Anal. Chem. 1997, 69, 5182-5185
Nonintrusively Measured Temperature Distributions as Evidence for Free Convection in Immunoassay Incubation Tom Beumer*,† and Brenda Timmerman‡
Organon Teknika BV, P.O. Box 84, 5280 AB Boxtel, The Netherlands, and Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands
Immunoassay incubation is classically performed at elevated temperatures to speed up reaction processes. This acceleration has long been assumed to be due to an increased association constant between reactants. Using an interferometric method to visualize temperature differences inside a small reaction cuvet, we demonstrate that the temperature profiles inside the liquid cannot be caused by conduction but only by thermal convection. Numerical simulations further support this experimental evidence. This paper conclusively demonstrates that thermally induced mass transport occurs in immunoassay incubation. Current data show that earlier estimates of heat transport coefficients in such incubations have been probably overestimated by a factor 3.
Ever since the introduction of immunoassays, an incubation is performed at elevated temperature.1 This heating does provide significantly higher signal levels. Recently we demonstrated2 that the increase is only to a limited extent due to the assumed improved association constant between reactants. Far more important is the thermal mixing that enhances the diffusion-limited reaction at low concentration levels. Even though our first report included direct measurements of the mixing kinetics of reactants, the temperature distribution data were of limited quality: the geometry was scaled up strongly and temperatures were recorded intrusively using thermocouples and on the symmetry axis only. Here, we report temperature difference distributions inside a cuvet that has similar dimensions and material properties as those used in ELISA immunoassays. The measurements are obtained nonintrusively using interferometry. This is reported to provide a more solid physical support behind the reality of mass and energy transport in immunoassay incubation and to obtain better estimates on the typical dimensions of these processes.
tion. Whichever dominates is governed by two characteristic dimensionless numbers: the Prandtl number (Pr) and the Grashof number (Gr). The Prandtl number is the ratio of mass transport by forced and free convection (Pr ) ηcp/R), and the Grashof number relates the buoyancy forces to viscous forces (Gr ) gL3∆T/ν2). If conduction is poor, Pr increases and local expansion is followed by mass transport with simultaneous heat transport. For water, the Pr is in the order of 5-10 (7 at 293 K), the Grashof number equals 10000∆T, or under typical conditions, Gr varies from 5000 to 30000. Above a critical Grashof number, temperature exchange no longer takes place solely by conduction but also by convection. 2. Interferometric Temperature Analysis. Interferometry is a classic and well-documented tool in, for example, thermal analysis.3,9 In short, the method works as follows: The velocity of light depends on the medium it passes through. When a collimated, monochromatic light beam traverses a medium with constant refractive index in the plane perpendicular to the propagation direction, the outcoming beam has a plane wave front. If, however, the refractive index is inhomogeneous, light rays will leave the medium at different phases. Therefore, it produces an integral projection of the refractive index field of the medium. If refraction is weak, then rays remain straight lines and the phase difference ∆φ(x,y) in the wave front directly reflects the integrated refractive index differences as3
∆φ(x,y) )
2π λ
∫ [n(x,y,z) - n ] dz L
0
∞
Combining such a disturbed light beam with one that has passed through a medium with constant refractive index results in an interference pattern with intensity I:
(x,y) ) Ibias(x,y) + Imod(x,y) cos∆φ(x,y) From this equation with three unknowns, the phase delay may
THEORY 1. Heating a Liquid. When liquid is heated from below, heat is transported by two mechanisms:10 conduction and convec†
Organon Teknika. Delft University of Technology. (1) Butler, J. F. Ed. Immunochemistry of solid phase immunoassay; CRC Press: Boca Raton, FL, 1991. (2) Beumer, T.; Haarbosch, P.; Carpay, W. Anal. Chem. 1996, 68, 1375-1380. (3) C. M. Vest, Holographic Interferometry; John Wiley & Sons: New York, 1979. ‡
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(4) T. A. W. M.; Lanen, P. G.; Bakker, P.; Bryanston-Cross, J. Exp. Fluids, 1992, 13, 56-62. (5) B.; Timmerman, D. Watt, Proc. 2nd Symp. Transit. Turbul. Compress. Flows, Hilton Head, 1995, 224, 169-176. (6) O. H.; Nestor, H. N. Olsen, SIAM Rev. 1960, 2 (3), 200-207. (7) B. H.; Timmerman, D. W. Watt, Meas. Sci. Technol. 1995, 6, 1270-1277. (8) B. H.; Timmerman, D. W. Watt, Proc. Optical Techniques in Fluid, Thermal, and Combustion Flow, San Diego CA, SPIE 1995, 2546, 287-296. (9) T. D.; Upton, D. W. Watt, Exp. Fluids, 1993, 14, 271-276. (10) J. Turner, Buoyance effects in fluids; Cambridge University Press: New York, 1973. S0003-2700(97)00661-6 CCC: $14.00
© 1997 American Chemical Society
be solved by generating at least three independent interferograms. For this, the optical path length in the undisturbed reference beam is modified. This is called phase stepping.4 Here, four interferograms are generated at different but constant phase steps. The field-induced phase difference is then calculated using the method of Carre´; that is followed by phase-unwrapping to obtain a continuous phase map. Knowing this phase information, the integrated variations of the refractive index in the medium are known.4,5,14 As the refractive index of a medium is closely related to its temperature (the average value for dn/dT equals -1.25 × 10-4/K in the 293-314 K interval3), the phase difference caused by heating of water is given by
∆φ(x,y) ) 1.25 × 10-4
2π λ
∫ ∆T(x,y) dz L
0
Thus, the integrated temperature distribution may be found from the measured phase map. MATERIALS AND METHODS Interference patterns of a disturbed and an undisturbed light beam, which only differ due to the field of interest, are generated using a holographic interferometer.3 The undisturbed light beam of a continuous-wave HeNe laser is recorded on a holographic plate by a reference beam. After photographic processing, the undisturbed beam is recreated by illuminating the holographic plate again with the reference beam. Next, another beam is sent through the same optics, but now including the field of interest. The reconstructed undisturbed beam and the real-time disturbed beam are then combined to form an interference pattern which may be digitized using a CCD camera and frame-grabber and stored in computer memory for further processing. To allow for the use of the proposed data analysis method, the process under study has to be slow compared to the time needed to record all four phase-stepped interferograms. Polystyrene and cylindrically shaped wells of a microtiter plate demonstrated optically inferior side wall properties and could not be used for these experiments. Instead, a 1 × 1 cm2 square cuvet made of quartz (Hellma) was used. Its dimensions closely match those of a microtiter plate well (diameter, 0.7 cm). Typically the liquid’s aspect ratio was set at ∼0.6. The bottom of the waterfilled cuvet was heated using a microplate heating device set at 310 K (Organon Teknika, Turnhout, Belgium). Before heating started, a phase map was obtained of the phase difference caused by the thermally equilibrated water-filled cuvet. This phase map was then subtracted from the phase maps that were subsequently obtained for heated situations in order to to find the phase delays due to the heated fields. To independently monitor the behavior of the heating device, temperature-time series were recorded using a Keithley 741 thermocouple scanner2 that was operated from a PC for data recording. Data were processed using Symphony spreadsheet software. (11) Handbook of Chemistry and Physics, 75th ed.; Lide, J., Ed.; CRC Press: Boca Raton, FL, 1994. (12) Numerical simulations of heat transfer and fluid flow on a personal computer, Kotake, S., Hijikata, K., Eds.; Transport processes in engineering 3. Elsevier: Amsterdam, 1993. (13) T. D.; Upton, D. W. Watt, Int.J. Heat Mass Transfer 1997, 40, 2679-2690. (14) B. H. Timmerman, Holographic interferometric tomography for unsteady compressible flows Ph.D. Thesis, Delft University of Technology, 1997.
Figure 1. Total phase difference in a cuvet as occurring in the raw interferograms and normalized by 2π as a function of time during cooling down phase. The drawn curve has been fitted to an exponential with a time constant of 2000s. X-axis, time (s); Y-axis, normalized phase difference.
As a theoretical reference, we used a software package CAVITYFL12 with which we numerically simulated the expected temperature and mass transport variations under the experimental conditions.
RESULTS AND DISCUSSION Numerical simulations of temperature and flow distributions at relevant Grashof and Prandtl values (5000-30000 and 0.5-10, respectively) predicted the following: that the temperature distribution tends to homogenize with a flame in the center, that the integral initially shows a steep decline close to the bottom, that, in a later stage, a second maximum close to the liquid surface appears, and that the effect of a slightly varying aspect ratio is minimal. The thermocouple data on the average liquid temperature in the cuvet showed the following: (1) The liquid temperature increases during heating at a constant rate of 0.055 K/s. This prohibits the use of our four-image interferometry method as we cannot assume thermally equivalent images within the 0.5 min time period for frame grabbing. (2) After the power is turned off, the element and the liquid in the cuvet cools exponentially with time, at a time constant of 1800 ( 100 s. (3) If the cuvet is removed from the heating element during cooling, the liquid cools much faster at a time constant of only 300 ( 20 s. Apparently, the average liquid temperature follows the heating element’s temperature. Considering item 2, we may assume that the temperature of the liquid varies only slowly during the cooling down period. A nice and convincing way to show that our interferograms do contain information on the temperature distribution in the liquid is shown in Figure 1, where the total temperature difference in the liquid phase decreases proportionally with the temperature difference between heating element and air. A second and conclusive argument to support the use of data from the cooling period instead of the heating period is that in both situations the direction of the thermal gradient proved to be the same, or in other words, in both cases the thermal conditions were caused by the temperature difference between bottom and top of the liquid. Analytical Chemistry, Vol. 69, No. 24, December 15, 1997
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The question that now still remains is whether or not the temperature distribution is caused by convection instead of conduction. Let us, for sake of argument, assume conduction the sole cause. In that situation, the temperature gradient dT/dy in the liquid would be constant and our interferograms would show equidistantly separated fringes. Quantitatively an even better argument can be obtained from our data: Typically in our experiments tw ) 5 mm and tg ) 1 mm; bw ) 0.6 W/m‚K and bg ) 0.9 W/m‚K.11 From these data, the maximum temperature difference in the water layer should be in the order of
[Te - Ta] )
[ ] [ ]
1 [T - Te] ) tg βw t 1+ tw βg 1 [T - Te] ≈ 0.9[Tt - Te] 1 0.6 t 1+ 5 0.9
or well over 10 K! Yet, in the cooling down period of our experiments, the temperature differences decreased only from 3 K to zero with decreasing (Te - Ta) values. The only plausible explanation for this large difference is to assume mass transport or free convection to reduce local temperature differences. Extrapolating these estimates to a polymer cuvet, with thermal conductivities in the same order as those of glass, is can be shown that in polymer cuvettes the same mechanisms will occur. Finally, considering equilibrium temperatures, the spatial temperature distribution in the water has been reconstructed (Figure 2) using the Abel inversion technique proposed by Nestor and Olsen.6 Here one assumes that the field is axially symmetric. The resulting pattern strongly resembles the theoretically predicted profile with a relatively large gradient near the cuvet bottom, a “curl” just below the liquid surface, and relatively small overall temperature differences in the order of 2-3 K. The temperature gradients vary from 100 K/m at the cuvet bottom to 20 K/m in the top area of the liquid. The question of whether such a gradient is plausible can only be answered by considering the 1 W/cm2 output power of the heating element. Assuming a thermal conductivity b ) 0.6 J/m‚K, the measured temperature gradients require a local heat flux to be in the order of 10-60 J/m2‚s. This by far exceeds the maximum heat flux that can be provided by the heating element, which makes heat transport by thermal convection even more likely. The currently found thermal gradients are a factor 3-5 smaller than those predicted from the scaled-up experiments.2 Even though mixing boundary layers must therefore be slightly thicker than predicted in,2 calculations on immunoassay incubation predict only marginal effects to reaction kinetics. A more considerable deviation from our earlier data is the finding of a second high-temperature region close to the liquid surface. One probable cause for these deviations is the erroneous use of scaling parameters in ref 2. This study was initiated to seek physical proof for the existence of thermal convection in a classic immunochemical analysis situation. Our data unambiguously show that (1) heat fluxes as estimated from density gradients by far exceed values caused by the heating element, (2) the number of fringes corresponds to the maximum temperature difference in the liquid, and maximum 5184
Analytical Chemistry, Vol. 69, No. 24, December 15, 1997
Figure 2. Reconstruction of the temperature distribution T(x,y,z) in the central plane of symmetry of the cuvet according to Abel’s method in equilibrated temperature situation. After numerical reconstruction, the image was processed to improve presentation quality. Temperatures are visualized in an arbitrary gray scale; the maximum temperature difference from bottom to top approximates 3.5 K.
temperature differences in the liquid are much smaller than could be expected from conduction based heat transport. The experimental technique was easy to use for versatile, realtime investigations on temperature differences inside liquids. Several improvements may still be thought of: (1) In situations of rapid temperature variation, one might record both the disturbed beam and the undisturbed beam on the holographic plate. Phase-stepping may then take place in a later stage. (2) Using the tomographic setup described in refs 7 and 8, even the instantaneous three-dimensional temperature distribution may be obtained. (3) A multichannel interferometer setup as in refs 9 and 13 could be used to follow the temperature distribution in time at video rate. (4) Isolation of the heating element could minimize image disturbances caused by convection in the surrounding air that in current work limited possible quantitative analysis. On the basis of these noninvasive thermal measurements on thermal convection-based mass transport we may now even more strongly emphasize the relevance of properly understood and wellcontrolled mass transport situations in immunoassay incubation.
SYMBOLS g
gravitational acceleration (m/s2)
Ibias
bias or background intensity
Imod
modulation intensity
L
width of the cavity (m)
refractive index (-)
coefficient of thermal expansion (-)
n∞
reference refractive index (-)
λ
width of the medium the beam went through (m)
dn/dT
variation of refractive index with temperature (1/K)
ν
kinematic viscosity of the fluid (N/m2‚s)
T
temperature (K)
∆φ
phase delay (rad)
∆T
temperature difference (K)
dT/dy
temperature gradient (K/m)
tw
water layer thickness (m)
tg
cuvet bottom thickness (m)
R
heat conduction coefficient (W/m‚K)
β
thermal conductivity (J/m‚K)
n(x,y,z)
Received for review June 24, 1997. Accepted October 8, 1997.X AC970661N X
Abstract published in Advance ACS Abstracts, November 15, 1997.
Analytical Chemistry, Vol. 69, No. 24, December 15, 1997
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