1. Introduction

Two dimensional (2D) nanomaterials (graphene, black phosphorus, transition atom chalcogenides, etc.) are an advanced trend in the science of nanostructures due to their peculiar topology and electronic features triggering unusual electron and heat transport properties. They also open a new manifold of possibilities for the design of nanostructured devices [1,2]. They show outstanding nonlinear optical properties like enhanced two-photon absorption and absorption saturation [3,4] providing new perspectives for application in laser technologies, optical computing and telecommunications [5].

Ongoing wave mixing studies in 2D materials are mainly focused on harmonic generation [6–8]. Four-wave mixing in near infrared was recently carried on a graphene monolayer [9], and revealed a third-order nonlinear susceptibility χ⁽³⁾ value of 2.1⋅10⁻¹⁵ m²V⁻² which is ca. 7 orders of magnitude larger than in bulk insulators like silica and BK7 glass, 3-5 orders of magnitude larger than in bulk semiconductors like silicon, germanium, cadmium and zinc chalcogenides, metal oxides, and ca. 10 times larger than in thin plasmonic gold films and nanoparticles [10]. Most recently, an even larger value of χ⁽³⁾ = 6.3⋅10⁻¹⁴ m²V⁻² was obtained in graphene nanoribbons at mid-infrared frequencies close to the transverse plasmon resonance [11]. Despite these not yet abundant but impressive advances of phase conjugation in graphene, the effect of 2D materials on stimulated Brillouin scattering (SBS) remains unknown. Meanwhile, this effect is fundamentally important in laser and fiber telecommunications, and currently attracts theoretical considerations [12–14] concerning bulk and composite semiconductor materials, including practical designs [15].

This motivates the investigation of SBS in 2D nanostructures. We report here on the peculiar character of SBS in liquid graphene suspensions. We observe a net SBS effect in water and N-methyl-2-pyrrolidone (NMP), which is a good solvent for nanocarbon, and SBS threshold energy changes upon graphene suspension addition into the solvents. The results show a strong quenching of SBS even using vanishingly small concentrations of graphene nanosheets, corresponding to an absorption coefficient in the order of 0.001 cm⁻¹ in the case of water. We attribute the observed effect to the interference of gratings formed in the liquid both by electrostriction and thermal expansion. By means of computer simulations of measured concentration dependences of SBS threshold energies we show that the thermal expansion is determined by carbon vapor bubble formation which strongly influences on the refractive index through density changes and acoustic wave damping through bubble compressibility. This allows the evaluation of effective bubble size in the suspensions under laser propagation conditions.

We also observed this effect in an aqueous suspension of single wall carbon nanotubes (SWCNT) at the dilution having absorption coefficient value of 0.016 cm⁻¹ at the excitation wavelength and found it lower than in the graphene suspension.

2. Experimental procedure

2.1 Graphene nanoparticles morphology

Graphene suspensions were prepared from graphite powder as described earlier [16]. Sodium cholate (NaC) surfactant was used to stabilize the aqueous suspension. Details of suspension preparation and absorption cross-section, σ_e, measurement are available in Appendix A. We evaluated σ_e = (0.80 ± 0.16)⋅10⁴ cm²g⁻¹ of graphene in the aqueous suspension using the density value measured by gravimetric method. Olympus BX53 optical microscope, FEI Tecnai G2 F20 TEM and Bruker Nano Inc. Dimension 3100 AFM were used to visualize graphene. Graphene flakes with transverse sizes from 0.2 to 2 μm were observed in a dried drop of aqueous suspensions (Gr_water) on a glass plate using both optical [Fig. 1(a)] and atomic force [Fig. 1(b)] microscopes. We could not obtain the same pictures of graphene from NMP (Gr_NMP), where separate flakes probably strongly aggregate during drying. For aqueous suspension, it turned out apparently possible because the surfactant prevents graphene aggregation by forming a sort of coat [high spikes in Fig. 1(b) at the flake edges]. The surfactant can also improve the visibility of graphene sheets, and this can be the alternative reason why we can see the flakes dried from water and do not see them from NMP. However, transmission electron microscopy images show that the graphene flakes in NMP also have several hundred nanometers average transverse size. A typical flake is shown in Fig. 1(c).

 

Fig. 1 (a) Optical and (b) AFM images with transverse profiles A and B of Gr_water; (c) TEM image of Gr_NMP; Raman G′ band spectra of (d) Gr_water and (e) Gr_NMP: curves A and B refer to the excitation wavelengths 488 nm and 514 nm respectively, dashed arrows show the corresponding maximum positions of this excitation-dependent band.

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Flakes thickness measured by AFM varies from 0.3 nm to 1.5 nm approximately, evidencing the few layers morphology with middle thickness around 1 nm corresponding to 3-layer graphene whose thickness is estimated as a sum of two 0.34 nm interlayer distances and two 0.17 nm van der Waals radii. The suspensions were further studied by Raman spectrum analysis. Raman studies were performed at the Research park of St. Petersburg State University, center for Geo-Environmental Research and Modeling “Geomodel”, using a Horiba Jobin-Yvon LabRAM HR800 spectrometer equipped with a microscope Olympus BX-41. Spectral resolution was better than 3 cm⁻¹. Raman spectra were excited by an Ar⁺ laser at 514 nm and 488 nm wavelengths with maximal power of 5 mW at the sample. A double Raman resonance G′ mode band of graphene dried from the aqueous suspension [see Fig. 1(d)] shows the spectral position and shape (a complex asymmetric band with maximum at the Raman shift of ca. 2688 cm⁻¹, and spectral width of ca. 60 cm⁻¹) which also correspond to the 3-layer graphene [17]. Graphene dried from NMP suspension [Fig. 1(e)] shows an intermediate shape of its G′ mode band between 2-layer and 3-layer graphene band shapes. Based on these data we suppose the preponderance of 3-layer graphene flakes in our suspensions.

2.2 Nonlinear optical setup

SBS observations were performed at room temperature using the second harmonic (λ = 532 nm) of 4 ns pulsed Continuum Minilite II Nd:YAG laser operating at 1Hz pulse repetition frequency. The setup is schematically shown in Fig. 2. Crossed polarizers P₁ and P₂ were used to change the laser pulse energy, together maintaining the laser beam polarization unchanged. A beam expander telescope (BE) was used to decrease the beam divergence angle 5 times from its initial divergence of θ_o = 1.8 mrad (measured by a ruler at the 4 m base).

 

Fig. 2 Schematic setup: Continuum Minilite II – laser source; P₁, P₂, P₃ – polarizers; BE – beam expander; BS – beam splitter; L₁, L₂ – lenses; F₁, F₂, F₃ – neutral filters sets; D₁, D₂, D₃ – detectors; S – sample cuvette; DP – diaphragm; dashed rectangle shows a light protection mask. Insets: A – X-axe beam intensity distribution, measured at L₁ focal position (solid) and its Gaussian fit (dash); B – image of SBS from pure NMP on the screen put on DP; C – time profiles of incident beam (solid) and backward scattered (dash); D – image of the interaction area from the cuvette with pure solvent in the SBS observation conditions; E – the same image from the graphene suspension when no SBS is observed.

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The beam was focused in a 3 cm-path quartz cuvette by 150 mm biconvex lens L₁. The beam spatial distribution after the lens being close to Gaussian in the focal point (see the inset A in Fig. 2), is influenced by diffraction at other Z positions. Z-scan study of the beam spot radius (W) performed in the air and approximations of its results by the test function W(Z) = W₀(1 + (Z/Z_R)²)^1/2 gave W₀ = 41 ± 7 μm waist radius and Z_R = 1.0 ± 0.2 mm Rayleigh-length.

The backward scattered beam was reflected by the beam splitter BS and observed on the screen placed after a 300 μm pinhole PH. Inset B in Fig. 2 shows a visual picture of the SBS effect which has specific threshold behavior. The backward beam polarization followed the polarization of the incident radiation. Time profiles measurements performed with Thorlabs DET 025AL/M 2 GHz silicon detector manifest a pulse-time reduction from τ_p = 3.7 ± 0.3 ns (FWHM) in the incident beam to τ_p = 2.6 ± 0.2 ns in the backward beam (inset C in Fig. 2) that evidences its nonlinear origin.

Energies of incident, back-scattered and transmitted beams were measured by silicon detectors Thorlabs PDA100A-EC calibrated both by Coherent J50MB-YAG-1561 and by Thorlabs ES111C pyroelectric heads to account for possible systematic errors related to the energy meters. The difference in the calibrations was within 5%. The maximal SBS beam energy was found to be ca. 0.9 mJ in organic liquids and ca. 0.05 mJ in water at the ultimate achievable input energy 4.5 mJ. No SBS was detected in pristine (not diluted) suspensions of graphene nanosheets in water and NMP. Then, graphene suspensions were added into pure solvents by small portions, evaluating its linear absorption coefficient α_e [exponent index in Lambert-Beer law: α_e = α₁₀⋅ln(10), where α₁₀ is instrumentally obtained absorption coefficient] by the dilution ratio.

3. Results of nonlinear optical measurements

3.1 Stimulated Brillouin scattering

SBS beam energy E_SBS was measured simultaneously with energy of the beam transmitted through the sample, E (optical limiting), vs. pump laser energy, E₀, for different linear absorption coefficients. The dependences obtained are shown in Fig. 3. In the studied diapason of E₀, they look linear, therefore SBS threshold energies, E_thr, can be numerically found as X-interсepts of linear fittings as shown in Figs. 3(a) and 3(b). The threshold determines the exponential SBS gain coefficient, G, according to the condition [18]:

 

Fig. 3 Incident energy dependences of: (a) back scattered energy from NMP-based liquids, (b) back scattered energy from water-based liquids, (c) transmitted energy through NMP-based liquids, and (d) transmitted energy through water-based liquids. Data correspond to (⬤) pure NMP, (⬛) pure water, (⬤) aqueous solution of NaC, and suspensions of different graphene concentration with following absorption coefficients α_e: () 0.013 cm⁻¹, (▼) 0.023 cm⁻¹, (▲) 0.032 cm⁻¹, (◆) 0.048 cm⁻¹, (◆) 0.001 cm⁻¹, (⬣) 0.002 cm⁻¹, (⬟) 0.004 cm⁻¹.

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Small additions of graphene suspensions in pure solvents remarkably decrease the SBS beam energy and increase its threshold. This is opposite to a reduction of the SBS threshold calculated for a graphene-clad tapered silica fiber [14]. In such structure graphene modulates the hypersonic wave which enhance SBS due to an appropriate ratio of photo-acoustic and electro-optic parameters of the core and the clad (acoustic velocity, Brillouin frequency and refractive index). In our case, we see a manifestation of another effect, taking place on the background of acousto-optical modulation of the bulk material by dispersed graphene nanoparticles and apparently originating from light absorption.

The obtained concentration dependence of the threshold energy reveals a rather linear character. Figures 4(a) and 4(b) show the dependences of E_thr on α_e, which look quite linear both in NMP and in water. Experimental errors in Fig. 4 are within the point’s size except the last point where error is large because of very small SBS energy values. The SBS threshold in water was shifted up even with the NaC addition, however, our measurements performed with higher concentrations of NaC (from 1 to 5 wt.%) did not reveal further threshold increase. Since the NaC concentration used in our suspension is in proximity of critical micellar concentration [19]: 0.68 wt.%, we consider the micellation as the reason of this SBS threshold change through an additional extinction both of optical and acoustic waves. It was extracted from the experimental points [see hollow squares in Fig. 4(b)] and only the effect connected with graphene was further considered.

 

Fig. 4 SBS threshold energy against absorption coefficient of graphene suspensions in (a) NMP and in (b) water; SBS gain factor against absorption coefficient of graphene suspensions in (c) NMP and in (d) water. Filled points: experimental results; empty points in (b): results corrected for the shift due to NaC (blue circle); lines in (a) and in (b): linear fitting; curves in (c) and in (d): simulations with simultaneous variations of ρ and Γ_B.

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We see from the Fig. 4 that SBS threshold is very strongly affected by graphene quantities which can hardly even be detected spectrometrically (α₁₀ < 0.01), and which can be considered as impurities. Thus, the SBS threshold measurement can be considered as a method for such impurity detection in pure liquids. The minimal detectable graphene concentration, ρ_{Gr min}, can be evaluated through the uncertainty of SBS energy at lower concentrations, ΔE_thr, which is biggest in case of water: ΔE_thr = 0.11 mJ. The minimal detectable graphene absorption is α_{e min} = ΔE_thr/b = 4⋅10⁻⁴ cm⁻¹, where b is the slope taken from the linear fitting, and ρ_{Gr min} = α_{e min} /σ_e = 5⋅10⁻⁸ g⋅cm⁻³. The value for NMP can be even smaller because α_{e min} consists only 2⋅10⁻⁵ cm⁻¹, however, since the absorption cross-section for the NMP-based suspension is not determined we cannot quote it here.

The Brillouin gain factor value obtained from the intercept of E_thr(α_e) fitting line by means of Eq. (1), g_B(0) = 6.4 ± 0.1 cm⋅GW⁻¹ for water, is in accordance with a theoretical one 6.4 cm⋅GW⁻¹, and however by 35% greater than an experimental one 4.8 cm⋅GW⁻¹ [20]. Although we could not find any published value of g_B for NMP, the plausible result for water convinces us in the correctness of our value g_B(0) = 18.6 ± 1.8 cm⋅GW⁻¹. It should be noted that the uncertainties indicated for g_B(0) are obtained from the spread of E_thr experimental points and do not include systematic errors. An analysis of the latter shows that the error of beam waist ΔW₀ dominates among them in Eq. (1) and consists in 34%.

Since the E_thr(α_e) dependence is linear, the g_B(α_e) one evaluated using Eq. (1) shows hyperbolic character [points in Figs. 4(c) and 4(d)].

3.2 Optical limiting

Figures 3(c) and 3(d) show the optical limiting curves manifesting moderate nonlinear behavior at energies below SBS thresholds (< 1 mJ for NMP and < 2 mJ for water) which has been considered as effective two-photon absorption (TPA) and fitted by parabolic functions shown in the figures as solid curves:

Table 1. Effective NLA coefficients of liquid systems with different content (absorption coefficient) of graphene obtained from fitting the experimental points with Eq. (2).

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We plotted the obtained coefficients against the linear absorption ones and fitted by straight lines as shown in Fig. 5. The line slope determines the nonlinear-to-linear absorption cross-sections ratio:

 

Fig. 5 Effective NLA coefficients of suspensions against graphene linear absorption (⬛) in NMP and (⬤) in water.

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The NLA cross-section for the suspension in NMP is found to be σ_eff = (5.59 ± 1.21) cm²GW⁻¹⋅σ_e, and in water: σ_eff = (42.0 ± 5.6) cm²GW⁻¹⋅σ_e. The data on TPA coefficients of pure solvents in visible range are scarce. For water at λ = 532 nm we have just an upper bound of it: β_eff < 0.004 cm GW⁻¹, obtained with picosecond pulses [21]. Our value is 20 times greater. Hence, it comprises other processes, enhancing the effect from the picosecond to nanosecond time scales. In view of the absence of linear absorption in visible range, we can exclude reverse saturable absorption effects related to long-lived states and consider the light scattering on increasing density fluctuations as the most realistic reason. These fluctuations follow the intensity fluctuations (speckles) due to the electrostriction forces even prior to the visible appearance of SBS [22]. At higher energies, around SBS thresholds the limiting curves in Figs. 3(c) and 3(d) show pits and subsequent linear behavior signaling additional reduction of the transmitted energy by SBS whose energy linearly depends on E₀ within our experiment accuracy.

On the other hand, for a graphene suspension in NMP with α_e = 0.223 cm^–1 we know a value of β_eff = 0.29 cm$\cdot$GW⁻¹ obtained from Z-scan measurements at λ = 532 nm, τ_p = 4 ns [23]. That should give us β_eff ~0.03 cm⋅GW⁻¹ for our suspensions with ten times smaller absorption. Our β_eff value for the graphene suspension in NMP: β_eff(α_e = 0.023 cm^–1) – β_eff(α_e = 0) = 0.14 cm $\cdot$ GW⁻¹ is about 5 times greater. Therefore, we think that the linear concentration dependence in Eq. (3) may saturate at higher graphene nanoparticle concentrations due to aggregation effects (flakes size redistribution).

4. Theory of the effect and simulations

4.1 Conceptual issues

Propagation of the SBS beam intensity I_S through a weakly absorbing substance is described by the equation [24]:

Somehow analogous effect was observed and described by McEwan and Madden [25] in experiments on degenerated four-wave mixing in carbon black suspensions. They showed that bubbles caused by local heating of carbon nanoparticles upon laser radiation, form a transient grating in the substance correlated to interference peaks of two incident laser beams. The fourth beam appears as a result of diffraction of the third beam on the grating. Graphene nanoparticles being good absorbers should form a similar grating from their thermally expanded microenvironment. The difference in our case is that this grating corresponds to the maxima of the incident beam intensity.

The sketch in Fig. 6 shows the processes taking place in the graphene suspensions under the conditions of SBS generation. First, electrostriction forces (red arrows) compress the matter and form the density peak (hence, the refractive index peak) grating which is responsible for the backward scattered wave amplification (the conventional SBS generation mechanism). At the same time, graphene nanoparticles form a “negative” grating of refractive index dips due to thermal expansion which is coherent with the first grating and destroys it.

 

Fig. 6 Sketch of the electrostriction – thermal expansion antagonism occurring in a graphene suspension upon irradiation and giving impact on SBS quenching.

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This background requires a consideration of the thermal expansion process that leads to an additive to the Brillouin gain factor of the same form as for the thermal-induced scattering but opposite in sign [24]:

Notations are: γ_e = ε₀ρ (∂ε/∂ρ)_S – electrostrictive coefficient; γ_T(α_e) = ε₀c²β_T v_acα_e/(C_pω₀) – thermo-optic coupling coefficient; β_T = (∂V/∂T)_ρ/V – thermal expansion coefficient; C_p – specific heat capacity at constant pressure, ρ – solvent density; S – entropy; n – refractive index.

4.2 Local heating of graphene flakes

A local temperature rise, ΔT, of absorbing graphene flakes drives the process, and the understanding of its scale is important. It can be upper estimated using the simple heat capacity definition relation: ΔT = ΔQ_abs/(m_Gr⋅C_p), where ΔQ_abs is the heat amount obtained by a graphene flake from the radiation absorbed, m_Gr is the flake mass, C_p = 0.7 J⋅K⁻¹g⁻¹ [26] for graphene.

Following the Lambert-Beer law at low absorption and neglecting the scattering: ΔQ_abs = E₀F(t)σ_em_GrL/V, where V is the irradiated suspension volume, F(t) = ∫₀^tI₀(τ)dτ/E₀ is the time dependent radiation dose, which can be easily calculated from the laser beam time profile (see Fig. 2, inset C). Considering the irradiated volume is that inside the caustic surface: V = 8πW₀²Z_R/3 for the Gaussian beam, we finally obtain that the temperature rise is not determined by the flake mass or size, but only by the absorption cross-section:

 

Fig. 7 Temperature rise of a graphene flake upon 1 mJ laser-pulse absorption (solid curve); laser beam temporal profile (dashed curve) in arbitrary units.

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For the first τ₁ = 1.34 ns a temperature rise of 4830 K is achieved which approximately corresponds to the graphite sublimation condition reported by McEwan and Madden [25]. The subsequent sublimation process can be traced using the relation:

Carbon atom and its ions have no remarkable absorption lines at λ = 532 nm and 266 nm, therefore a further temperature growth after the sublimation can be determined either by carbyne forms in the carbon vapor or by plasma electrons. The plasma is apparently getting generated in the bubbles due to a partial oxidation of the carbon atoms. These processes are hardly evaluable and seem unimportant in the framework of the present study.

It is important to understand that at a laser pulse energy around the SBS threshold all dispersed graphite nanoparticles become evaporated after 4.65 ns. The characteristic time of SBS is the inverse acoustic wave dumping coefficient (2π/Γ_B), and for most of the studied liquids it is equal to 3-4 ns, which means that the graphite evaporation begins to interfere with the SBS process. For greater pulse energies this time scale is shorter. Hence, in the presence of graphene flakes, SBS is developing in a medium which represents an emulsion of carbon vapor bubbles and has non-steady temperature fields. Thermodynamic characteristics of such medium can significantly differ from those of pure liquid. The increasing of graphene concentration will also increase the bubble concentration and the material characteristics will change drastically.

4.3 Simulation of g_B(α_e) dependence

Only thermo-expansive part of the Brillouin gain factor [see Eq. (7)] shows an explicit dependence on α_e, which, however, does describe neither the derivative dg_B(α_e)/dα_e observed, nor the hyperbolic form of the g_B(α_e) dependence. Thus, we must assume an influence of graphene flakes on SBS parameters in Eqs. (6) and (7).

We performed simulations of g_B(α_e) using one of these parameters, P, changed by small (less than 20%) variations δP:

Some of the parameters depend on others, such as refractive index and acoustic velocity on density. In our simulations we considered the former using the Lorentz-Lorenz equation:

Table 2. Physical properties of the solvents at T = 293 K and standard pressure.

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Electrostrictive coefficients were also evaluated using Eq. (11), which gives a well-known relation: γ_e = ε₀(n²–1)(n² + 2)/3. The Brillouin line widths Γ_B (α_e = 0) were taking those to fit the observed values of g_B(0) in calculating by Eq. (5) at α_e = 0: Γ_B = 2.545 GHz (NMP) and Γ_B = 3.475 GHz (water). The heat capacity variation δC_p in γ_T(α_e) functionally gives the same behavior as δβ_T with the opposite sign, so that the latter can be considered as a variation of the thermo-optic coupling coefficient in total: δβ_T = δγ_T.

The simulated g_B(α_e) dependences due to different parameter variations are superimposed onto the experimental points in Figs. 8(a) and 8(b). The efficiency of each parameter towards the relative change of g_B is illustrated by dependences in Figs. 8(c) and 8(d): more rapid curve at the (1,1) point indicates more influenced parameter. The efficiency curves make it clear that the thermal expansion gives a very small contribution to the Brillouin gain factor, which in case of water is only ca. 10⁻³ of g_B^e values in all the small variations range. In case of NMP, g_B^T still has nonzero impact, which has been considered mostly in simulations at other parameter variations. The simulated curves of g_B(α_e) due to δγ_T have too strong curvature [curve E in Fig. 8(a) and curve С in Fig. 8(b)] and cannot describe the effect observed.

 

Fig. 8 SBS gain factor of (a) NMP and (b) water against linear absorption of graphene suspensions; points: experimental results; curves: simulations; varied parameters are: (a) n (curve A), ρ (curve B), v_ac (curve C), Γ_B (curve D), γ_T (curve E), (b) n and ρ (curve A), v_ac and Γ_B (curve B), γ_T (curve C). Relative changed SBS gain factor of (c) NMP and (d) water against corresponding varied parameters: (c) n (curve A), ρ (curve B), v_ac (curve C), Γ_B (curve D), γ_T (curve E), (d) n (curve A), ρ (curve B), v_ac and Γ_B (curve C), γ_T (curve D).

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We see that all the simulated curves in Figs. 8(a) and 8(b) do not follow the experimental points sufficiently well. However, the proximity of each curve to them indicates reliability of the parameter influence. Since the dependence is more nonlinear for NMP, Figs. 8(a) and 8(c) are more indicative. The changes of v_ac and Γ_B, both positive (increase), give similar contributions to g_B(α_e), even equal for water where the g_B^T part independent on v_ac is negligible. But for NMP we can see the difference between them: the simulation with δv_ac [curve C in Fig. 8(a)] does not reproduce the curvature of the experimental points. On the other hand, the positive variation δΓ_B gives the best approaches to them. This makes Γ_B more important parameter in the SBS quenching as compared to v_ac.

The negative variations of ρ and n (their decrease) give even stronger influence on the effect observed and reproduce the true sign of curvature [curves A, B in Fig. 8(a) and curve A in Fig. 8(b)]. The absorptive increment (decrements) providing the best fit of the experimental points are given in Table 3. The sensitivity of g_B(α_e) curves to these values defines their uncertainties which are within the last valid digits.

Table 3. Relative absorptive changes (in cm) of refractive index, density, acoustic velocity, Brillouin line width and thermal optic coefficient providing the observed decrease of g_B.

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Lower Δ_αP value shows stronger influence of the corresponding parameter. Despite n is the most influenced parameter, it is not independent, and its variation can be contributed from both density and mass refraction changes. An analysis of the mass refraction contribution given in Appendix II demonstrates the prevailing of density-determined nature of the refractive index changing. It can be calculated that values Δ_αρ = –11.1 cm (NMP) and –32 cm (water) is obtained when n(ρ) dependence is taken into account.

Therefore, the simulation results denoted a density decrease and an increase of Brillouin line width as the key changes leading to the SBS quenching, whereas the effect is very low sensitive to the thermo-optic coupling coefficient. Analyzing parameters that determine Δ_αΓ_B in Eq. (8), we can see that its growth is unlikely determined by the Brillouin frequency shift (ω₀ – ω_B), whose change should be rather negative upon irradiation [29].

Thermodynamic studies show a remarkable and monotonous decrease of viscosity (both shear and bulk) of liquids [36] and steam [35] in a large range of temperature and pressure growth. Hence, if the Eq. (8) remains adequate for the inhomogeneous media, the growth of Γ_B should be attributed to a drop of acoustic velocity. Such kind of decrease is anticipated in a bubbly liquid due to a high compressibility of bubbles [30]. The effect can be considered as an acoustic wave extinction in growing bubbles that increase the acoustic absorption coefficient α_ac = Γ_B/v_ac of the liquid, which is most likely the main parameter in the SBS quenching along with density.

Apparently, both the density and acoustic absorption factors take place in the real process acting simultaneously. For this reason, we performed a fitting of the experimental g_B(α_e) points by a simulation with simultaneous variation of ρ and Γ_B values. The curves obtained are shown in Figs. 4(c) and 4(d) and manifest a perfect coincidence with the measured dependence with correlation coefficients 0.992 (NMP) and 0.991 (water). Corresponding parameter changes are: Δ_αρ = –1.55 cm (NMP), –5.4 cm (water) and Δ_αΓ = 50 cm (NMP), 90 cm (water).

4.4 Bubble size scaling

Since the density reduction is determined by the bubble formation: δρ = ρ (1 – f_C), where f_C is the bubble filling factor, the density decrement allows to estimate the average bubble radius:

The bubble radius found is comparable with those obtained in SWCNT suspension in binary water-PEG solvent under nanosecond laser irradiation: 1.2 μm [30]. Therewith, the referenced study confirms a weak dependence of the initial bubble size on the laser pulse energy. Our one-point measurement of the SBS quenching in SWCNT suspension: Δg_B/Δα_e = –94 cm²GW⁻¹ is less than our result for graphene that will also give a lower value for the bubble size in case of nanotubes.

Simple evaluations [see Eqs. (14) and (15) in Appendix B] using the same density decrement value give the ratio of bubble-to-flake volume V_b/V_f = 1⋅10⁵ without data of flake surface size, but with the flake thickness. In our case it does not improve the uncertainty, but at some conditions the thickness can be made more uniform and precisely measured by AFM or Raman spectroscopy than the flake area. In this case the volume ratio can be more indicative.

In both cases the absorption cross-section of nanomaterial should be known. For graphene flakes numerous experiments give values which are several times different depending on the suspension preparation conditions [38–41]. Although similar σ_e values were demonstrated for different organic solvents [38], the value found in aqueous suspensions was twice less than that in NMP-based suspensions after the same treatment procedure [38,39], presumably, due to the surfactant influence. If, following this reason, we perform similar evaluations for NMP in assumption of the graphene absorption cross-section being twice lager than in water, we obtain values: r_b = 1.7 μm and V_b/V_f = 8⋅10⁴, which are comparable with the values obtained for aqueous suspension.

Despite the demonstrated possibility of the SBS method in obtaining the bubble size, the simultaneous fitting performed reveals a wide minimum which is crucial especially for Δ_αρ due to its small value and can increase the uncertainties of the characteristics found. Independent measurements of the acoustic absorption by photo-acoustic methods or Brillouin spectroscopy must be done to obtain the full image of the mechanism of light-matter interaction in conditions of the SBS type phase conjugation in liquid suspensions of strongly absorbing nanoparticles. Once the acoustic absorption coefficient is fixed, the dimension of vapor bubbles can be determined more precisely.

5. Conclusion

In summary, we have studied the stimulated Brillouin scattering in NMP and water in the presence of nanoscale graphene flakes and have measured its energetic characteristics. We found a strong SBS quenching effect in liquids by graphene with low concentrations of the nanoparticles that give no remarkable absorption in the material. Established linear dependences of SBS threshold on graphene absorption coefficient (i.e., concentration) can be used for the detection of small nanomaterial quantities in liquid media down to 5⋅10⁻⁸ g⋅cm⁻³ which was shown for the aqueous suspension.

Computer simulations of the Brillouin gain factor show the efficiency of different thermodynamic, electrooptic and photoacoustic parameters in the SBS quenching. The role of density and compressibility among them which change due to carbon vapor bubble formation is found to be decisive. Effective bubble size has been estimated, which can be specified in combination with acoustic wave absorption measurements providing more information on bubble properties in the nanosecond time scale. From this point of view, the SBS method can be used as a tool for the nanosecond-resolved bubble formation scaling in nanoparticle contained media.

The studied effect can be used for SBS suppression in situations where it is undesirable (both in laser technology and optical telecommunication networks). We expect an appearance of this effect in transparent solids containing traces of nanoparticles and believe that the method can be applied to other kinds of 2D-nanomaterials and maybe other nano-sized linear or nonlinear absorbers.

Appendix A Suspension, preparation, and absorption cross-section measurements

Graphene suspensions were prepared from graphite powder (Sigma-Aldrich 332461) by the way like that described earlier [16]. NMP (99.5%, Sigma-Aldrich 328634) and double deionized water were used as solvents. Graphite powders were added to the solvent with initial concentration of 5 mg/ml. Sodium cholate (NaC) hydrate > 99% (Sigma-Aldrich C6445) surfactant was added to the aqueous sample with initial concentration 0.03 mg/ml. Mixtures were then sonicated using Sonics Vibra-cell VCX-750 W ultrasonic processor at the amplitude 38% for 1 h. Then the suspensions were centrifugated in Rotofix 32A centrifuge at 4000 rpm for 1 h twice with 15 h interval. After that supernatant portions were taken, and their absorption spectra were measured.

The precipitate was dried in an evacuated drying box at ca. 1 Torr pressure and T = 90°C with a periodical control of weight using Mettler Toledo XS105 analytic balance. The drying ended when the weight stopped changing. The total weight of the dry precipitate was subtracted from the initial weight to evaluate the carbon concentration ρ_C in the suspension. For aqueous suspension we obtained ρ_C = 56.3 ± 11.2 μg $\cdot$cm⁻³. The uncertainty is mainly determined by the weight error ± 0.5 mg appearing from poorly controllable water vapor adsorption by dry carbon powder. No weight loss was found in carbon precipitate from NMP comparing to the initial carbon powder weight in several attempts of suspension preparation and drying during weeks. This apparently means that the procedure doesn’t allow to eject NMP molecules adsorbed by carbon. After spectral measurements 0.47 mg cm⁻³ of NaC was added to the aqueous suspension for better stability. The total concentration 0.5 wt.% of NaC in the suspension is close to its critical micelle concentration [19] so that we can expect that graphene flakes are totally surrounded by the surfactant. Both suspensions (NMP- and water-based) were stable, and no sedimentation was seen during more than 6 months.

Absorption spectra of the studied suspensions were measured in 1 cm path cell using PerkinElmer Lambda 750 dual-beam UV/VIS/NIR spectrometer, just after taking supernatant. The spectra were measured with reference to the solvent placed in the reference beam. The results are shown in Fig. 9.

 

Fig. 9 Absorption spectra of the as-prepared graphene suspensions in water and NMP.

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The NMP suspension was approximately 5 times denser; its spectrum in the Fig. 9 is divided by 10 to compare with that of aqueous suspensions. Two main factors determine the absorption spectra shape. First, it is correlated with graphene density of states DOS(λ), which in case of electron-hole symmetry is determined by the complete elliptic integral of first kind [1]:

(13)$$\begin{array}{l}DOS(\lambda )=\frac{8}{{\pi }^{2}hc}\frac{{\lambda }_h^2}{\lambda }{Z}_0^{1/2}{\displaystyle \underset{0}{\overset{\pi /2}{\int }}\frac{d\phi }{Z_0-Z_1{\sin }^2\left(\phi \right)}},\\ Z_0={\left(1+\frac{{\lambda }_h}{\lambda }\right)}^2+\frac{1}{4}{\left[{\left(\frac{{\lambda }_h}{\lambda }\right)}^2-1\right]}^2,Z_1=4\frac{{\lambda }_h}{\lambda }.\end{array}$$

Here λ_h is the peak position of DOS which corresponds to twofold nearest-neighbor hopping energy in graphene (E_h ≈ 2.8 eV), h is the Plank constant, c – the speed of light in vacuum. The calculated DOS(λ) function using Eq. (13) is shown in the Fig. 9 by a dotted curve with maximum at λ_h = 221.4 nm. The other dotted curve with maximum at λ_h = 270 nm is a Lorentz function which is a good approximation for a surface plasmon absorption cross-section [42]. The transverse plasmon resonance frequency in graphene takes place at ~5 eV [43].

It can be seen in the Fig. 9 on the example of aqueous suspension spectrum, that each of these dependences do not describe it separately. The Lorentzian can describe only its red part for wavelengths λ > 500 nm with a constant adduct (red dashed curve). Both DOS and Lorentzian together reproduce the experimental curve down to λ ≈ 300 nm (magenta dash-dotted curve). The constant adduct is also necessary to fit the spectrum, and it is apparently connected with the linear Dirac electronic spectrum of graphene, which leads to the constant absorption coefficient equal only απ per layer, where α = 1/137 is the fine structure constant, in all NIR and visible spectral region down to 450 nm [43].

The peak shape in Fig. 9 may be strongly influenced by light scattering which is small in visible for the nanoparticles concentrations studied, but becomes remarkable in UV not only due to diffraction enhancement, but also due to dramatic changes in dynamic dielectric permittivity through the factor [ε(ω) – ε_NaC(ω)].

These considerations assume a complicated character of the spectrum determined by different electron movements in graphene. In a real graphene flake electron-hole symmetry can be broken due to defects and a confinement effect that leads to a shift of the DOS peak position and distortion of its shape. The real spectrum of graphene suspension is even more complicated being a sum of different particles absorption, where the size and layer thickness distribution as well as defect structures are crucial.

Thus, the total absorption coefficient of graphene in water measured at λ = 532 nm is α₁₀ = 0.1954 ± 0.0011 cm⁻¹ that taking into account ρ_C value obtained gives the absorption cross-section σ_e = α₁₀ ⋅ ln(10) /ρ_C = (0.80 ± 0.16)⋅10⁴ cm²g⁻¹. It is a little less than the value 1.39⋅10⁴ cm²g⁻¹ obtained for low-concentrated aqueous graphene suspensions [39], which we explain by difference in layer thickness and defect structures of our graphene flakes compared to those studied in the reference.

Appendix B Carbon filling factors and mass refraction

Fast carbon evaporation during 4 ns pulse leads to an increase of total volume of bubbles V_b. The bubble filling factor, which is the ratio of its volume to the caustic volume can be then evaluated: f_C = V_b/V = r_b³N_b/(2Z_RW₀²), where r_b is the bubble radius and N_b is the number of bubbles in the caustic region. The latter coincides with the number of graphite flakes in the same volume. We can calculate it from carbon density ρ_C, V, and the mass of one particle m_Gr, which can be obtained only by rough estimations. Assuming three-layer thickness in average, we can accept: m_Gr = 3d²ρ_s, where ρ_s = 7.7⋅10⁻⁸ g⋅cm⁻² is the surface density of graphene [37]. Then, considering that α_e = ρ_Cσ_e, we obtain:

$$N_{\text{b}}=\frac{8\pi W_0^2{Z}_{\text{R}}{\alpha }_e}{9d^2{\sigma }_e{\rho }_{\text{s}}}\text{,}$$

which gives the value N_b = 1.5⋅10⁶⋅α_e(cm⁻¹) with the assumption of the average characteristic flake size based on the AFM image scale to be around d = 600 nm. Its uncertainty comes mainly from the size distribution standard deviation SD(d), and apparently consists a few tens of percent. Thus, for the bubble filling factor we have:

$$f_{\text{C}}=\frac{4\pi \cdot r_{\text{b}}^3{\alpha }_e}{9d^2{\sigma }_e{\rho }_{\text{s}}}\text{,}$$

or f_C = 5⋅10¹¹⋅[r_b(cm)]³⋅α_e(cm⁻¹) in water with the said uncertainty.

At that, the initial graphene flakes filling factor f₀ can be estimated more precisely because it does not require the particle size, but only its thickness h:

(14)$$f_0=\frac{\text{h}\cdot {\alpha }_e}{3{\sigma }_e{\rho }_{\text{s}}}\text{.}$$

Assuming the middle thickness of graphene flakes in our suspension h = 1 nm, we obtain f₀ = 5.4⋅10⁻⁵⋅α_e(cm⁻¹) for water. At the average number of layers equal to three, its standard deviation cannot be more than one that implies the 33% upper estimation for the Δh uncertainty. With the 20% uncertainty of density ρ_C it gives Δf₀/f₀ = 39%.

From the definition of filling factors also follows that their ratio is just a ratio of the average bubble volume to the average flake volume:

(15)$$\frac{f_{\text{C}}}{f_0}=\frac{V_{\text{b}}}{V_{\text{f}}}\text{.}$$

Mass refraction is a ratio of the Lorentz-Lorenz function to the mass density:

(16)$$R\equiv \frac{n^2-1}{n^2+2}\frac{1}{\rho }\text{,}$$

and it is a measure of polarizability a₀ of a particle with the mass M₀:

(17)$$R=\frac{4\pi }{3}\frac{a_0}{M_0}\text{.}$$

In application to the substances under study, R was determined from Eq. (16) for NMP: R = 0.271 cm³g⁻¹ and for water: R = 0.206 cm³g⁻¹ using data presented in Table 2. For the graphene mass refraction, a refractive index value n_3layer = 2.27 for the three-layered graphene [44], and a simple estimation of graphene density: ρ_Gr = 3ρ_s/h = 2.31 g cm⁻³ were used: R_Gr = 0.251 cm³g⁻¹.

Refractive index of the suspension is not practically affected by the mass refraction of graphene in view of its very low density:

$$n_{\text{susp}}={\left[\frac{1+2\left(R_{\text{solv}}{\rho }_{\text{solv}}+R_{\text{Gr}}{\rho }_{\text{C}}\right)}{1-\left(R_{\text{solv}}{\rho }_{\text{solv}}+R_{\text{Gr}}{\rho }_{\text{C}}\right)}\right]}^{1/2}\text{,}$$

n_susp = 1.472 (suspension in NMP) and 1.333 (suspension in water) at maximal concentrations used.

The mass refraction of carbon in bubbles is evaluated from Eq. (17) and literature data on carbon atom polarizability [45]: R_C = 0.374 cm³g⁻¹ and found even greater than all other refractions. Thus, carbon vaporization could increase the refractive index in changing the term R_Grρ_C, which is negligible, by R_Cρ_C/f_C = R_C/(5⋅10¹¹⋅r_b³σ_e) which can be larger than solvent’s ones at small bubbles. Since we know from the experiment and related computer simulations that the refractive index of the suspension heated by laser pulse is decreasing, we can establish the conditions on the bubble radius value: r_b >> 10 nm. In that case R_Cρ_C/f_C << 0.1, and the decreasing of the refraction is assured only by the solvent density reduction due to bubbles formation:

(18)$$\text{δ}\left(R\rho \right)=R_{\text{solv}}{\rho }_{\text{solv}}\left(1-f_{\text{C}}\right)\text{.}$$

From here we can see that f_C = Δ_αρ⋅Δα_e, where Δ_αρ the absorptive decrement of density introduced by Eq. (10). The substitution of Eq. (14) into Eq. (15) and then into Eq. (18) gives Eq. (12) for the bubble size.

The expression is exact, i.e., obtained from the geometric consideration with the only assumption of the square shape of the flake. In case of other shape types d² should be replaced by the corresponding area value. Therefore, the bubble radius depends on the flake surface area, absorption cross-section and absorptive decrement of density.

Roughly supposing the relative standard deviation of the flake size distribution to be SD(d)/d = 30%, the fractional error δ_dr_b due to flake size uncertainty is 23%. The 20% uncertainty of the absorption cross-section gives the fractional error δ_σr_b = 7%. The error in Δ_αρ obtained from the experimental data fitting is not large and consists only few percent, however, other minima in the fitting related to other Δ_αΓ values are possible. In total, we evaluate δr_b error by 25%. Therefore, simultaneous measurement of acoustic absorption will help to obtain more reliable data of average bubble size.

Funding

National Natural Science Foundation of China (NSFC) (61675217, 61522510); the Strategic Priority Research Program of Chinese Academy of Sciences (CAS) (XDB16030700); the Key Research Program of Frontier Science of CAS (QYZDB-SSW-JSC041); the Program of Shanghai Academic Research Leader (17XD1403900); President’s International Fellowship Initiative of CAS (2017VTA0010, 2017VTB0006, 2018VTB0007).

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