannot be employed. In the latter case we also

F = (P2 2 − Patm 2)Tov /[(a1 + a 2 + a3 + a4)Patm]

(2)

This equation is the result of the integration of the Poiseuille equation (without slip at wall correction).12 Here P2 is the pressure at the outlet of the pneumatic system of GC displayed at its panel Figure 1; Patm is the atmospheric pressure measured by a separate manometer (incorporated in GC); Tov is the temperature in the GC oven. Finally, αi = 16 μ (Ti) LiTi/(πri4) i = 1, 2, 3 are the constants of the stretches of tubing connecting the GC pneumatic system with the inlet of the column: a1 is the guard column BA in Figure 1; a2 is the heated transfer line connecting the guard column with the first port of the six-port valve in Figure 1; and a3 is the other tubing in the GC oven. These have the following parameters: Li and ri are the length and inner radius of the tubing, respectively; Ti is its 22507


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The Journal of Physical Chemistry C temperature; and μ(T) is the dynamic viscosity of the carrier gas at the temperature T. Finally, a4 represents the flow resistance of the packed column in eq 2. Of the first three constants, a1 with parameters L1 = 5.196 m, r1 = 222.5 μm, and T1 = Tov is dominant. Parameters of a2 are L2 = 0.8 m, r2 = 380 μm, and T2 = 150 °C. The parameters of a3 are L3 = 0.5 m, r3 = 219.4 μm, and T3 = Tov. These parameters determine F0, the volumetric flow rate without a packed column (a4 = 0 in eq 2). The value of F0 was calibrated by adjusting r1 in such a way that the calculated value of F0 coincided with that measured by the bubble meter at Tov = 35 and 100 °C. When an MSD detector is employed, the bubble flow meter which measures F is attached to the open outlet of the splitter S in Figure 1. Thus, it measures not the total flow from the column F but the difference between F and FMSD. To evaluate FMSD, the value of F0 is measured at the splitter by a bubble flow meter, and from the measured value the value calculated by eq 2 is subtracted. The value of FMSD somewhat depends on the oven temperature because part of the restrictor capillary is at that temperature. At Tov = 40 °C, FMSD = 0.89 mL/min, and at Tov = 100 °C ,FMSD = 0.86 mL/min. To determine a4, one can measure F in a system with the column and then subtract from it the value of F0. Then one can calculate a4 from the following equation (F0 − F )/F0 = a4 /(a1 + a 2 + a3 + a4)

saturator T in Figure 1 which actually decreases the value of P2 which drives the carrier gas into the system. There is no such peak in Figures 3 and 4 because the detector current of FID is

Figure 4. MSD response to injection of D2O by saturator in the system with chromatographic column packed with 3 g of glass fiber SSA = 0.2 m2/g. Oven temperature, 40 °C; saturator temperature, 15 °C.

proportional to FC(t) (eq 1), whereas in MSD it is proportional only to C(t). In the latter case, Fd in eq 1 is the volumetric flow rate in the restrictor capillary, FMSD, which depends not on F but on atmospheric pressure. Concentration (partial pressure) of the solute at the surface of glass fibers is lower than that in the saturator because the pressure of the carrier gas decreases along its way from the saturator to the column. The ratio y of the concentration of the solute to the pressure of the carrier gas p does not change if solute is not adsorbed along its way with the carrier gas. However, the pressure of the carrier gas drops as it moves through the tubing and the column and the concentration of the solute drops correspondingly:

(3)

After that, one can determine the pressure drop over the column P − Patm from the following equation P 2 − Patm 2 = Fa4Patm/Tov

(4)

The pressure drop over the column is usually evaluated in the gas chromatography by the James−Martin compressibility factor j=

2 3 (P /Patm) − 1 2 (P /Patm)3 − 1

(5)

The value of j is essential for calculation of equilibrium adsorption concentration in section 3.1. The value of F0 calculated from eq 2 with α4 = 0 is somewhat different in the LOAD position of the six-port valve in Figure 1 when pure He flows through the system and the INJECT position when a mixture of He with solute (D2O or n-decane) flows through the system. In the latter case, the dynamic viscosities, μ, of the carrier gas which enters the constants αi of eq 2 are different. Those viscosities were evaluated for He/ water mixture by the Wilke equation.13 For the partial pressure of water vapor corresponding to the saturator temperature of 15 °C, the mole fraction of water in the He/water mixture is about 0.013. The viscosity of the He/water mixture at such a concentration of water according to the Wilke equation should be about 2% lower than that of pure He. The mole fraction of decane in the He/decane mixture at the same saturator temperature is only 8 × 10−4. In this case we do not expect a noticeable change of the mixture viscosity in comparison with pure He. Accordingly, we would not expect the initial increase of F when switching from INJECT to LOAD position. In reality, it slightly (about 2%) increases for n-decane. This can be seen in Figure 2 or in Figure 5 as a small peak of the detector current at about 15 min when switching from INJECT to LOAD position. (At first current drops, probably, because of drop of flow rate at the moment of switching and then increases.) We ascribe this to some pressure drop on the

C(x) = yp(x)

(6)

Here x is the distance along the way of the carrier gas. For example, one may put x = 0 at the saturator, then the concentration of solute at the outlet of the column (Co(ts) in eqs 8 and 9) will be CsatPatm/P2 where Csat is the concentration of the solute in the saturator. Or one can put x = 0 at the inlet of the column. Then C(x) will be the variation of the equilibrium concentration along the column considered in section 3.1.

3. DYNAMIC VOLUMETRIC METHOD Usually after the electron multiplier of the MSD is switched on, the initial position of the six-port valve in Figure 1 is LOAD, which means that pure He flows into the system through the first port of the valve. Correspondingly, the detector signal in Figures 2−5 is at the background level. In 0.5 min, the six-port valve switches to the position INJECT and a mixture of He with a solute (n-decane or D2O) flows into the system. The quantity of the solute which enters the system per unit of time is FiCi where Fi is the volumetric flow rate of the carrier gas at the inlet of the system (first port of the six-port valve in Figure 1) and Ci is the concentration of solute at the same place. One need not know the values of either Fi or Ci. The only important point is that they do not depend on time. The quantity of solute that flows out of the system in the unit of time is FCo(t) 22508


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The Journal of Physical Chemistry C where F and Co(t) are the volumetric flow rate and the concentration of solute at the outlet of the system (inlet of the restricting capillary for MSD in Figure 1 or inlet of FID not shown in Figure 1), respectively. Thus, the rate of the solute accumulation in the system is FiCi − FCo(t) and the quantity of solute accumulated in the system by the time t (at the moment of switch t = 0) N(t) is N (t ) =

∫0

local balance equation and the ideal chromatography model and we do not consider FACP here. Nevertheless, eq 10 is useful when a volume of an empty ampule is measured by the dynamic volumetric method (see section 3.2). 3.1. Adsorption Isotherm in the Dynamic Volumetric Method. The front part of the pulse in Figure 4 is a breakthrough curve of FA chromatography. Its profile can be determined by solution of a basic partial differential equation of chromatography based on the local mass balance in the column. To make it tractable, the partial differential equation should be presented in a model form and solved numerically.9 In contrast to that, eq 9 based on the global mass balance does not include model approximations and allows one to determine the quantity of solute (D2O in Figure 4) adsorbed by the system or the system’s effective volume with accuracy limited only by experimental errors. However, it does not allow one to determine the profile of the breakthrough curve. To determine the quantity of solute adsorbed on the surface of glass fiber one has to extract from Veff in eqs 9 and 10 the volume of the gas phase in the system. The latter consists of the volume of the system tubing mentioned above and that of the gas volume in the column, which equals the volume of the empty column minus the weight of the fiber divided by its density. The remaining part of Veff should by multiplied by some concentration to determine the adsorbed quantity. Thus, both the dynamic and static volumetric methods determine adsorbed quantity as an effective gas volume, Veff, multiplied by some concentration. In the static case, it is an equilibrium gas concentration which corresponds to adsorption. In the dynamic case, it is the concentration at the outlet of the column Co(ts) (eqs 9 and 10). However, in contrast with the static case, Co(ts) is not the equilibrium concentration that should correspond to the adsorbed quantity determined by the volumetric dynamic method. This is because the concentration of the solute (adsorptive) in the carrier gas is proportional to the gas pressure (eq 6), and the latter changes along the column because the carrier gas is compressible. (This problem does not arise for adsorption from incompressible liquids.) Correspondingly, the local specific adsorption also depends on the position, x, along the column. Thus, the total quantity of adsorbate, N, contained in a homogeneously packed column is given by the equation

t

dt ′[FC i i − FCo(t ′)]

(7)

One can see from Figures 2, 4, and 5 that Co(t) (detector current) steadily increases but it cannot exceed Ci. Thus, at some moment of time, ts, the steady state will be reached when FC i i ≅ FCo(ts)

(8)

Theoretically, the steady state can be reached only after infinitely large interval of time (ts = ∞), but practically ts is finite (cf. Figures 2, 4, and 5). (One can observe in Figure 4 the slight decrease of the detector signal in the interval of time where one would expect the steady state; this is the defect of MSD detector shown in Figure 3.) Insertion of eq 8 into eq 7 gives Veff = N (ts)/Co(ts) = F

∫0

ts

[1 − Co(t ′)/Co(ts)] dt ′

(9)

Here the left-hand side is the effective volume of the system which includes the volume of the guard column BA in Figure 1 and other tubing. The latter (2.1 mL) can be determined by the run in Figure 2. The other part is either effective volume of the packed column in Figure 4 (13.9 mL) or that of the empty ampule (22.84 mL) in Figure 5. The integrand in the right-hand side of eq 9 is similar to that in eq 85 and eq 10.6 The difference is that in eq 9 both numerator and denominator of the concentration ratio refer to the outlet of the column, whereas in refs 5 and 6, instead of that ratio stands a transmission: the concentrations at the inlet divided by that at the outlet of the column. However, because the concentration at the inlet of the column is assumed to be constant and the global mass balance equation is implied, there is no principal difference between the above-mentioned5,6 equations and eq 9. When the system practically reaches the steady state (at the moment ts), one can switch the six-port valve back in the position LOAD and the solute stops entering the system. Now one can determine the total quantity N(ts) eq 7 in the system at Co(ts) simply by measuring the total quantity of solute that eluted from the system. Thus, the effective volume of the system can also be measured as Veff = N (ts)/Co(ts) = F

∫t

s

N=

L

a(C(x))ρS dx

(11)

Here a(C(x)) is the local equilibrium adsorption isotherm per the unit of weight as a function of equilibrium concentration which varies together with pressure p(x) (eq 6) along the column; ρ is the density of column packing; S and L are the column cross section and length, respectively. The function p(x) is well-known in gas chromatography

∞

Co(t )/Co(ts) dt

∫0

(10)

p(x ) =

Both eq 9 and eq 10 are rigorous and can be employed for measuring effective volumes and adsorption. For example, in Figure 4, eq 9 would use the front of the pulse and eq 10 its tail. However, one can see that the front and tail of the pulse in Figure 4 have different profiles, the front being considerably steeper than the tail. This situation is typical for packed columns, and for them, application of eq 10 is inconvenient. Instead the tail of the pulse in Figure 4 could be used to obtain the whole isotherm of adsorption instead of only one point by the method called FACP9 (FA by characteristic point). However, FACP is an approximate method based on the

pi2 − (pi2 − po2 )x /L

(12)

Here pi = p(0) and po = p(L) are the pressures at the column inlet and outlet, respectively. Now the problem is to find the function a(C) (equilibrium adsorption isotherm) which provides the left-hand side of eq 11. Expand a(C(x)) in eq 11 at C(L) = Co = ypo: a(C(x)) = a(ypo ) + a′(ypo )y(p(x) − po ) + 22509

1 a″(ypo )y 2 (p(x) − po )2 + ··· 2

(13)


Article

The Journal of Physical Chemistry C Here y (eq 6) does not depend on either time or x in the steady state. Insert eqs 12 and 13 in eq 11, integrate, and use eq 6. The result is

loosely packed columns with values of James−Martin factor close to unity, which is usually the case for glass fibers. For more tightly packed columns with lower values of James− Martin factor, eq 17 should be used. 3.2. Empty Ampule in Place of Packed Column. Figure 5 presents the same FA run as in Figure 4 when the packed

⎛C ⎞ 1 N = a(Co) + a′(Co)⎜ o − Co⎟ + a″(Co)Co 2j1 + ··· 2 ρSL ⎝ j ⎠ (14) 2

j1 = [(pi /po ) + 3]/2 − 2/j

(15)

Here j is the James−Martin compressibility factor eq 5. In eq 11, C varies between ypo and ypi. Assume that in this interval of C the isotherm is approximately linear. Then, retaining only two first terms in the right-hand side of eq 14 gives N = a(Co/j) ρSL

(16)

The denominator in the left-hand side of this equation is the weight of the column packing. Thus, the left-hand side of this equation is the specific adsorption and Co/j is the equilibrium concentration. The next approximation which retains the quadratic terms in eqs 14 and 16 can be presented as ⎡ ⎛ 1 − j ⎞2 ⎤ N 1 2⎢ − a″(Co)Co j1 + ⎜ ⎟ ⎥ = a(Co/j) ⎢⎣ ρSL 2 ⎝ j ⎠ ⎥⎦

Figure 5. FID detector response to injection of n-decane by saturator in system with empty volume of 22.84 mL instead of packed column. Oven temperature, 100 °C; saturator temperature, 15 °C.

column is substituted for an empty ampule. There are two obvious distinctions of this figure from Figure 4: (i) the front of the pulse in Figure 4 is steeper than that in Figure 5 and (ii) whereas the front and the tail parts of the pulse in Figure 4 have different symmetry, those in Figure 5 are symmetric. That symmetry is displayed in Figure 6. The solid line in Figure 6 is the normalized tail of the pulse in Figure 5 (integrand in eq 10). The symbols in Figure 6 are the integrand of the righthand side of eq 9 for the front of pulse in Figure 5. Moreover, both the solid line and the symbols in Figure 6 can be described by a simple equation

(17)

Here j1 in the bracket of eq 17 comes from the third term on the right-hand side of eq 14, and the next term comes from the quadratic term of expansion of the right-hand side of eq 16 at Co. Equation 17 cannot be used directly because it requires the second derivative of the yet unknown adsorption isotherm a″(Co) . Thus, eq 17 should be employed to calculate an adsorption isotherm iteratively. First one calculates the isotherm in the first approximation by eq 16 at several points around Co and evaluates a″(Co) and then uses eq 17 to refine the first approximation. A typical example of application of eq 17 to the BET equation (which is used in section 4) is the following. The BET equation10 and its second derivative are a(x) = amC BETx /{(1 − x)[1 + (C BET − 1)x]} a″(x) =

(18)

2amC BET[2 − C BET + 3(C BET − 1)x + (C BET − 1)2 x 3] (1 − x)3 [1 + (C BET − 1)x]3 (19)

Here x = p/ps is the relative pressure (ps is saturated vapor pressure); for the ideal gas, pressure can be substituted for the gas concentration: p/ps = C/Cs. In eqs 18 and 19, CBET is a constant which in the literature is known as the C-constant of the BET equation.10 Now take x = 0.065, CBET = 76.5, and pi/po = 1.05 which correspond to the lowest point in Figure 8, section 4. Then j = 0.975, j1 = 8.10−4, a/am = 0.900, and the second term on the left-hand side of eq 17 will be 0.03% of the first term. Thus, the error in adsorption in the second approximation eq 14 is about 0.03%, which is much less than other experimental errors. For the points at higher relative pressures (in Figure 8), that error is even less. However, for pi/po = 1.3 with j = 0.86 which corresponds to tighter column packing, that error would have been already 1.2%, which is comparable with other experimental errors. Thus, eq 16 can be used for relatively

Figure 6. Solid line, normalized tail of pulse in Figure 5; symbols, integrand of eq 9 for the front of that pulse.

⎡ ⎛ t − t 0 ⎞⎤ ⎟ y(t ) = y0 exp⎢ −⎜ ⎣ ⎝ τ ⎠⎥⎦

(20)

Here, the left-hand side is the plot in Figure 6 (solid line or symbols), t0 being the time lag of the plot. When the six-port valve in Figure 1 switches from position INJECT back to position LOAD, the solute ceases to enter the system and its quantity in the system begins to decrease. However, the 22510


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Table 2. Specific Surface Areas (m2/g) of Various Glass Fibers

detector current does not start to decrease immediately after the switch. At first it drops, then increases again, stays about constant for short time, and then begins to decrease (cf. Figure 5 and solid line in Figure 6). It is that decreasing part of the plot in Figure 6 that is approximated by eq 20. One can understand eq 20 as follows. The ampule of 22.85 mL which substitutes the packed column is a short cylinder of about 2 cm diameter. The carrier gas enters it through a narrow (445 μm) capillary of the guard column (BA in Figure 1) and considerably expands. As a result of that expansion, the concentration of the solute becomes homogeneous inside the ampule. Now the filling of the ampule by the solute can be described by the following differential equation: dN /dt + N /τ = FC i i

fiber E-glass E-glass E-glass E-glass E-glass E-glass

1 2 3 4 5 6

BET surface area measured by D2O adsorption

BET surface area measured by Kr adsorption

0.33 0.42 0.19 0.18 0.27 0.23

0.321 0.402 0.197 0.116 0.135 0.107

geometric surface area 0.364 0.375

0.13

(21)

where Fi and Ci are the volumetric flow rate and the solute concentration at the inlet of the volume V (volume of the ampule). The solution of eq 21 is Co(t ) = C i(1 − e−t / τ )

(22)

where τ = V/Fi and N(t) = VCo(t). In the experiment of Figure 5, there was V = 22.8 mL and Fi = 14.5 ml/min; which gives τ = 1.6 min. The solution of eq 21 is compared with the experiment in Figure 7. The difference with the experimental value of τ is probably due to the time lag, t0, in eq 20 which is not taken into account in eq 21. Figure 8. BET plot of D2O adsorption on glass fiber 4 at 40 °C. Symbols, experimental points; line, least-squares fit.

where P is the partial pressure of D2O in the carrier gas, Ps the saturated vapor pressure of D2O at 40 °C, and a the adsorption per gram of the fiber. The three experimental points in Figure 8 were obtained by the dynamic volumetric method described above at the temperatures of the saturator 0.53, 8.75, and 15.17 °C. Each point was obtained on a preliminary cleaned surface (see section 2). The saturated vapor pressures at these temperatures and 40 °C were calculated by an empirical equation.14 The concentrations corresponding to those partial pressures (which are required for calculation of a were calculated by the ideal gas law with the second virial coefficient correction. The latter was taken for water,15 but the corrections were negligibly small. The deviation of the symbols in Figure 8 from the straight line can be evaluated by the correlation coefficient, which is 0.9996. The intercept and the slope of the straight line determine the monolayer capacity and the CBET constant of the BET model.10 For the straight line in Figure 8, CBET = 76.5. The above values of the correlation coefficient and the CBET constant are usually considered to be satisfactory for reliable determination of the monolayer capacity by the BET method. (Generally it is believed that for successful application of the BET method CBET should be more than 50 and less than 150 (see page 103 in ref 10).) To determine SSA, one has to multiply the monolayer capacity determined from the straight line in Figure 1 by the area of a water molecule. The latter is chosen as 0.09 nm2 (see below). Thus, the BET SSA for fiber 4 is determined as 0.182 m2/g. By the same method the SSAs presented in Table 2 measured by D2O adsorption were determined. The SSAs of the same fibers were determined by the static volumetric adsorption technique with Kr on ASAP 2020 (Micromeretics). In the last column of Table 2 are geometric SSAs which were

Figure 7. Solid line: the same as in Figure 6. Symbols: y(t) (eq 20) with t0 = 0.7 min, τ = 1.23 min, and y0 = 0.823.

Finally, one can see from eq 22 that the steady-state concentration, Ci, will be reached only in the infinitely long time: Ci = Co(∞). However, already in 14 min, Co(t) will deviate from Ci by less than 0.1%.

4. RESULTS AND DISCUSSION The glass fibers 3, 4, 5, and 6 were commercially prepared, whereas fibers 1 and 2 were prepared in the laboratory in a single-tip bushing. The fibers were not coated or surface treated in any way. All of the fiber compositions fall under the E-glass designation, but vary in the levels of B, Ca, and Mg. The BET plot for adsorption of D2O on glass fiber 4 (see Table 2) is shown in Figure 8. The BET coordinate in the plot is10 P /Ps a(1 − P /Ps) (23) 22511


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The Journal of Physical Chemistry C determined from the diameters, d, and the densities, ρ, of glass fibers as 4/dρ. The accuracy of measuring surface areas from adsorption data is believed to be no more than ±10% (see page 104 in ref 10). In the range of that estimate, the data presented in the two last columns of Table 2 actually coincide. In fact, this just confirms a well-known fact that usually the BET surface area measured by nitrogen or krypton gives a geometric surface area. However, there may be doubts whether such small SSAs as those of glass fibers can be measured from Kr adsorption on ASAP 2020 with above-mentioned accuracy. Our result confirms the reliability of those measurements. In the first three rows of Table 2, the BET SSAs measured for the same fiber either by D2O or Kr deviate from each other by less than 5%. This is less than the standard accuracy of the measuring surface areas from adsorption data mentioned above. However, the SSAs presented in the second column of Table 2 were determined by multiplication of the monolayer capacity that actually gives the BET method by the area of a D2O molecule. We chose that area to be 0.09 nm2 to minimize the difference between SSAs presented in the second and third columns of the first three rows of Table 2. The molecular areas of a water molecule that can be found in the literature vary from 0.2 nm2 (see page 274 in ref 10) to 0.105 nm2 calculated from the liquid density of water (see eq 2.27 in ref 10). Thus, the closeness of SSAs for the first three fibers in Table 2 is partly based on our choice of molecular area for water. That molecular area is close to that calculated from the density of water and allows one to bring to agreement the BET surface areas obtained from Kr and water adsorption for three different glass fibers. The main result shown in Table 2 is the SSA of fibers 4, 5, and 6 measured by D2O. They are much larger than those measured by Kr. The most popular method for measuring SSA is the BET method.10 The most suitable adsorptives for BET on high surface area solids are nitrogen and krypton. The latter is typically used for SSA less than 5 m2/g (see page 283 in ref 10). A small number of adsorptives other than nitrogen and krypton are sometimes used for SSA determination (see pp. 73−83 and 283 in ref 10), but water is not one of them. Comparison of the SSA values determined by nitrogen and water for a series of Cr-oxide gels prepared in several ways displays both reasonable agreement and drastic discrepancy (see page 238 in ref 10). This was explained by the fact that a water molecule, being considerably smaller than nitrogen, can penetrate into pores which are too narrow to admit nitrogen molecules. Adsorption isotherms of water on a silica gel change qualitatively upon heat treatment (see page 269 in ref 10). This implies that the BET SSA determined by water would drastically change with heat treatment whereas BET SSA determined by nitrogen stays constant. In this case, the geometrical surface of the adsorbent (evaluated by nitrogen) does not change, but the monolayer capacity of water changes because of a decrease of concentration of adsorption sites for water (silanol groups) in the course of heat treatment. These examples show how comparison of the BET SSA determined by a standard BET probe versus water can uncover chemical or structural differences on the surface which would otherwise not be detected. Thus, the geometric surface areas of glass fibers that can be measured by Kr adsorption may be insufficient for the description of water adsorption. The surface may contain micropores inaccessible for Kr but accessible to a smaller water

molecule, or an adsorption activity which is homogeneous for Kr but perhaps heterogeneous for water. There might be other, yet unknown, reasons why the geometric surface areas of glass fibers are not adequate for the characterization by water adsorption. This is already evident in the results presented in the last rows of Table 2. We cannot definitively establish at this stage why SSAs determined from water adsorption are considerably larger than the geometric surface areas found by adsorption of Kr. We only report this fact based on our development of a dynamic volumetric method for measuring the SSA of glass fibers by water adsorption.

5. CONCLUSIONS The dynamic volumetric technique considered above is a generalization of frontal analysis chromatography. FA is based on the local mass balance equation (mass balance in each cross section of a chromatographic column) which allows one to calculate the profile of the breakthrough curve which is of main interest for FA chromatography. In addition to that, the integration of the breakthrough curve allows one to determine the isotherm of adsorption within the packing of a column. Although the breakthrough curve is not absolutely accurate because (at present) it can be obtained only from some approximate model of chromatography, its integration can give an accurate adsorption isotherm. This is because the integral of the breakthrough curve (adsorption) obeys another equation: the global mass balance equation (GMBE). The DV technique is based on GMBE that determines the rate of mass change in the system through the in- and out-flows of mass at its ends. It cannot determine the profile of breakthrough curve of a packed chromatographic column but can accurately determine adsorption in that column versus partial pressure (concentration) averaged over the column. GMBE can also resolve some problems where the local mass balance equation of chromatography is inapplicable. Such a problem arises when one substitutes a packed chromatographic column by an empty ampule. The “breakthrough curve” in this situation has an origin different from that in a packed column. Application of GMBE makes it possible to determine the profile of that curve. The DV technique allows one to determine the volume of the ampule or an effective volume of a packed column by a dynamic (mass transfer facilitated by carrier fluid) method which justifies the name of the technique. The application of this DV technique for measuring adsorption isotherms on glass fibers makes it possible to determine the specific surface area (SSA) of a fiber by the BET method using water. Usually, the BET SSAs of glass fibers are measured by adsorption of Kr which determines the geometric surface areas (surface area of a cylinder). The BET SSAs determined by water (in fact, D2O) adsorption are, in some cases, close to the geometric values determined by Kr adsorption, but deviate considerably in other cases. This reveals the fact that the geometric surface areas of fibers may be insufficient for the description of water adsorption. The surface may contain micropores inaccessible for Kr but accessible to a smaller water molecule, or an adsorption activity which is homogeneous for Kr but perhaps heterogeneous for water. Thus, it is not just the geometric SSA of a fiber but the SSA determined by water that is a more meaningful parameter for characterization of the interaction of water with a glass fiber. 22512


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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported by PPG Industries. REFERENCES

(1) Doremus, R. H. Glass Science, 2nd ed.; John Wiley & Sons: New York, 1994. (2) Bakaev, V. A.; Pantano, C. G. Inverse Reaction Chromatography. 2. Hydrogen/Deuterium Exchange with Silanol Groups on the Surface of Fumed Silica. J. Phys. Chem. C 2009, 113, 13894−13898. (3) Surface Measurement Systems. www. surfacemeasurementsystems.com. (4) Beebe, R. A.; Evans, P. L.; Kleinsteuber, T. C. W.; Richards, L. W. Adsorption Isotherms and Heats of Adsorption by Frontal Analysis Chromatography. J. Phys. Chem. 1966, 70, 1009−1015. (5) Huang, J.-C.; Forsythe, R.; Madey, R. Gas-Solid Chromatography of Methane-Helium Mixtures: Transmission of a Step Increase in the Concentration of Methane through an Activation Carbon Adsorber Bed at 25 °C. Sep. Sci. Technol. 1981, 16, 475−486. (6) Madey, R.; Photinos, P. J.; Rothstein, D.; Forsythe, R. J.; Huang, J. C. Adsorption Interference in Mixtures of Adsorbate Gases Flowing through Activated Carbon Adsorber Beds. Langmuir 1986, 2, 173− 178. (7) Jaroniec, M.; Lu, X.; Madey, R.; Choma, J. Comparison of the Equilibrium Adsorption Isotherms Measured by the Dynamic and Static Methods for Hydrocarbons on Microporous Activated Carbons. Carbon 1990, 28, 737. (8) Guiochon, G.; Felinger, A.; Shirazi, D. G.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography, 2nd ed.; Elsevier: Amsterdam, 2006; p 141. (9) Andrzejewska, A.; Kaczmarski, K.; Guiochon, G. Theoretical Study of the Accuracy of the Pulse Method, Frontal Analysis, and Frontal Analysis by Characteristic Points for the Determination of Single Component Adsorption Isotherms. J. Chromatography A 2009, 1216, 1067−1083. (10) Gregg, S. J.; Sing, K. S. W. Adsorption Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. (11) Chirico, R. D.; Nguyen, A.; Steele, W. V.; Strube, M. M. Vapor Pressure of n-Alkanes Revisited. New High-Precision Vapor Pressure Data on n-Decane, n-Eicosane, and n-Octacosane. J. Chem. Eng. Data 1989, 34, 149−156. (12) Fryer, G. M. A Theory of Gas Flow through Capillary Tubes. Proc. R. Soc. London, Ser. A 1966, 293, 329−341. (13) Wilke, C. R. A Viscosity Equation for Gas Mixtures. J. Chem. Phys. 1950, 18, 517−519. (14) Harvey, A. H.; Lemmon, E. W. Correlation for the Vapor Pressure of Heavy Water from the Triple Point to the Critical Point. J. Phys. Chem. Ref. Data 2002, 31, 173−181. (15) Harvey, A. H.; Lemmon, E. W. J. Correlation for the Second Virial Coefficient for Water. J. Phys. Chem. Ref. Data 2004, 33, 369− 376.

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A Dynamic Volumetric Method for Measuring Adsorption of Water on

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