Anal. Chem. 1998, 70, 4822-4826
Solid-Phase Microextraction in Headspace Analysis. Dynamics in Non-Steady-State Mass Transfer Jiu Ai*
U.S. Tobacco Manufacturing Company, Inc., 800 Harrison Street, Nashville, Tennessee 37203
Previously proposed models (Ai, J. Anal. Chem. 1997, 69, 1230-6; 3260-6) dealing with the dynamics of solidphase microextraction (SPME) were based on the assumption that a steady-state mass transfer is in effect throughout the SPME extraction process. In reality, a steady-state mass transfer may not be established at the initial stage of headspace SPME, which involves mass transfer across two interfaces. The rate of analyte transfer at the condensed phase/headspace interface may not be equal to the extraction rate at the headspace/SPME polymer film interface. An improved model is proposed in this report to handle the situation of the non-steadystate mass transfer for the headspace SPME. A mathematical solution is obtained for the dynamic process of the non-steady-state mass transfer by correlating the variation of the analyte concentration in the headspace with the analyte extraction rate. This solution provides an equation relating the extracted amount of analyte to the extraction time a little more complicated than the previous models. It also gives a better description of experimental observations. Solid-phase microextraction (SPME) developed by Pawliszyn and co-workers3,4 has great advantages for analyzing organic chemicals in the headspace over a condensed phase. During a static headspace analysis, chemicals with high affinity toward the SPME polymer film are concentrated in the film5-10 and the extraction results in a higher sensitivity than conventional static headspace analysis. SPME is a partition process if the polymer film acts like a liquid phase. Once the partition equilibrium is attained, the extracted amount of analyte can be expressed as follows: * Present address: Corporate Research Center, International Paper, 1422 Long Meadow Rd., Tuxedo, NY 10987; (phone) (914) 577-7489; (e-mail)
[email protected]. (1) Ai, J. Anal. Chem. 1997, 69, 1230-6. (2) Ai, J. Anal. Chem. 1997, 69, 3260-6. (3) Arthur, C. L.; Pawliszyn, J. Anal. Chem. 1990, 62, 2145-8. (4) Zhang, Z.; Yang, M. J.; Pawliszyn, J. Anal. Chem. 1994, 66, 844A-53A. (5) Zhang, Z.; Pawliszyn, J. J. High Resolut. Chromatogr. 1993, 16, 689-92. (6) Field, J. A.; Nickerson, G.; James, D. D.; Heider, C. J. Agric. Food Chem. 1996, 44, 1768-72. (7) MacGillivray, B.; Pawliszyn, J.; Fowlie, P.; Sagara, C. J. Chromatogr. Sci. 1994, 32, 317-22. (8) Pelusio, F.; Nilsson, T.; Montanarella, L.; Tilio, R.; Larsen, B.; Faccetti, S.; Madsen, J. J. Agric. Food Chem. 1995, 43, 2138-43. (9) Zhang, Z.; Pawliszyn, J. J. High Resolut. Chromatogr. 1996, 19, 155-60. (10) Matich, A. J.; Rowan, D. D.; Banks, N. H. Anal. Chem. 1996, 68, 4114-8.
4822 Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
n∞ )
KfhKhsVfVs C KfhKhsVf + KhsVh + Vs 0
(1)
Here n∞ is the amount of analyte extracted by the SPME polymer film when partition equilibrium is attained. Kfh and Khs are equilibrium partition constants for the analyte between the headspace and the polymer film and between the condensed phase and its headspace, respectively. Vf, Vh, and Vs are volumes of the SPME polymer film, the headspace, and the sample matrix. C0 is the initial concentration of the analyte in the sample matrix. The dynamic process of the SPME before partition equilibrium has also been investigated. Pawliszyn and co-workers used a diffusion equation to describe the SPME extraction process.11-13 Based on the boundary conditions of SPME, the analytical solution is very complicated.13 Direct application of the model to the experimental results is a difficult task. A simple dynamic model2 with the focus on the mass transfer at the two interfaces was proposed to deal the dynamic process of headspace SPME. This model avoided the mathematical treatment for the second-order partial differential equation, and a simple analytical solution was obtained:
n ) n∞[1 - exp(-aht)]
(2)
Here n is the amount of analyte extracted by the SPME polymer film before partition equilibrium, t is the extraction time, and ah is a complicated parameter that determines how fast the equilibrium can be reached. The extracted amount before partition equilibrium is proportional to the amount that can be extracted at equilibrium. There is an exponential term between them. As time goes to infinity, this term vanishes. This model of eq 2 fits the experimental extraction time profiles as shown in the previous study.2 The above model is derived on the basis of a steady-state kinetics that assumes that the mass-transfer rate at the condensed phase/headspace interface is equal to the mass-transfer rate at the headspace/SPME film interface. Analyte concentration in the headspace, Ch, remains at a constant level. This assumption may (11) Louch, D.; Motlagh, S. Pawliszyn, J. Anal. Chem. 1992, 64, 1187-99. (12) Zhang, Z.; Pawliszyn, J. Anal. Chem. 1993, 65, 1843-52. (13) Pawliszyn, J. Solid-Phase Microextraction, Theory and Practice; Wiley-VCH: New York, 1997. 10.1021/ac980642t CCC: $15.00
© 1998 American Chemical Society Published on Web 10/10/1998
function.14,15 The parabolic model of the concentration profile in solid films is better than the linear diffusion approximation, which is common in treating diffusion-controlled mass transfer.16 The average concentration of the analyte in the polymer film is thus (Cf|x)0 + 2Cf|x)δ)/3 following the parabolic approximation. Therefore, the extracted amount of analyte is the average concentration times the volume of the SPME polymer film
n ≈ Vf
Figure 1. Schematics of the SPME polymer fiber and headspace interface. The concentration profile of the analyte inside the polymer film follows a parabolic simulation.
not describe the headspace SPME process precisely. Once the SPME polymer film is exposed to the headspace over a condensed phase, analyte concentration in the headspace is disturbed since the mass transfer from the condensed phase to the headspace is not an instant process. In this report, an improved theoretical model is built to deal with the situation where the steady-state mass transfer between the two interfaces is not established. The analyte concentration in the headspace varies as the headspace extraction starts. The rate of variation of the analyte concentration in the headspace is the difference between the rate of analyte evaporation from the condensed phase and the rate of analyte extraction by the SPME polymer film. By correlating the rate of analyte extraction and its rate of variation in the headspace, an expression for extracted amount n can be obtained as a function of extraction time t. Unlike eq 2, it contains two exponential terms, with each one similar to eq 2. The new model fits experimental data better than eq 2 and it also provides an explanation for quantitative headspace SPME in a non-steady-state mass-transfer process. THEORETICAL TREATMENT As described in the previous study,2 headspace SPME involves mass transfer in three phases and across two interfaces. Mass transfer in the headspace (gas phase) is considered a very fast process. The dynamics at the two interfaces (condensed phase/ headspace and headspace/SPME polymer film) is the focus of the study. At the headspace/SPME film interface, the rate of analyte extraction can be expressed according to Fick’s first law of diffusion from the SPME polymer film surface to its inner layers.
( )
∂Cf 1 dn ) -Df | ) mf(Cf|x)0 - Cf|x)δ) Af dt ∂x x)0
Cf|x)0 + 2Cf|x)δ 3
(4)
On the basis of eq 4, mf(Cf|x)0 - Cf|x)δ) in eq 3 can be written as a function of Cf|x)0 and n:
(|
3 mf(Cf|x)0 - Cf|x)δ) ) mf Cf 2
x)0
-
n Vf
)
(5)
At the SPME polymer surface with the headspace, the partition equilibrium exists for the analyte. Therefore we have
Cf|x)0 ) KfhCh
(6)
Ch is the analyte concentration in the headspace. Substituting eqs 5 and 6 into eq 3, we have
dn 3Afmf 3 n ) AfmfKfhCh + dt 2Vf 2
(7)
Equation 7 correlates the extracted analyte n with its concentration in the headspace Ch. We can express Ch in terms of n and dn/dt using this equation. Differentiating eq 7, we have
dCh 2 d2n 1 dn ) + dt 3AfmfKfh dt2 KfhVf dt
(8)
At the condensed phase/headspace interface, the evaporation rate of the analyte from the condensed phase to its headspace is assumed to be proportional to the deviation of its headspace concentration from the equilibrium value as is described in the previous study.2 Therefore we have
r ) k(C0h - Ch) ) k(KhsCs - Ch)
(9)
(3)
Af is the surface area of the SPME polymer film, Df is the diffusion constant of the analyte inside the SPME polymer film, Cf is the analyte concentration in the polymer film, and mf is the masstransfer coefficient of the analyte, mf ) Df/δ. δ is the thickness of the SPME polymer film. Figure 1 shows a schematic description of the headspace/SPME polymer film interface. As shown in Figure 1, the analyte concentration profile inside the SPME polymer film can be simulated with a parabolic
r is the evaporation rate of the analyte and k is the evaporation rate constant. Ch0 is the headspace concentration of the analyte at equilibrium. It is equal to KhsCs, and Cs is the analyte concentration in the condensed phase. Cs can be expressed as follows:2,12 (14) Liaw, c. H.; Wang, J. S. P.; Greenhorn, R. A.; Chao, K. C. AIChE J. 1979, 25, 376. (15) Yang, R. T. Gas Separation by Adsorption Processes; Butterworth: Boston, 1987. (16) Crank, J. The Mathematics of Diffusion; Clarendon Press: Oxford, U.K., 1975.
Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
4823
Cs ) C 0 -
VhCh n Vs Vs
(10)
Therefore, eq 9 can be rewritten in terms of n and Ch:
r ) kKhsC0 -
(
)
KhsVh + Vs kKhs n-k Ch Vs Vs
(11)
(12)
)
KhsVh + Vs dCh kKhs 1 dn ) kKhsC0 - k Ch ndt Vs Vs Vh dt
(13)
Substitute Ch from eq 7 and dCh/dt from eq 8 into eq 13, we have
d2n dn 3 +p + qn ) AfmfkKfhKhsC0 dt 2 dt2
(14)
with
(
p)k
)
(
)
KhsVh + Vs KfhVf + Vh 3 + A fm f Vs 2 VfVh
(15)
and
KfhKhsVf + KhsVh + Vs 3 q ) Afmfk 2 VfVs
(16)
Equation 14 is a second-order nonhomogeneous linear differential equation. Its general solution is
n ) C1e-at + C2e-bt +
(18)
b ) 1/2(p - xp2 - 4q)
(19)
C1 and C2 are integration constants. Apply the initial condition n|t)0 ) 0, we have
C1 + C 2 +
KfhKhsVfVs C )0 KfhKhsVf + KhsVh + Vs 0
or
R+β)
KfhKhsVfVs C ) n∞ KfhKhsVf + KhsVh + Vs 0
KfhKhsVfVs C (17) KfhKhsVf + KhsVh + Vs 0
4824 Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
(20)
with R ) -C1 and β ) -C2. Equation 17 can be rewritten as
n ) R[1 - exp(-at)] + β[1 - exp(-bt)]
The first term in eq 12 is the analyte evaporation rate from the condensed phase. It has a positive sign since analyte evaporation increases the headspace concentration. The second term is the SPME extraction rate expressed as the depletion rate of the analyte in the headspace with a negative sign. The variation rate as described in eq 12 is not equal to zero. Substitute eq 11 into eq 12, we have
(
a ) 1/2(p + xp2 - 4q) and
In the previous study in dealing with the headspace SPME,2 it is assumed that the evaporation rate r is equal to the extraction rate described in eq 3. If a steady-state mass transfer from the condensed phase to the SPME polymer film is established through the headspace, the assumption is true and the analyte concentration in the headspace is a constant. But, before the steady-state mass transfer is established, the analyte concentration in the headspace varies with time. The variation rate of the headspace concentration is the difference between the evaporation rate and the extraction rate:
dCh 1 dn )rdt Vh dt
with
(21)
The amount of extracted analyte n can be expressed as a function of extraction time t in a form with two exponential terms. According to eq 20, R and β should be proportional to C0. Therefore we have n ∝ C0 in eq 21 once t is held constant. The n ∝ C0 relation meets the key requirement for quantitative analysis. The amount of sample extracted by SPME is proportional to the initial concentration of the material in the sample matrix. As extraction time goes to infinity, n is equal to R + β and eq 21 becomes eq 1, which is derived from the thermodynamic partition process. EXPERIMENTAL SECTION The SPME fibers used in this study were purchased from Supelco (Bellefonte, PA). They were silica fibers coated with 85µm polyacrylate (PA) film and 100-µm poly(dimethylsiloxane) (PDMS) film. The chemical used in this study was 2-octanol purchased from Matheson Coleman & Bell (East Rutherford, NJ). It was used without further purification. The condensed phase, which was an aqueous solution of 2-octanol, was prepared by diluting 2-octanol 5000 times with saturated NaCl and few drops of methanol. A 10-mL aliquot of the solution were put in a 50-mL flask with a headspace volume of 45 mL. The flask was put on the heating plate for room-temperature and above room-temperature headspace SPME. For below room-temperature headspace SPME, the flask was immersed in a beaker containing ice water. The flask was sealed with a rubber lid, and the headspace SPME was carried out by punctuating the rubber lid and inserting the fiber into the headspace for a certain period of time. Headspace SPME time profiles were measured in this way with the condensed phase at different temperatures. GC/MS was used as the detection device. HP5890 II GC was interfaced with a Finnigan MAT TSQ700 mass spectrometer. A 30 m × 0.25 mm with 0.25-µm film DB5 column (J&W Scientific,
Figure 3. SPME headspace extraction time profiles of 100-µm PDMS fiber over 10 mL of 2-octanol (1:5000 diluted) aqueous solution in a 50-mL flask: condensed-phase temperature (b) 1, (O) 22, (1) 29, and (3) 45 °C. Table 1. List of Parameters p and q Derived from the Regression of Experimental Data with Eq 21 for Headspace SPME with Condensed Phase at Different Temperatures and Different Polymer Fibers
Figure 2. SPME headspace extraction time profiles of 100-µm PDMS fiber (a) and 85-µm PA fiber (b). The condensed phase was 10 mL of 2-octanol (1:5000 diluted with saturated NaCl solution) and its temperature was 22 °C. Vh was about 45 mL. The solid lines are eq 2 fit, and dotted lines are eq 21 fit.
Folsom, CA) was used as the analysis column. The injection port temperature, which was also the desorption temperature for the SPME fiber, was set at 250 °C and desorption time was set at 2 min. The GC oven was kept at 200 °C. Helium was used as the carrier gas in GC, and its flow rate was 1.0 mL/min. Full mass scan from 35 to 135 amu was acquired in the electron impact ionization mode (EI ) 70 eV). The ion source temperature was 150 °C, and the manifold temperature was 70 °C. The base peak (m/z 45+) was integrated, and the integrated intensity was used as the SPME extraction intensity. RESULTS AND DISCUSSION In the previous study,2 eq 2 fitted experimental data with good precision. A linear proportional relationship between n and C0 was observed when the sampling time was far shorter than that required to reach a partition equilibrium. Matich et al.10 observed that eq 2 was not the best form to describe their experimental results in using headspace SPME for volatile chemicals in apples. They fitted their data with a two-term exponential equation in the same form as eq 21 empirically and obtained much better results. In this report, eq 21 is theoretically derived from the dynamic process of headspace SPME. Figure 2 shows both eqs 2 and 21 fitting the experimental data of headspace SPME at room temperature (22 °C). It is obvious that eq 21 describes the experi-
SPME fiber
temp (°C)
p (s-1)
q (s-2)
100-µm PDMS 100-µm PDMS 100-µm PDMS 100-µm PDMS 85-µm PA
1 22 29 45 22
0.0074 0.028 0.050 0.56 0.031
3.6 × 10-6 1.0 × 10-4 4.3 × 10-4 9.0 × 10-3 1.9 × 10-5
mental measurements for both PDMS and PA fibers more precisely. This is an expected result because eq 21 is derived under a situation more close to the reality during a headspace extraction. Through the regression using eq 21, two parameters a and b are obtained. According to eqs 18 and 19, we have p ) a + b and q ) a × b. Therefore, the values of two important parameters p and q are obtained. Parameter p has two terms (eq 15) corresponding to the mass transfer at the two interfaces (condensed phase/headspace and headspace/polymer fiber, respectively). The first term of p is proportional to k, which is the rate constant of analyte evaporation from the condensed phase to its headspace. It is also dependent on the Khs, the partition constant of the analyte between the condensed phase and its headspace. Both k and Khs are expected to be highly dependent on the temperature of the condensed phase. The second term of p is proportional to mf, the mass-transfer coefficient of the analyte inside the SPME polymer film. It is expected to be dependent on the nature of the polymer film. Parameter q is proportional to both k and mf and is also related to Khs as described in eq 16. If the temperature of the condensed phase is raised, both k and Khs values will increase and we would expect to see larger p and q. Figure 3 shows the extraction time profiles of the PDMS fiber at four different temperatures (1, 22, 29, and 45 °C). Table 1 lists the parameters p and q obtained from the regressions using eq 21. As expected, both p and q values increase as the temperature of the condensed phase increases. As temperature was increased from 1 to 22 °C, the q value increased about 2 orders of magnitude. If the evaporation rate constant follows an Ahrrenius’ form Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
4825
than that of 85-µm PA fiber during a room-temperature headspace SPME. The PDMS film is a liquid phase and the PA film is partially cross-linked, according to the manufacturer (Supelco Co). Mass transfer inside the PDMS film is expected to be easier than in the PA film. The larger q for PDMS film compared to PA film is seen as predicted. The difference between the previous model (eq 2) and the new model (eq 21) is dependent upon how fast the analyte molecules can evaporate from the condensed phase to its headspace. If the analyte evaporation is a fast process, a steady-state mass transfer at the two interfaces can be quickly reached. In this case, eq 21 will reduce to eq 2, which describes the dynamic process of headspace SPME with a steady-state mass transfer. As the temperature of the condensed phase increases, the evaporation rate of analyte increases, and there should be less difference between eq 21 and eq 2. Figure 4 shows the experimental data fitted with both eq 2 and eq 21. At low condensed-phase temperature, the difference between these two models is large (Figure 4a). As the temperature of the condensed phase increased to 45 °C, there is a very small difference between these two models (Figure 4c). As far as the relationship between n and C0 before partition equilibrium is concerned, the previous study2 showed that there is an excellent linear relationship between them. Equation 21 also has the n ∝ C0 relationship. This relationship indicates that quantification is feasible using SPME for headspace analysis even before a steady state of mass transfer is reached. This study provides the theoretical base for fast quantitative headspace analysis using SPME. Figure 4. SPME headspace extraction time profiles of 100-µm PDMS fiber with three different solution temperatures: (a) 1, (b) 22, and (c) 45 °C. The solid and dotted lines are regressions with eq 2 and eq 21, respectively.
(k ) koe-Ea/RT), for every temperature increase of 10 °C, k will increase 2-3-fold. Khs is also expected to increase as temperature increases, following a similar form (Khs ) K0e-∆H/RT). A 2-3-fold increase of Khs is also expected for every 10 °C temperature increase. Therefore, a 2 orders of magnitude increase in q is reasonable for a temperature increase from 1 to 22 °C combining the increases of k and Khs. As for parameter p (eq 15), only the first term of p is proportional to k, which will make the p value increase relatively small as the temperature of the condensed phase increases. Since parameter q is proportional to mf according to eq 16, different SPME polymer films should have different q values when the same experimental conditions are applied. Table 1 shows that the q value for 100-µm PDMS fiber is more than 5 times larger
4826 Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
CONCLUSION Compared to the dynamic model for headspace SPME proposed previously,2 the new model provides a better description of the experimental measurements, especially when the evaporation rate of analyte from the condensed phase to its headspace is slow. As is predicted by the new model, the parameters (p and q) derived from the experimental data vary as the temperature of the condensed phase (aqueous solution) changes. As the temperature of the condensed phase increases and the evaporation rate of analyte molecules increases, the new model also reduces to the previous proposed model, as predicted. The difference in SPME polymeric materials shows that mass transfer in the PDMS film is faster than in the partially cross-linked PA film.
Received for review June 11, 1998. Accepted September 3, 1998. AC980642T
