**6.3. Temperature measurement equations for Rhodamine B dye solution in the context of microfluidics**

The temperature-sensitive nature of molecular fluorescence is suitable for measurement of temperature in small volume fluids like in microfluidic channels (Lou et al., 1999; Gallery et al., 1994; Sakakibara et al., 1993; Ali et al., 1990; Kubin et al., 1982). The temperature measurement is based on measuring fluorescence intensity ratios. The fluorescence intensity is typically measured at a known reference temperature, which is later used to normalize the intensity measured after heating the medium to an unknown temperature. The temperature is extracted by substituting the normalized intensity into a calibration curve. For lab-on-a-chip application, the use of dilute solutions of a temperature-sensitive fluorescent dye, particularly Rhodamine B (RhB), has become very popular for optical measurement of temperature distributions. The RhB is a water-soluble fluorescent compound with an absorption peak at 554 nm, an emission peak at 576 nm, and a temperature dependent fluorescence quantum yield (Ferguson et al., 1973). In addition to high temperature sensitivity, its other properties such as negligible pressure sensitivity and nominal pH-independent absorption (above a pH value of 6) are attractive for measuring temperatures in microfluidic channels with high spatial and temporal resolution (Ross et al., 2001). The RhB solution has been employed: to examine in-channel temperature and flow profiles at a T-shaped microchannel intersection during electrokinetic pumping (Erickson et al., 2003) and to characterize the temperature field resulting from resistive microheaters embedded in a poly(dimethylsiloxane) (PDMS) microchip (Fu et al., 2006). Even though the RhB has been primarily used for temperature measurement in aqueous environment, the absorbed RhB dye molecules in a PDMS thin film can be used for whole chip temperature measurement (Samy et al., 2008). The calibration equations used for computing temperature using fluorescent dyes relate the fluorescence intensity at an unknown temperature to the intensity at only one particular reference temperature. Such relations are not directly usable to applications requiring a different reference or initial temperature. The existing single-dye calibration equations have been generalized for extending their use to fluorescence intensity data normalized to reference temperatures other than those for which the original calibration equations were derived (Shah et al., 2009). Two methods have been described in detail: one is approximate, while the other, based on solution of a cubic equation, is an accurate mathematical treatment that does not incur errors beyond those already inherent in the calibration equations.

#### *6.3.1. Generalization of RHODAMINE B temperature equations*

190 The Development and Application of Microwave Heating

equipment.

(Figure 7(*a*)) to convert from the microstrip geometry to the coaxial geometry of the test

**Figure 7.** (a) A picture of the integrated microfluidic device for generating microwave-induced

The method described here for fabricating conductors is easily transferable to other microfluidic substrates such as glass and PDMS. Furthermore, the one-step method for conductor fabrication obviates the need for electroplating, which is typically required following thin-film deposition to achieve sufficient conductor thickness. This method also provides easy bonding of the top and bottom cover plates to create enclosed channel structures, which has proven challenging for thermoplastic materials (Shah et al., 2006). In contrast to previously published reports (Shah et al., 2007a; Booth et al., 2006; Facer et al., 2001), the transmission line structure isolates the fluid from the metal conductors making these devices suitable for a variety of biochemical applications in which reagent

**6.3. Temperature measurement equations for Rhodamine B dye solution in the** 

The temperature-sensitive nature of molecular fluorescence is suitable for measurement of temperature in small volume fluids like in microfluidic channels (Lou et al., 1999; Gallery et al., 1994; Sakakibara et al., 1993; Ali et al., 1990; Kubin et al., 1982). The temperature measurement is based on measuring fluorescence intensity ratios. The fluorescence intensity is typically measured at a known reference temperature, which is later used to normalize the intensity measured after heating the medium to an unknown temperature. The temperature is extracted by substituting the normalized intensity into a calibration curve. For lab-on-a-chip application,

temperature gradients. (b) A cross-sectional view of the microwave heating device.

contamination due to electrolysis or corrosion is undesirable.

**context of microfluidics** 

Let *S*(*T*) represent the signal received from a fluorescence detection system observing a small volume of fluorescent species at temperature *T* and let *IRT(T) = S(T)/S(RT)* represent the fluorescence intensity ratio normalized to the signal measured at nominal room temperature (RT).


**Table 1.** Properties of Different RhB Solutions Used by Three Different Authors for Fluorescence-Based Temperature Measurements.

The fluorescence intensity acquired from an image captured at an elevated temperature, *S*(*T*), is normalized by the intensity acquired from an image captured at nominal room temperature, *S(RT)*, to obtain *IRT(T) = S(T)/S(RT).* The temperature is then obtained by a least-squares adjustment of the constants in the equation

$$T = A\_0 + A\_1 I\_{RT} \left( T \right) + A\_2 I\_{RT} ^2 \left( T \right) + A\_3 I\_{RT} ^3 \left( T \right) \tag{7}$$

to fit the measured IRT(T) for different values of T.

Table 1 includes the values of *A*0 to *A*3 that were reported by Fu et al., 2006; Ross et al., 2010; and Samy et al., 2008. The temperatures *T*0 given by eqn. 7 when *I*RT(*T*) = 1.0, as well as the RT used by each of the authors for normalization purposes are also shown in Table 1. These temperatures differ slightly from the normalized temperatures used by the different authors because their calibration equations were not constrained to produce RT when the intensity ratio was 1.0.

**Figure 8.** Comparison of the relative fluorescence-intensity versus temperature calibrations published by Fu et al., 2006 (─); Ross et al., 2010 (--); and Samy et al., 2008 (┅┅), for different rhodamine B chemistries

The calibration curves from all three authors are compared in Figure 8. The general trend of all three curves is similar. The difference between the curve of Samy et al., 2008 and the other two authors can be explained by the different physical medium and local environment used in the measurements.

#### *6.3.2. Generalized calibration equation*

Measurement of fluorescence intensity at a reference temperature in the vicinity of 23°C, for normalization purposes, is not always possible. For example, in applications requiring rapid temperature cycling of fluidic solutions the cycling temperatures of interest might be significantly different from 23 °C. In these situations, it is useful to calculate the fluorescence intensity ratio at a convenient reference temperature, *T*1, in the temperature range of interest as

$$I\_{T1} = I\_{T1}(T) = S(T) / S(T1) \tag{8}$$

While it might appear plausible to use *IT*1 (T) in eqn. 7 instead of *I*RT(*T*) and then add (*T*<sup>1</sup> − *T*0) to the result to estimate the temperature, this process will add some additional error to the calculated temperatures obtained based on the calibration equations of Fu et al., 2006; Ross et al., 2010; and Samy et al., 2008 (Shah et al., 2009). The additional error introduced by this procedure is zero when *T*0 = *T*1 and larger than ±1 °C in certain temperature regions for some values of *T*1 for all three calibration equations.

With a different approach, it was possible to eliminate all additional error except that inherent to the calibration equations themselves. For this approach, the normalized fluorescence intensity is generalized by rewriting it as

$$I\_{RT}\left(T\right) = \frac{S\left(T\right)}{S\left(RT\right)} = \frac{S\left(T\right)}{S\left(T\_1\right)}\frac{S\left(T\_1\right)}{S\left(RT\right)} = I\_{T\_1}(T)I\_{RT}\left(T\_1\right) \tag{9}$$

where *T*1 is any convenient known reference temperature. Therefore, if *IT*1(*T*) data have been measured where *T*1 is not the reference temperature used in deriving the calibration equation, then *I*RT(*T*) can be calculated for use in eqn. 8 from eqn. 9, where *I*RT(*T*1) can be obtained from the real solution(Terry et al., 1979) of the cubic equation

$$0 = A\_3 I\_{RT}{}^3 \left( T\_1 \right) + A\_2 I\_{RT}{}^2 \left( T\_1 \right) + A\_1 I\_{RT} \left( T\_1 \right) + A\_0 - T\_1 \tag{10}$$

with the values of *An* as listed in Table 1 and

$$\begin{aligned} I\_{RT} \left( T\_1 \right) &= \frac{-A\_2}{3A\_3} + (R + \sqrt{D})^{\frac{1}{3}} + (R - \sqrt{D})^{\frac{1}{3}} \\ D &= \mathbf{Q}^3 + R^2 \\ \mathbf{Q} &= \frac{3A\_1A\_3 - A\_2^{-2}}{9A\_3^{-2}} \end{aligned}$$

and

192 The Development and Application of Microwave Heating

ratio was 1.0.

used in the measurements.

*6.3.2. Generalized calibration equation* 

to fit the measured IRT(T) for different values of T.

 2 3 01 2 3 *RT RT RT T A AI T AI T AI T* (7)

Table 1 includes the values of *A*0 to *A*3 that were reported by Fu et al., 2006; Ross et al., 2010; and Samy et al., 2008. The temperatures *T*0 given by eqn. 7 when *I*RT(*T*) = 1.0, as well as the RT used by each of the authors for normalization purposes are also shown in Table 1. These temperatures differ slightly from the normalized temperatures used by the different authors because their calibration equations were not constrained to produce RT when the intensity

**Figure 8.** Comparison of the relative fluorescence-intensity versus temperature calibrations published by Fu et al., 2006 (─); Ross et al., 2010 (--); and Samy et al., 2008 (┅┅), for different rhodamine B chemistries

The calibration curves from all three authors are compared in Figure 8. The general trend of all three curves is similar. The difference between the curve of Samy et al., 2008 and the other two authors can be explained by the different physical medium and local environment

Measurement of fluorescence intensity at a reference temperature in the vicinity of 23°C, for normalization purposes, is not always possible. For example, in applications requiring rapid temperature cycling of fluidic solutions the cycling temperatures of interest might be significantly different from 23 °C. In these situations, it is useful to calculate the fluorescence intensity ratio at a convenient reference temperature, *T*1, in the temperature range of interest as

While it might appear plausible to use *IT*1 (T) in eqn. 7 instead of *I*RT(*T*) and then add (*T*<sup>1</sup> − *T*0) to the result to estimate the temperature, this process will add some additional error to the calculated temperatures obtained based on the calibration equations of Fu et al., 2006; Ross

1 1 /1 *T T I I T ST ST* (8)

$$R = \frac{9A\_1A\_2A\_3 - 27\left(A\_0 - T\_1\right)A\_3^{\;\;2} - 2A\_2^{\;\;3}}{54A\_3^{\;\;3}}\tag{11}$$

When *I*RT(*T*) is calculated from eqn. 9 with measured *IT*1(*T*) data and *I*RT(*T*1) obtained from the solution to eqn. 10 that is given in eqn. 11, no error is introduced into the result beyond that already inherent in eqn. 7. A treatment similar to that described above can be applied to generalize calibration equations based on linear (Gallery et al., 1994) and second order polynomial (Erickson et al., 2003) fits to normalized *I*(*T*) data to a convenient reference temperature.

#### **6.4. Device characterization**

Experiments were performed to evaluate the performance of the CPW devices for heating in the microchannel environment. The device is characterized for its frequency response in order to obtain absorption ratios with empty and fluid-filled microchannels. The fluid temperature is measured at various microwave frequencies. The results obtained from the

first experiment are used to derive a power absorption model to find the distribution of the incident microwave power in different absorbing structures of the device.

The CPW device frequency response is characterized by scattering (S) parameter measurements. The S-parameters relate the forward and reflected traveling waves in a transmission medium and can be used to understand the power flow as a function of frequency. The S-parameters are used in conjunction with the conservation of energy to model absorption of microwave power and to predict the fluid temperature based on the absorbed power. The predicted temperature is then fitted to the measured temperature to determine heating efficiency. The experiments for the results presented below were performed with deionized (DI) H2O and with fluids of two different salt concentrations: 0.9% NaCl solution and 3.5% NaCl solution.

#### *6.4.1. S-parameter measurements*

The amplitude of the reflection coefficients (S11) and transmission coefficients (S21) from 0.3 GHz to 40 GHz for the device with an empty channel as well as for the fluid-filled channels are shown in Figure 9(a) and Figure 9(b), respectively. In comparison with the empty channel device, S11 is reduced for the device with fluid-filled channel and it approaches that of an empty channel for frequencies above 10.5 GHz. The decrease in S11 of the fluid-filled devices below 10.5 GHz which indicate good impedance matching conditions are apparently fortuitous. The S11 is also found to be almost independent of the ionic concentration of the fluid. The localized peak and trough features observed in the S11 are likely interference effects due to reflections at the probe-CPW interfaces and the CPW/air-CPW/fluid interfaces (Facer et al., 2001). The S21, as seen from Figure 9(b), decreases with

0.9% NaCl solution and 3.5% NaCl solution.

*6.4.1. S-parameter measurements* 

first experiment are used to derive a power absorption model to find the distribution of the

The CPW device frequency response is characterized by scattering (S) parameter measurements. The S-parameters relate the forward and reflected traveling waves in a transmission medium and can be used to understand the power flow as a function of frequency. The S-parameters are used in conjunction with the conservation of energy to model absorption of microwave power and to predict the fluid temperature based on the absorbed power. The predicted temperature is then fitted to the measured temperature to determine heating efficiency. The experiments for the results presented below were performed with deionized (DI) H2O and with fluids of two different salt concentrations:

The amplitude of the reflection coefficients (S11) and transmission coefficients (S21) from 0.3 GHz to 40 GHz for the device with an empty channel as well as for the fluid-filled channels are shown in Figure 9(a) and Figure 9(b), respectively. In comparison with the empty channel device, S11 is reduced for the device with fluid-filled channel and it approaches that of an empty channel for frequencies above 10.5 GHz. The decrease in S11 of the fluid-filled devices below 10.5 GHz which indicate good impedance matching conditions are apparently fortuitous. The S11 is also found to be almost independent of the ionic concentration of the fluid. The localized peak and trough features observed in the S11 are likely interference effects due to reflections at the probe-CPW interfaces and the CPW/air-CPW/fluid interfaces (Facer et al., 2001). The S21, as seen from Figure 9(b), decreases with

incident microwave power in different absorbing structures of the device.

**Figure 9.** (A) Measured reflection coefficients ( |S11| )of the device. (B) Measured transmission coefficients ( |S21| )of the device. (■) Empty microchannel, (●) microchannel filled with deionized H2O, (▲) microchannel filled with 0.9 % NaCl, (♦) microchannel filled with 3.5% NaCl.

increasing frequency for the devices with both the empty and fluid-filled channels. It is believed that the apparent low transmission coefficient of the empty-channel device is likely due to the smaller than optimum thickness of the CPW conductors (0.5 *μ*m). The difference in S21 between the empty and fluid-filled channels becomes more pronounced at higher frequencies (> 5 GHz). The S21 of the fluid-filled devices is smaller compared to the emptychannel device due to the additional attenuation caused by the absorption of the microwave energy in the water.

Figure 10 shows percent absorption ratios (the fraction of the incident power absorbed by the device) as a function of frequency for the devices with empty and fluidfilled microchannels. The absorption ratio, *A*, is calculated from the S-parameter data shown in Figure 9 and using the equation: *A = 1 – R – T* where *R* (the reflection coefficient) = *│S11│*, *T*  (the transmission coefficient) *= │S21│*, and *Sij (dB) = 10 log10│Sij│*. It should be noted that the absorption ratio is dependent on the position of the microchannel over the length of the waveguide. In other words, S21 would not equal to S12 and the device would function as a non-reciprocal two-port network unless the microchannel is precisely centered over the length of the waveguide. The absorption ratio obtained for the device with empty channel (■) is due to ohmic dissipation in the thin-film CPW, which can be modeled by an attenuation constant, αcpw, which corresponds to 2.86 dB/cm at 10 GHz. The attenuation constant of PDMS is assumed to be negligible because of the relatively low values of loss tangent (the ability of a material to convert stored electrical energy into heat) at microwave frequencies (Tiercelin et al., 2006). An increase in *A* observed with increasing frequency for the empty channel device (■) is expected from the dependence of skin depth on frequency. The absorption ratio for the fluid filled devices (•, ▲, and ♦) is greater for all frequencies

measured in comparison with the empty channel device (■), due to microwave absorption in water and also ionic absorption in the ionic solutions. The data also exhibit a dependence of *A* on ionic concentration at lower frequencies as would be expected due to ionic conductivity, while the data at higher frequencies show that *A* is approximately independent of ionic concentration as would be expected due to dielectric conductivity. This trend is in agreement with theory (Wei et al., 1990).

**Figure 10.** The percent absorption ratios (the fraction of the incident microwave power absorbed by the device) as a function of frequency. The absorption ratio, A, was calculated from the measured transmission and reflection coefficients using *A = 1 – R – T.* (■) Empty microchannel, (●) microchannel filled with deionized H2O, (▲) microchannel filled with 0.9% NaCl, (♦) microchannel filled with 3.5% NaCl.

#### *6.4.2. Power absorption model*

It seems from Figure 10 that the simplest approximation to the fraction of the incident microwave power absorbed in the fluid is the difference between the power absorbed by the full-channel and empty-channel devices (for the case of water-filled device, *AH2O – Aempty* in Figure 10). However, various microwave power absorption models described in (Geist et al., 2007) show that this simple approximation greatly underestimates the actual fraction of incident microwave power absorbed in the fluid. For this reason, a simple model, alpha absorption model, is chosen to extract the fraction of the incident power absorbed in the fluid. This model is also used to differentiate microwave heating of the fluid from conductive heating due to ohmic heating of the CPW conductors. As shown in Figure 11, the model is constructed by assuming that the PDMS completely covers the CPW and by partitioning the CPW into three regions: a center region that interacts with the fluid in the

trend is in agreement with theory (Wei et al., 1990).

NaCl.

*6.4.2. Power absorption model* 

measured in comparison with the empty channel device (■), due to microwave absorption in water and also ionic absorption in the ionic solutions. The data also exhibit a dependence of *A* on ionic concentration at lower frequencies as would be expected due to ionic conductivity, while the data at higher frequencies show that *A* is approximately independent of ionic concentration as would be expected due to dielectric conductivity. This

**Figure 10.** The percent absorption ratios (the fraction of the incident microwave power absorbed by the

It seems from Figure 10 that the simplest approximation to the fraction of the incident microwave power absorbed in the fluid is the difference between the power absorbed by the full-channel and empty-channel devices (for the case of water-filled device, *AH2O – Aempty* in Figure 10). However, various microwave power absorption models described in (Geist et al., 2007) show that this simple approximation greatly underestimates the actual fraction of incident microwave power absorbed in the fluid. For this reason, a simple model, alpha absorption model, is chosen to extract the fraction of the incident power absorbed in the fluid. This model is also used to differentiate microwave heating of the fluid from conductive heating due to ohmic heating of the CPW conductors. As shown in Figure 11, the model is constructed by assuming that the PDMS completely covers the CPW and by partitioning the CPW into three regions: a center region that interacts with the fluid in the

device) as a function of frequency. The absorption ratio, A, was calculated from the measured transmission and reflection coefficients using *A = 1 – R – T.* (■) Empty microchannel, (●) microchannel filled with deionized H2O, (▲) microchannel filled with 0.9% NaCl, (♦) microchannel filled with 3.5%

**Figure 11.** Top view of a CPW integrated with an elastometric microchannel consisting of three regions, center region with the microchannel and the two end regions without the microchannel.

microchannel and two end regions that have no microchannel over them. The lengths of each of the three regions are defined by the center region, Z2, 0.36 cm long and two end regions, Z1 and Z3, each 0.57 cm long. The reflectance at the interface between the regions is assumed to be negligible, and the transmission coefficient, *T,* is modeled as follows:

$$T = \left(1 - R\right) \times e^{-2a\_{cpw}z\_1} \times e^{-2a\_{cpw}z\_2} \times e^{-2a\_wz\_2} \times e^{-2a\_{cpw}z\_3} \tag{12}$$

The derivation of this equation is based on exponential attenuation of microwave power in the direction of propagation where the rate of decay with distance is described by the attenuation constant, *α*. Because of the presence of water in the center region, it should be noted that the attenuation due to water, *αw*, is added to the attenuation due to CPW, *αcpw*. Eqn. 12 is then used to derive *αcpw* and *α<sup>w</sup>* as follows. First, *αcpw* is derived for the empty channel device by setting *α<sup>w</sup>* equal to zero into eqn. 12 and substituting the measured values of *T* and *R* for the empty channel device. Next, this value of *αcpw* is substituted into eqn. 12 along with the measured values of *Tf* and *Rf*, which are *T* and *R* for the water filled device, respectively. The resulting equation is solved for *αw*, which is found to be 3.68 dB/cm at 10 GHz. The absorption ratio for the central region of the water filled channel, *A2,* is calculated as

$$A\_2 = (1 - R\_f) \times (e^{-2a\_{cpw}z\_1}) \times (1 - e^{-2(a\_{cpw} + a\_w)z\_2}) \tag{13}$$

Finally, the absorption ratios of the water, *Aw*, and CPW conductors, *Am*2 , in the center region are calculated by eqn. 14 and eqn. 15, respectively.

$$A\_w = \left[\frac{\alpha\_w}{\alpha\_{cpw} + \alpha\_w}\right] A\_2 \tag{14}$$

$$A\_{m2} = A\_2 - A\_w \tag{15}$$

**Figure 12.** The distribution of the incident microwave power in different absorbing structures of deionized H2O filled device as obtained from the alpha absorption model. *Tf, Rf* are the transmission and reflection coefficients of the water-filled device, respectively; *Am, Am2, Aw* are the absorption ratio of the CPW conductors, the absorption ratio of the CPW conductors in the region with the microchannel, and the absorption ratio of the water, respectively.

Eqns 12 through 15 are first order approximations. For impedance matched conditions, the equations show that *Aw* is dependent of position of the microchannel along the length of the CPW with a maximum occurring at the source end because the transmitted power attenuates exponentially along the transmission line. Further, *Aw* decreases exponentially along the length of the microchannel. Therefore, there is a design tradeoff between the microchannel length, its position relative to the source, and the frequency of operation to obtain a uniform temperature rise and efficient absorption of microwave power. Figure 12 shows the distribution of the incident microwave power in different absorbing structures of deionized H2O filled device. It shows that *Aw* increases with frequency in agreement with theory because the power absorbed by water per unit volume is proportional to *f*ε '' as calculated from Franks (Franks, F., 1972) . It also shows that the absorption by the metal in the central region (*Am2*) competes with the absorption by the water (*Aw*). At low frequencies (< 3.5 GHz), *Am2* is slightly higher than *Aw*. The difference between the two becomes negligible as the frequency increases further, and *Aw* starts to dominate as the frequency exceeds approximately 8 GHz. This can be explained by the differences in the attenuation constants. For instance, *α<sup>w</sup>* is 1.3 times as high as *αcpw* at 10 GHz. It can be observed from Figure 12 that the fraction of the incident power absorbed by the CPW conductors, *Am*, is noticeably high over the entire frequency range measured. This results in ohmic heating of the CPW conductors, which is also expected to contribute to the temperature rise of the fluid. However, based on the incident power absorbed by the CPW conductors in the center region, *Am2*, a worst-case first order analytic calculation (Carslaw et al., 1959) of the contribution of the metal heating to water heating shows that the power dissipated in the CPW contributes less than 20 % of the total heating observed in the microchannel.

#### *6.4.3. Temperature measurements*

198 The Development and Application of Microwave Heating

and the absorption ratio of the water, respectively.

**Figure 12.** The distribution of the incident microwave power in different absorbing structures of deionized H2O filled device as obtained from the alpha absorption model. *Tf, Rf* are the transmission and reflection coefficients of the water-filled device, respectively; *Am, Am2, Aw* are the absorption ratio of the CPW conductors, the absorption ratio of the CPW conductors in the region with the microchannel,

Eqns 12 through 15 are first order approximations. For impedance matched conditions, the equations show that *Aw* is dependent of position of the microchannel along the length of the CPW with a maximum occurring at the source end because the transmitted power attenuates exponentially along the transmission line. Further, *Aw* decreases exponentially along the length of the microchannel. Therefore, there is a design tradeoff between the microchannel length, its position relative to the source, and the frequency of operation to obtain a uniform temperature rise and efficient absorption of microwave power. Figure 12 shows the distribution of the incident microwave power in different absorbing structures of deionized H2O filled device. It shows that *Aw* increases with frequency in agreement with theory because the power absorbed by water per unit volume is proportional to *f*ε '' as calculated from Franks (Franks, F., 1972) . It also shows that the absorption by the metal in the central region (*Am2*) competes with the absorption by the water (*Aw*). At low frequencies (< 3.5 GHz), *Am2* is slightly higher than *Aw*. The difference between the two becomes negligible as the frequency increases further, and *Aw* starts to dominate as the frequency exceeds approximately 8 GHz. This can be explained by the differences in the attenuation constants. For instance, *α<sup>w</sup>* is 1.3 times as high as *αcpw* at 10 GHz. It can be observed from Figure 12 that the fraction of the incident power absorbed by the CPW conductors, *Am*, is noticeably high over the entire frequency range measured. This results in ohmic heating of the CPW conductors, which is also expected to contribute to the temperature rise of the fluid. However, based on the incident power absorbed by the CPW conductors in the center The points (■) in Figure 13 show the fluid (aqueous solution of Rhodamine B) temperature measured at various microwave frequencies. The applied power was kept constant at 10 mW. The temperature was measured ~ 250 ms after turning on the microwave power, which was approximated to be within 5% of quasi-thermal equilibrium. The error bars indicate the pooled standard deviation over all measurements for two instances at each frequency added with the estimated standard deviation (0.5 °C) of the room temperature (22.5 °C). The observed temperature rise was 0.88 °C/mW at 12 GHz and 0.95 °C/mW at 15 GHz.

**Figure 13.** The measured temperature (■) of an aqueous solution of 0.2 mmol/L Rhodamine B in a 19 mmol/L carbonate buffer as a function of frequency. The solid line indicates predicted temperature calculated from the alpha absorption model, and the dashed line was calculated by employing the power difference model.

The temperature can also be calculated from the energy absorbed in the water during the heating period *dt* using equation 16

$$dT = \frac{P\_v dt}{\rho \mathcal{C}\_p V} = \frac{K\_c I A\_w dt}{\rho \mathcal{C}\_p V} \tag{16}$$

Where ρ and *Cp* are the density and heat capacity of water, respectively, at appropriate temperature, *I* is the incident microwave power, *Aw* is the fraction of the incident power absorbed in the water as shown in Figure 12, and *V* is the volume of water in the microchannel. *Ke* is the channel-heating efficiency, which is defined as the fraction of energy

absorbed in the water during the time *dt* that remains in the water. The rest of the energy absorbed in the water during the time *dt* is conducted into the substrate. The value of *Ke* cannot be easily obtained from the geometry and thermal properties of the channel and substrate due to the unknown contact thermal resistance (Kapitza resistance) between the water and the hydrophobic surface of the substrate (Barrat et al., 2003). Rather than attempt to calculate the value of *Ke*, it was adjusted in a least-squares fit of the predictions of eqn. 16 to the measured data points (■) in Figure 13. The solid line in Figure 13 indicates the predicted temperature calculated using *Aw* obtained from Figure 12 (alpha absorption model), and the dashed line indicates the predicted temperature calculated using *Aw*  obtained from the power difference model (*AH2O – Aempty*). It is clear from the results of fitting the predictions of the two different absorption models that the alpha absorption model provides a better fit to the measured data. Further, the heating efficiency obtained from the alpha absorption model indicates that only 5 % of the total heat (time integral of the absorbed power) was stored in the fluid while the rest was lost to the surroundings (PDMS and glass). Because the ratio of stored heat to lost heat increases with decreasing heating time, it is possible to confine most of the heat to the fluid and heat it to a higher temperature by increasing the microwave power and decreasing the heating period simultaneously.

#### *6.4.4. Nonlinear temperature gradients*

The choice of COC and Cu tape for the microfluidic cell (Figure 7) offers several advantages for producing ntegrated microfluidic devices for microwave heating. The high glass transition temperature of COC (*Tg* = 136 °C) as well as its chemical compatibility with acids, alcohols, bases and polar solvents make it suitable for photolithographic procedures. While the low thermal conductivity of COC (0.135 W m−1 K−1) has a negative impact for contact heating approaches, it offers a significant advantage for direct volumetric-based heating strategies by minimizing undesired heat losses, so that a larger fraction of the incident power is contained in the fluid during heating. Additionally, the low dielectric constant (*ε<sup>r</sup>* = 2.35) and the low loss factor (tan *δ* ∼1·10−4 at 10 GHz) of COC make it suitable for highfrequency applications. The use of electro-deposited Cu compared to metal alloys as in the low-melt solder fill technique (Koh et al., 2003) e.g. a combination of indium-bismuth-tin alloy (Yang et al., 2002) for forming the transmissionline electrodes permits high-frequency operation of the devices due to the high electrical conductivity (*σCu* ∼5.51×105 S cm−1, *σ*In-alloy = 0.19×105 S cm−1) of Cu. On the other hand, if the conductor thickness (which is 35 *μ*m in the device of Figure 7) exceed the thickness (3*δ* = 2 *μ*m at 10 GHz for Cu, where *δ* is the skin depth) needed to sufficiently suppress ohmic losses due to the skin effect, then the greater-than-required thickness of Cu (33 *μ*m in this case) acts as a thermal heat sink, limiting the maximum achievable temperature in the microchannel.

An electromagnetic simulation of a geometrical structure similar to that shown in Figure 7(*a*) was performed using Sonnet Software with nominal properties for copper, acrylic, COC and water. The design parameters were varied to optimize microwave power absorption in the fluid since absorption governs the maximum attainable temperature. A trade-off relation was found to exist between the power absorbed in the fluid and the ratio of the channel height to the total substrate thickness (the sum of cover plates and COC thickness). A smaller channel height to substrate thickness ratio reduced the absorbed power for a given fluid volume and incident microwave power. The coupling of the microwave power from the amplifier (less than, but approximately equal to 1W) to the transmission line and the microchannel was characterized theoretically and experimentally by measuring the reflection coefficient (*S*11) and the transmission coefficient (*S*21) and by calculating the absorption ratio (*A* = 1 − |*S*11| |*S*21|), which describes the fraction of the incident power absorbed by the device, where *Sij* (dB) = 10 log10|*Sij*|.

200 The Development and Application of Microwave Heating

*6.4.4. Nonlinear temperature gradients* 

absorbed in the water during the time *dt* that remains in the water. The rest of the energy absorbed in the water during the time *dt* is conducted into the substrate. The value of *Ke* cannot be easily obtained from the geometry and thermal properties of the channel and substrate due to the unknown contact thermal resistance (Kapitza resistance) between the water and the hydrophobic surface of the substrate (Barrat et al., 2003). Rather than attempt to calculate the value of *Ke*, it was adjusted in a least-squares fit of the predictions of eqn. 16 to the measured data points (■) in Figure 13. The solid line in Figure 13 indicates the predicted temperature calculated using *Aw* obtained from Figure 12 (alpha absorption model), and the dashed line indicates the predicted temperature calculated using *Aw*  obtained from the power difference model (*AH2O – Aempty*). It is clear from the results of fitting the predictions of the two different absorption models that the alpha absorption model provides a better fit to the measured data. Further, the heating efficiency obtained from the alpha absorption model indicates that only 5 % of the total heat (time integral of the absorbed power) was stored in the fluid while the rest was lost to the surroundings (PDMS and glass). Because the ratio of stored heat to lost heat increases with decreasing heating time, it is possible to confine most of the heat to the fluid and heat it to a higher temperature by increasing the microwave power and decreasing the heating period simultaneously.

The choice of COC and Cu tape for the microfluidic cell (Figure 7) offers several advantages for producing ntegrated microfluidic devices for microwave heating. The high glass transition temperature of COC (*Tg* = 136 °C) as well as its chemical compatibility with acids, alcohols, bases and polar solvents make it suitable for photolithographic procedures. While the low thermal conductivity of COC (0.135 W m−1 K−1) has a negative impact for contact heating approaches, it offers a significant advantage for direct volumetric-based heating strategies by minimizing undesired heat losses, so that a larger fraction of the incident power is contained in the fluid during heating. Additionally, the low dielectric constant (*ε<sup>r</sup>* = 2.35) and the low loss factor (tan *δ* ∼1·10−4 at 10 GHz) of COC make it suitable for highfrequency applications. The use of electro-deposited Cu compared to metal alloys as in the low-melt solder fill technique (Koh et al., 2003) e.g. a combination of indium-bismuth-tin alloy (Yang et al., 2002) for forming the transmissionline electrodes permits high-frequency operation of the devices due to the high electrical conductivity (*σCu* ∼5.51×105 S cm−1, *σ*In-alloy = 0.19×105 S cm−1) of Cu. On the other hand, if the conductor thickness (which is 35 *μ*m in the device of Figure 7) exceed the thickness (3*δ* = 2 *μ*m at 10 GHz for Cu, where *δ* is the skin depth) needed to sufficiently suppress ohmic losses due to the skin effect, then the greater-than-required thickness of Cu (33 *μ*m in this case) acts as a thermal heat sink,

An electromagnetic simulation of a geometrical structure similar to that shown in Figure 7(*a*) was performed using Sonnet Software with nominal properties for copper, acrylic, COC and water. The design parameters were varied to optimize microwave power absorption in the fluid since absorption governs the maximum attainable temperature. A trade-off relation

limiting the maximum achievable temperature in the microchannel.

Figure 14 shows the simulated and measured absorption ratios for the empty-channel and water filled device. Close agreement is found in both the amplitude and shape for the fullchannel device, but only in shape for the empty-channel device with the correlation coefficient of 0.98 for the empty-channel device and 0.96 for the water filled device. The amplitude deviation in the experimental absorption ratio for the empty-channel device can be attributed to imperfections in the as-fabricated conductor. However, this difference is much smaller for the water-filled device apparently because fluid absorption dominates the absorptive process. The *S*-parameter information was also used to select frequencies where large sinusoidally shaped temperature gradients were expected. Specifically, it was found that constructive interference exists between the traveling and reflected waves at a frequency corresponding to local maxima in the absorption ratio curve for the water-filled

**Figure 14.** Comparison between simulated and measured absorption ratios, the fraction of the incident microwave power absorbed by the device, as a function of frequency. The simulated and measured responses are compared for the empty channel as well as the water-filled device. The curves are constructed from S-parameter data according to A = 1 - |S11| - |S21|, where S11 is the reflection coefficient and S21 is the transmission coefficient. The upper dotted line is simulated full channel response, the upper solid line is measured full channel response, the lower solid line is measured empty channel response, and the lower dotted line is the simulated empty channel response.

device. The absorbed power increases with increasing frequency, and the peaks in the absorbed power exist at a variety of frequencies (Figure 14). However, the amplitude of the absorbed power at the peaks for frequencies lower than 12 GHz is significantly smaller than that at 12 GHz and above. Here, the results of temperature measurements at the lowest (12 GHz) and highest (19 GHz) frequencies that gave relatively large peaks in the absorption ratio data are shown.

Figure 15 shows the experimental temperature profile for the excitation frequency of 19 GHz. The curve in Figure 15 was constructed by using the calibration curve of Rhodamine B dye to convert the fluorescence intensity into temperature. A nonlinearly modulated profile extending along the length of the microchannel was observed. The temperatures measured at different positions was compared with the model of temperature gradient generation by performing nonlinear least-squares fitting of equation 6 to the measured data points using Origin software (solid line in Figure 15) and a good agreement (*R*<sup>2</sup> = 0.98) was found. The parameter estimates and the associated standard errors are listed in Table 2.


**Table 2.** Results from nonlinear least-squares fitting of the temperature gradient model (equation 6) to the measured data points shown in Figures 15 and 16 for one device. A standard error of zero indicates that this value was fixed during the fit.


**Table 3.** The average and standard deviations of the fitting parameters extracted from nonlinear leastsquares fitting of measured temperature gradient data to the theoretical model shown in equation (6). A standard deviation of zero indicates that this value was fixed during the fit as described in the text.

ratio data are shown.

that this value was fixed during the fit.

device. The absorbed power increases with increasing frequency, and the peaks in the absorbed power exist at a variety of frequencies (Figure 14). However, the amplitude of the absorbed power at the peaks for frequencies lower than 12 GHz is significantly smaller than that at 12 GHz and above. Here, the results of temperature measurements at the lowest (12 GHz) and highest (19 GHz) frequencies that gave relatively large peaks in the absorption

Figure 15 shows the experimental temperature profile for the excitation frequency of 19 GHz. The curve in Figure 15 was constructed by using the calibration curve of Rhodamine B dye to convert the fluorescence intensity into temperature. A nonlinearly modulated profile extending along the length of the microchannel was observed. The temperatures measured at different positions was compared with the model of temperature gradient generation by performing nonlinear least-squares fitting of equation 6 to the measured data points using Origin software (solid line in Figure 15) and a good agreement (*R*<sup>2</sup> = 0.98) was found. The

*a* (V mm-1) 6.986 0.051 6.31 0.024 *b* (V mm-1) 0.501 0.041 -0.281 0.017 β (deg mm-1) 29.11 0 46.35 0.974 θp (deg) 264.8 3.25 154.1 7.071 α (mm-1) 0.015 0.002 0.01 0.001 Chi-square 1.193 - 0.257 - *R*2 0.975 - 0.981 - **Table 2.** Results from nonlinear least-squares fitting of the temperature gradient model (equation 6) to the measured data points shown in Figures 15 and 16 for one device. A standard error of zero indicates

*a* (V mm-1) 6.912 0.049 6.427 0.035 *b* (V mm-1) 0.502 0.04 0.092 0.025 β (deg mm-1) 29.106 0.04 46.123 1.833 θp (deg) 259.4 3.229 193.6 14.22 α (mm-1) 0.011 0.002 0.012 0.002 Chi-square 1.142 - 0.712 - *R*2 0.971 - 0.923 - **Table 3.** The average and standard deviations of the fitting parameters extracted from nonlinear leastsquares fitting of measured temperature gradient data to the theoretical model shown in equation (6). A standard deviation of zero indicates that this value was fixed during the fit as described in the text.

12 GHz 19 GHz Value SD Value SD

12 GHz 19 GHz Mean Average SD Mean Average SD

parameter estimates and the associated standard errors are listed in Table 2.

**Figure 15.** The measured temperature versus distance along the microchannel of an aqueous solution of 0.2 mmol L-1 Rhodamine B in a 19 mmol L-1 carbonate buffer at the microwave excitation frequency of 19 GHz. The solid line represents a theoretical temperature fit to the measured data points shown by squares. The measurement frequency was selected based on a local maximum in *A* for the full channel device. At 19 GHz, S11 = 0.040, S21 = 0.343 and *A* = 0.617 for the full channel device.

**Figure 16.** The measured temperature of an aqueous solution of 0.2 mmol L-1 Rhodamine B in a 19 mmol L-1 carbonate buffer as a function of position along the microchannel at the microwave excitation frequency of 12 GHz. The solid line represents a theoretical temperature fit to the data points shown by squares. The measurement frequency was selected based on a local minimum in S11 (not shown here) and a local maximum in *A* for the full channel device. At 12 GHz, S11 = 0.063, S21 = 0.490 and *A* = 0.447 for the full channel device.

For a given geometrical structure, the nonlinear temperature profile (Figure 15) can be altered by changing the frequency of the microwave signal. This is demonstrated in Figure 16, which shows the spatial temperature profile obtained for 12 GHz excitation frequency. Here, a nonlinear profile representing a sinusoidal wave extending along the length of the microchannel was observed. The data also resulted in a quasilinear temperature gradient with a slope of 7.3 °C mm−1 along a 2 mm distance. Linear temperature gradients with comparable slopes have been used for DNA mutation detection (Bienvenue et al., 2006), phase transition measurements in phospholipid membranes (Saiki et al., 1986), singlenucleotide polymorphism (SNP) analysis (Baker., 1995) and continuous-flow thermal gradient PCR (Liu et al., 2007). As before, nonlinear least-squares fitting of the theoretical model (solid line in Figure 16) to the experimental data shows good agreement (*R*2 = 0.98). Due to limitations of the temperature gradient model, the fit to the 12 GHz data was not able to estimate the attenuation constant, *α*, with reasonable uncertainty because of multicollinearity (all of the coefficients of the covariance matrix were *>*0.8). Therefore, the fit to 12 GHz data by fixing *β* (Table 2) has been performed. The value of *β* was obtained from the fit to 19 GHz data by assuming that dispersion was negligible and scaling appropriately for the ratio of the two frequencies. One noteworthy application of a rapid, nonlinear temperature (electric field) gradient is electric field gradient focusing (EFGF) of charged molecules. It has been suggested theoretically that the peak capacity and resolution of EFGF and other related methods, such as TGF (Lagally et al., 2004), could be increased by using a nonlinear field (temperature) gradient provided that the first portion of the gradient is steep, the following section is shallow and that the sample components can be moved from the first portion to the second after focusing in the first portion (Chaudhari et al., 1998; Yoon et al., 2002). To demonstrate the efficacy of the technique for generating temperature gradients, measurements on several different devices were performed. The general shape of the temperature gradient curve was found to reproduce well for all of the measurements. The statistical deviations shown in Table 3 come primarily from variations in the geometrical dimensions of the device introduced during the fabrication process. Even though Rhodamine B dye was used successfully to demonstrate the presence of temperature gradients, accurate generation of the temperature gradient profile warrants improvements to the temperature detection method. In fact, a number of researchers have reported the specific absorption of Rhodamine B in PDMS substrates (Pal et al., 2004; Kopp et al., 1998 ). For the present device structure, the Rhodamine B was found to absorb into acrylic adhesive resulting in temperatures that were representative of channel surface rather than that of the bulk fluid. Additionally, after repeated use the absorption resulted in non-uniform fluorescence intensity across the length of the microchannel indicating that the absorption was non-uniform in space, which limits the reusability of the devices. Hence, these results demonstrate that while the method for generating spatial temperature gradients is robust as shown from the repeated measurements and low statistical error (Table 3), the frequent use of devices results in the channel surface becoming saturated with Rhodamine B dye limiting their overall use.

It should be possible to prevent the absorption of Rhodamine B dye on the microchannel surface by modifying those surfaces with appropriate surface treatments as is typically done for analyte separations in microfluidic devices (Schneegass et al., 2001). Further, a two-step process could be utilized to eliminate potential interactions of Rhodamine B dye with chemicals of interest. As the first step, a set of devices would be used with a dye solution to calibrate the temperature difference versus the microwave power characteristic of the device, and an identical device would be later used without the dye solution for performing biological or chemical studies. Alternatively, an electronic temperature sensor such as a thermocouple or resistance thermometer could be integrated into the device at a convenient reference location along the channel for optical calibration. Figure 17 plots the temperature as a function of time at one location along the microfluidic channel for a 1 s duration pulse of approximately 1 W of microwave power applied to the device. Substantially, more power (about 1 W from the amplifier) was required to raise the temperature of the fluid to 46 ◦C in 1 s than would be required to hold it at this temperature for an additional second as might be required in practical applications. It would be much easier to add feedback control of the microwave power if the temperature at a reference location was measured electronically rather than optically even if Rhodamine B or some other fluorescent dye was compatible with the other chemicals in the microchannel.

204 The Development and Application of Microwave Heating

their overall use.

For a given geometrical structure, the nonlinear temperature profile (Figure 15) can be altered by changing the frequency of the microwave signal. This is demonstrated in Figure 16, which shows the spatial temperature profile obtained for 12 GHz excitation frequency. Here, a nonlinear profile representing a sinusoidal wave extending along the length of the microchannel was observed. The data also resulted in a quasilinear temperature gradient with a slope of 7.3 °C mm−1 along a 2 mm distance. Linear temperature gradients with comparable slopes have been used for DNA mutation detection (Bienvenue et al., 2006), phase transition measurements in phospholipid membranes (Saiki et al., 1986), singlenucleotide polymorphism (SNP) analysis (Baker., 1995) and continuous-flow thermal gradient PCR (Liu et al., 2007). As before, nonlinear least-squares fitting of the theoretical model (solid line in Figure 16) to the experimental data shows good agreement (*R*2 = 0.98). Due to limitations of the temperature gradient model, the fit to the 12 GHz data was not able to estimate the attenuation constant, *α*, with reasonable uncertainty because of multicollinearity (all of the coefficients of the covariance matrix were *>*0.8). Therefore, the fit to 12 GHz data by fixing *β* (Table 2) has been performed. The value of *β* was obtained from the fit to 19 GHz data by assuming that dispersion was negligible and scaling appropriately for the ratio of the two frequencies. One noteworthy application of a rapid, nonlinear temperature (electric field) gradient is electric field gradient focusing (EFGF) of charged molecules. It has been suggested theoretically that the peak capacity and resolution of EFGF and other related methods, such as TGF (Lagally et al., 2004), could be increased by using a nonlinear field (temperature) gradient provided that the first portion of the gradient is steep, the following section is shallow and that the sample components can be moved from the first portion to the second after focusing in the first portion (Chaudhari et al., 1998; Yoon et al., 2002). To demonstrate the efficacy of the technique for generating temperature gradients, measurements on several different devices were performed. The general shape of the temperature gradient curve was found to reproduce well for all of the measurements. The statistical deviations shown in Table 3 come primarily from variations in the geometrical dimensions of the device introduced during the fabrication process. Even though Rhodamine B dye was used successfully to demonstrate the presence of temperature gradients, accurate generation of the temperature gradient profile warrants improvements to the temperature detection method. In fact, a number of researchers have reported the specific absorption of Rhodamine B in PDMS substrates (Pal et al., 2004; Kopp et al., 1998 ). For the present device structure, the Rhodamine B was found to absorb into acrylic adhesive resulting in temperatures that were representative of channel surface rather than that of the bulk fluid. Additionally, after repeated use the absorption resulted in non-uniform fluorescence intensity across the length of the microchannel indicating that the absorption was non-uniform in space, which limits the reusability of the devices. Hence, these results demonstrate that while the method for generating spatial temperature gradients is robust as shown from the repeated measurements and low statistical error (Table 3), the frequent use of devices results in the channel surface becoming saturated with Rhodamine B dye limiting

It should be possible to prevent the absorption of Rhodamine B dye on the microchannel surface by modifying those surfaces with appropriate surface treatments as is typically done

**Figure 17.** Transient temperature response of the integrated microfluidic device. A temperature of 46 0C was obtained at some locations in the microchannel with 1W of microwave power at the output of power amplifier for 1s. The fact that the temperature did not reach a steady state value in this time shows that considerably less power would be required to hold the temperature constant for a second. The addition of feedback control and a higher power amplifier would facilitate higher temperatures, a faster rate of increase in temperature, as well as the capability to hold the fluid temperature constant for a short period of time without raising the device temperature significantly.

Finally, the growing concern that exposure to microwaves can be harmful to living cells may limit the ability to operate highly integrated lab-on-a-chip devices containing living microorganisms in conventional microwave ovens. On the other hand, the microwave field decreases rather rapidly away from a properly designed microscale microwave generator, potentially allowing live organisms and microwave transducers to co-exist on a lab-on-achip device. The approach explained to establish temperature gradients appears to be especially well suited for thermal gradient focusing methods for analyte separations of cell metabolites in lab-on-a-chip devices. Other potential applications of integrated microwave heaters include cell lysis and PCR (Liu et al., 2002), as mentioned previously. The localized nature of on-chip microwave heating means that separate microwave heaters optimized for these different tasks could also co-exist on a single lab-on-a-chip device. Therefore, it is believed that the technique outlined here will facilitate the application of microfluidics to other biological and chemical applications requiring spatial temperature gradients as well as to temperature gradient generation.
