A ~ I Chem. .
ism, 61, 77-83
have a very small elution difference. Higher order differentials can be used to deconvolute more complicated chromatographic situations. Although the technique has conceptual simplicity and relative ease of-implementation, it is capable of s u b s k i d spectral improvement and could even be generated in real-time during data collection. LITERATURE CITED (1) Wlngs. J. C.; Davis, J. M. Anal. Chem. 1983, 55, 418-424. (2) Biller, J. E.; Blemann, K. Anal. Lett. 1974, 7 , 515-526. (3) Dromey, R. G.; Steflk, M. J.; Rlndflelsch, T. C.; DuffleM, A. M. Anel. Chem. 1978, 48, 1368-1375.
77
(4) Knwr, F. J.; Futrell, J. H. Anal. Chem. 1979, 57, 1236-1241. (5) Sharaf, M. A.; Kowalskl, B. R. Anel. Chem. 1982, 5 4 , 1291-1296. (6) Fell, A. F.; Scott, H. P.; Gill, R.; Foffat, A. C. J . Chromafogr. 1983. 273. 3-17. (7) Ghosh, A.; Morlson, D. S.; Anderegg, R. J. J . Chem Educ. 1988, 65, A 154-A 156. ( 8 ) Talsky, G.; Mayrlng. L.; Kreuzer, H. Angew. Chem., I n t . Ed. Engl. 1987. 17, 765-799. (9) O'Haver, T. C. Anal. Chem. 1979, 51, 91A-100A.
for review June 24, 1988- Accepted October 11,
1988.
Model for a Thermoelectric Enzyme Glucose Sensor Michael J. Muehlbauer,*JEric J. Guilbeau, and Bruce C. Towe Department of Chemical, Bio and Materials Engineering, Arizona State Uniuersity, Tempe, Arizona 85287
A new calorimetric sensor that detects glucose through the change In enthalpy produced by an enzymatlc reaction Is mathsinatkallymodeled. The sensor Is a thennoplle attached to a membrane of immobllized glucose oxidase and catalase enzymes. The model predicts the temperature rise that occurs wlthln the membrane In response to the concentratlon of glucose. The dependence of the modeled temperature rlse on varlous condltlons such as the oxygen concentration, external heat and mass transfer, and enzyme loading Is sup ported wlth experhnentai data from a prototype sensor. The sensHlvlty of the sensor to glucose is characterized In terms of two quantttles: Its Initial value at low concentrations of glucose; and the range of response, specHled as the concentratlon of glucose at which the sensltlvlty devlates from Its lnltlal value by 10%. Each quantity Is represented slmply as a functlon of only two variables: an approxknate Thlele modulus and the mass transfer Blot rrumber. Each affects the two quantltles In opposing directlons. The lnltlal sensltlvity increases wlth an increase In Thlele modulus or a decrease in Blot number, whereas the range of response dimlnlshes.
INTRODUCTION Recent investigations with a t h i n - f i i thermoelectric g l u m sensor have renewed interest in calorimetric enzyme probes (I). These probes operate by measuring the temperature rise produced in response to the heat evolved during an enzyme-catalyzed reaction of a specific substrate. Previous calorimetric probes were fabricated by using thermistors as the temperature-sensing element (2-5). Although extremely sensitive, these devices suffered due to poor rejection of common mode thermal signals and self-heating. Consequently, they could only be operated in controlled flow and temperature environments. The present sensor is thermoelectric and generates a passive signal with a much improved thermal common mode rejection ratio. No environmental control has been found necessary. The new sensor employs a thin-film thermopile to measure the evolved heat. Alternate thermoelectric junctions are coated with a membrane of immobilized glucose oxidase and 'Present address: Cytogam, Inc., 3498 N. San Marcos Pl., Suite 7, Chandler, AZ 85224. 0003-2700/89/0361-0077$01.50/0
-
catalase. These enzymes catalyze the respective reactions glucose + O2 + H20 H202 gluconic acid + 80 kJ and H202
-
1/202
+
+ H20 + 100 kJ
Each mole of glucose consumed produces 180 kJ of total heat. In operation, glucose diffuses to and reacts at the enzyme surface, producing heat that generates a temperature gradient across the thermopile. The result is a Seebeck voltage produced in proportion to the concentration of glucose. The efficient operation of the glucose sensor depends on the development of a substantial temperature rise inside the enzyme membrane in response to glucose. Likewise, to provide responsiveness to glucose, the concentration below which glucose remains the limiting reactant must be a reasonably high value. The various kinetic and transport parameters that control these two effects, however, often affect them in opposing directions. The optimum design of a reliably functioning glucose sensor therefore depends on the judicious selection of these parameters. For this purpose, a mathematical model that solves the descriptive energy and mass balances to provide an indication of the temperature response to glucose is essential. Previous mathematical models concerned with describing the transport and reaction of glucose inside membranes containing glucose oxidase have disregarded the energy balance (6, 7). These models were primarily concerned with the reaction of glucose as it related to the output of polarographic or potentiometric glucose Sensors. Hence the small variation of temperature was of no concern. In a new mathematical model described here, the heat balance is solved and the temperature arising a t the extreme end of the membrane, closest to the thermoelectric sensor, is of direct consequence with regard to the sensor output. To ascertain the validity of the model, results are compared with the experimental output from prototype thermoelectric sensors. As diagrammed in Figure 1,each sensor consists of a thin-film thermopile constructed on Mylar and mounted at the tip of a 3-mm-diameter catheter. The conductors of the sensor face the inside of the catheter, and the enzyme is immobilized on the external surface. Heat generated upon placement in a flow stream containing glucose is conducted across the Mylar for detection by the thermopile. When 0 1988 American Chemical Society
ANALYTICAL CHEMISTRY. VOL. 81, NO. 1. JANUARY 1. 1989
78
THERMOPILE
3 m
-&;WIELDED
WIRE & L Y E T H I L E M TUBING
\
4POXY
r
ENZYM
catalyzed reaction is assumed to he rate-limiting, and the above rate form is not affected. This assumption is valid hecaw appreciable catalase activity is normally provided by small amounts of the enzyme. With the addition of catalase, u,deereases from 1 to 0.5, and the enthalpy change of reaction AH is altered from -80 to -180 kJ/mol. The concentration profile of each component in the membrane ia specified hy a mass halance equation that describes the mass transport and reaction of that component. In dimensionless form, these equations appear as
iRDLLED MYLAR WITH PMYURETHANE FOAM INTERIOR
Flour0 1. Thermoelectric enzyme glucose sensw. SOLUTION GLUCOSE
PRODUCTS HEAT
02
4I
I
v
I
t
ENZYME MEMBRANE HEAT
d'C;/dX
MYLAR
t
FOAM . . . . . . . .INSULATION . . . . . . . .. .. . . . .. .. . . . ... .. . . . .
2. schematic Nuslration of Ihe heat and m85 transfer amund Ihe enzyme membrane.
operated at room temprattm, t h e sensors t y p i d y generate zero voltage at zero g l u m concentration. The signal-to-noise ratio is generally greater than 25 a t 5 mM g l u m . In addition, the response time is extremely rapid, often less than 6 s.
THEORY Figure 2 illustrates a mom seetion of the enzyme membrane into which glucose and oxygen diffuse and react.. Products from the reaction diffuse in the opposite direction from the membrane into solution. Heat generated by the reaction dissipates from the membrane by two pathways: both into solution and towards the sensor. The reaction rate a t a specific location inside the membrana depends on the local concentrations of glucose and oxygen. In dimensionless form, the rate is given hy
+ l/c, + K,/CJ
R = @'/(I/$
(3)
which is the rate expreeaion for the reaction catalyzed by g l u m oxidase. Here, C, and C. are the dimensionlessg l u m and oxygen concentrations, defined as cg
cg/Cgb
co = D&o/a&gb where e;, with i = g or 0, is the concentration of either g l u m or oxygen, and the additional subscript b refers to values in the bulk solution; D;is the effectivediffusivity of a component in the membrane; and no is the ratio of stoichiometric coefficients for oxygen relative to glucose. The dimensionless constants 4'. K,. and K~ are the reaction parameters, defined
as
4' = L2V-/D&, 4 = K,/% =o
= Dd(./ad&
where L is the membrane thickness and Vu, K,, and KO represent respectively the maximum rate and the respective reaction constants for glucose and oxygen in the rate expression. The dimensionless parameter 4' is the square of an approximate Thiele modulus, which provides by its magnitude an indication of the influence of diffusion on the enzymatic reaction in the membrane. If catalase and gluoxidase are present in the membrane, catalase is always present in excess. The glucose oxidase
=R
(4)
Again, i equals either g or o to represent either glucose or oxygen. X is the dimensionless depth into the membrane, defined as
x =r/L where x is the depth into the membrane from the solution interface. The temperature profile is specifiedby the energy balance equation, describing the generation and transfer of heat within the membrane. In dimensionless form, the equation is -d2(A0)/dX2 = R
(5)
Here, the dimensionless temperature A 0 is defined 88
A0
A(Al)/DgCgb(-ahn
where A T is the difference between the membrane temperature a t a specific location and the bulk solution temperature. A is the effective membrane thermal conductivity. For the solution of eq 4 and 5, houndary conditions are specified at both the membranesolution interface and the memhrane-thermopile interface. At these interfaces, the diffusional flux of either mass or heat is equated with the external convective flux. Hence, a t the membranesolution interface dC,/dX = -Bi;(C,% - Ck)
X=0
(6)
and
X
d(A0)/dX = BiJA0J
=0
(7)
In the first expression, Si; is the mass transfer Biot number for either glucose or oxygen, defined as
Si; = k;L/D; For each component, ki representa the mass transfer meficient in the external solution, and the Biot number indicates the relative membrane-to-solution mass transfer resistance. Likewise, the heat transfer Biot number Bi. a t the memhranesolution interface
Si, = h&/A indicates the relative membrane-to-solution heat transfer resistance, in which h. is the heat transfer coefficient in the external solution. The subscripts refers to membrane-solution interface values, and h represents bulk solution values, as before. Across the membrane-thermopile interface, only the transfer of heat may occur. Hence, the appropriate boundary conditions a t this interface are dC;/dX=O
X=1
(8)
and d(AB)/dX = -Bi,(A04
X=1
(9)
A new heat transfer Biot number is defined at the thermopile interface, represented hy
ANALYTICAL CHEMISTRY, VOL. 81, NO. 1, JANUARY 1, 1989
79
Bit = h t L / h in which the effective heat transfer coefficient ht accounts for heat loss from this interface. The subscript t above refers to values at the membrane-thermopile interface. The mathematical solution to each of the differential equations described in eq 4 and 5 is simplified by the existence of algebraic relationships between the various membrane profiles. Hence, we need only solve one profile to obtain the others. Generally, eq 4 is solved for the glucose concentration profile by employing an analytical solution to the differential equation which appears in the form of a definite integral. The solution then proceeds with a simple numerical integration preceded only by a search routine to determine the appropriate limits to the integral. Details of the solution are provided in the appendix. From the mathematical solution, the most significant result for determining the performance of a thermoelectric glucose sensor is the value obtained for the temperature AT,. This value is a prediction of the actual temperature difference experienced by the thermopile in response to the concentration of glucose in the bulk solution. The Mylar support is sufficiently thin and thermally conductive so that its heat transfer resistance may be neglected. Plots of ATt versus Cgb, as displayed later, represent the full range of the sensor response to glucose. In dimensionless parameters, the glucose concentration in the bulk solution is represented nondimensionally as the inverse of Cob:
1/ Cob =
ffcPDgCgb/DoCob
(10)
The dimensionless temperature response, corresponding to ATt, is represented by l/cobmultiplied by AeV A@t/cob
= %J(ATt)/D&ob(-hH)
(11)
Here, the parameter Aet (previously specifying the dimensionless temperature of the membranethermopile interface) assumes a new meaning as the nondimensionalized glucose sensitivity, in which the response ATt is divided by the input Cgb:
= X(ATJ/DgCgb(-W
(12)
From the previous analysis, the temperature profile of the membrane, and consequently the sensitivity of the sensor to glucose, is clearly a function of the following variables:
Aet = f(@",Kg,
KO,
Big, B ~ o&, , Bit, cob)
(13)
This function is simplified by realizing that K~ can often be neglected from the denominator of the rate expression, especially at low values of the glucose concentration. Furthermore, we may assume that Big,Bi,, and Bi, are proportionally related. Also, K~ and Bit are essentially constant values. Therefore
net = f(@,Biw
cob)
(14)
EXPERIMENTAL SECTION Sensor Fabrication. Fifty-couple bismuth-antimony thermopiles were constructed by sequentially evaporating the two metals through complementary metal masks onto a 1.5-mil-thick sheet of Mylar. The reverse side of the Mylar had an adhesive surface attached to a removable backing. Completed thermopiles were clipped from the sheet in 12 X 10 mm sections, and shielded copper lead wires were connected to the thermopile contacts with Super Glue and silver print. The Mylar was then bent into cylindrical form and inserted inside a 3-mm4.d. silicone tube such that the thermopile face inward. The shielded lead wire was threaded inside a 3-mm-0.d. polyethylene tube, which was inserted at one end of the silicone tube. Polyurethane foam insulation was sprayed in through the opposite end of the tube. After the
AT
m
OOL
0
I
I
I
I
20
40
60
80
'00 IO0
DISTANCE, pm Flgure 3. Simulated profiles for glucose, oxygen, and temperature in a glucose oxidase membrane, calculated for cpb= 5 mM and c* = 1.3 mM.
foam dried, the silicone tube was removed and the seams of the Mylar were sealed with epoxy. Enzyme Immobilization. Enzyme was cross-linked to the active junctions of the sensor with albumin by using glutaraldehyde after first applying a film of segmented polyurethane. All reagents used were purchased from Sigma Chemical. One milligram of glucose oxidase (137 units/mg), 0.1 mg of catalase (11000 units/mg), and 10 mg of bovine serum albumin were dissolved in 50 FL of distilled water. After addition of 1 p L of 25% glutaraldehyde and rapid mixing, the solution was applied to the sensor with a paint brush. The resultant gel was allowed to set for 10 min, and the completed sensor was stored at 4 O C in 0.5% sodium benzoate. Sensor Testing. The prototype sensors were tested in a flow environment at room temperature without any additional thermostatic control. Each sensor was fitted inside a plastic Y-connector with the catheter and lead wires emerging from one of the two split legs of the connector. The other legs of the connector were connected between 0.313-in.-i.d. Tygon tubing so that the sensor tip was positioned inside. Through this tubing, 0.01 M phosphate buffer, pH 7.4,was delivered at 1.2 L/min by a Masterflex pump that drew on a 20-L tank. Driven by the same pump drive, a Fluid Metering, Inc., (FMI) pump injected a 10% glucose solution into the flow stream. Adjustment of the FMI pump stroke volume provided variation of the glucose concentration from 0.25 to 20 mM. The external leads of the sensor were connected to a microvoltmeterand recorder, with low-pass fdtering of 10 Hz. RESULTS Membrane Profiles. Representative membrane profiles for glucose, oxygen, and temperature are illustrated in Figure 3. Here, the bulk solution has a glucose concentration of 5 mM and is saturated with oxygen for an equilibrium value of 1.3 mM. The mass transfer resistance presented by the bulk solution, as represented by finite values for the mass transfer coefficients, reduces the concentration of glucose and oxygen at the membrane-solution interface to respective values of 2.7 and 0.79 mM. The concentration is reduced further at greater distances inside the membrane by virtue of the enzymatic reaction. Even though glucose is present in almost 4-fold excess in the bulk solution, oxygen is not completely consumed,as a result of its greater difhivity. Indeed, at the furthermost end of the membrane, oxygen achieves a greater concentration (0.46 mM) than glucose (0.45mM). The temperature profile in the figure is almost uniform. This is due to a small value for Bi,, which reflects the fact that the
80
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989
Table I. Values for Model Parameters L
0.01 cm 1.3 X IO” mol/cm3 0.5 -180,000J/ mol 120 x 10” mol/cm3 0.84 X 10“ mol/cms 1.9 x 10” cmz/s 6.5 X 10“ cm2/s 1.7 X W/(cm K) 5.4 x IO4 cm/s 12.4 X lo-’ cm/s 0.08W/(cm2K)
cob a0
AH K B
KO
D B
DO
x
k, k0
h,
1
2.51
Ob=1.31mM 2.0
0
-
E
3 la E
1.5
IO
15
20
Figure 5. Illustration of the dependence of the sensor output on enzyme loading, with experimental data indicated as points.
0
w’
5
GLUCOSE CONC., rnM
-
10-
a
5I-
0.5-
00 0
5
10
15
20
25
GLUCOSE CONC., mM Figure 4. Illustration of the dependence of the sensor output on oxygen concentratbn, with experimental data indicated as points: V, = IO-& (mot/s)/cm3.
membrane thermal resistance is small compared to the solution resistance. The temperature at the membrane-thermopile interface represents the temperature difference sensed by the thermopile and is proportional to the bulk glucose concentration. Comparison of the Model and Experimental Data. The values listed in Table I represent the best estimates for the model parameters. Unless stated otherwise, these were the values used in the comparison between the model results and the experimental data obtained from the prototype sensors. The value for V-, not listed in the table, varied from sensor to sensor, dependent on the quantity of active enzyme immobilized. Individual values were derived from activity assays performed on the sensors. The heat transfer coefficient h, at the membranesensor interface was generally given a finite value comparable to that of h,. Figures 4, 5, and 6 illustrate the qualitative agreement between the model and experimental results. In these f i i e s , the sensor output is represented by the individual points. Generally measured in microvolts,the output is converted to a temperature difference in millidegrees Celsius by dividing by the value of the thermoelectric emf, typically about 4 pV/m°C. These experimental points are compared with the model prediction for AT, as a function of concentration, represented as a smooth line. Figure 4 illustrates the dependence of the sensor output on the oxygen concentration. The upper curve is the response in an oxygen-saturatedsolution with an oxygen concentration of 1.3 m M the lower curve is for an air-saturated solution with a concentration of 0.27 mM. Both curves have the same initial sensitivity (0.24m°C/mM), but the range of response is extended for the oxygen-saturated case. The ultimate response
oov 0
I
1
I
5
IO
15
I
20
GLUCOSE CONC., mM Figure 6. Illustration of the dependence of the sensor output on the Reynolds number, with experimental data indicated as points: V, = 2.5 X 10“ (mol/s)/cm3. At the lower Reynolds number, ail external transport coefficients are reduced by a factor of 0.65.
in each case is directly proportional to the concentration of oxygen. The dependence of the response on enzyme loading is indicated in Figure 5 where V- increases in moving from the lowermost to the uppermost curve. The sensitivity increases from 0.04 to 0.11 to 0.16 m°C/mM as enzyme loading is increased. These values are comparable to those measured with thermistor probes (5). Unfortunately, as V,, increases, the range of response is reduced. Figure 6 illustrates the result of adjusting the flow rate delivered past the sensor. Here, the Reynolds number Re is defined for flow through a circular annulus. By increasing the Reynolds number, both heat and mass transfer in the external solution are increased. Hence, although heat generation is increased by the supply of an additional flux of readants, heat dissipation is also increased. The overall effect on the sensor output is a decrease in ATv In Figure 6, the sensitivity decreases from 0.20 to 0.14 m”C/mM as the Reynolds number increases. Conversely,the range of linear response increases from 11 to 12.5 mM. Dimensionless Curves for Sensor Characteristics. As the previous three figures indicate, the performance of an individual sensor is described by two important characteristics: ita initial sensitivity to glucose and ita range of linear response. The dimensionless plots of Figures 7 and 8 indicate the effect
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989
81
effects an increase in sensitivity and a concomitant decrease in range. Increasing Big reduces the sensitivity while simultaneously increasing the range.
v)
8
01
10
I
2
L
IO0
Vmax / D g K g
A€)/ plotted versus the square of the approximate Thiele modulus for various values of the Biot number for glucose, Big = k& IDv For this plot, Bi, = 2/3Big; B!, = '/@Ig; 61, = 0;KO = 0.048 and l / K g = 0. Flguro 7. Dimensionless initial glucose sensitivity
-'"I
I
n
L L V r n d Dg K g
Figure 8. Dimensionless range of linear response (l/C,,,,k, plotted versus the square of the approximate Thlele modulus for various vaiues of the Blot number for glucose, Big = k g L / D g . Conditions are the same as in Figure 7.
that the various kinetic and transport properties have on these characteristics. As previously stated, the dimensionless temperature Aet at the membrane-thermopile interface also represents a dimensionless glucose sensitivity. At low values of the dimensionless glucose concentration 1/Cob, the sensitivity Ahe, maintains a constant value. Alternately stated, the dimensionless temperature response Aet/C,b is initially linear with 1/c,b. As 1/c,b continues to increase, however, a point is reached where the sensitivity begins to decline. Ultimately, AOt and 1/c& become inversely proportional, and the response A8,/Cob is constant and unaffected by further increases in glucose concentration. The value of A e t at low values of 1/C,b is defined as the initial sensitivity and is represented by Ae;. The value for l/Cob a t which A€+ is reduced by 10% of this initial value is defined as the range of linear response, represented by (1/ C&)Enea. These dimensionless characteristics are plotted, respectively, in Figures 7 and 8 as functions of the square of the Thiele modulus d2 and with the Biot number Big as a parameter. In these plots, K, = 0.048 and Bit = 0. The plots indicate the inverse relationship between the initial sensitivity and the range of linear response. Generally, increasing d2
DISCUSSION The plots of Figures 7 and 8 provide a useful guide for the design of a calorimetric glucose sensor. Generally, one wants to maximize the sensitivity of the sensor while maintaining an adequate range of response. Figure 7 reveals that the sensitivity is maximized by increasing the square of the Thiele modulus d2. Typically, this is achieved by immobilizing a higher density of enzyme in the membrane, thereby increasing Vmm. The effect of increased enzyme loading on sensitivity is verified experimentally in Figure 5 and is clearly a result of enhanced heat generation produced by a greater rate of reaction. As enzyme loading is increased without limit, the reaction in the membrane becomes limited by the transport in the external solution. Under this circumstance, the sensitivity approaches the ultimate value of Big/Bi,, or 6 in Figure 7 . Also evident from Figure 7 is that the sensitivity increases by reducing the Biot number Bi,. Generally, this is effected through a reduction in the Reynolds number, which reduces the transport coefficients in the external solution. As assumed in developing the functionality of eq 14, the reduction in Big coincides with a proportional reduction in both Bi, and Bi,. The effect of reducing the Reynolds number is therefore 2-fold. Whereas external mass transfer is diminished, thereby reducing heat generation in the membrane, heat dissipation is also reduced due to a decrease in external heat transfer. The two effects oppose one another-the first causes a reduction in sensitivity; the latter, an increase. However, the overall effect is an increase in sensitivity as predicted in Figure 7 and supported experimentally in Figure 6. The apparent reason lies in the effect that reducing the Reynolds number produces on the overall transport outside of and within the membrane. As the Reynolds number is reduced, each of the Biot numbers-Big, Bi,, and Big-is reduced. In the external solution, therefore, the mass transfer resistance (to both glucose and oxygen) and the heat transfer resistance are each increased, in direct proportion to one another. Within the membrane, however, the transport resistance to both mass and heat remains constant. Now, consider that the mass transfer Biot numbers, Big and Bi,, are respectively 6 and 4 times as large as the heat transfer Biot number Bi,. Hence as each is lowered, the ratio of membrane-to-solution resistance for mass transfer remains substantially more significant than the corresponding ratio for heat transfer. Thus although the mass transfer resistance of the external solution is increased, its overall value is less affected because of the continued significance of the membrane resistance. For heat transfer, the reduction in overall resistance is more greatly affected. Therefore as each Biot number is reduced, heat dissipation is reduced more substantially than heat generation, and the sensitivity increases. Figure 8 reveals that in order to enhance the range of linear response, one must either reduce d2or increase Big, in direct opposition to the requirements for increasing the sensitivity. In Figure 5, the reduction of d2is effected by lowering V,, by means of reduced enzyme loading and results in an increase in the range of response, as predicted. In Figure 6, the Biot number increases with increasing Reynolds number and also produces the expected increase in the range of response. Indeed as the enzyme loading is reduced or the Reynolds number increased, the reaction within the membrane approaches more closely reaction-controlled conditions. Under this circumstance, the rate of glucose transport begins to exceed ita rate of chemical reaction. Hence, the rate of glucose consumption is more closely guided by the rate expression in
82
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989
eq 7 and is less disguised by transport limitations. In the limit, as the reaction becomes completely kinetically controlled, the range of response approaches its highest value. For this occurrence, the range of response (if K~ is neglected) is given by (1/ Cob)linear = 1/ (9Ko)
(15)
In Figure 8, for which K~ = 0.048, (l/Cob)linear = 2.3 as $2 approaches zero. In contrast, the membrane reaction becomes more diffusion-limited as the enzyme loading is increased or the Reynolds number is reduced. The limit appears when the reaction of glucose and oxygen occurs instantaneously on the membrane-solution interface. In this situation, the range of response is established when oxygen is reduced to a concentration of zero at the interface. The range of response then becomes equal to Bio/Bir In Figure 8, as 42 approaches infinity, the range of linear response approaches a value of 0.67. The effect that increasing the membrane thickness L has on the sensitivity and the range of linear response is not immediately apparent. Both C#J~and Big are increased and exert opposing influences. A 10-fold increase in thickness, however, increases Big by the same factor, whereas $2 is increased 100-fold. Hence, as may be perceived in Figures 7 and 8, the initial sensitivity increases and the range of linear response declines. GLOSSARY Bi mass transfer Biot number heat transfer Biot number at membrane-solution Bi, interface heat transfer Biot number at membrane-thermoBit pile interface C concentration, mol/cm3 C dimensionless concentration D membrane diffusion coefficient, cm2/s function f AH enthalpy change of reaction, J/mol h heat transfer coefficient, W/(cm2 K) k mass transfer coefficient, cm/s K reaction constant, mol/cm3 L membrane thickness, cm R dimensionless reaction rate per volume Re Reynolds number for an annulus AT temperature difference, K maximum reaction rate per volume, (mol/s)/cm3 V,, X membrane depth, cm X dimensionless membrane depth
GREEK LETTERS stoichiometric ratio P dummy variable 4 approximate Thiele modulus Y dummy variable K dimensionless reaction constant x membrane effective thermal conductivity, W/(cm K) Ae dimensionless temperature difference
ground that inspired the mathematical model.
APPENDIX To determine the various membrane profiles for each component and the temperature, an alternative to solving each of the differential equations described in eq 4 and 5 separately is to derive appropriate algebraic relationships. In this case, only one profile-that of glucose-needs to be solved to determine the others. The relationship that exists between the glucose and oxygen profiles is found to be Cox = Cob - (Big/Bi0)(1- Cgs)- (C, - Cgx) (16) in which the subscript x refers to the value a t a specific membrane depth. The relationship between the temperature and glucose is A@, = A0, + (C,, - Cgx) (A8t - A0, + C, - C,)X (17) C, and C, are the concentrations of glucose at the solution and thermopile interfaces, respectively. The corresponding interface temperatures A8, and AOt are Bi,(c, + Big(1- CgJ AOt = (18)
+
c,, Bi, + Bit + Bi,Bit
and
Bi,(l A@, =
+ Bi,)(l
- Bit(C,, - C,) Bi, + Bit + Bi,Bi, - C,)
(19)
The membrane profile for glucose is now solved by using a method suggested by Bischoff (8). Here, eq 4 is transformed into an implicit expression for C , in terms of the depth X
The correct limits to the above integral are found from a computer search routine that finds the unique set of values for C,, and C , which satisfy the following two conditions. First, the boundary condition at the solution interface must be satisfied:
Second, X must approach unity when C , =C , .
ff
SUBSCRIPTS b bulk solution g glucose linear linear range 0 oxygen S membrane-solution interface t membrane-thermopile interface X membrane depth x SUPERSCRIPTS i initial ACKNOWLEDGMENT The corresponding author is indebted to Tim Cale at Arizona State University for providing the theoretical back-
To aid in the search for C , and C, a range of candidate values is established within which the search is conducted. The criterion that oxygen clearly cannot assume a negative value of concentration delineates the range. The range of candidate values for C, is therefore
and the range for C, is max(0, C,, - Cos)
C,
< C,,
(24)
For the sake of brevity, the actual computer code is not listed but is available from the corresponding author upon request.
LITERATURE CITED Soc.Artif. Intern. Organs 1987. X X X I I I , 329-335. (2) Weaver, J. C.; Cooney, C. L.; Fulton, S. P.; Schuler, P.: Tannenbaum, S. R. Biochim. Biophys. Acta 1976, 452, 285-291. (1) Gullbeau, E. J.; Towe, B. C.; Muehlbauer, M. J. Trans. Am.
Anal. Chem. 1989, 61, 83-85 (3) Tran-Mlnh, C.; Vallln, D. Anal. Chem. 1978, 5 0 , 1874-1878. (4) Rich. S.; Ianniello, R. M.; Jespersen, N. D. Anal. Chem. 1979. 5 1 , 204-206. (5) Fulton, S. P.; Cooney. C. L.; Weaver, J. c. Anal. C h m . 1980, 5 2 , 505-508. (6) Leypoklt. J.; &ugh, D. Anal. Chem. 1984, 5 8 , 2896-2904. (7) Cares, S.; Petelenz, D.; Janata, J. Ana/. Chem, 1985, 5 7 , 1920-1923.
83
(8) Bischoff, K. AIChEJ. 1965, 1 1 , 351-355.
RECEIVED for review November 10,1987. Resubmitted March 29, 1988. Accepted October 11, 1988. This work was performed with SUPPOd by Pants from the Whitaker Foundation and the Arizona Disease Control Research Commission.
CORRESPONDENCE High-Resolution Ion Partitioning Technique by Phase-Specific Ion Excitation for Fourier Transform Ion Cyclotron Resonance Sir: The intrinsic versatility of Fourier transform ion cyclotron resonance (FT-ICR) for analytical studies has been widely demonstrated. The utility of ion traps arises from the ability to manipulate ions that are stored in the trap ( I ) . Ion manipulation is accomplished in FT-ICR by ion ejection (e.g., ion selection for chemical studies (2) and ejection of high abundance ions for enhancement of dynamic range ( 3 , 4 ) ) .Ion ejection is achieved by accelerating unwanted ions (by resonant radio frequency (rf) excitation) until their radii exceed the confines of the ICR cell. During a prolonged ejection sweep, signal loss can occur due to ion-neutral collisions or ion evaporation (5),therefore rapid sweeping or “chirp” excitation is utilized. Because the excitation magnitude decreases as the inverse of the sweep time (6),rapid sweep rates often lead to a loss of selectivity with respect to ion excitation (3). Ions of interest are inadvertently accelerated resulting in (i) ejection of reactant ions and (ii) complication of ion-molecule reaction studies. It has also been demonstrated that the phase relationship of translationally hot ions complicates ion detection (7). Owing to the importance of ion selection in FT-ICR, there has been much work in high-resolution ion isolation (8-10). Marshall and co-workers have demonstrated that the ejection sequence can be performed rapidly while maintaining high selectivity by utilizing an inverse Fourier transform (11). Although this procedure significantly reduces the problems incurred, there can be undefined excitation of ions of interest as well as incomplete ejection (12). We recently demonstrated that ions can be selectively partitioned (in terms of m / z and kinetic energy) in a twosection cell (13). Unwanted ions are excited to radii that exceed the dimensions of the conductance limit orifice and are selectively discriminated against upon partitioning from the source to the analyzer regions of the two section cell. Because the ions are not translationally excited to large radii, ion selection is highly selective. Because ion cyclotron frequency shifts occur as ions reach large orbital radii, any method of ion ejection loses selectivity. Thus, even perfectly selective excitation sources are limited in resolution by the ICR frequency differences between the center and edge of the ion cell. The two-section ion cell partitioning experiment is potentially advantageous because the ions do not have to be totally ejected in order to be removed. Although the excitation magnitude is substantially decreased by accelerating unwanted ions to radii which need only exceed the dimensions of the conductance limit, tailing of an excitation sweep can still result in inadvertent excitation of the ion of interest. We describe here a technique for ion
selection that combines phase-specific excitation and ion discrimination on partitioning in a two-section FT-ICR cell. This method greatly enhances ion isolation by applying the relationship of the excitation phase angle to ion acceleration. The radial velocity of ions is modulated by phase-specific excitation. By control of the radial velocity (i.e., cyclotron radius), selected ions can be manipulated to produce high ion selectivity upon partitioning.
EXPERIMENTAL SECTION All experiments were performed on a prototype Nicolet Analytical Instruments FTMS-1000 spectrometer equipped with a 3-T superconducting magnet. The vacuum system has been modified to accommodate a two-sectioncell. The two-section cell consists of two cubic cells (3.81 X 3.81 X 3.81 cm) mounted collinearly along the central axis of the magnetic field. The two cells share a common trap plate that also serves as a conductance limit for the differential pumping system. The orifice in the conductance limit has a radius of 2 mm. The vacuum in both sections of the differentially pumped system is maintained by oil diffusion pumps. Background pressures for both sections of Torr or less. Gaseous reagents the vacuum system were 1 X were admitted to the vacuum system by variable leak valves (Varian Series 951). Ionization was performed by electron impact (50-eV electrons). Detection of the ions in either the source or the analyzer region of the cell was performed by electronically switching the rf excite pulses between the cell regions. RESULTS AND DISCUSSION To illustrate the capabilities of this technique, nominal isobars of Fe2(C0)3+and Fe(CO)S+ ( m / z 195.8546 and 195.9095, respectively) were isolated from one another. Separation of these ions requires a mass resolution of ca.4000. The corresponding cyclotron frequencies (in a 3.028-T field) for these ions are 243.0847 and 243.0165 kHz yielding a frequency difference of ca. 50 Hz. Contained in Figure 1A is a mass spectrum of Fe(C0)5. The peak at m / z 196 corresponds to the isobar of Fe2(C0)3+and Fe(C0)5+. This system was selected because isolation of one of these ions for chemical studies by a simple “chirp” ejection sweep is difficult without inadvertent excitation of the ion of interest. By use of phase-specific ion excitation it is possible to select one of the isobars for either enhancement of dynamic range or chemical studies (Figure 1B). Ion selection is better illustrated when a narrow mass range is used such that the selection of specific peaks can be observed. Contained in Figure 2 is the mass spectrum of Fez(CO),+ and Fe(CO)5+with isotopes at m / z 194,196, and 197. The spectrum of isolated Fez(C0)3+contained in Figure 3 was obtained by using phase-specific ion selection. An initial
0003-2700/89/0361-0083$01.50/00 1988 American Chemical Society
