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J. Phys. Chem. C 2010, 114, 10132–10139
Gold Nanoparticles Allow Optoplasmonic Evaporation from Open Silica Cells with a Logarithmic Approach to Steady-State Thermal Profiles Aaron G. Russell,† Matthew D. McKnight,† Adam C. Sharp,‡ Jamie A. Hestekin,† and D. Keith Roper*,†,§ Ralph E. Martin Department of Chemical Engineering, 3202 Bell Engineering Center, and Department of Industrial Engineering, 4207 Bell Engineering Center, UniVersity of Arkansas, FayetteVille, Arkansas 72703, and Department of Materials Science and Engineering, 304 CME, UniVersity of Utah, Salt Lake City, Utah 84112 ReceiVed: February 26, 2010; ReVised Manuscript ReceiVed: April 20, 2010
In this work, plasmonically heated solid-state gold nanoparticle (AuNP) arrays are investigated under novel conditions that include large (>35 °C) steady-state (SS) temperature increases (∆T) dominated by conduction in open environments that allow vapor-liquid phase change. Evaporative cooling from the open system decreases SS ∆T of the system by as much as (11.6 ( 0.33) °C (45%), consistent with predictions from an energy balance model expanded in this work to account for evaporative cooling and associated decreasing thermal mass. Comparing dynamic and steady temperature profiles from water evaporating from a AuNPcoated Si cell at 50 mW laser irradiation with the model yielded an average accumulated residual sum of squares of 2.95 °C2 over 200 s. Temperature increases that distribute nonuniformly across sample cell surfaces due to high laser power (e150 mW) and conductive heat transfer are accurately and uniformly (17-fold and >10-fold reduction in system equilibration time constants for air and water cells, respectively, and an increase in laserto-heat transduction efficiency from (3.12 ( 0.40)-(9.92 ( 2.06)% (colloidal AuNPs, water) to (96.92 ( 8.94)% (AuNP thin film, air cell) and (99.22 ( 2.51)% (AuNP thin film, water cell). System SS ∆T per incident laser power (I), ∆T/I, increased from 24.51 ( 4.45 °C W1- to 157.07 ( 21.68 °C W1-. The enhanced thermal properties arise from resonant near-field interactions that increase LSP resonance (LSPR) absorption at the excitation wavelength as well as from improved energy transfer within the system.35 Fluids within these OPCs were closed to the environment.
10.1021/jp101762n 2010 American Chemical Society Published on Web 05/14/2010
AuNP in Optoplasmonic Evaporation from Silica
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TABLE 1: Physical and Optical Properties of AuNP Sample Cells
This study examines dynamic and steady-state thermal behavior of OPCs under long-term (3-4 min), CW excitation at powers sufficient for moderate to high system SS ∆T (7-37 °C) in open cells containing either air or water. At these conditions, large-scale evaporation concurrent with decrease in thermal mass occurred in the open water cell. Terms representing these phenomena were integrated into the energy balance model validated previously in closed systems to characterize resulting temperature profiles in open OPCs at large SS ∆T. Predictable steady-state temperature profiles are observed on OPCs in which conduction dominates heat transfer. The ability to predict and control these effects at the scales demonstrated in this work could have many important uses in the above-referenced applications. 2. Experimental Methods 2.1. Sample Cell Fabrication. The samples for this work were fabricated according to a previously developed electroless (EL) plating and thermal transformation method.35–37 The internal walls of a chemically etched, borosilicate glass rectangular capillary were sensitized by a tin (Sn2+) solution and activated with ammoniacal AgNO3, which was then galvanically replaced by Au reduced from a gold sulfite solution, Na3[Au(SO3)2]. Each solution was inserted into an open end of the capillary and was allowed to incubate before removal. Sensitization, activation and galvanic replacement steps were repeated in 1 min, 30 s, and 30 s intervals for total exposure times of 3 min, 2 and 4 min, respectively. After each interval, the solution insertion end was alternated to increase plating uniformity. Excess metal residue was removed between steps by washing the internal walls of the capillary with room temperature distilled, deionized and degassed water (D3-H2O). AuNPs were then formed from the EL plated Au film by subjecting it to successive thermal annealing steps at 350 and 750 °C. Heating ramp times at these temperatures were 1 h and
30 min with exposure times of 3 h and 30 min, respectively. The furnace was allowed to cool naturally for ∼2 h before the samples were removed. The furnace was purged with N2 gas for ∼3 min before heating. The capillary dimensions are given in Table 1. Example capillary pictures and scanning electron microscope images in Table 1 illustrate the characteristic change of color that occurs as a result of thermal transformation from Au film to AuNPs. 2.2. Thermal Evaluation. The sample was placed orthogonally to a CW Ar+ gas laser beam (514.5 nm Highlight, Coherent, Palo Alto, CA). The laser beam contacted a neutral density filter, two mirrors and a beam splitter prior to contact with the sample. The beam splitter decreased incident laser power by 50%. Decreases in incident laser power were measured using a power meter (Melles Griot). Reported laser power values represent actual incident values after correcting for any decreases. The laser beam spot size was 2.4 mm. Temperature changes of the outer capillary wall were monitored using a thermocouple (TC) and recorded by a digital thermometer (HH509R, Omega, Stamford, CT) and computer. Figure 1 depicts the equipment setup and laser path and provides a picture of the sample cell and mount. Laser power was measured immediately before and after sample irradiation to ensure its consistency throughout the experiment. The power meter was positioned to measure the transmitted laser power during the experimental run, and the data was recorded by hand and streamed to a data acquisition card (NI USB-6009, National Instruments, Austin, TX). The laser was modulated for half the experimental runs with a 30slot mechanical chopper (model SR540, Stanford Research Systems, Sunnyvale, CA) operating at 2 kHz, resulting in a 50% laser power reduction. Laser power was attenuated to the same level for the unchopped runs by adjusting the neutral density filter. Temperature profiles were recorded for 30 s prior to exposing the capillaries to the laser. The heating response was
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Figure 1. Experimental setup used to measure thermal responses of laser irradiated AuNP sample cells.
measured for 4 min. This was sufficient time to reach steadystate temperature. A one minute cooling cycle was also recorded by the TC before the experimental run was concluded. The temperature sampling rate was 0.5 Hz. The ambient temperature and humidity were recorded using a “Humidity Alert” monitor (445814, Extech Instruments, Waltham, MA), and the sample was shielded from air currents by placing it inside a plexiglass enclosure. This procedure was performed at two different positions on the capillary. Position 1 (P1) and position 2 (P2) were located approximately 3.5 mm and 1.5 mm from the laser spot, respectively. The procedure was repeated at P2 using the water cell. The cell remained open at both ends allowing evaporation to occur, and the cell was refilled after each experimental run. The procedure was repeated a minimum of eight times for each type of cell, four chopped and four unchopped at three laser powers: 50 mW, 100 mW and 150 mW. An alternative setup was used to measure spectral characteristics of the system. A DC regulated, fiber optic white light (WL) source (EW-41720-40, Cole Parmer, Vernon Hills, IL) was focused using an objective lens and passed through the beam splitter onto the same axis as the laser beam. The WL was focused to a spot size completely contained within the area of the laser beam on the sample. This made it possible to record the spectra of only AuNPs that were excited by the laser. After passing through the cell, the laser beam and WL were focused by two additional objective lenses through a notch filter (514.5 nm HSPF-514.5-1.0, Kaiser Optical Systems, Inc.) and into a collimator. The notch filter reduced the laser beam power by 6 orders of magnitude to protect the spectrometer. The captured light was then passed from the collimator through a fiber optic cable to the spectrometer (USB2000, Ocean Optics, Dunedin, FL) in which the transmission signal was recorded. The AuNP cell exhibited a LSPR wavelength (λLSPR) of 531.8 nm. The λLSPR
Figure 2. Temperature profile of the outer wall of an AuNP plated borosilicate capillary during a 240 s continuous irradiation period by a 100 mW Ar Ion laser. Results for both chopped and unchopped laser beam are shown.
was red-shifted slightly in the water cell to 534.0 nm due to the difference in refractive index. Extinction spectra and values for the cell at these wavelengths and at the wavelength of excitation (λexc, 514.5 nm) are shown in Table 1. 3. Results and Discussion Substrate Thermal Profiles Exhibit Characteristic Dynamic and Steady-State Regions. Figure 2 displays a plot of outer wall temperature changes with respect to ambient temperature versus time for the air cell at a laser power of 100 mW. The shape of the thermal profiles shownsa brief transient period of rapid temperature increase followed by a plateau when steady state is reachedsis typical of all the samples. Similar plots were generated, with and without laser modulation, for the air cell at P1 and P2 and for the water cell at P2. These
AuNP in Optoplasmonic Evaporation from Silica profiles were then fit to a previously established energy balance.33,34 This balance was used to calculate system time constants, τs, and transduction efficiencies, ηT, for the two types of cells. This model was expanded to account for energy loss by liquid evaporation during irradiation. Limitations of this model were considered, and an alternative approach was developed. Laser Modulation at 1-4 kHz Chopping Rates Has Minimal Effect on Solid-State AuNP Thermal Behavior. One aspect of this work focused on investigating the effect of laser modulation on the thermal behavior of solid-state AuNPs arrayed on silica exposed to different fluid environments. Ahn et al. demonstrated that modulation of the incident laser beam by mechanical chopping increased the transduction efficiency of colloidal suspensions of 20 nm AuNPs, centrifuge-concentrated to 0.092 g of AuNPs per 100 mL of D3-H2O, greater than 2-fold.33 Chopping at a rate faster than diffusional collision of adjacent, suspended AuNPs reduced aggregation and subsequent precipitation. Richardson et al. reported that laser modulation did not affect the transduction efficiency of aqueous solutions of 20 nm AuNPs with a concentration of 7 × 1010 particles/ cm3 in a suspended droplet,32 a result attributable to their substantially lower AuNP concentration. The effects on solidstate AuNP arrays were investigated by the method detailed above: utilizing the neutral density filter to tune actual incident laser power to a consistent level regardless of whether or not the mechanical chopper was included in the laser path. The chopping rate was varied between 1 and 4 kHz. When only statistically insignificant differences were observed for example, the chopping rate was kept constant at 2 kHz for all data points. The typical results of chopped versus unchopped thermal profile comparison can be seen in Figure 2. System equilibration time constants, which characterize the transient behavior of the system, varied by an average of less than 5% and one standard deviation for chopped runs compared with unchopped runs. Differences in SS ∆Τ of chopped runs compared to unchopped runs amounted to an average of less than 1.5% of the total change. In general, the magnitude of the temperature change was slightly greater for unchopped runs; however this was not always the case. Differences in both transient and steady-state thermal behavior were on the order of those that occurred due to environmental variations. AuNPs in this study are covalently bound to silica walls via electroless plating, preventing any diffusional collisions that could result in aggregation. Negligible observed effect of laser modulation on optothermal behavior of fixed AuNPs supports the hypothesis that free-solution diffusional collisions, rather than slow relaxation of excited surface plasmons, are responsible for light-induced reductions in thermal efficiency of suspended AuNPs. Given this result, the remaining discussions will focus on the unchopped data alone. SS ∆Τ Increases Disproportionately in Open, Water-Filled Cells and in Air Cells at High Power. Figure 3 shows the average SS ∆T values for each cell at each level of laser power. The air cell at P1 (P1A) exhibits average SS ∆T values of 6.9 °C, 11.7 °C, and 18.1 °C for incident laser powers of 50 mW, 100 mW, and 150 mW, respectively, consistent with prior reports that SS ∆T increases in proportion to incident laser power.35 The remaining data points did not exhibit a consistent proportional response from 50 to 150 mW. The air cell at P2A displayed the expected response for the increase of 100 mW and 150 mW with SS ∆Ts of 25.7 and 37.2 °C, respectively. However, when the power was increased from 50 mW to 100 mW, the SS ∆T for P2A increased by a factor of 2.8, from 9.2
J. Phys. Chem. C, Vol. 114, No. 22, 2010 10135
Figure 3. Steady-state temperature change with respect to ambient temperature as a function of incident, unchopped laser power. Points represent air cells at P1 (diamond) and P2 (square) and a water (triangle) cell at P2. Error bars represent one standard deviation.
to 25.8 °C. The opposite effect was observed in the water (P2W) cell. The P2W cell experienced an increase in SS ∆T of approximately the expected factor of 2 between 50 mW and 100 mW. However, between 100 mW and 150 mW, it showed an increase in SS ∆T by greater than a factor of 2 rather than the expected value of 1.5. Evaporative cooling and exponentially decreasing thermal profiles in these large, open capillaries were found to explain these results, as discussed in the following sections. Evaporative Cooling Causes Reduction in SS ∆T for Water Cells. Figure 3 shows that SS ∆T increases less in proportion to increasing laser power in cells filled with a liquid that can evaporate from its open ends. At 50 mW, the P2W cell exhibits an average SS ∆T reduction of 1.85 °C. The difference in SS ∆T between air and water cells increases substantially to 11.62 °C at 100 mW due to the increased rate of evaporation. Increasing irradiation power from 100 mW to 150 mW reverses the previous trend: evaporative cooling lowers SS ∆T by 6.93 °C for the P2W cell. Energy Balance for a Closed, Isothermal System. The data were initially evaluated using an energy balance for a closed system at a uniform temperature that was introduced by Roper et al.,33 evaluated by Hoepfner and Roper for applicability at large scales,34 and adopted by others.32,35,38 An overall energy balance was applied to the system of the form
[QI - U(T - Tamb)] dT ) dt miCp,i
∑
(1)
i
where the i terms in miCp,i represent the thermal masses of the components of the system (sample cell and fluid), T is the uniform temperature of the system, Tamb is the temperature of the ambient environment and t is time. QI represents energy input to the system, i.e. the transduced laser power defined as
QI ) I(1 - 10-Aλ)ηT
(2)
where I is the incident laser power, Aλ is the measured AuNP extinction at the wavelength of excitation, λ, and ηT is the transduction efficiency. The value of ηT represents the fraction of extinguished laser power that is transduced to thermal energy. U is an overall heat transfer coefficient that includes the different heat transfer areas and is defined as
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U≡
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kAcond + hconvAs + Asεσ(T2 + Tamb2)(T + Tamb) L
(3) where k is the thermal conductivity of the material which conducts heat away from the cell, Ac is the contact area perpendicular to conduction, L is the characteristic length for conduction to the environment, hconv is the heat transfer coefficient specifically for convection, As is the surface area of the sample cell exposed to air, ε is the emissivity of AuNPs and σ is the Stefan-Boltzmann constant. Design of this particular experimental system allowed QI to be alternately defined as
QI ) (I - IT)ηT
(4)
where IT is the transmitted laser power measured by the power meter. Calculations were performed using both methods to determine consistency. Equation 1 can be solved by two methods: a numerical calculation of ηT can be made by fitting data such as in Figure 2 to eq 1, or a convenient linearization can be made that allows a direct analytical determination of ηT. The analytical solution involves the defining of an overall system time constant, τs,
τs ≡
∑ miCp,i i
hA
(5)
In this equation, h functions as a constant, overall heat transfer coefficient and is equivalent to U from eq 3 divided by the total surface area of the system. Thus, total heat transfer from the system becomes linear in T and the quartic dependence of radiation may be neglected at typical values of T. This time constant is determined by fitting the solution of the linearized energy balance to the experimentally obtained thermal profiles. It is then used to solve for ηT, which is given by
ηT )
hA(Tmax - Tamb) I(1 - 10-Aλ)
(6)
To be applicable to an open system experiencing evaporative cooling whose internal temperature may not equilibrate much faster than energy exchanges with the environment, these solutions derived for closed systems with uniform temperatures required modifications that are outlined in the following sections. Accounting for Evaporative Cooling Allows Prediction of SS ∆T for Water-Filled Cells. Applicability of the model in eqs 1-6 can be extended to systems experiencing evaporating cooling by expanding eq 1 to account for both the associated energy loss mode and the change in thermal mass as the evaporation occurs:
[QI - U(T - Tamb) - m ˙ ∆Hv,f] dT ) dt miCp,i - (mf - m ˙ t)Cp,f
∑
(7)
i
where m ˙ is the evaporation rate of the liquid, ∆Hv,f is the latent heat of vaporization of the liquid, and Cp,f is the heat capacity of the liquid. As the large-scale system temperature remained
Figure 4. Measured (data points) and predicted (lines) thermal profiles of air and water sample cells as function of time with 50 mW laser irradiation power. Predicted profiles were calculated using the evaporative cooling model.
below the boiling point of the liquid, the evaporation was determined to be convective, giving
m ˙ ) AE(MWf)kc∆Cf
(8)
where AE is the area available for evaporation, MWf is the molecular weight of the liquid, kc is the convective mass transfer coefficient, and ∆Cf is the difference in liquid concentration between the liquid boundary and the environment. The term kc was given by
kc )
NA ∆Cf
(9)
where NA is the experimentally determined average flux of the liquid from the substrate surface. Figure 4 shows the thermal profiles predicted by the evaporative cooling model (ECM) given by eqs 7-9, compared to the experimental data for the P2A and P2W cells at 50 mW. Parameter values used in the ECM were hconv ) 11.9 W m-2 K-1, calculated from a natural convection model,39 ε ) 0.8,34 and ηT ) 1.09 taken from the analytical solution. The value of kc was determined to be 0.061 m s-1 for the P2W cell. Rather than estimate the thermal conductivity, k, and conduction length, L, governing heat conduction to the sample mount, the parameter k/L was used to fit the P2A cell SS ∆T, yielding a value of 112.5 W m-2 K-1. This value was used to predict SS ∆T for the water cell; all other parameters used were determined a priori, as listed above. The contributions to U in eq 7 for the air cell due to conduction, convection and radiation are 2.25 × 10-3 W K-1, 1.17 × 10-3 W K-1 and 4.97 × 10-4 W K-1, respectively: conduction dominates the heat transfer. This differs from previous work in which radiation has been the dominant heat loss mode and results from the large contact area between the cell and the optical mount (see Figure 1). Figure 4 shows the thermal profile predicted by the ECM agrees with P2W data in both the transient and steady-state regions giving an average residual sum of squares (RSS) of 2.95 °C2 over 200 s of 50 mW irradiation. The SS ∆T of the P2W cell is predicted to be 7.34 °C compared to a measured value of 7.33 °C, a difference of less than 0.14%. The values of τs for the ECM and the experimental data were 18.9 s and (15.56 ( 1.58) s, respectively. This represents a larger deviation than is seen with SS ∆T, but the actual transient temperature values remain close as is shown
AuNP in Optoplasmonic Evaporation from Silica in Figure 4. Validation of the ECM for other liquids (e.g., ethanol, isopropanol, butanol) is the subject of ongoing investigation. The value of SS ∆T calculated by the ECM accounting for dynamic thermal mass and evaporation energy losses is a pseudo-steady-state value. Evaporation rate reaches a maximum when a relatively constant temperature change is reached in the water cell (∼100 s). The instantaneous evaporation rate of water at this point is 2.08 × 10-3 mg s-1 or a loss of 0.021% of the total initial mass of the liquid per second. Instantaneous evaporative power loss for the P2W cell is 5.1 mW or 16.4% of the total measured extinguished laser power. The relative magnitude of these percentages and the accuracy of the model in comparison to experimental results demonstrate that plasmonic heating of these structures can achieve moderate, predictable pseudo-steady-state temperature increases on AuNP plated surfaces while rapidly evaporating liquid from the substrate. Nonuniform Cell Temperatures Produce ηT > 1 If Calculated Assuming a Uniform Cell Temperature. The values of τs obtained from the analytical model were calculated using the cooling portion of the curve at the point when ∆T had been reduced by 98%. The average calculated values for the P1A, P2A and P2W cells were (6.53 ( 0.41) s, (7.36 ( 0.74) s and (15.56 ( 1.58) s, respectively. These values are similar to those previously reported for these types of systems.35 As expected for the P2W cell, τs increases significantly due to the large increase in total thermal mass of the system. However, transduction efficiencies, ηT, calculated using eqs 1-6 which assume a uniform cell temperature resulted in values >1. Previously reported values of ηT were approximately 1 for systems resembling these, indicating that virtually all of the absorbed laser power was transduced to thermal energy.35 Richardson et al. also reported ηT values remarkably close to 1 for their systems.32 The analytical solution yielded values relatively consistent with these results for all P1A data points (0.95 ( 0.039, 0.92 ( 0.045 and 0.95 ( 0.066 at 50 mW, 100 mW and 150 mW, respectively), one P2A point (1.13 ( 0.091 at 50 mW) and two P2W points (1.09 ( 0.063 and 1.00 ( 0.055 at 50 mW and 100 mW, respectively). For the remaining data points, values of ηT e 2 were calculated. Measuring temperature values across the length of the capillary revealed that values of ηT >1 are produced by a nonuniform temperature distribution across the cell. The consistency of τs values given by this model with those previously reported demonstrates the robustness of the analytical model’s ability to describe the dynamic behavior of these systems even when a nonuniform temperature distribution is exhibited. The data points in the inset of Figure 5 show the center line steady-state temperatures of five points along the half the length of the air cell for three laser powers with the laser positioned in the center of the cell (0 mm on the x-axis). As incident laser power increases, the cell temperature clearly becomes nonuniform, approaching an exponential distribution. The values of ηT > 1 result from the uniform temperature model’s inability to account for an exponential SS temperature profile. Lower laser powers and increased thermal conductivity (e.g., water-filled cells) yield more uniform temperature distributions. The uniform temperature model describes the P1A cell relatively well because the thermocouple is located farther from the laser spot and measures a lower temperature closer to the average for the cell than at P2A. Approximation of nonuniform temperature profiles is possible using a well-established model for an infinite fin.
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Figure 5. Comparison of nondimensionalized steady-state temperature data points, a single infinite fin distribution and uniform temperature model distributions along half the cell length during laser irradiation of 50 mW (triangles), 100 mW (squares), and 150 mW (diamonds) located at the center of the cell (x ) 0). Inset shows the temperature data points and individually calculated infinite fin distributions before nondimensionalization.
Infinite Fin Model Describes Nonuniform Temperature Distributions in Cells. As an initial description of these systems’ behavior, we fit the data points shown in inset of Figure 5 to an adaptation of the infinite fin equation given by C¸engel39 as
T(x) - T∞ ) e-ax Tb - T∞
(10)
where T(x) is the temperature at position x along the fin, T∞ is the ambient temperature, Tb is the fin base temperature (temperature of the capillary at the center of the laser spot), and a is a property of the system given by
a)
hP kfinAc
(11)
where h in this case is the overall heat transfer coefficient defined in eq 5, kfin is the thermal conductivity of the fin, P is the perimeter of the fin cross-section and Ac is the cross-sectional area of the fin. The calculations were done for a solid borosilicate fin with the same cross-sectional area as the capillary. The values of a in eq 10 for the 50 mW, 100 mW and 150 mW data given by the fit were 377.2, 400.1, and 402.9. As laser power is increased, the temperature distribution of the cell more closely corresponds to that of the infinite fin model and, thus, calculated values of a approach an asymptotic value of ∼403. At lower levels of laser irradiation, there is not sufficient energy input to the system to adhere to the requirements of the infinite fin model: ambient temperature is reached at or very near the end of the fin. The resulting distributions using these values of a are shown as lines in the Figure 5 inset. The R2 values of the fits were 0.99, 0.99, and 0.92 for the 150 mW, 100 mW and 50 mW data points, respectively. The smaller R2 value for the 50 mW data results from the outlier point at 1.5 mm from the capillary center. The data points for all three powers at this position were the original points used in the analytical calculations and were taken under a different system arrangement than the other four points. Exclusion of this outlier from the 50 mW fit increases the R2 value to 0.99. Figure 5 shows the nondimensionalized data points, the infinite fin distribution using the asymptotic value of a and the
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TABLE 2: Overall Heat Transfer Coefficients for Numerical, Analytical and Infinite Fin Models at Incident Laser Powers of 50, 100, and 150 overall heat transfer coefficient (h, W m-2 K-1)
deviation from fin h (%)
power (mW)
numerical
analytical
fin
numerical
analytical
50 100 150
34.42 41.99 37.07
34.36 42.04 36.97
36.3 41.15 41.75
5.17 2.05 11.21
5.34 2.16 11.45
distributions used in the uniform temperature model for all three laser powers. It is clear from comparison of these profiles why values of ηT >1 were calculated for the 100 mW and 150 mW data, but not for the 50 mW data. The SS ∆T used in the analytical calculations for the 50 mW data (9.18 °C) fell very close to an average SS ∆T calculated from the infinite fin distribution (9.71 °C) by integrating the dimensionless temperature profile over the dimensionless length of the capillary, dividing by the total distance, entering the resulting value into the right side of eq 10 and solving for the numerator of the left side. Average SS ∆Ts for 100 mW and 150 mW calculated by this method were 17.47 and 23.64 °C, respectively, which differed significantly more from the experimental values of 25.75 and 37.24 °C. If the experimental values are taken to be constant along the length of the capillary, much more input energy would be required than was actually used. The values of ηT for the 100 mW and 150 mW data can then be better reconciled with expected values near 1 by entering the average SS ∆Ts calculated from the infinite fin distribution into the analytical solution in place of the measured SS ∆Ts. This yields ηT values of 1.29 ( 0.060 and 1.02 ( 0.048 for the 100 mW and 150 mW data, respectively, compared to the original values of 1.9 ( 0.088 and 1.61 ( 0.076. The consistency in nondimensionalized data points at 50, 100, and 150 mW suggests that the single parameter, a, in eq 11 accounts for all measurable cooling effects in open, optoplasmonically heated cells under the conditions shown, provided that the temperature at the capillary’s end is near enough to Tamb to produce an infinite fin. As Figure 5 demonstrates, the single asymptotic value of a produces an accurate fit for each of the three laser powers. Therefore, this value of a provides an excellent initial estimate of the SS temperature profile of these systems. The values of the overall heat transfer coefficient, h, from eq 11, from the analytical solution in eq 5 and from the numerical solution given by dividing U in eq 3 by the total surface area of the system are shown in Table 2 for air cells at all three laser powers. The fin values of h, evaluated from the steady-state temperature distribution of the cell, and the analytical and numerical values, calculated from the dynamic thermal evolution of a single point on the cell, agree within 2.05 to 11.45%. This confirms the model independence of this parameter, h, that includes all modes of heat transfer from the system allowing prediction of either the steady-state or dynamic thermal behavior of the system from evaluation of the other. The consistency of this technique will be a subject of future work. 4. Conclusions The dynamic and steady-state thermal behavior of plasmonically heated solid-state gold nanoparticles on silica in millimeterscale systems was investigated for processes including high (>35 °C) steady-state (SS) temperature increases (∆T), large conductive heat transfer, and liquid-vapor phase change. Modulation of the laser at 1-4 kHz was determined to have a minimal effect
on either the dynamic (
