Anal. Chem. 2000, 72, 4221-4229
A Mathematical Model for Kinetic Study of Analyte Permeation from Both Liquid and Gas Phases through Hollow Fiber Membranes into Vacuum Alexey A. Sysoev*
Moscow State Engineering Physics Institute (Technical University), Kashirskoe sh. 31, Moscow 115409 Russia
A mathematical model and a Matlab-5 computer code have been developed to study the dynamic response of the hollow fiber membrane probe. The depletion layer formation at the sample/membrane interface is taken into consideration by the mathematical model for the liquid mobile phase. The code produces concentration profiles within a sample feed stream and in the membrane. Flux values at the vacuum side of the membrane can also be calculated as a function of time. The method can be applied both for gas and liquid feed streams. Concentration profiles in a mobile phase and the flux of analytes through the hollow fiber membrane inlet have been studied with this simulation technique as a function of the liquid-phase flow rate. The influence of the formation of a layer of the analyte depletion during the dynamic response has been considered. The shape of the depleted layer and selectivity of permeation from a liquid mobile phase through the membrane into the vacuum are shown to be dependent on the mobile-phase flow rate. In addition, for studied conditions, formation of a depletion layer is demonstrated to be fast compared with membrane diffusion. Thus, if a homogeneous aqueous sample is coming through the inlet cross-section of a hollow fiber membrane containing pure water, the response time mostly depends on analyte diffusivity in the membrane. However, if the aqueous sample is coming through the inlet cross-section of a hollow fiber membrane containing clean air, response time also depends on equilibrium analyte concentration in the depletion layer. Membrane inlet mass spectrometry (MIMS) has become an increasingly popular analytical technique for trace analysis of organic compounds directly from water and air.1,2 A flow-through capillary direct membrane probe3 or a sheet membrane probe4 can be used either for water or for air analysis. However, there are significant differences in permeation of compounds from liquid phase and from gas phase. The diffusion coefficients of organic * Corresponding author: (telephone) +7-095-323-9163; (fax) +7-095-3242111; (e-mail)
[email protected] (1) Kotiaho, T.; Lauritsen, F. R.; Choudhury, T. K.; Cooks, R. G. Anal. Chem. 1991, 63, 875A-883A. (2) Lauritsen, F. R.; Kotiaho, T. Rev. Anal. Chem. 1996, 15(4), 237-264. (3) Bier, M. E.; Cooks, R. G. Anal. Chem. 1987, 59, 597-601. (4) Bier, M. E.; Kotiaho, T.; Cooks, R. G. Anal. Chim. Acta 1990, 231, 175190. 10.1021/ac991388n CCC: $19.00 Published on Web 08/03/2000
© 2000 American Chemical Society
compounds in gases are much greater than those in a membrane, and there is no concentration gradient in the mobile phase. But, in a liquid phase, the diffusion coefficients of organic compounds are comparable with the ones in a membrane. This and poor mixing in the sample/membrane interface results in a layer of analyte depletion next to the sample/membrane interface, which affects both the steady- and nonsteady-state permeation. Therefore, the response of a mass spectrometer depends on a sample flow rate, and quantification errors can be observed if a constant flow rate is not used for analysis of samples and standard solutions. Influence of depleted layer formation on the dynamics of MS response should also be considered in the case of rapid analysis. Flow dynamics has been intensively studied both experimentally and theoretically in MIMS5-7 and other membrane-related analytical methods.8-10 However, the influence of the layer of analyte depletion on response during liquid-phase analysis has not been theoretically researched. The author has found only a few articles where this phenomenon has been analytically studied. A thickness of the oxygen-depleted layer in front of a sheet membrane has been calculated in ref 11 and the static of mass transfer from blood was researched in refs 12, 13. The present paper provides an analytical tool to fill this gap and provides means for better understanding the flow-rate-dependent phenomena and their influence on the MIMS response. The permeating flow dynamics can be studied using an analytical solution to the diffusion equation only in the cases of transitions between steady-state processes, when the intermediate time boundary condition is not too complex. However, the cases in which there is a transition between two nonsteady-state processes are of particular interest due to their practicality. In such conditions, nonsteady-state rise is immediately followed by nonsteady-state fall of the sample stream without a steady-state (5) LaPack, M. A.; Tou, J. C.; Enke, C. G. Anal. Chem. 1990, 62, 1265-1271. (6) Tsai, G.-J.; Austin, G. D.; Syu, M. J.; Tsao, G. T.; Hayward, M. J.; Kotiaho, T.; Cooks, R. G. Anal. Chem. 1991, 63, 2460-2465. (7) Overney, F. L.; Enke, C. G. J. Am. Soc. Mass Spectrom. 1996, 7, 93-100. (8) Yang, M. J.; Adams, M.; Pawliszyn, J. Anal. Chem. 1996, 68, 2782-2789. (9) Luo, Y. Z.; Adams, M.; Pawliszyn, J. Anal. Chem. 1998, 70, 248-254. (10) Luo, Y. Z.; Adams, M.; Pawliszyn, J. Analyst (Cambridge, U.K.) 1997, 122, 1461-1469. (11) Lundsgaard, J. S.; Petersen, L. C.; Degn, H. In Measurement of Oxygen; Degn, H., Baslev, I., Brook, R., Eds.; Elsevier Scientific Publishing Co.: Amsterdam, 1976; 168-183. (12) Davis, H. R.; Parkinson, G. V. Appl. Sci. Res. 1970, 22, 20-30. (13) Baumgardner, J. E.; Neufeld, G. R. Ann. Biomed. Eng. 1997, 25, 858-869.
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Figure 1. Inlet geometry and definitions used in the simulations. Mobile phase flows through the lumen of the membrane, while the outer surface is exposed to the MS vacuum. z is a membrane axis, inlet cross section corresponds to z ) 0, outlet cross section corresponds to z ) L. r represents a radial distance from the membrane axis, r < a corresponds to the mobile phase, a < r < b corresponds to the membrane phase.
condition being reached and vice versa. These problems cannot be solved analytically and require application of numerical methods. In this study, a computer code has been developed to simulate mass transfer processes through a mobile phase, in a membrane, and into the vacuum of a mass spectrometer. The code was developed for a capillary membrane probe. The finite difference method14 of numerical analysis has been chosen in order to implement all possible sample modulation scenarios. The Crank-Nicolson method15 has been applied to solve the corresponding partial differential equations. For the nonsteady-state processes studied, a sample stream modulation is provided by a step function. However, the code allows the use of any type of a sample modulation function if particular application requires it. THEORY Inlet Geometry and the Main Assumption of the Model. The membrane inlet used in this study is a direct-insertion flowthrough-type device utilizing a hollow fiber membrane, in which the mobile phase is flowing inside the capillary (Figure 1). The outside surface of the capillary membrane is exposed to the vacuum of a mass spectrometer. The concentration of analyzed compounds is small enough so that a solution-diffusion model16,17 can be applied, and the flux through the membrane is small compared with the flow rate of sample through the capillary membrane. The liquid-phase flow rates used in the simulations were chosen to be similar to the ones used in MIMS. The inside and outside diameters of the membrane used in the simulations were 0.0305 and 0.0635 cm, correspondingly, which are diameters of a capillary membrane typically used in MIMS (Silastic medicalgrade tubing from Dow-Corning Corp.). The membrane material is poly(dimethylsiloxane). The length of the membrane capillary is 2.5 cm. Since most of the volatile organic compounds have much larger diffusion coefficients in air than in a silicone membrane, their (14) Samarsky, A. A.; Popov, J. P. Finite difference methods for gas dynamics, 3rd ed.; Nauka: Moscow, 1992 (in Russian). (15) Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon Press: Oxford, 1975. (16) Stern, S. A. Membrane Separation Technology; Elsevier: Amsterdam, 1995. (17) Skelland, A. H. P. Diffusional Mass Transfer; John Wiley and Sons: New York, 1974.
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diffusion in a gas phase can be considered as very fast. Because of this, no concentration gradient in a gaseous mobile phase will exist, and the concentration at the sample/membrane interface can be considered constant during the passage of the sample through the membrane probe. However, the diffusion coefficients in a liquid mobile phase are comparable with the ones in a silicone membrane. Therefore, poor mixing at the sample/membrane interface leads to formation of a layer of analyte depletion next to the sample/membrane interface, which, in turn, causes a decrease in sample flux into the vacuum. To obtain better mixing of the mobile phase and decrease the analyte depletion layer thickness, turbulent flow conditions should be used. Transition from laminar flow to turbulent flow occurs at the Reynolds number values of 2000-3000.18 To achieve these values, water (25 °C) flow rates more than 25 mL/min are required for the membrane geometry used in this study.5 However, the flow rates typically used in MIMS are below 10 mL/min,1,2 and therefore, only flow rates up to 10 mL/min were used in the simulations. Under these conditions, the flow of liquids is laminar, mass transport in the axial direction has convective character, and mass transport in the radial direction has diffusion character. Gaseous Mobile Phase. In the case of gas analysis, the analyte concentration in the mobile phase is constant, which makes it possible to consider only the processes in a membrane. Permeation through a membrane can be described by the Fick’s first and second laws, which can be written for a capillary membrane as
∂c ∂r
f ) -D
1 ∂ ∂c ∂c )D r ∂t r ∂r ∂r
( )
(1) (2)
where c is a local concentration in a membrane, D is a diffusion coefficient in a membrane, and r is a radial coordinate. Using the Henry’s law, it is convenient to introduce the phase distribution coefficient5
K)
c(a) cm(a)
) SPt
(3)
where cm(a) is a volume concentration of the compound in the layer of mobile phase next to a sample/membrane interface, S is a solubility constant for the compound in a membrane, Pt is a sample pressure. The flow through a hollow fiber membrane can be written as5
F)
2πLDKcm(a) ln(b/a)
(4)
where L is the length of the capillary, a and b are inside and outside radii of the membrane, respectively. Equation 4 can be applied both for gas and liquid mobile phase.19 (18) Perry, R. H.; Green, D. Perry’s Chemical Engineer’s Handbook, 6th ed.; McGraw-Hill Book Co.: New York, 1984; Chapter 5. (19) Lee, C. H. J. Appl. Polym. Sci. 1975, 19, 83-95.
process, and eq 2 can be expressed as
∂c exp(-2X) ∂2c ) ∂T b2 ∂X2
(7)
which presents the equation of diffusion in the planar symmetric anisotropy environment. The numerical solution of the diffusion equation for the planar symmetric isotropy environment has been described in detail previously.7 Here, the same finite difference derivative function was applied but the network was Xi ) ln(a/b) + (i - 1)δX, Tj ) (j - 1)δT, where i ) 1,2...N, j ) 1,2...M, δX ) ln(b/a)/(N - 1), δT ) Dtm/(M - 1). Defining ci,j ) c(Xi,Tj) and using the Crank-Nicolson implicit scheme,15 it is possible to write a network approximation of eq 7
ci,j+1 - ci,j exp(-2Xi) ) {ci-1,j - 2ci,j + ci+1,j + ci-1,j+1 δT 2(δX)2b2 2ci,j+1 + ci+1,j+1} (8) with boundary conditions on the inside and outside surfaces of the membrane capillary c1,j ) ca,j, cN,j ) 0. After some transformations one obtains eq 9
-fici-1,j+1 + (1 + 2fi)ci,j+1 - fici+1,j+1 ) fici-1,j + (1 - 2fi)ci,j + fici+1,j (9)
Figure 2. Calculated variation of the toluene concentration profiles inside the membrane at various time intervals during gas-phase analysis. The time interval between successive profiles is 3.6 s, i.e., line number 1 is after 0.3 s and line 9 is after 29.1 s. (a) Concentration in the mobile phase is modulated by a step function from 0 to 1 ppm. (b) Concentration in the mobile phase is modulated by a step function from 1 to 0 ppm.
(br)
T ) Dt
dcm(r,z) dr
m fm r (r,z) ) -D ‚
m fm z (r,z) ) c (r,z)vf(r)
Making a substitution of variables
X ) ln
where fi ) δT/2(δX)2b2 exp(-2Xi), which can be solved numerically. The method described was applied in development of a computer code in order to simulate non-steady-state permeation processes. The Matlab-5 software was used for the code development. The code allows researching dynamics of permeating flow from a gas mobile phase into vacuum through a capillary membrane and calculating a concentration map inside a membrane as a function of time. Figure 2 illustrates an example of the possibility of how to study nonsteady-state diffusion in a membrane. One shows toluene concentration profiles inside of a silicone capillary membrane when the mobile-phase composition is modulated by a step function from 0 to 1 ppm and a step function from 1 to 0 ppm. Liquid Mobile Phase. The model expects that the flow of liquids is laminar, which is expected to be true at flow rates used in the simulations. Because of the concentration gradient, diffusion causes mass flow in the radial direction in the mobile phase and convective transfer causes mass flow in the axial direction.20 Therefore, flow density will be as follows
(5) (6)
there, X ∈ [ln(a/b),0], T ∈ [0,Dtm], where tm is the time of the
(10) (11)
m m where f m r (r,z), f z (r,z), and c (r,z) are a local flow density in the mobile phase in the radial direction, a local flow density in the mobile phase in the axial direction, and a local concentration of
(20) Sysoev, A. A. Ph.D. Thesis, Moscow State Engineering Physics Institute (Technical University), Moscow, 1999.
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analyte in point [r,z] of the mobile phase, respectively, and vf(r) is a local linear velocity of laminar liquid flow at the same point. A local linear velocity of convective liquid flow in the capillary can be written as:21
vf (r) )
2V0 πa
2
( ) r2 1- 2 a
∂cm + div cmb v)0 ∂t
(13)
In a cylindrical geometry one will get m m 1 ∂(rc vr) ∂(c vz) + r ∂r ∂z
(14)
Following the assumptions made earlier it is possible to write
vz ) vf (r) vr )
fm r c
m
)-
( ) ∂cm ∂r
(12)
where V0 is a mobile phase pumping rate. To find the time dependence of the permeation flux through a membrane using the custom-made computer code, there is a need to know concentrations in the layer of the mobile phase next to a sample/membrane interface at every time step. To find the concentrations in the mobile phase, the differential equation of indissolubility was used14
v) div cmb
between radial flow of a compound from liquid phase into polymer and permeating flow of the compound inside polymer. After some transforms, it can be derived to the expression20
(15)
Dm ∂cm cm ∂r
)-
r)a
D K cm(a,z,t) Dm a ln(b/a)
(21)
The eqs 18-21 present boundary conditions for the eq 17, which now can be solved numerically both for steady-state and nonsteady-state cases. To solve eq 17 in the nonsteady-state case, the threedimensional network ri ) δr(i - 1), zi ) δz(j - 1), tk ) δt(k - 1), m ci,j,k ) cm(ri,zi,tk) was used, where i ) 1,2...N, j ) 1,2...M, k ) 1,2...K, δr ) a/(N - 1), δz ) L/(M - 1), δt ) tm/(K - 1). In the steady-state case, this three-dimensional network is simplified to a two-dimensional one. Defining gi ) (2V0/(Dmπa2))ri(1 - δr2i /a2) and using the fivepoint pattern “cross” (and the Crank-Nicolson implicit scheme in the nonsteady-state case), it is possible to write a finite difference approximation of eq 17 that can be derived to the following equation for the steady-state case m ci-1,j
(
ri
δr
2
-
)
(
)
2ri ri 1 1 m m - ci,j + ci+1,j + + 2 2δr δr δr2 2δr gi gi m m - ci,j+1 ) 0 (22) ci,j-1 ‚ ‚ 2δz 2δz
(16) and to the equation
where Dm is a diffusion coefficient of the compound in the liquid phase. Inserting eqs 12, 14-16 in eq 13, one obtains
( )
2V0 r ∂cm r2 ∂cm ∂2cm ∂cm + r 1 ) r ∂r Dm ∂t ∂r2 Dmπa2 a2 ∂z
(17)
The concentration at inlet cross section of the capillary is defined by a type of analysis
cm(r,0,t) ) c0m(t)
∂cm ∂r
r)0
)0
(19)
It is also possible to write the following condition at the outlet cross section of the capillary
( ) ∂cm ∂z
z)L
)0
(20)
The last boundary equation can be found from the need of balance (21) Loytsiansky, L. G. Mechanics of gas and liquids, 6th ed.; Nauka: Moscow, 1987 (in Russian).
4224
( ) (
) (
)
Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
(
ri ri ri ri 1 m m + ci,j,k - m - ci+1,j,k + 2 2 4δr 2δr δr D δt 2δr2
( )
)
gi gi ri 1 1 m m m + ci,j+1,k ) ci-1,j,k+1 - ci,j-1,k 4δr 4δz 4δz 2δr2 4δr m ci,j,k+1
ri
δr
2
+
ri m
)
D δt
m + ci+1,j,k+1
(
ri
2δr
2
+
gi 1 m + ci,j-1,k+1 4δr 4δz gi m ci,j+1,k+1 (23) 4δz
(18)
Because of symmetry one may write
( )
m ci-1, j,k
for the nonsteady-state case. Defining B ) D/Dm K/a ln(b/a), the boundary condition (eqs 18-21) in the steady-state case can be derived to the expressions as follows m ci,1 ) cm 0 , i ) 1,2,...N
(24)
m cm 1,j ) c2,j, j ) 2,3,...M - 1
(25)
m m ci,M ) ci,M-1 , i ) 1,2,...N
(26)
m m ) (Bδr + 1)cN,j , j ) 2,3...M - 1 cN-1,j
(27)
In the nonsteady-state case, the boundary condition (eqs 18-
21) at any time step k can be presented as m m ci,1,k ) c0,k , i ) 1,2,...N
(28)
m m c1,j,k ) c2,j,k , j ) 2,3,...M - 1
(29)
m m ci,M,k ) ci,M-1,k , i ) 1,2,...N
(30)
m m cN-1,j,k ) (Bδr + 1)cN,j,k , j ) 2,3,...M - 1
(31)
To simulate both the steady-state and the nonsteady-state permeation processes, the method derived above was embedded into a custom-made computer code. The Matlab-5 software was used for the code development. In the steady-state case, the concentration distribution in the layer of a mobile phase next to a sample/membrane interface was achieved by the numerical solution of eq 17 that was used to calculate the permeation flux. In the nonsteady-state case, the concentration distribution in a mobile phase was obtained by a numerical solution of eq 17 for different scenarios of changing the concentration in a mobile liquid phase. Then the concentration distribution in the layer of a mobile phase next to a sample/membrane interface was used as initial data for an earlier described numerical solution of eq 2. Setting a boundary condition at any time step in an inlet cross section of a membrane tubing, it was possible to simulate any scenario of analysis and to simulate every possible conversion between nonsteady-state processes. The custom-made computer code allows one to study dynamics of permeating flow from a liquid mobile phase into vacuum through a capillary membrane and to calculate a concentration map inside a mobile phase and in the membrane as a function of time. As an example, Figure 3 shows the benzene concentration profiles in different cross sections of a mobile phase flowing at 1 mL/min through the membrane inlet. The initial concentration of the benzene solution was 1 ppm. RESULTS AND DISCUSSION The developed models were used to study both the dynamics and the static of permeation of different compounds (toluene, benzene, ethylbenzene, and ethanol) through a capillary membrane into vacuum. The values of diffusivity in poly(dimethylsiloxane) membrane used in the study were obtained from the paper of LaPack et al.22 The values of diffusivity in water were obtained from a handbook.23 The values of distribution coefficients were calculated using the data obtained from ref 22. The parameters are shown in Table 1. Steady-State Permeation. In gas-phase analysis, a permeation flux is defined only by properties of a membrane, and the flux is calculated by eq 4. However, in liquid-phase analysis, an analyte molecule in the bulk of the liquid phase must first diffuse through a layer of analyte depletion. The concentration profile in this layer is expected to be dependent on both a mobile-phase flow rate and diffusion properties of the compound. The dependence of the layer of analyte depletion on the mobile-phase flow rate is presented in Figures 4 and 5. Figure 4 shows the concentration profile in the (22) LaPack, M. A.; Tou, J. C.; McGuffin, V. L.; Enke, C. G. J. Mem. Sci. 1994, 86, 263-280. (23) Physics Constants Handbook; Grigoriev, I. S., Meylihova, E. Z., Eds.; Energoatomizdat: Moscow, 1991, Chapter 17 (in Russian).
Figure 3. Calculated concentration profiles in cross sections of mobile phase at z equals 0.1 cm (1), 0.5 cm (2), 1 cm (3), 1.5 cm (4), 2 cm (5), and 2.5 cm (6). z is a membrane axis. z ) 0 corresponds to the inlet cross section. z ) L ) 2.5 cm corresponds to the outlet cross section (see Figure 1). An aqueous solution of 1 ppm of benzene (5.6 × 10-8 moles/cm3) was pumped through the inlet at a 1 mL/min flow rate.
layer of a mobile phase next to a liquid/membrane interface as a function of z for benzene-water solution at flow rates ranging from 0.5 to 10 mL/min, whereas Figure 5 shows the concentration profile in the depleted layer as a function of r in the outlet cross section of a capillary membrane. The results presented were obtained using the eqs 22, 24-27, and benzene concentration in the solution was 1 ppm (1 ppm benzene is 5.6 × 10-8 moles/ cm3). The width of the depleted layer at 95% concentration level changes approximately from 0.015 mm at a 10 mL/min flow rate to 0.046 mm at a 0.5 mL/min flow rate. Benzene concentration at the interface layer in the outlet cross section changes approximately from 0.43 × 10-8 mole/cm3 at a 0.5 mL/min flow rate to 1.1 × 10-8 mole/cm3 at a 10 mL/min flow rate. The results of permeation flux measurement agree quite well with the experimental data obtained by5 LaPack et al., who showed that the flux of toluene into vacuum increases more than 3 times when a flow rate increases from 1 to 10 mL/min. In the present study, with the same kind of a membrane as the one used in ref 5, the toluene flux into vacuum ranges from 2.5 × 1013 molecules/s at 1 mL/m to 5.3 × 1013 molecules/s at 10 mL/min, the increase being a factor of 2.1. However, under real experimental conditions it can be quite difficult to ensure laminar flow conditions and, therefore, the experimental dependence of the flux into vacuum can be a bit higher than the simulated one due to some local turbulent phenomena. Note that even with the flow rate of 10 mL/ min, concentration of benzene in the layer of analyte depletion is a few times smaller than its concentration in the bulk sample, 5.6 × 10-8 mole/cm3 (Figure 5). That is why, if unexpected local sample flow turbulence occurs at higher pumping rates, one can increase an analyte concentration in the depleted layer and additionally raise permeation flux through a membrane. Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
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Table 1. Diffusion and Phase Distribution Coefficients of Organic Compounds22,23 Used for the Evaluation of the Model and Permeation Flux of These Compounds through a Capillary Polydimethylsiloxane Membrane into Vacuuma permeation flux from H2O distribution coefficients
substance
diffusivity in water (×106)
diffusivity in a membrane (×106)
from N2
benzene toluene ethylbenzene ethanol
7.5 6.4 6.1 10
4.9 3.5 1.7 0.4
212 576 1895 2122
from H2O
from N2 (×10-12)
at flow rates 2 mL/min (×10-12)
at flow rates 10 mL/min (×10-12)
52.7 171.8 565.1 0.510
0.6 1.2 1.9 0.5
32 32 32 0.15
50 53 53 0.15
a The units for diffusivity are cm2/s, the units for phase distribution coefficients are (mole/cm3)/(mole/cm3), and the units for permeation flux are molecules/s. Concentration of individual compounds was 1 ppm either in N2 or H2O.
Figure 4. Calculated concentration profiles in the contact zone of the mobile phase as a function of sample flow rate. Flow rate varies from 0.5 mL/min (curve 1) to 10 mL/min (curve 11). Sample is a 1 ppm (5.6 × 10-8 moles/cm3) aqueous solution of benzene.
Figure 5. Concentration profiles calculated in the outlet cross section of the mobile phase in the capillary membrane as a function of sample flow rate. Flow rate varies from 0.5 mL/s (curve 1) to 10 mL/s (curve 11). Sample is a 1 ppm (5.6 × 10-8 moles/cm3) aqueous solution of benzene.
The sample flow rate can also have an effect on the permeation selectivity during analysis of liquid samples. This can be clearly observed from the calculated permeation flux data of different compounds presented in Table 1. Fluxes of all the compounds were simulated for samples of 1 ppm concentration level. It can be also seen from Figure 6, where the dependencies of permeation flux of toluene, ethylbenzene, and ethanol on liquid sample flow rate are presented. Fluxes of toluene and ethylbenzene were simulated for samples of 1 ppm concentration level in H2O. Flux of ethanol was simulated for samples of 100 ppm concentration level in H2O. In the case of gaseous mobile phase, the steadystate flux of ethylbenzene exceeds 1.6 times that of toluene (Table 1). Whereas, fluxes of these compounds are about the same in the case of liquid mobile phase. However, the selectivity between toluene and ethanol changes from 1.3 × 102 at a 0.5 mL/min flow rate to 3.5 × 102 at a 10 mL/min flow rate. Selectivity is defined as a ratio of permeation fluxes of two compounds measured at the same membrane geometry and the same sample concentration. For gas-phase analysis, selectivity is equal to membrane
permselectivity of these two compounds. For liquid-phase analysis, their diffusivities in liquid also have an effect on the selectivity. These results demonstrate that the depleted layer is an additional separation factor that affects the quantitative results obtainable by membrane inlet mass spectrometry. Figure 6 also demonstrates how flow rate instability can cause quantification errors. 20% fluctuation at a 9 cm3/min sample flow rate will cause 7% variation of permeating flux for toluene, 7% variation of permeating flux for ethylbenzene, and 0.01% variation of permeating flux for ethanol. If the flow rate fluctuation results in different standard and sample flow rates, quantification errors are observed. Figure 6 also shows that higher sample pumping rates are better due to increasing permeating fluxes and smaller quantification errors caused by pumping-rate instability. Temperature variation can also result in quantification errors, because diffusion and distribution constants are temperaturedependent. Usually the temperature of a membrane inlet is kept very stable during MS analysis. However, if the temperature of
4226 Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
Figure 6. Permeation fluxes for ethylbenzene (1), toluene (2), and ethanol (3) as functions of liquid sample flow rate. Fluxes of toluene and ethylbenzene were simulated for samples of a 1 ppm concentration level. Flux of ethanol was simulated for samples of a 100 ppm concentration level.
the membrane inlet/sampling system and that of the sample are not the same, the membrane temperature can be different than that of the membrane inlet. Particularly, limited heat conductivity and membrane heating from an ion source can cause a difference between measured membrane inlet temperature and the membrane temperature. In this case, membrane temperature and permeating fluxes are sample-flow-rate-dependent, which can result in additional quantification errors, especially if the sample and standard mixtures have different initial temperatures and they are not heated properly before entering the membrane inlet. Nonsteady-State Permeation. Under the nonsteady state conditions, the developed model was used to calculate permeation curves for compounds with different diffusion and distribution coefficients. As an example, Figure 7a illustrates a case in which a gas sample containing toluene, ethylbenzene, and ethanol is injected (sample introduction time, 100 second) into the continuous gas stream flowing through the membrane inlet. From Figure 7a it can be observed that only the toluene flux reaches steady state. After the 100-second sample introduction time, the permeation fluxes of ethylbenzene and ethanol correspond approximately to 99 and 53% of the steady state level, respectively. It is also observed that the fluxes of ethylbenzene and ethanol are rising even though their concentration in the mobile phase is zero. This effect is seen clearly in the experimental data measured for ethanol, which shows that the time delay between the transition point (1 ppm f 0) and the maximum permeation flux is 34 s. This can be explained by redistribution of concentration inside the membrane, which takes place during transition between two nonsteady-state modes. However, when the time of sample introduction is 350 s, it is noticed that the permeation fluxes of all the compounds reach steady state (Figure 7b).
Figure 7. Permeation flux dynamics for toluene (1), ethylbenzene (2), and ethanol (3) (1 ppm of each in the air sample) is modulated by (a) 100-s and (b) 350-s step functions.
The dynamics of the permeation process are also affected by the existence of the depleted layer, because its formation is a timedependent process. Two different cases were studied. The first one is gas-liquid transition occurring when a liquid sample is pumped through a capillary containing air and the second one is a liquid-liquid transition occurring when a liquid sample is pumped through a capillary containing clear water. Simulations of these transitions (gas-liquid sample and clean water-liquid sample) are shown in Figure 8. The concentration of benzene in the interface layer of a mobile phase versus z axis at different time steps illustrates dynamics of depleted-layer formation for the Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
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Figure 9. Normalized flux versus time profiles for (1) gas sample analysis and (2) gas-liquid transition.
Figure 8. Calculated variation of the benzene concentration profiles in the layer of a sample next to the liquid/membrane interface at different time intervals. The time interval between successive profiles is 0.05 s. The sample flow rate is 2 mL/min. (a) Clean air followed by an aqueous 1 ppm (5.6 × 10-8 moles/cm3) benzene sample (gasliquid transition). (b) Pure water followed by an aqueous 1 ppm (5.6 × 10-8 moles/cm3) benzene sample (liquid-liquid transition).
gas-liquid transition (Figure 8a) and liquid-liquid transition (Figure 8b) at a 2 mL/min sample flow rate. The time interval between different curves is 0.05 s. From the results obtained it is observed that the process of depleted-layer formation is much faster than that of diffusion through a membrane, and it can be considered to be an instantaneous process. The concentration profiles calculated using the longest times intervals in both cases are the ones which correspond to the steady-state distribution of concentration (see curve 3 in Figure 4). 4228 Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
In Figure 9, the permeation curves are presented for the cases in which a gas sample or a liquid sample, surrounded from both sides by clean air, is analyzed by MIMS. In the case of the gassample analysis (Figure 9, line 1), the concentration of analyte in a gas/membrane interface changes instantaneously from zero to cm 0 and the rise time of the permeation curve reflects only analyte diffusivity in a membrane. But in the case of a liquid sample analysis (Figure 9, line 2), the concentration of analyte in a liquid/ membrane interface changes instantly from zero to cm 0 and rapidly falls to cm(a,z), reflecting the membrane and liquid phase diffusion equilibrium. Because of this, the permeation flux at the beginning of the permeation process will reflect a higher concentration of analyzed compounds in the liquid/membrane interface than that when the steady state condition is reached. This also explains why the response time in the case of a liquid sample analysis is smaller than that in the case of a gas sample analysis (Figure 9). Results presented show that the response time in the case of the analysis of a liquid sample surrounded by clean air will not reflect directly the analyte diffusivity in a membrane. In this case, the response time also depends on equilibrium analyte concentration in the depleted layer that, in turn, depends on liquid sample pumping rate. Note also that the rise time for a liquid sample surrounded from both sides by clean water is expected to be approximately the same as that for a gas sample surrounded by clean air (Figure 9, line 1). In this case, the behavior of rise time depends on analyte concentration in a liquid/membrane interface, which changes rapidly from zero to cm(a,z), reflecting the membrane and liquid phase diffusion equilibrium. This process is very fast and therefore it is expected that it will not considerably affect rise times, because those mostly depend on the analyte diffusivity in a membrane. CONCLUSION The developed models and computer algorithms have allowed theoretical studies of statics and dynamics of mass transfer phenomena during permeation of various analytes through a
capillary membrane into the ion source of a mass spectrometer. The model can be applied both for gaseous and liquid samples. The research on statics of permeation flux from liquid phase makes it possible to estimate quantification errors caused by fluctuation of sample flow rate. Permeating flow dynamics simulations has shown that the depleted layer has weak influences on the response times. It should also be noted that mobile-phase flow rate variation could cause a change in membrane inlet temperature conditions, which, in turn, can result in variation of both dynamics and statics of the permeation flux. When the paper had been completed, the author was informed about the work of Dr. Friedrich Lennemann24 in which extensive theoretical study of MIMS related membrane permeation processes was made on the basis of numerical simulation. ACKNOWLEDGMENT I wish to thank Dr. Tapio Kotiaho for valuable discussion and very important help with the manuscript preparation. I am grateful to Prof. Valentine D. Borisevich and Dr. Viacheslav B. Artaev for great help with the manuscript translation. This work was funded by the Russian Ministry of Education within the framework of the program “Conversion and High Technology 1997-2000” (project 49-1-10). A travel grant from the Academy of Finland is also gratefully acknowledged. LIST OF SYMBOLS a
inside membrane radius [cm]
b
outside membrane radius [cm]
c ) c(r,t)
volume concentration in a membrane [mole/cm3]
c0m ) c0m(t)
volume concentration in inlet cross section of mobile phase [mole/cm3],
cm ) cm(r,z,t)
volume concentration in mobile phase [mole/ cm3]
D
diffusion coefficient in a membrane [cm2/s]
Dm
diffusion coefficient in mobile phase [cm2/s]
F
steady-state permeating flux through the membrane into vacuum [mole/s]
(24) Lennemann, F. Ph.D. Thesis. Hamburg-Harburg Technical University, Hamburg, Germany, 1999.
f ) f(r,t)
local flow density in membrane phase [mole/ cm2/s]
frm ) frm(r,z,t) local flow density in mobile phase in radial direction [mole/cm2/s] fzm ) fzm(r,z,t) local flow density in mobile phase in axial direction [mole/cm2/s] K
phase distribution coefficient [(mole/cm3)/(mole/ cm3)]
L
membrane length [cm]
Pt
sample pressure [cmHg]
r
radius from membrane axis [cm]
S
solubility coefficient in membrane phase [cm3/ cm3/cmHg]
t
time [s]
tm
duration of a numeral experiment [s]
V0
mobile phase pumping rate [cm3/s]
vf ) vf(r)
local linear velocity of laminar liquid flow inside infinite impermeable capillary [cm/s]
v ) v(r,z,t)
local linear velocity of analyte in mobile phase [cm/s]
vr ) vr(r,z,t)
local radial velocity of analyte in mobile phase [cm/s]
vz ) vz(r,z)
local axial velocity of analyte in mobile phase [cm/s]
z
axial coordinate [cm]
SUPPORTING INFORMATION AVAILABLE A listing of Matlab-5 script files is available as Supporting Information. Current ordering information is found on any masthead page. This information is available free of charge via the Internet at http://pubs.acs.org.
Received for review December 1, 1999. Accepted May 4, 2000. AC991388N
Analytical Chemistry, Vol. 72, No. 17, September 1, 2000
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