Thermal Analysis of the Role of Condensation in PCR with Capillary Tubes

Thermal profiling provides the understanding needed to enhance the reliability of Polymerase Chain Reaction (PCR) systems, but is difficult to perform experimentally when the reagents are housed in capillary tubes. The use of 3D numerical simulation with COMSOL here showed that with capillary tubes attached to a slider that moved over 3 differentially heated blocks without slots, the central axis temperature points of the tube provided poorer estimates of the thermal performance of the scheme by as much as 8.5 °C compared with the average cross-sectional temperature. It also showed that a 2-mm-thick slider would be able to improve the thermal response characteristics of a 16.5-mm-thick slider by up to 12 °C. Despite this, the method provided the best immunity to condensation effects in PCR. With the standard stationary heated block method, the temperature profiles in the capillary tubes were found to be highly affected by the amount of water condensate present in the slots. Since condensation is random, this would result in fluctuating PCR thermal behavior. When the tilt method is conducted with slot-guided capillary tubes placed heated blocks, the condensate heat transfer problem remained with stiction also affecting the movements of the capillary tubes.


Introduction
The development of laboratory instruments has progressed in tandem with prodigious advancements made in science.
Not surprisingly, what began as bespoke physical solutions to problems encountered in the laboratory was quickly externalized to create an industry that is currently valued at around $30 billion globally.Due to the intrinsic complexity of analytical measurements made within a laboratory setting, the factors that can affect instrument functionality and performance have to be carefully considered.Condensation is the physical change that happens when substances in the gaseous state alters to the liquid state as a result of energy diminution at the molecular level due to heat loss or pressure applied.It can manifest as a film or a series of drops on surfaces [1].As condensation occurs naturally when saturated air is cooled to its dew point, its presence on instruments can affect performance outcomes in various applications.These include biofiltration [2], gaseous separation in chromatography [3], and vapor ionization in mass spectrometry [4].Even minute volumes of condensate can hamper X-ray scattering [5] and atomic force microscope (AFM) measurements [6].Depending on operating conditions, it is possible for the air inside any instrument that is closed to the ambient environment to contain supersaturated vapor.Condensation will occur when vapor in this state contacts any cooler surface within the instrument [7].These condensates, if present as overhanging drops in the instrument, can through mechanisms such as dripping and capillary filling, create liquid bridges between the narrow spaces of solid bodies [8], or in this case between the capillary tubes and heating blocks.Hitherto, there have been no known investigations done to assess the effect of such liquid bodies, which has its genesis from water condensation, on the heating cycles in PCR, even though it is widely acknowledged to be a potential cause of problem for the method.The Polymerase Chain Reaction (PCR) technique involves amplification of DNA target sequences and is widely applied in healthcare by virtue of its enhanced detection sensitivity [9][10][11].The PCR thermal cycling process necessitates subjecting the reaction mixture to three distinct temperatures to accomplish DNA denaturation, annealing, and extension.The method of placing vials containing the reagent sample in a solid metal block whereby the required temperature is achieved by electrical heating is the most widely adopted in commercial instruments [12] although heated gas approaches have also been attempted [13].When smaller volume samples are used, which has the advantage of permitting more tests to be conducted with less biological samples and reagents, the heated block approach has been adapted for use with microfluidic channels [14] or capillary tubes [15].
The classical approach to perform thermal cycling in PCR is by moving the DNA samples in repeated cycles between three water baths maintained at different temperatures [16].A cardinal advantage with this approach lies with the reaction vessel being moved between media that possess high thermal inertia and this allows the reagent to be stably kept at specific temperatures for the required period of time without the need for complex temperature control schemes.An alternative technique of moving the capillary tubes containing the reagent samples by programmed tilts using gravity between thermal blocks as surrogates of water baths has recently been demonstrated.The earlier version of the programmed tilt prototype involved slot-guided translation of capillary tubes across differentially heated blocks [17].An improved prototype employed slider-mediated translation of the capillary tubes across differentially heated blocks without slots [18].With the use of thermal insulation material in the latter scheme, very low input power (within 18.5 W) was needed to conduct the PCR, making it amenable for incorporation into drones [18].
Profiling the temperature characteristics of thermal cycling systems will enhance understanding of the mechanics of heat transfer and temperature control to optimize PCR efficiency and reliability.In this work, 3D numerical studies are conducted to reveal the thermal characteristics of key components in the sliding tilt and stationary block PCR systems (Fig. 1).The thermal effects of water film condensates (which have higher heat transfer effects than drops) trapped in the slots of a stationary heated block for holding sample-containing capillary tubes are investigated.This is then contrasted with results obtained with the sliding tilt method to move the capillary tubes across differentially heated blocks with a slider [18].This scheme by virtue of its mode of operation is expected to circumvent unwanted thermal effects from water condensate that forms on surfaces.The use of 3D numerical analysis here is justified by the lack of visual access to the capillary tubes and its liquid contents (that is, the PCR reaction samples) during the thermal cycling process.

Physical Descriptions
For both the sliding tilt and stationary block methods, each capillary tube is assumed to be made of quartz glass and have dimensions of 75 mm length, 1.6 mm outer diameter, and 1.1 mm inner diameter.The reagent is assumed to completely fill the capillary tube and to have the properties of water.The ends of the capillary tube are sealed with wax putty.The small diameter (relative to its length) meant that heat transfer from the capillary tube can be taken to be insignificant.It is also assumed that all material properties are constant with temperature and time.The reagent in the capillary tube is taken to be Newtonian, stationary, nonviscous, and incompressible.The tube and reagent sealed within it are initially kept at 25 °C.
In the stationary block method, a single aluminum block with dimensions of 110 × 70 × 5.75 mm is used.The block has twelve machined slots of 3.25 mm depth along its length, each slot having rounded 1.25 mm radii instead of sharp bottom edges.Sample-containing capillary glass tubes are housed in the slots, wherein condensate filling to heights ranging from 0.1 to 0.8 mm in the slots are investigated.A high wattage polyamide film heater is attached to the bottom of the block to cause the temperature changes and a digital temperature sensor incorporated to provide signal feedback.The thermal cycles of the stationary block are assumed to be perfectly attained via a well-tuned programmable proportional-integral-derivative (PID) controller with no occurrence of temperature going above or below the pre-set value.

Mathematical Methodology
A differential equation of heat conduction can be used to describe the heat transfer in solids, or Separate heat transfer equations have to be formulated for the capillary tube, heating block, and reagent inside the capillary tube.Under the assumption of negligible thermoelastic damping, the heat transfer taking place across the capillary tube as it is travelling can be described as If the capillary tube is stationary, the heat transfer can be described using This then means that the heat transfer in the block is governed by On the other hand, the heat transfer of the reagent inside the capillary tube is given by ( 1) 1 Schematic depictions of the a sliding tilt and b stationary block methods.In the former, the capillary tubes with reagents are moved between heated acetal blocks that are kept at specific temperatures.The tubes are attached to a shuttle slider and are moved in repeated cycles through all three blocks via programmed tilting.This movement of the tubes on the surface overcomes film condensates from affecting the heat transfer characteristics.In the latter (with only the right side illustrated), the temperature of the aluminum block is altered through power supplied to the resistive heaters to achieve the different thermal cycles.The sample-containing capillary tubes reside within slots created in the block.The formation of film condensates in these slots can affect heat transfer characteristics with consequent impact on PCR outcomes It is reasonable to assume that there is insignificant heat transfer in the air that is enclosed between the capillary tube and the slots on the thermal block.Hence, Eq. ( 5) can be simplified to An initial temperature of 25 °C is imposed on the domains of air, capillary tubes and the reagents within.This of course corresponds to the condition where no heat transfer should occur when t = 0 in the simulation.All the outer walls of the geometry are taken to have an adiabatic boundary condition.To depict the actual heating configurations, fixed temperature boundary conditions are applied on all the bottom surfaces of the 3 thermal blocks.Under a global definition, these are designated as T1, T2 and T3 being 94 °C, 58 °C and 72 °C as the temperatures for denaturation, annealing, and extension respectively.It is also expected that the thermal blocks are ( 5) isothermal and that there is no fluctuation or variation in temperature over time.

Computation Methods
COMSOL Multiphysics 5.5 was used to conduct the numerical study.Apart from all the coupled heat conduction and convection equations being solved via the heat transfer module of the software, the segregated approach was also applied [19,20].The latter meant that the computational problem was broken down into individual independent steps, wherein each step would address the particular attendant physics.Under this approach, the steps could be computed sequentially within a single iteration, thereby reducing the demands of computational memory.The drawback of this approach was that a greater of number of iterations is typically needed to arrive at convergence.Due to the geometrical conditions to be handled, a deformed mesh approach was deemed most suited to simulate this problem [21].In order to obtain results with higher sensitivity, graded meshes with tetrahedral shaped finite elements were used.
The tilt method has domains that interface with each other in a more complicated manner (see Fig. 2).Hence, it required the meshing density to be more domain dependent Fig. 2 The essential domains modelled in the tilt method include the a heater blocks, b shuttle slider, c capillary tubes, d reagents inside the capillary tubes and e surrounding air (see Fig. 3).Overall, higher mesh resolutions were used for the glass and liquid domains as opposed to the air and heating block domains.
A mesh independence study was performed for five different mesh settings to ensure that the simulation results were sufficiently grid-independent.The average temperature of liquid inside the capillary tubes was taken to be the key parameter to ensure that it remains relatively invariant with any refinement of the mesh.By monitoring the liquid average temperature, it was possible to establish relative percentage errors to determine convergence.The convergence criterion for the temperature field was set at 0.6% for all three PCR steps (denaturation, annealing and extension) in order to obtain the desired accuracy.
In the stationary block method, simulation was performed for half of the original model due to geometric symmetry to save on computational time.A solid wall boundary condition was ascribed to the aluminum block and predefined pressure, thermal conductivity, density and heat capacity were applied.The initial temperature was assumed to be 25 °C for all the components.A thermal insulation boundary condition was applied on all the sides of block such that there is no heat lost to the environment from them.The PCR reagent inside capillary tubes is assumed to be Newtonian, stationary, non-viscous and incompressible.The air surrounding the tubes and present was taken to be stationary.Guided by a previous simulation work done [22], tetrahedral elements were used to mesh the elements of the liquid and solid domains with sizes ranging from 0.0165 to 3.85 mm.In contrast, the air domains meshed more coarsely with sizes ranging from 1.98 to 11 mm.

Experimental Assessment of "Condensation" Effects on Capillary Tube Movement
Selected slots (for holding the capillary tubes) in the previously developed tilting platform setup [17] were filled with 50 μL of water in each block to mimic the random effects of condensation.The platform was then tilted and videos of the relative movement of the capillary tubes recorded and analyzed.

Results and Discussion
The sliding tilt method is assumed to be free from thermal effects from condensation films on the heated surfaces as they would be cleared away by movement of the slider with capillary tubes (Fig. 1a).With this method, the mesh independence study on the numerical model was performed to determine and select the most suitable model for the simulation.The parameter for mesh convergence was chosen based on the average temperature of the reagent for all three stages of PCR thermal cycling process.The termination criterion for convergence of results was chosen to be less than 0.6% for the denaturation, annealing and extension phases.The outcomes of the mesh independence study of the sliding tilt method are summarized in Table 1.As expected, the percentage error was reduced with higher resolution meshes.When mesh D (472,628 mesh elements) was used, the percentage errors in the temperature dropped below 0.6% for all three thermal blocks.Hence, mesh D with 472,628 elements was considered to provide sufficient grid independence in this study.It is noteworthy that the percentage error was higher with the block kept at the annealing temperature when mesh setting B and C were used.This behavior is attributed to this block being at the center of the assembly (i.e.Acetal block 2 in Fig. 1a).Since there are narrow air gaps of 2.5 mm between the blocks as well as the presence of sharp edges, insufficient resolution will result in higher levels of discretization errors at these regions [23].The block in the middle interfaces with two such regions, unlike the other two end blocks that encounter only one each (Fig. 4).
The simulation model for the sliding tilt method was tested according to the temperatures indicated in Table 2, which were thermal cycling parameters previously shown to be suitable for successful PCR [17,18].In the simulation, it is assumed that the time taken for movement from one heated block to the next is 1 s.The manner in which the temperature distributions alter when the capillary tubes move from one heated block to another block is important.This is illustrated the tubes were transferred from blocks 1 to 2 (Fig. 5), and from blocks 2 to 3 (Fig. 6) respectively.It can be seen that temperatures of the former required 3 s to attain uniformity as opposed to 0.5 s with the latter.Obviously, the smaller temperature differential between blocks 2 and 3 help to reduce the time needed to reach temperature equilibration.
From Figs. 5 and 6, it can be seen that a thermal gradient exists across the whole cross section area from the heater block to the slider shuttle at the top.This could engender wide discrepancies if temperatures are determined from the central axis point versus taking the average temperature of the entire cross-sectional space within the capillary tube.This differential thermal profile is shown in Fig. 7 as the reagent equilibrates temporally over the heating processes occurring at the various thermal blocks (which are kept at different phase temperatures).A clearer depiction of this behavior can be obtained by calculating the departure temperature (ΔT) with respect to time t using where T p is the phase temperature (denaturation, annealing or extension) at the specific point in time.From Fig. 7, it can be seen that the magnitude of temperature deviation is greater when the average cross-sectional temperature is used.At certain instances in the time space, the variation was found to be in excess of 8.5 °C.Hence, although it is easier to estimate the temperature of a point at the central axis of the tube, it is clear that this metric does not provide ( 7) a sufficiently accurate representation of the thermal state of the reagent.
It is also noteworthy that while the average cross-sectional temperature distribution follows a recurring pattern with time in accordance with the thermal cycling process, the average departure temperatures manifest decreasing magnitude with time.This behavior reflects a role played by the thermal inertia of the slider.Logically then, the use of a thinner slider is expected to reduce the contribution of this factor.This is demonstrated in the simulated departure temperature traces with time using sliders that were 16.3 mm and 2 mm thick (see Fig. 8).At certain points in the time Fig. 5 The temperature distributions obtained at the cross section corresponding to the front end of the capillary tube with the sliding tilt method.The start state depicts the point when the tube is just about to leave block 1 (aimed at 94 °C).The tube is positioned over block 2 (aimed at 58 °C) within 1 s.Thereafter, equilibration towards the temperature of block 2 is observed 3.5 s after departure from block 1.The thickness of the slider block was 16.3 mm Fig. 6 The temperature distributions obtained at the cross-section corresponding to the front end of the capillary tube with the sliding tilt method.The start state depicts the point when the tube is just about to leave block 2 (aimed at 58 °C).The tube is positioned over block 3 (aimed at 72 °C) within 1 s.Thereafter, equilibration towards the temperature of block 3 is observed 3.5 s after departure from block 2. The thickness of the slider block was 16.3 mm 1 3 Fig.7 Simulated reagent thermal profile comparing the average cross-sectional temperature and temperature at the central axis point of the tube against time with the sliding tilt method.The departure temperature (ΔT), calculated using Eq. ( 6) in both cases, are also presented.The thickness of the slider block was 16.3 mm Fig. 8 Comparison of average temperatures (of the reagent in the capillary tube) against time based on a simulation run with a thick (16.3 mm) and thin (2 mm) slider used with the tilt method.The departure temperatures (ΔT), calculated using Eq. ( 6), in both cases are also presented space, improvements by as much as 12 °C could be attained with the thinner section.
The relative advantage of a thinner slider to achieve improved thermal characteristics is, however, tempered by the concomitant reduction in weight of the slider, which in turn hampers its ability to be moved by the action of gravity when the system is tilted.One way to circumvent this is to introduce an air gap into the slider such that its bottom section is of much lower thickness compared to its top section.This design also affords the ability to incorporate a denser material such as brass (density = 8500 kg/m 3 ) over PEEK (density = 1320 kg/m 3 ) at the top section of the slider to improve its movement by gravity when tilted without compromising the improved thermal characteristics when conducting PCR.In the quest to facilitate movement this way, it is pertinent to consider the consequence on payload that a heavier object would have if the method is to be incorporated for use with drones.This has been recently been highlighted in another context [24].
In the case of the stationary block method, similar thermal gradients are evident across the whole cross section area (see Figs. 9 and 10) but with different distribution patterns (see Figs. 5 and 6).This is likely due to the capillary tubes enclosed entirely within the slots of the stationary block and thus imposing limits on temperature distribution variation.Despite this, the impact of the amount of condensation present in the slots (depth of 0.1 mm or 0.8 mm) on temperature distribution is evident.As expected, thermal variation is Fig. 9 The temperature distributions obtained at a cross section of the capillary tube with the stationary block method as a function of time in which the depth of film condensate in the slot was a 0.1 mm and b 0.8 mm when heating was targeted at 58 °C Fig. 10 The temperature distributions obtained at a cross section of the capillary tube with the stationary block method as a function of time in which the depth of film condensate in the slot was a 0.1 mm and b 0.8 mm when heating was aimed at 72 °C more pronounced when the pre-set temperature differentials are higher (from 92 to 58 °C in Fig. 9 as opposed to from 58 to 72 °C in Fig. 10).The extent of this variation becomes more evident when the average cross-sectional temperature is traced against time in the thermal cycling process (see Fig. 11).It should be noted that the extent of condensation on the stationary block is typically random (i.e. the amount of condensate in each slot will be different).This will then result in the capillary tubes experiencing different temperature versus time distributions.It is also worthy of mention that simulations in the presence of temperature overshoots and undershoots typically encountered in practice with block heaters were not performed [25].These transients would be expected to exacerbate the temperature variations even further.
The application of the sliding tilt method with slots in separate heater blocks stably maintained at different temperatures is expected to eliminate these transients [17].Yet, the Fig. 11 Simulated reagent thermal profile of the average cross-sectional temperature with the stationary block method at various depths to which the condensate fills the slots in the block Fig. 12 Experimental images of capillary tubes (each with red end tips) placed in slots on differentially heated blocks [15] when kept in the a horizontal position and after b tilting at 40° to the horizontal.The unsynchronized displacement of the capillary tubes in (b) with water dispensed in the corresponding slots demonstrates the adverse effect of condensation can have on thermal cycling by tilting capillary tubes in slots thermal precision of such a system can be adversely affected by condensation.Since the capillary tubes need to be translated across the blocks for thermal cycling, the presence of condensation can lead to unsynchronized tube movement between slots due to stiction (see Fig. 12).This in turn can significantly alter the temperature versus time distributions from one capillary tube to another.
Clearly then, the sliding tilt method where capillary tubes are translated across thermal blocks by movement of a slider provides the best immunity from the effects of condensation.This method obviates the use of slots within the thermal block to hold the capillary tubes and the sliding action minimizes the likelihood of condensation forming.It is also worth noting that the use of a plurality of sliders will allow the capillary tubes to be mounted in a process that is independent of the thermal cycling process.This offers an avenue to improve overall processing throughput.
Finally, a common misconception made relates to requiring the saturation vapor pressure P s (in kPa), related to the temperature (T) by to correspond with the partial vapor pressure P s , related to the relationship humidity (RH) at any temperature given by for condensation to occur.If the temperature at a cover over the experiment is taken to be 25 °C and the air underneath it is 58 °C (the temperature at annealing), condensation should indeed not happen as depicted in Fig. 13.However, it is possible in practice for the cover to be cooled (by airflow etc.) and for the air temperature just below it to be significantly lower if the heated surface is sufficiently separated from the cover.Hence, if the former and latter were now assumed to be 20 °C and 25 °C respectively, condensation drops would start to appear when the relative humidity in the closed setup exceeds 70%.

Conclusions
Here, a numerical 3D study was applied successfully to reveal the thermal characteristics of a sliding tilt system to conduct PCR where sample-containing capillary tubes are translated by gravity via programmed tilts across three acetal (polyoxymethylene) blocks kept at specific temperatures.A grid independence study found that 472,628 graded meshes in the model performed within the required 0.6% error benchmark.It was found that temperature stabilization across the entire cross-sectional space of the capillary tube was more quickly achieved when the phase temperature was lower.A departure temperature metric developed showed that temperature readings taken from a point at the central axis of the tube gave deviant estimates by up to 8.5 °C in the time space compared with thermal profiles derived from average cross-sectional temperature.By reducing the thermal inertia effect, a 2-mm-thick slider offers to improve Fig. 13 Plots of the saturation vapor pressure (P s ) against temperature, as well as the partial vapor pressure (P p ) against relative humidity at for temperature 58 and 25 °C.The horizontal dashed lines delineate the partial vapor pressure (P p ) and relative humidity conditions needed to cause condensation when the surface temperatures are kept 25 and 20 °C the thermal response characteristics compared with that of a 16.5-mm-thick slider by up to 12 °C at certain time instances.Simulations also show that the sliding tilt system obviates adverse effects of condensation on the thermal profile of the liquid within the capillary tubes.In contrast, the stationary block method in which the capillary tubes are located in slots cycled through different temperatures was prone to condensation effects.Temperature profiles in the tubes were found to be highly affected by the amount of condensate filling in the slots.Since this filling is random, it will engender uncontrollable temperature versus time distributions.Finally, when the tilt method is conducted by sliding capillary tubes in slots across differentially heated blocks, the condensate filling problem is retained as well as an added limitation of the movement of capillary tubes affected by stiction.

2 r 1 T
Abbreviations ρDensity, kg/m 3 U i Flow velocity, m/s μ Dynamic viscosity, Pa s β Thermal expansion coefficient, 1/K g i Gravitational acceleration, m/s 2 T Fluid temperature, K T w Cylinder surface temperature, K T ∞ Ambient temperature, K Pr Prandtl number C p Specific heat capacity at constant stress, J/(kg K)

Fig. 3
Fig. 3 An example meshing of the simulation model for the tilt method.The a block domain is specified together with the air that is enclosed within the gaps of the block.A close up view of a section of

Table 1
Relative percentage errors at five different mesh resolutions in the tilt method When 472,628 mesh elements were used, the percentage error was below 0.6% for all three block temperatures

Table 2
The PCR thermal cycling protocol used to run the simulations