Photothermal Imaging of Transient and Steady State Convection Dynamics in Primary Alkanes

: This paper presents a photothermal spectroscopy technique that effectively images convective heat ﬂow in molecular liquids resulting from localized laser-induced heating. The method combines aspects of thermal lensing and photothermal deﬂection. A high-energy infrared laser is used to induce a thermal lens in the sample, and a divergent visible laser is used to probe the entire region of the excitation beam within the sample. This approach allows for the observation of the convective ﬂow of the liquid above the excitation beam. The study focuses on the liquid primary alkanes, from n -pentane to n -pentadecane. The paper provides experimental results, including dynamical data for the propagation of the thermal plume, a transient feature, in these alkanes and the exploration of dependence on excitation laser power.


Introduction
This article presents experimental results for a variant of photothermal spectroscopy [1][2][3].This technique is an imaging method that is particularly good at detecting convective heat flow in liquids arising from localized heating by a laser beam.As a proof-of-principle experiment and for characterization of this technique, the primary alkanes ranging from n-pentane (C 5 H 12 ) to n-pentadecane (C 15 H 32 ) are studied.The aim of this work is to study the thermal plume head dynamics for this family of important solvents.The primary contributions of these experimental results are (i) a variant in the experimental setup and (ii) direct visual and quantitative measurement of the thermal plume head position during the early part of its rise.Correlation is made with carbon chain length, viscosity, and Prandtl number.
Photothermal spectroscopies are non-destructive analytical techniques based on the transfer of light energy (often from a laser) into heat energy within a sample [1][2][3][4].When a sample is irradiated with light, it absorbs the light and converts it into heat, leading to spatiotemporal temperature gradients.Temperature gradients can be (indirectly) detected by various means, such as changes in refractive index, thermal expansion, or generation of acoustic waves, depending on the specific photothermal technique employed.The intensity of the photothermal signal is directly proportional to the light absorption coefficient of the sample.The three primary branches of these methods are photoacoustic, thermal lensing, and photothermal deflection spectroscopies.The method discussed in this work primarily has aspects of thermal lensing and photothermal deflection.
Over the years, these techniques have evolved and diversified, leading to the development of various methods, each with its unique advantages and applications.Very recent work includes the development of photothermal microscopy [5], study of heterogeneous systems [6,7] use in photobiology to study channelrhodopsins [8] and other solvated proteins [9,10], and in medicine such as for photodynamic cancer therapy [11,12].For a thorough review from 2022, the reader is directed to the work of Proskurnin et al. [4].
Thermal lensing, which developed in the late 1960s and early 1970s, is based on the creation of a thermal lens in the sample due to the spatially non-uniform heating caused by the absorption of a focused laser beam [1,4,[13][14][15][16][17][18][19].The thermal lens effect can be observed as a change in the refractive index of the medium, which can be measured using a probe beam.This technique is highly sensitive and can be used to measure very low absorption coefficients, making it particularly useful in the study of transparent materials and liquids.As mentioned above, in the case of the current work, the primary alkanes are studied: n-pentane (C 5 H 12 ) through n-pentadecane (C 15 H 32 ).These substances are liquids at room temperature and transparent to both the excitation beam (1030 nm for this work) and the probe beam (650 nm for this work).
Photothermal deflection spectroscopy, also known as the mirage effect, is a technique in which an excitation beam is used to heat the sample, creating a temperature gradient in the surrounding medium, which leads to a change in its refractive index [20,21].A probe beam deflected due to this change is then detected to measure the absorption of the sample.This technique has high sensitivity and has been widely used for the study of thin films and surfaces.
The photothermal technique discussed in this work combines the thermal lensing effect and the mirage effect.A collimated, rather than focused, high-energy IR laser irradiates a sample, inducing a thermal cylindrical lens along its path through the sample cuvette.A visible (red, in the current case) divergent laser illuminates the sample perpendicularly to the excitation beam.The divergent nature of the beam allows it to probe the entire length of the excitation beam in the sample as well as many millimeters above and below it.Beyond the sample, the probe beam is captured by a camera.The advantage of this technique is that one is able to observe the convective flow of the liquid above the excitation beam.Figure 1 shows a schematic of the experimental setup.The method is reminiscent of that of Longaker and Litvik, which they published in 1969 [22].They, too, used an excitation-perpendicularto-probe beam arrangement to irradiate the sample and corresponding camera.The study of heat transfer in molecular liquids is of great importance due to its extensive use in various fields such as chemistry, pharmacology, medicine, electronics, and computer technology.With the aim of device miniaturization and process efficiency at an industrial scale, improving the performance of heat transfer fluids has been a long-standing objective.Broadly speaking, heat transfer occurs via two distinct modes of transport: conduction and convection.Much of the theory and experimental investigation developed to date for photothermal phenomena focuses on conduction.Lately, groups such as that of Goswami et al. [23][24][25][26][27], along with others [10,28], have studied the role of convection, particularly in thermal lensing.This recent work builds upon earlier work such as that of Buffett and Morris [29].
The underlying physics of the traditional photothermal spectroscopies has been that of conduction (and photoacoustic waves).These traditionally considered effects are symmetric in character.For the current case of a collimated cylindrical laser beam through a cuvette, these effects would have cylindrical symmetry about the beam position.Convection, on the other hand, is inherently asymmetric because the heated region of the sample will rise vertically.
Convection can be categorized into two main types: free convection and forced convection [30].Free convection occurs when the fluid motion is caused by buoyancy forces that result from density variations due to temperature changes in the fluid.When a fluid is heated, it expands and becomes less dense than the surrounding cooler fluid.The less-dense heated fluid then rises, and the cooler, denser fluid sinks.This movement sets up a circulating flow of fluid, which transfers heat from the warmer area to the cooler area.Forced convection, on the other hand, occurs when the fluid movement is induced by an external source, such as a pump, fan, or wind.In this case, the heat transfer does not rely on the natural buoyancy-induced flow of the fluid but rather on the mechanical movement imposed on it [30].Only free convection is considered in the present work.
Free convection is of great importance in engineering and the physical and life sciences.For example, understanding the role of convection in liquids is an active area of study in microfluidics.Convective heat transfer is more difficult to characterize than conduction because no single intrinsic property of the material, such as thermal conductivity for conduction, can be defined to scale the process.Both the thermal properties of fluids and hydrodynamic characteristics of flow govern heat dissipation by convection.In practice, convective heat transfer is often analyzed empirically.The current work is focused on the experimental results of this new technique.A satisfactory theory (at least a phenomenological model) will require further investigation.
In particular, the technique reported in this work nicely tracks the dynamics of thermal plume heads resulting from rapid heating by a laser.In general, thermal plumes, also referred to as convective plumes, are upward flows of fluid propelled by differences in temperature.In the current context, thermal plumes involve the transport of heat and mass.Studies of thermal plumes began in the 1960s [14,31] and continued through the 1980s and early 1990s [32][33][34].Within the last dozen years, work on thermal plumes continued [35][36][37] and is continuing today with the very recent work of Wang et al. [38].
The Reynolds number (Re) and the Prandtl number (Pr) are two essential dimensionless numbers characterizing thermal plume formation and dynamics.Plume heads are not expected to form in fluids with Re < 20,000 [35].The Prandtl number provides an idea about the relative thickness of the momentum and thermal boundary layers.A Prandtl number less than 1 indicates that heat diffuses faster than momentum, while a Prandtl number greater than 1 means momentum diffuses faster than heat.The liquid primary alkanes have Prandtl numbers exceeding 1 and given in Table 1.A heuristic description of the experimental results that follow presentations for high Prandtl number fluids [35,36,39] and, in particular, work in the area of geophysics [39][40][41] is given in the Section 4 below.The abbreviations are defined as follows: MW (g/mol): molecular weight, ρ (g/cm 3 ): density, η (cP): dynamic viscosity, Pr: Prandtl number, n: refractive index, MP/BP (°C): melting point/boiling point, and a WF (cm/s 2 ): wavefront acceleration.Values for the physical properties were obtained from Wolfram Alpha [42] via ChatGPT 4.0 [43].

Alkane Series and Power Study Protocol
The following experimental protocol was applied consistently for all samples and power levels.An individual run consisted of 2.0 s of background collection.Here, the shutter was held in the closed position.This was followed by an active phase of 4.0 s.A recovery phase of 2.0 s completed the run.
For each sample and power level, 10 runs were taken sequentially with a 4 s rest interval between runs, i.e., a given run began its background phase 4 s after the recovery phase of the previous run.During data analysis, careful observation for systematic trends across the 10-run set was made to ensure that the overall sample was not heating over the course of the runs.
Lastly, the calibration from pixel to vertical distance was obtained by replacing the sample with a 1.45 mm opaque obstruction.This casts a shadow onto the camera.Due to unavoidable diffraction, a judgment was made on which pixels represented the edges of the obstruction.The resulting value used throughout is 3.54 µm/pixel.

Pulse Study Protocol
For the study of the effect of a short-duration heat supply from the laser, the above protocol was followed with the following modification.Rather than a constant active phase of 4 s in which the laser is on the sample, active phase durations of 2.0 s, 1.5 s, 1.0 s, 0.50 s, 0.25 s, and 0.10 s were used.The sample used was n-pentane.

Excitation and Probe Lasers
The setup, as shown in schematic form in Figure 1, consists of an excitation laser, QSL103A Q-Switched Picosecond Microchip Laser System (ThorLabs, Newton, NJ, USA).The output is at 1030 nm and is delivered as approximately 500 ps pulses with an approximately 9 kHz repetition rate.The spatial profile of the beam is roughly Gaussian.(Note, in terms of the time scale of the various thermal processes, the laser is considered to be a continuous wave).
The laser power is controlled discretely by the introduction of neutral-density filters.The power entering the sample (but mostly not absorbed) is measured via PM100D Digital Optical Power and Energy Meter (ThorLabs) with a S140C powerhead (Thorlabs).This has an approximately 7% uncertainty at 1030 nm per the specification datasheet.Further, it was measured that approximately 7% of the excitation laser reflects off of the front surface of the cuvette.Thus, the powers reported in the Section 3 below are 93% of those measured directly out of the laser.A computer-controlled, solenoid-driven shutter is used to block and release the beam as needed to record the background (shutter closed), active (shutter open), and recovery (shutter closed) phases of the experiment.
The probe beam is a simple diode laser (HiLetgo, Shenzhen, China) that outputs at 650 nm.The laser is positioned within a passive aluminum heat sink.Further, the laser is connected in series to a potentiometer to offer controllable irradiation of the sample.While the powers are inconsequential to the sample, control is important for optimizing camera exposure.

Camera
The camera is a DCC1240M CMOS monochrome camera (ThorLabs); the camera offers 1280 × 1024 pixel resolution, and frame rates of about 12 fps are achieved.Data are saved as avi files under the MJPG codec.A timestamp file is dumped with each acquisition video so the frames are precisely timed.That is, the timestamp file contains the machine times when the particular frames are written to the file.

Acquisition and Analysis Software
The entire acquisition phase of the experiment is controlled by a homewritten program.Data processing takes place in three steps and is also done with homewritten programs.First, a new video is produced in which the background phase frames are averaged, and this is subtracted from all the frames from the raw video file.This allows for better visualization of both the transient and stationary state components of the material response.The transient feature is the thermal plume, while the stationary component is the region where the laser is on the sample.The next stage isolates the transient response by producing a third video file in which the ith frame is produced by taking the ith frame minus the (i − 1)th frame of the second video file.Finally, a fourth video file is produced by applying threshold values to the third video, which identifies the wavefront and trailer of the transient response.These points are fit to a second-order polynomial to produce smooth-curve representations of the wavefront and trailer.It is from these curves that the velocity and acceleration of the transient wave are determined.It is appropriate to acknowledge the use of ChatGPT 4.0 in assisting with the development of the programs to acquire and analyze data [43].

Primary Alkane Samples
The samples were obtained from either Fisher Scientific, TCI, or Aldrich and used without further purification.Relevant physical properties of the primary alkanes used in this experiment are collected in Table 1.The samples were in a glass fluorimeter cell (1 cm deep and 1 cm wide) and taken to be equilibrated in temperature with that of the laboratory, which remained between 19.8 and 19.9 • C during the course of the data run.

Results
The results presented here are from one continuous roughly 5 h data collection period.Both the excitation and probe lasers were warmed up and left continuously running during the collection period.Power measurements of the excitation laser were taken at regular intervals to monitor for slow power drift.Likewise, the ambient temperature was continuously monitored for slow drift.The only aspects of the setup that changed were the sample cuvettes and neutral-density filters to control the excitation laser power when appropriate.
Regarding experimental reproducibility, the uncertainties associated with the runs accumulated during the data collection period are reported below with their corresponding results.Further, in addition to the data included in this paper, many preliminary data sets were obtained.This is particularly the case for n-pentane and n-hexane.Similar quantitative and qualitative results were consistently observed.

Concrete Example of Video Data: Hexane
By way of example, consider one run involving neat n-hexane at full excitation power.These data sets included raw data, background subtracted data, and transient isolated data as described in Section 2. As depicted in Figure 2, each of these frames was taken at an identical time point, approximately 3.4 s into the run, which is 1.4 s after the shutter opens.This time point was chosen because the transient signal (thermal plume head) has clearly separated from the stationary signal.
In the raw data frame, presented on the left in Figure 2, the laser is visibly active and located near the bottom.This laser positioning and activity are also discernible in the middle frame, which represents the background-subtracted data.At this particular instant in the experiment, the transient wavefront has advanced about three-quarters of the way up the frame.This progression is indicative of the changes occurring in the sample due to free convection.
The middle frame, representing the background-subtracted data, offers a clear and detailed view of both the transient and stationary responses.The subtraction technique employed here allowed the authors to distinctly observe the changes in the sample induced by the laser, separate from the background probe light.The position of the laser beam is seen as a dim horizontal streak at about 1/8 of the way up from the bottom of the frame (more visible online than in print).This is the steady-state feature.The transient is seen as the brighter horizontal streak about 3/4 of the way up from the bottom of the frame.
The rightmost frame in Figure 2 shows the transient-isolated data.This frame provides a view that isolates only the transient response.The isolation of the transient response facilitates the study of laser-induced changes in the sample apart from the stationary elements.Continuing with the concrete example of the neat n-hexane run of Figure 2, shown are captured and analyzed still frames from the transient isolated data files.These frames, as shown in Figure 3, illustrate the progression of the transient signal with wavefront tracking, providing a quantitative view of the process.This allows for the determination of the velocity and acceleration of the transient part of the signal.
In each frame, three distinct curves are superimposed to track and analyze the transient signal's progression.The yellow curve represents a second-order polynomial best fit to the wavefront.This curve provides a mathematical representation of the wavefront's shape and progression, facilitating the analysis of its behavior.This is the main source of the results presented in this article.
The cyan curve, similar to the yellow curve, represents a second-order polynomial best fit but tracks the trailing edge of the transient signal (bottom of the plume head).This curve provides additional insights into the widening of the transient signal, an essential aspect of understanding the overall transient response.The red curve represents the center of mass motion of the transient signal.This curve tracks the 'average' position of the transient signal over time, providing an additional picture of the signal's overall movement and progression.The cyan and red curves are of secondary help for data analysis and not the source of the results presented in this article.

Alkane Series
An interesting observation is that the wavefront of the plume head accelerates upwardly from the position of the laser beam.These accelerations of the wavefront, a WF , can be measured by tracking the yellow marker shown in Figure 3. Representative examples are shown in Figure 4 where the aggregate positions of the wavefront as a function of time are shown for n-pentane, n-nonane, and n-tetradecane.These data were collected for the remaining alkanes but not shown in the figure in the interest of visual clarity.The acceleration is most pronounced for n-pentane, where a clear parabolic curve can be seen in the data.The parabolic curve for, say, n-tetradecane is much more subtle.In fact, there is an additional dynamical feature very early on that causes a slight downward curvature at early times.This is seen consistently across the higher alkanes.The a WF are collected in the last column of Table 1 and shown in Figure 5.These aggregate data along with the remaining alkanes, were fit to produce the results listed in Table 1 and Figure 5.  x as a function of viscosity, η, (fit funtion: a WF = 0.036 η ).The 95% confidence interval on the fit parameter is (0.033, 0.039) In addition, because the Prandtl number is nearly linear with viscosity, it, too, is nearly of the form 1 x (fit function: a WF = 0.71 Pr ), confidence interval (0.67, 0.76)) The error bars represent the 95% confidence interval for the fit parameters.

Pulse Duration
Varying the duration of the time the laser is on the sample provides additional insight.As noted above, the laser is considered to be a continuous wave despite being pulsed at 9 kHz.Therefore, presumably, one can refer to the duration of the time the laser is on the sample simply as a pulse.Four pulse durations, 2.0 s, 1.0 s, 0.25 s, and 0.10 s, were used in this part of the study.The first two represent the case when the laser is on during the full time for the transient feature to exit the top of the video frame.Conversely, the last two pulse durations represent an "impulse" situation in which the transient feature travels almost its entire way across the viewing frame without the laser being on beneath it.The results of this aspect of the study are shown in Figure 6.The magnitude of the signal is largely independent of pulse duration."Magnitude of the signal," means the total signal in the frame of the background-subtracted video just after the opening of the shutter.Further, this signal is normalized to unity for n-pentane, and all other primary alkanes are referenced to n-pentane.As seen in Figure 6, the overall signal decreases with increasing alkane.
In the cases where the laser is on while the wavefront is in view of the camera (from t = 2.0 s to beyond approximately 3 s), the wavefront experiences acceleration.That acceleration is independent of total pulse duration.For the case of n-pentane, an acceleration of a WF = 0.120 ± 0.006 cm/s 2 was measured for a 2.0 s pulse.Likewise, an acceleration of a WF = 0.12 ± 0.04 cm/s 2 was measured for a 1.0 s pulse.Note: The equivalence of these two values makes sense because, in each case, the laser is on during the determination of the acceleration.
However, for the cases in which the laser is on only briefly, the wavefront rises with constant (terminal) velocity.This velocity is independent of the pulse duration.A 0.25 s pulse gave a v WF = 0.16 ± 0.01 cm/s, while a 0.10 s pulse gave a v WF = 0.18 ± 0.01 cm/s.Here, too, these values are the same within mutual uncertainty.

Power Study
The nature of the photothermal response was measured as a function of laser input powers.It is clear that the transient response is more sensitive to input power than the steady-state response.This is true both in terms of the overall signal strength and the wavefront propagation accelerations/velocities. Figure 7 shows the process of creating what is referred to here as row marginals.They are the total binned rows of a video frame.Figure 8 shows plots of the row marginals for the different laser powers studied in this work.Times were chosen such that the transient and stationary features were well separated.A ratio, r, of areas under the transient feature to the steady state feature was calculated and is plotted in the bottom, right frame of Figure 8.A direct and apparently linear relationship is seen for the range of powers used (see the bottom right panel of Figure 8).Figure 9 shows the results of the power study for the acceleration of the transient feature of n-pentane.The acceleration is seen to be linearly proportional to laser power.The right panel of the figure shows a few representative aggregate wavefront position data sets.Much like for the longer alkanes (e.g., n-tetradecane in Figure 4), the low power case shows an initial downward curvature before a very subtle parabolic behavior.

Video Data
Much of the physical insight gained with this technique comes from viewing the resulting raw and processed video files.Unfortunately, much of the dynamic information that is well conveyed by a video does not come across in static paper form.Figures 2 and 3 are an attempt to convey some of the video information.All of the raw and processed video files can be found at https://www.darinulness.com/research/data/primary-alkanes(accessed on 12 June 2023).

Discussion
The primary conclusion from the above results is the following.A cylindrical region of heated solvent rises in the form of a propagating thermal plume head (transient feature) with a clear wavefront (top of the plume head).If the laser remains incident upon the sample, the wavefront will accelerate (a WF ).Conversely, if the laser is blocked before the sample, the wavefront will quickly reach terminal velocity.Finally, the magnitude of the transient signal and the acceleration of the wavefront depend on the incident laser power.The authors were able to discern these results are new and made readily observable by the approach of the experimental setup.A phenomenological discussion of the underlying physics is presented in the remainder of this section.This discussion is motivated by the work of Whitehead et al. [35], albeit that work was for a much higher Prandtl number regime.

Conjecture about the Physical Processes
A narrative emerges when the results presented in the previous section are taken together.On a timescale shorter than that which is observed in the experiment, the laser heats a cylindrical region of the sample.This is similar to the well-known laser temperature jump spectroscopy methods used in biochemistry [44].Evidently, a thermal equilibrium is reached quickly.This is supported by the results displayed in Figure 6, which show that the magnitude of the photothermal signal is rather insensitive to pulse duration down to 0.10 s.
These data indicate that a dynamic thermal equilibrium establishes a modest yet conspicuous local temperature change compared to the bulk solvent.While it is not possible from the results of the pulse experiment to obtain an accurate temperature change, it is certain to be less than 17 • C because there was no indication of local boiling in the n-pentane sample.Local boiling would be readily observed with this technique as the vapor bubbles would have a large difference in the index of refraction.
Because the excitation beam is fully nonresonant for all the primary alkanes used, the energy delivered to each sample as heat is similar.A reasonable approximation is that the relative energy delivered comparing two samples is given by the ratio of their respective indices of refraction.For example, using the data from Table 1, n-pentadecane would receive about 1.431  1.358 = 1.05 times more power than that of n-pentane.Consistent with other photothermal methods is the guiding physical principle that the local heating results in a local density change (consequently a local change in the index of refraction) in accordance with the new higher temperature.Photoacoustic waves propagate out symmetrically, and thermal diffusion also symmetrically distributes heat.Though these processes are happening, the technique used in the current work is sensitive to the bulk motion of the transient feature and the convective flow.The results of the previous section give insight into these processes.
The observation of the intact motion of the transient feature suggests a stark boundary between the heated cylindrical region, which becomes the plume head, and the bulk surrounding it during the approximately 1 to 2 s that the feature is in view of the camera.A sharp boundary is not unexpected, as was suggested by observing dropping a cold drop of food coloring into a glass of warm water.The drop stays intact and in terminal velocity motion nearly until the conduction of heat equilibrates the temperature.This is also consistent with the results of Davaille et al. [36].
The results presented in the previous section for the set of pulse-duration experiments show that when the laser is off during the rise of the transient feature, the wavefront moves at a constant (terminal) velocity.This is in direct analogy with the simple-minded experiment of the drop of cold food coloring mentioned above.The underlying physics are the same in both cases.That is, an upward buoyant force is present because of the difference in density, which is balanced by a gravitational force and a drag force, both directed downward.
The acceleration of the transient feature that occurs when the laser is on is quite interesting.While this phenomenon awaits a full hydrodynamic theoretical description, the following conjecture is offered.This description follows closely the model presented by Turner [14].As the heated cylindrical region rises, some backfilling must come from the cold sides along with the hot laser beam region below (rising straight above the beam).This backfill is evidently enough to allow for a visible trailing edge to the transient component.
As the transient component separates from the beam region, an upward directed slipstream, also called column or conduit [35], is produced as a steady state influx from the sides and from the beam region.See Figure 2 of reference [35] for a visual representation of the temperature and slipstream field lines for a similar situation as described.
Acceleration occurs because the lamina moving around the upward-moving cylindrical region is no longer behaving according to the Stokes drag law.This results in the necessary imbalance of force to cause acceleration.Admittedly, this argument needs a much more thorough hydrodynamic analysis to properly identify whether or not the imbalance in force is due to an "effective reduction" in drag or some sort of "convective push" from the rising slipstream.The results of the effect on the acceleration of the transient feature in n-pentane as a function of incident laser power are displayed in Figure 9.The fact that acceleration decreases with decreasing laser power supports the conjecture that the uprising convective flow is responsible for the imbalance of the forces.

Conclusions
This work introduces a variant technique for observing free convection arising from localized heating via a laser beam.A perpendicular divergent probe beam enabled a viewing field surrounding the excitation laser to be collected via a camera.The experiments were performed on primary alkanes.Convection occurs with a transient and steady-state response.Dynamical data for the propagation of the transient wavefront were determined for these alkanes.Excitation-power dependence was also explored.This revealed that the transient feature rises upwardly at a constant terminal velocity if the excitation laser is blocked from the sample.Further, when the laser remains incident upon the sample, the transient feature accelerates.The magnitude of acceleration is linearly proportional to input power and inversely proportional to alkane length and, thus, viscosity.A logical next step would be to use this technique to investigate other molecular liquids.Preliminary work indicates that this technique is effective with many other liquids.Perhaps equally interesting is that some liquids do not give a good response to this technique.Studying mixtures of alkanes would be another natural direction.However, this would also require a careful study of the viscosities of the mixtures.Other possible directions for future work could include investigating convection in constrained environments, such as narrow cuvettes or cuvettes with solid glass obstacles.

Figure 1 .
Figure 1.Schematic of the experimental setup.The excitation beam is from a QSL103A-1030 nm Q-Switched Picosecond Microchip Laser System (Thorlabs).The probe laser is from a 5V 650 nm laser diode (HiLetgo).The shutter is driven by a solenoid controlled by the computer.The camera is a DCC1240M High-Sensitivity CMOS Camera, 1280 × 1024 pixel array sensor (Thorlabs).The camera is controlled by the computer.

Figure 2 .
Figure 2. Still frames from the raw (left), background subtracted (middle) and transient isolated (right) data files for one particular run of neat n-hexane at full excitation power.Each of the three frames is at precisely the same time (approximately 3.4 s into the run or 1.4 s after the laser first enters the sample).At this time in the run, the laser is on and towards the bottom of the frame (best seen in the left and middle images).The transient wavefront (top of thermal plume head) has progressed about 3/4 of the way up the frame.Note that the middle image clearly shows both the transient and stationary responses, while the rightmost image isolates only the transient response.

Figure 3 .
Figure 3. Still frames from the transient isolated data files for one particular run of neat n-hexane at full excitation power.From left to right, the three frames show the progression of the transient signal with wavefront tracking.The yellow curve represents a second-order polynomial best fit to the wavefront; the cyan curve is similar except for the trailing edge of the transient signal.The red is the center of mass motion of the transient signal.

Figure 4 .
Figure 4. Aggregate wavefront position values from the 10 runs of n-pentane (blue), n-nonane (green),and n-tetradecane (red).The remaining alkanes are not shown for visual clarity.Very slightly for n-nonane and more pronounced for n-tetradecane, one sees an initial downward curvature.This is followed by a parabolic curvature showing upward acceleration.It is very slight for n-tetradecane.These aggregate data along with the remaining alkanes, were fit to produce the results listed in Table1and Figure5.

Figure 5
Figure 5 indeed shows a diminished acceleration with increasing carbon number (left panel).This behavior is roughly linear.The right panel of Figure 5 shows the behavior of the acceleration versus viscosity and versus Prandtl number.Curiously, both are fit well by a simple y = A x model.

Figure 5 .
Figure 5. Acceleration values, a WF , for the primary alkanes (except for n-pentadecane) plotted as a function of alkane (left panel), viscosity (middle panel) Prandtl number (right pannel).The measured acceleration is roughly linear with number of carbons, N, in the alkane (fit function: a WF = −0.015N+ 0.21).The 95% confidence intervals on the fit parameters are (−0.017,−0.012) and (0.18, 0.24), respectively.It is nearly of the form 1x as a function of viscosity, η, (fit funtion: a WF = 0.036 η ).The 95% confidence interval on the fit parameter is (0.033, 0.039) In addition, because the Prandtl number is nearly linear with viscosity, it, too, is nearly of the form 1x (fit function: a WF = 0.71 Pr ), confidence interval (0.67, 0.76)) The error bars represent the 95% confidence interval for the fit parameters.

Figure 6 .
Figure 6.Initial averaged (10 runs) signal strength for various pulse durations for n-pentane (•), n-hexane ( ), and n-heptane ( ).The signals are all normalized such that n-pentane with a 2.0 s pulse is unity.Error bars represent plus and minus the standard deviation of the signal strength over the 10 run averaging.The signal is a combination of the transient and steady-state features.To within error, the signal is roughly constant.

Figure 7 .
Figure 7. Process of row binning to create the row marginal spectra shown in Figure 8.Each point on the spectra is the average value for the full row in the video frame.There are 1024 rows that are converted to a 1024-point spectrum to create the row marginal.

Figure 8 .
Figure 8. Row marginals for times when the transient and steady-state features are well distinguished.Position refers to the position on the camera with pixels converted to centimeters (0.000354 cm/pixel).The laser power, P L , for the top row from left to right is 0.59 W, 0.43 W, and 0.30 W. The power for the bottom row, left to right, is 0.21 W and 0.14 W. Per the specification sheet of the Thorlabs S140C powerhead, the uncertainty in the power is approximately 7%.The bottom rightmost graph is the linear best fit to the ratio of the transient component to the stationary state component.The ratio, r, is calculated as the area under the transient part of the marginal (right side of graph) over the area under the stationary part of the marginal (left side of graph).The best-fit line is r = 4.2P L + 0.6.The 95% confidence intervals on the fit parameters are (3.1, 5.3) and (0.2, 1.0), respectively.

Figure 9 .
Figure 9. Acceleration of the transient feature for n-pentane as a function of incident laser power.(Left panel): The acceleration is linearly proportional to the incident laser power.Fit parameters: y = 0.27x − 0.008.The 95% confidence intervals on the fit parameters are (0.21, 0.31) and (−0.03, 0.01), respectively.The error bars represent the 95% confidence interval for the a WF fit parameter.(Right panel): Representative aggregate wavefront position data from which the a WF in the left panel are calculated.Blue, 0.59 W; green, 0.30 W; red, 0.14 W. Per the specification sheet of the Thorlabs S140C powerhead, the uncertainty in the power is approximately 7%.

Table 1 .
Physical properties and wavefront dynamical information of the alkanes used in this work.