Extreme thermodynamics in nanolitre volumes through stimulated Brillouin–Mandelstam scattering

Examining the physical properties of materials—particularly of toxic liquids—under a wide range of thermodynamic states is a challenging problem due to the extreme conditions the material has to experience. Such temperature and pressure regimes, which result in a change in the refractive index and sound velocity, can be accessed by optoacoustic interactions such as Brillouin–Mandelstam scattering. Here we demonstrate the Brillouin–Mandelstam measurements of nanolitre volumes of liquids in extreme thermodynamic regimes. This is enabled by a fully sealed liquid-core optical fibre containing carbon disulfide. Within this waveguide, which exhibits tight optoacoustic confinement and a high Brillouin gain, we are able to conduct spatially resolved measurements of the local Brillouin response, giving us access to a resolved image of the temperature and pressure values along the liquid channel. We measure the material properties of the liquid core at very large positive pressures (above 1,000 bar) and substantial negative pressures (below –300 bar), as well as explore the isobaric and isochoric regimes. The extensive thermodynamic control allows the tunability of the Brillouin frequency shift of more than 40% using only minute volumes of liquid. Material characterization of liquids in extreme thermodynamic conditions is a challenging technical problem. Brillouin scattering metrology in an optical fibre design with a sealed liquid core now enables spatially resolved temperature and pressure measurements, using carbon disulfide as an example.

Examining the physical properties of materials-particularly of toxic liquids-under a wide range of thermodynamic states is a challenging problem due to the extreme conditions the material has to experience.Such temperature and pressure regimes, which result in a change in the refractive index and sound velocity, can be accessed by optoacoustic interactions such as Brillouin-Mandelstam scattering.Here we demonstrate the Brillouin-Mandelstam measurements of nanolitre volumes of liquids in extreme thermodynamic regimes.This is enabled by a fully sealed liquid-core optical fibre containing carbon disulfide.Within this waveguide, which exhibits tight optoacoustic confinement and a high Brillouin gain, we are able to conduct spatially resolved measurements of the local Brillouin response, giving us access to a resolved image of the temperature and pressure values along the liquid channel.We measure the material properties of the liquid core at very large positive pressures (above 1,000 bar) and substantial negative pressures (below -300 bar), as well as explore the isobaric and isochoric regimes.The extensive thermodynamic control allows the tunability of the Brillouin frequency shift of more than 40% using only minute volumes of liquid.
Understanding the physics at extreme thermodynamic points such as high pressures or high temperatures has been challenging for several decades 1,2 .An illustrative example is the exploration of the negative-pressure regime that has vast implications for the understanding of diffusion processes in the micro-and nanoscopic world, being particularly relevant for mineral transport in plants 3 and body cells 4 , as well as for potential advances in material engineering 5 .The lack of reconfigurable and reproducible measurement schemes at negative pressures has caused debates about the phase diagram of water up to our modern age 6 and has not let much room for generalizing the negative-pressure concept to molecular systems other than water to date.In general, the experimental study of liquids under extreme conditions can be exceptionally demanding due to the need for thermodynamic stabilization 7 and the handling of liquids 8 , particularly when the liquids involved pose risks to health and the environment.
Non-contact measurements that encapsulate the material of interest from the environment are, therefore, highly desirable and can be performed by measuring optical [9][10][11] or acoustic [12][13][14] parameters of the

Brillouin-Mandelstam scattering in CS 2 -filled LiCOF
Brillouin-Mandelstam scattering is an optoacoustic effect where a strong pump laser is coherently scattered by travelling acoustic waves, resulting in a backscattered Stokes and anti-Stokes signal.The Stokes signal is frequency downshifted by the BFS Ω B /2π and shows a Lorentzian shape.A comprehensive tutorial on Brillouin scattering can be found elsewhere 31 .The BFS is described as where both speed of sound v ac (p, T) and mode refractive index n eff (p, T) are dependent on temperature T and pressure p inside the fibre 32 .In contrast to solid-core fibres, in LiCOFs, externally applied local stresses distribute throughout the core in the form of changes in global pressure.Indeed, due to the fluid state of the core material, the pressure can be assumed to be constant along the fibre as a potential pressure gradient is eventually equalized through the flow.In contrast, the temperature is a local property (T = T(z)) as a change in temperature at location z = z 1 does not impose the temperature to change at z = z 2 .For the mode refractive index, we use the extended Sellmeier equation 26 to describe pressure and temperature dependencies.We assume that the speed of sound can be separately described by a linear combination of contributions from the pressure and temperature, with the pressure contribution taken from the model in ref. 25 and the temperature contribution from that in ref. 33.In this case, n(CS 2 , 1 bar, 20 °C) = 1.588 is the refractive index and v ac (CS 2 , 1 bar, 20 °C) = 1,242 m s -1 is the speed of sound for CS 2 .In our experimental system, liquid CS 2 is sealed within a fused silica capillary by direct hot splicing to glass fibres.The high-index core liquid embedded in the lower-index silica forms a close-to-ideal circular step-index optical and acoustic waveguide, whereas the fibre has been engineered for efficient coupling to standard single-mode fibre (SMF) pigtails via bridge fibres (Methods).
Controlling the temperature in the capillary can be straightforwardly done by external heating or cooling, but directly controlling the pressure is more involved.An increase in temperature leads to the expansion of CS 2 , following at 20 °C (ref.34).If the liquid fills the complete capillary, it cannot expand further, which then translates to an increase in pressure.Following another work 11 and extending it for partially heated fibres, the relation between a temperature change and a pressure change is determined by where κ = 9.38 × 10 −10 Pa −1 is the isothermal compressibility of CS 2 (ref.34) and L H is the heated length, assuming α V ΔT ≪ 1.With this relation, T being a local property and p being a global property of CS 2 , one can change the pressure in the complete LiCOF by heating one part of the fibre, which does not change the temperature of the other part.In contrast to non-sealed systems 35,36 , this relation provides full control of the pressure without requiring open access to the liquid, by partially heating or cooling the fibre and keeping the temperature of the other region constant.
material.In particular, combining the advantages of both optics and acoustics via stimulated Brillouin-Mandelstam scattering (SBS)-a nonlinear interaction of light and sound waves-allows to probe changes in temperature, pressure and density with high precision.SBS has been demonstrated in liquids 15 , gases 16 and optical fibres 17 .In particular, optical fibres are a very common platform for SBS applications such as microwave photonics 18 , signal processing [19][20][21] and sensing 22 because of their low optical loss, long interaction lengths, easy handling and tight optoacoustic confinement.However, the use of conventional glass materials in optical fibres prevents the platform to unfold its potential for material studies.Using liquids as a guiding core medium opens up this potential by providing access to the thermodynamics of fluids and reducing sample amounts to nanolitre volumes and preserving the metre-long interaction lengths and high field confinement of optical fibres.Such liquid-core optical fibres (LiCOFs) have already found applications in supercontinuum generation 23 , lasing 24 , temperature sensing 25 or signal processing 26 .Also, first experiments of SBS in open LiCOFs have been performed 27 .Yet, access to extreme thermodynamic ranges of liquids, such as very high or even negative pressures, has so far been beyond reach with fibre optics.Moreover, the full description of such dynamics requires a method sensitive to thermodynamic parameters like the refractive index and speed of sound, ideally spatially resolved along the fibre axis.Here SBS provides both by exploiting the interaction of optical and acoustic waves.
In this work, we present an optoacoustic, fully fibre-integrated photonic platform that allows access to extreme thermodynamic regimes of liquid materials.This framework exploits SBS in fully sealed and low-loss LiCOFs with unprecedented capabilities to sense temperature and pressure dynamics of liquids in a spatially resolved and highly reproducible fashion within a waveguide environment that simultaneously confines light and sound.We demonstrate our concept using only a few nanolitres of volatile and toxic carbon disulfide (CS 2 ) in an experimental case study.
Our all-fibre setup allows for the local and global control of the thermodynamic state of CS 2 .The characterization of the optoacoustic response reveals a substantial Brillouin gain of (32.2 ± 0.8) W −1 m −1 .In particular, we perform spatially resolved measurements of the optoacoustic interaction using Brillouin optical correlation domain analysis (BOCDA) with centimetre resolution.This investigation allows us to determine the local and global influences of temperature and pressure changes.
Our framework allows us to probe up to pressure values larger than 1,000 bar and unprecedented negative pressures down to -300 bar.With these insights, we are able to perform the thermodynamic control of SBS, allowing us to tailor the Brillouin frequency shift (BFS) by more than 40% of its initial value with high precision.We extend our measurements from the isochoric (constant volume) to the isobaric (constant pressure) regime of the liquid.Studying the latter, we show that the absence of the pressure component leads to a more than sevenfold enhancement in temperature sensitivity of the BFS compared with standard solid-core fibres.In this case, SBS is extremely well suited, since the air gap formation inside the fibre prevents any transmission measurements.Additionally, the isochoric state at the boundary to the isobaric state allows stable and reproducible access to the regime of negative pressure.This metastable state is associated with a stretched liquid, where attractive forces between the molecules prevent the formation of an energetically more favourable state of separated liquid and gas phase 28 .This regime is normally only accessible with high experimental expenses and has rarely been achieved in liquids other than water and at such high negative-pressure values 29 .Indeed, probing those boundaries of metastable thermodynamic regimes is of particular importance for verifying the equations of states in common molecular models 30 .Our work uniquely showcases the use of optoacoustics as a tool for material sciences to advance our To illustrate the thermodynamic behaviour of this system, a schematic of the LiCOF geometry is shown in Fig. 1c.Two regimes are depicted in Fig. 1c: first, the isochoric regime, in which the fibre is fully filled and the CS 2 volume is constant (left).In this regime, the fibre has optical transmission and an increase in temperature also increases the internal pressure.Second, the isobaric regime, where a gap is present and the pressure in CS 2 is constant (right).In this regime, the fibre does not transmit light but is accessible via backward SBS from one side.

Integrated Brillouin-Mandelstam analysis
To gain a fundamental understanding of the Brillouin-Mandelstam response of the LiCOF, we perform an integrated, thermally seeded SBS analysis at a pump wavelength of λ = 1,549 nm.Extended Data Fig. 1 shows a simplified sketch of the setup.The SBS spectrum at room temperature is shown in Fig. 1b, where we observe peaks at 2.25, 2.58 and 3.24 GHz.Numerical simulations reveal that the peaks are due to high optoacoustic overlap between the fundamental optical mode and the first three radially symmetric longitudinal-pressure acoustic modes, namely, L 01 , L 02 and L 03 (Fig. 1d), with BFSs at 2.25, 2.62 and 3.24 GHz, respectively.Therefore, the simulations qualitatively confirm the origin of the rich mode spectrum due to multiple acoustic modes only, in the absence of higher-order optical modes.We find a small deviation in the BFS of the second peak from the simulated value, which is probably due to the idealized geometry of the capillary in the numerical model.It should be noted that the simulation uses the refractive index n(CS 2 , p = 1 bar, T = 20 °C) = 1.588 and the speed of sound v ac (CS 2 , p = 1 bar, T = 20 °C) = 1,242 ms −1 .For the peak at 2.25 GHz, we find a total gain of (9.29 ± 0.23) W −1 m −1 , and a gain of (32.2 ± 0.8) W −1 m −1 when compensating for the coupling losses of 5.4 dB with a linewidth of 65 MHz.This value is two orders of magnitude higher than the values of a standard SMF at around (0.2-0.3) W −1 m −1 (refs.37,38).Supplementary Section 1 provides a detailed description of how the gain is measured, whereas the underlying physics is comprehensively described elsewhere 17,31 .
In Fig. 2a, the temperature response of the LiCOF is investigated by measuring the Brillouin spectrum and heating a part of the capillary from room temperature to 130 °C in steps of 5 °C on a hotplate.Above room temperature, the peaks associated with the first-and second-order longitudinal-pressure acoustic modes split up and move to higher frequencies with increasing temperature.To identify the peaks in the following, we number them from 1 to 4 by increasing the frequency.The observed splitting is explained in Fig. 2b by the additive but opposite influence of global pressure (positive) and local temperature (negative) via the thermo-and piezo-optic coefficients of CS 2 .First, the global pressure influence, following equation (2), upshifts the BFS inside and along the whole core (Fig. 2b, green solid arrows).Second, the local temperature influence downshifts the BFS only in the heated part of the capillary relative to the upshifted one (Fig. 2b, red dashed arrows).In the integrated spectrum, the local Brillouin spectra along the whole fibre are summed up 17 , resulting in the observed splitting (Fig. 2a).We can now identify each peak with the influences it is subjected to: although all the four peaks are subjected to a BFS upshift due to pressure changes (green arrows marked with green circles), only peaks 1 and 3 are subject to an additional BFS downshift due to temperature changes (red arrows marked with circles having a redgreen gradient), as shown in Fig. 2b.This allows us to discriminate the influences of both temperature and pressure.

Spatially resolved Brillouin-Mandelstam analysis
To experimentally confirm the local and global nature of the pressure and temperature influence on the different peaks, we employ direct-frequency-modulated BOCDA 39,40 for a distributed analysis of the fibre.Extended Data Fig. 2 shows a sketch of the setup.Following the scheme shown in Fig. 3a, a frequency-modulated, counterpropagating pump in combination with a probe creates a localized acoustic response inside the fibre.By changing the frequency of the modulation, the correlation position can be moved throughout the fibre to This yields a resolution of Δz = 6.4 cm and sensing distance d m = 146 m for a Brillouin gain bandwidth of Δν B = 65 MHz, and a group velocity of v g = c/n with an assumed refractive index of n = 1.47 arising from the simulations in 'Spatially resolved Brillouin-Mandelstam analysis' section.Due to the longer measurement times of the distributed approach and to reduce spontaneous scattering, shorter fibres of L = (0.60 ± 0.05) m are used in this section.
Similar to previous measurements, we heat several fractions of the fibre and record the BFS response.Figure 3b,e shows the typical results, with either the half at the incoupling of the probe (Fig. 3b) or half at the incoupling of the pump (Fig. 3e) heated to 80 °C.The shaded parts denote the SMF and ultrahigh-numerical-aperture (UHNA) fibre.The SMF response at these positions is not shown, due to the BFS of ν B = 10.8GHz.The dashed white box denotes the region that is heated; the pump direction is always towards the right side of the figure.Calibration measurements reveal a room-temperature BFS of ν B = 2.2 GHz, in agreement with the integrated data.In contrast to the integrated measurements, only the dominant SBS mode is visible, due to the reduced signal-to-noise ratio of the distributed approach.This is caused by the reduced measurement length and the corresponding exponentially smaller gain.Furthermore, all the measurements exhibit a strong asymmetric background, which is a common feature for direct-frequency-modulated BOCDA 41 and arises from the non-instantaneous amplitude response of the frequency-modulated laser source.To treat this, we conduct numerical simulations for the Brillouin gain following other work 41,42 (Fig. 3c,f).Neglecting the pump loss inside the fibre, the simulations clearly reproduce the experimental data.The simulations retrieve the initial, undisturbed Brillouin gain distribution (Fig. 3d,g).Additionally, in the simulations, we also suppress the correlation beat noise, which is independent of the laser source.We find that the BFS of the heated regions (cancelling local T and global p influence) is steady at ν B = 2.2 GHz for all the temperatures, whereas the BFS of the non-heated regions (global p influence only) is upshifted.This is in agreement with the findings from 'Integrated Brillouin-Mandelstam analysis' section and confirms our theory of local temperature and global pressure influence in the fibre.

Extreme thermodynamic regimes
As stated in equation ( 2), the pressure variation inside the fibre can be achieved using partial heating of the waveguide as an indirect method of control.To validate this assumption, the fibre is heated along different fractions and the BFS of peaks 1 and 2 are measured.Peak 1 is dependent on temperature and pressure, whereas peak 2 is only dependent on pressure inside the capillary.In Fig. 4a, the BFS of peak 2 is plotted against the pressure change.The dominating uncertainty of the data points in Fig. 4a is caused by the uncertainty in the heated length.The absolute value of the BFS differs from Fig. 2 because of different sample parameters (2.100 μm diameter instead of 1.125 μm diameter and 2 m length).The overlap for the different heated fractions indicates that the pressure can be tuned independent of the temperature by changing the length of the heated fraction of the capillary.We then complete this picture by including peak 1, which is dependent on the pressure as well as temperature (Fig. 4b).With the full understanding of the pressure and temperature dynamics using equation (1), we are able to tailor the BFS of the fibre over a large range of almost 1.00 GHz (0.97 GHz).This is an unprecedented tunability of more than 40% of the initial value, using only fractional heating with a standard hotplate, offering a tenfold enhancement compared with the temperature tunability of the standard solid-core SMF 43 .This large tunability in combination with the high gain of our system is highly interesting for optical fibre sensing and microwave photonics applications, such as highly tunable notch filters based on SBS 44 .
In the next step, the fibre is cooled to temperatures below 5 °C (Fig. 4c) resulting in a pressure reduction.In this measurement, most of the fibre is located on a Peltier element except for a short section that is attached on a hotplate.We, therefore, use peak 1, which is influenced by both temperature and pressure.Initially, the fibre is fully filled and thus follows isochoric cooling, and therefore, during cooling down, it shows the expected decrease in the BFS.When the inside of the capillary reaches the vapour pressure of CS 2 (ref.45), the BFS decrease continues with a steady slope from the positive-to the negative-pressure regime.The latter state may seem less intuitive as one might expect the pressure to be always positive.However, as previous works show, a liquid in tight confinement, such as capillaries or micropores, can be stretched due to adhesion to the vessel boundaries, opening a new thermodynamic metastable regime 29,30 .In this case, the liquid column is completely filled and the attractive forces between the molecules (cohesion) lead to an inner tension that is expressed as a negative pressure as soon as the system is cooled below the temperature where it would reach vapour pressure.Under this condition,   Article https://doi.org/10.1038/s41567-023-02205-1 the energetically more favourable phase separation (isobaric state) is suppressed by the attractive cohesion force between the molecules 46 .This metastable state of the liquid column remains intact until an inhomogeneity in the system or a fluctuation of the environment leads to nucleation (bubble formation).Here, as soon as a small perturbation is present, the liquid column breaks up and a liquid-gas interface forms that transfers the system into an isobaric state.The gas phase then expands until the vapour pressure and cohesive forces are in balance again.The metastability of this particular thermodynamic regime is described by the classical nucleation theory 47 , after which an energy barrier must be overcome to break the balance between the forces, eventually leading to homogeneous nucleation.In the experiment, this statistical process is usually triggered by perturbations in the liquid or the experimental system 29 , which is called heterogeneous nucleation.Additionally, it was shown 48 that the smaller the capillary, the lower the pressure can get at which heterogeneous nucleation occurs.In our case, the small volume of CS 2 , high quality of the capillaries and high mechanical strength of the splices allowed to access negative pressures of less then -300 bar following equation (2) without homogeneous nucleation.
To initiate nucleation in a controlled way, a small fraction at the end of the fibre is heated above the boiling point of CS 2 .At this point, the transmission of the sample is lost, confirming the breakup of the liquid column and thus the onset of the isobaric regime.Furthermore, a strong jump in BFS is visible, caused by the sudden pressure change from large negative pressure to the vapour pressure of CS 2 inherent with nucleation.Next, the liquid is heated in the isobaric regime.The liquid expands with increasing temperature without relevant pressure influence on the BFS 45 .This leads to a large negative slope of the BFS of −7.5 MHz °C −1 (Fig. 4c, red curve), which is a more than sevenfold enhancement compared with standard SMF (0.95 MHz °C −1 ) (ref.43).The BFS decreases continuously with a linear behaviour until it reaches the recombination point of 28.7 °C.Here the transmission of the sample is restored as the bubble is closing again.The fibre transitions back from the isobaric to isochoric regime, which are indistinguishable at the recombination point.The exact temperature of the recombination

Article
https://doi.org/10.1038/s41567-023-02205-1point depends on the filling ratio of the fibre as well as the ratio of the heated to cooled length.For temperatures above this recombination point, the behaviour is described in 'Integrated Brillouin-Mandelstam analysis' section.

Conclusion
In summary, we have investigated extensive positive-and negativepressure regimes in the isobaric and isochoric liquid states of CS 2 with the help of integrated and spatially resolved Brillouin-Mandelstam scattering experiments in fully sealed CS 2 -filled capillary fibres.Pressures of more than 1,000 bar and less than −300 bar were reached, which enables extraordinary control of the Brillouin response with a record tunability of 40% using only a few nanolitres of sample volume of toxic and volatile liquid CS 2 .Distributed Brillouin measurements with centimetre spatial resolution allowed us to experimentally distinguish the local response of temperature changes and the global response of different pressure values.In the isobaric state, we were able to observe the purely temperature-related BFS change of −7.5 MHz °C −1 .
In contrast to previous studies 24,49 , our work opens entirely new optoacoustic insights into the thermodynamic properties of a liquid.Backward SBS provides unique features such as being able to measure in the isobaric regime where no optical transmission is possible and to provide a spatially resolved measurement of the refractive index and sound velocity.
From a materials science point of view, our approach is suitable to access states of matter at conditions that are beyond those commonly reached, one example being unexplored points in phase diagrams.Some phase transitions that can typically only be seen using invasive approaches might be accessible when studying optoacoustics in fully sealed LiCOFs.Moreover, it can be used to determine difficult-to-measure material constants such as the piezo-optic coefficients 11 , which are hard to determine otherwise.
This work also marks the successful integration of LiCOF in a fully fibre-based circuitry, opening up an extended application area for fibre optics-the investigation of matter under unexplored conditions of pressure and temperature.The fully sealed fibre system-the overall embedding of the liquid sample into a reproducible micrometre compartment-enables operation in a closed thermodynamic system at the lowest experimental complexity.It enables the use of volatile and toxic liquids in scarce quantities of material.The cleanliness, nanometre-stable size of the core and the reproducibility and cleanliness of the fibre samples preventing organic contamination at the capillary's inner wall lead to a very reliable and immaculate environment.The approach is also transferable to other fibre types, such as selectively filled air-clad fibres 50 and is thus principally extendable to liquids with a lower index than silica, such as water.
Our results might, moreover, pave the way for a future generation of liquid-core Brillouin photonics, tailorable by thermodynamic parameters.Consequently, example applications are highly sensitive fibre-based monitoring devices, all-optical switching via nonlinear optical interaction, controlled nanolitre chemistry and widely tunable radio-frequency applications.

Fig. 1 |
Fig. 1 | Sample geometry and analysis of the integrated Brillouin spectrum at room temperature.a, Artist's view of the LiCOF sample with a laser.b, Measured spectrum (blue) and simulation (red) of SBS spectrum with 1.125 μm core diameter.c, Geometry of the LiCOF sample spliced on both sides with a UHNA fibre for coupling and sealing.SMF is used for integrability.If the capillary is fully filled, the volume (V) stays constant (isochoric regime), whereas if a bubble is present, the pressure (P) stays constant (isobaric regime).d, Numerical simulations of the corresponding longitudinal, radially symmetric pressure acoustic modes L 01 , L 02 and L 03 .Panel a credit: Long Huy Dao.

Fig. 2 |
Fig. 2 | Explanation of the observed splitting in integrated Brillouin measurements.a, Map of SBS response to partial heating.A splitting of the optical peaks related to the first (second) acoustic mode into two peaks, namely, 1 and 2 (3 and 4), is observed.b, Global influence of pressure (upshift, green)

Fig. 3 |
Fig. 3 | Distributed Brillouin measurements.a, Scheme of BOCDA.A frequencymodulated, counterpropagating pump and probe wave is launched into the FUT, causing a time-and position-dependent frequency beating.Only at specific correlation positions, the acoustic wave is stimulated and continuously emitted, leading to a continuous-wave, localized Brillouin response.b,e, Position-resolved measurement of BFS in partially heated LiCOF; the Brillouin intensity is colour coded.The heated length is denoted by the dashed white box.The grey-shaded area denotes the system parts of the SMF.The pump direction is from the right.BFS in the non-heated part is upshifted compared with the heated part by approximately Δν = 200 MHz.c,f, Numerical simulations of the heated LiCOF BFS response.d,g, Initial Brillouin gain distribution retrieved from simulations.

Fig. 4 |
Fig. 4 | Thermodynamic control of stimulated Brillouin scattering (sample with 2.1 μm diameter and 2 m length).a, BFS of peak 2 over relative pressure increase for differently long heated fractions of the fibre.b, Thermodynamic control of BFS of peaks 1 and 2, which originate from the same resonance, by almost 1.00 GHz (0.97 GHz), changing the temperature and pressure inside the fibre core.c, BFS of peak 1 is observed in the isobaric (red curve) and isochoric (blue curve) regime by first cooling down the LiCOF followed by switching to