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Anal. Chem. 1991, 63, 2460-2465
Theoretical Analysis of Probe Dynamics in Flow Injection/Membrane Introduction Mass Spectrometry Gow-Jen Tsai, Glen D. Austin, Mei J. Syu, and George T. Tsao
Laboratory of Renewable Resources Engineering, Purdue University, West Lafayette, Indiana 47907 Mark J. Hayward, Tapio Kotiaho,l and R. Graham Cooks*
Department of Chemistry, Purdue University, West Lafayette, Indiana 47907
The dynamic response of the membrane probe in flow injection/membrane introduction mass spectrometry has been studied theoretkaily. A mathematical model is formulated to obtain analyte concentratton profiles in the flow chamber and indde the membrane and to calculate the analyte flux across the membrane. For ideal input perturbations such as block and step inputs, analytkai soluHons are p r o w for nonlinear input perturbations an efficient and robust computational procedure for obtalning numerical solutions is presented to account for mixing of the analyte sample in the flow conduit before entering the flow chamber. The effect of soma system parameters and operational variables on the dynamic behavior of the probe is investigated. This information should help In the design of new membrane probes and in the selection of new membranes.
This work presents a theoretical analysis of the dynamics of the operation of the membrane probe in the FIA/MIMS system. Although theoretical analyses of membrane transport and flow injection sampling has been intensively studied in the past (16-19), these topics have been dealt with separately and the combination of both phenomena in the FIA/MIMS system has not yet been analyzed. Therefore, this paper provides analytical tools to fill this gap and so provide a better understanding of the time-dependent phenomena associated with use of this type of probe. A mathematical model has been formulated and solved analytically for block and step inputs (defined in Figure 1) and numerically for any arbitrary inputs. The method of Laplace transforms was employed to obtain the analytical solution, and the orthogonal collocation (20) and Gear’s stiff method (21) were used to obtain the numerical solutions. The effect of each system parameter on the dynamic responses is investigated and discussed.
INTRODUCTION Membrane introduction mass spectrometry (MIMS) has received considerable attention recently for identification and quantification of organic compounds in aqueous media ( 1 4 ) . The technique is based on the preferential pervaporation of analytes as opposed to the solvent across the membrane. When the membrane probe is inserted directly into the mass spectrometer ion source, better performance is achieved in terms of response time and sensitivity compared to membranes that are located at a greater distance (7-11). For process monitoring and control, rapid and continuous measurements of the concentrations of target compounds in the analyte solution are essential. The use of MIMS with flow injection analysis (FIA) sampling is appropriate to achieve this requirement. This technique has been demonstrated successfully for on-line monitoring and control of fermentation processes for the production of optically active isomers of 2,3-butanediol (12-15). The membrane probe has two main parts; a flow chamber and the membrane (Figure 1). The flow chamber allows direct contact of the analyte with the membrane, while the membrane itself allows pervaporation, a process which includes adsorption of analyte onto the membrane surface, diffusion through its body, and release from the inner surface into vacuum. The major concerns in the selection of polymeric materials for the membrane probe are achievement of the maximum flux across the membrane and the time required to complete the analysis of one sample. The first determines the response sensitivity and the second determines the sampling frequency. The desired membrane will be the one that gives a high sampling frequency with high sensitivity to all the analytes, while simultaneously rejecting solvent.
THEORETICAL CONSIDERATIONS Figure 1 shows a simplified schematic diagram of the membrane probe. The analyte solution is pumped through the flow chamber, in which the analyte is dissolved, and analyte diffuses through the membrane. Obviously, the concentration of analyte in the flow chamber largely affects its mass flux across the membrane, especially when the sample volume is comparable to the volume of the chamber and mixing in the chamber is negligible. Therefore, we also consider the variation of analyte concentration in the flow chamber in obtaining results from the model. The model shown below is the simplest one, which assumes that the fluid inside the chamber is well mixed and the analyte transport across the membrane is one-dimensional. Also, the f i model is employed to account for mass transfer between the fluid phase and the membrane phase. If we further assume that the effective diffusivity (0) in the membrane and the partition coefficient (HIbetween solution and the membrane surface are constant over time, then the governing equations can be expressed as follows:
* Corresponding author.
On leave from the Technical Research Center of Finland, Chemical Laboratory, Biologinkuja 7, 02150 Espoo, Finland. 0003-2700/9 1/0363-2460$02.50/0
-dC- - u(Ci - C) - Kea, dt
-ac, = D - a2cm at ax2 with initial and boundary conditions t = O
c = o cm=o
(3)
x = L cm=o (5) where C = concentration of analyte in the chamber, C, = concentration of analyte inside the membrane, Ci = concen0 1991 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 63, NO. 21, NOVEMBER 1, 1991
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Laplace domain, S, through the following transformation:
Input Perturbations
C(S) = $me-st C(t) dt'
(10)
C,(x,S) = Jme-st Cm(x,t) dt
(11)
0
(c) Arbitrary
\
I
/
u
r\
0
The transformed equations are then solved simultaneously to obtain the following Laplace domain solution: - e-bq(
-(e*S G o
S C(S) =
-1
h 1-
(F K s / H tanh
tanh
6
6
L +l)(S
L + 1)
+ V) + K,u,
1-
L
Flgure 1. Schematic diagram of a membrane probe.
tration of analyte in the inlet, H = partition coefficient, Cm/C, K , = interphase mass-transfer coefficient, cm/s, a, = ratio of total membrane surface to chamber volume, cm-l, D = effective diffusivity of sample solute in the membrane, cm2/s, t = response time, s, x = spatial coordinate, cm, L = thickness of membrane, cm, u = ratio of inlet liquid flow rate to chamber volume, s-l. Note that eq 5 holds only when the downstream side of the membrane is under vacuum, which is true for the system in which we are interested. The sample concentration in the inlet can be any arbitrary input perturbation, i.e. Ci(t) = cjof(t) (6) in which f(t) is the normalized input perturbation. The function f(t), which accounts for mixing of sample solution between the sample injection port and the flow chamber, is experimentallymeasurable. For ideal input perturbations such as the step or the block function, eq 6 becomes in which u, and ub are unit step functions starting from t = a and t = b, respectively. When b approaches infinity, the input perturbation becomes a step change. Equations 1and 2 give concentrationprofiles of analyte in the flow chamber and at each position of the membrane. However, the quantity in which we are interested is the total analyte flux at the downstream side of the membrane, i.e.
,.
(g
+ l ) ( S + v) + Ksa,
tanh $ L
Then, the time domain solution is obtained by inverting eqs 12 and 13 using the inversion theorem for Laplace transform and Cauchy's residue theorem. Since eqs 12 and 13 are analytic in the S domain except the poles at S = 0 and S, = -(D/L2)Xm2(n = 1,2, ..., =J),the analytical solution is simply the summation of the residues evaluated at these poles. Note that A, is the eigenvalue at pole S,, which was obtained from the following characteristic equation:
g
(S, + v)( A, cot A, - 1 = -
1
S,
+ 1) + Ksa, I
(14)
+ v + K,a,
Notice that eq 14 was derived by letting the denominator of eq 12 or 13 equal zero. After complicated mathematical manipulations, the time domain solution is obtained as follows: 05 t
