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[Ken Takazawa](https://orcid.org/0000-0002-2169-5169), [Jun-ichi Inoue](https://orcid.org/0000-0001-6743-4258)

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[Subsonic phase front propagation in martensitic transitions of 1,2,4,5-tetrabromobenzene single crystals](https://mdr.nims.go.jp/datasets/aacec52c-f20c-4c6c-981f-411b2dd646b4)

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communications chemistry ArticleA Nature Portfolio journalhttps://doi.org/10.1038/s42004-026-02119-ySubsonic phase front propagation inmartensitic transitions of 1,2,4,5-tetrabromobenzene single crystalsCheck for updatesKen Takazawa 1 & Jun-ichi Inoue1Martensitic transitions in organic crystals,which are diffusionless structural phase transitionsbetweenpolymorphs, are topics of current interest. These transitions proceed by fast propagation of the phasefront (phase boundary) across the crystals and induce fascinating phenomena, such as thermosalienteffect and superelasticity, which have potential applications in molecular devices. Revealing themechanism of these transitions is of crucial importance for understanding the origins of thesephenomena and for their device applications. To this end, here we observe the spatio-temporalevolution of the phase front in temperature-induced martensitic transition in 1,2,4,5-tetrabromobenezen (TBB) crystals. We observed subsonic phase front propagation across thecrystals with the velocity of up to ~800m/s. The analysis using a nonlinear lattice model based onLandau’s theory of phase transitions showed that thephase frontmotion is describedasapropagatingsolitary wave in a non-conservative system. Our results revealed the essential factors governingorganic martensitic transitions and novel insights into the transition mechanism.Structural phase transitions between polymorphs of organic crystalsinduced by external stimuli, such as temperature and pressure, haveattracted increasing attention. These transitions give rise to fascinatingphenomena, such as the thermosalient effect, superelasticity, and shapememory effect, which have potential applications in organic crystal-baseddevices1–32. Crystallographic, thermodynamic, and spectroscopic investiga-tions of these phase transitions have demonstrated that they are first-orderdiffusionless transitions that proceed with small, albeit cooperative, motionof the constituent molecules. Therefore, they are referred to as organicmartensitic transitions by analogy with martensitic transitions in Fealloys33,34. The progression of the martensitic transition is characterized bynucleation followedby thepropagationof the interfacebetween thenewandoriginal phases (the phase front). This is in contrast to transitions accom-panied by diffusive atomic/molecular motions, which proceed with thenucleation of multiple small domains followed by the growth of thesedomains (the nucleation and growth mechanisms).Although numerous studies on martensitic transitions in metals andalloys have been performed for many years, their organic counterparts areemerging research areas, and themechanisms of the transitions are topics ofongoing research. A promising approach to gain crucial information forunderstanding the mechanisms is direct and real-time observation of thephase front propagation. The velocity, propagation directionwith respect tothe crystallographic axes, and width of the phase front are closely related tothe thermodynamic and elastic properties of the crystal as well as to thepacking and interactions of molecules in the crystal. All these propertiestogether govern the dynamics of phase transitions. Therefore, the detailedanalysis of phase front dynamics by employing an appropriate theoreticalmodel reveals the essential factors governing the phase transitions.However, phase frontpropagationhas beendirectly observed for only afew organic martensitic crystals to date13–15,21 because of the followingexperimental difficulties: for most martensitic transitions, the new phasecannot be clearly distinguished from the original phase with standardoptical measurements. Because the molecular structure of individualmolecules remains nearly unchanged throughout the transition, the colorand/or optical density of the new phase is also almost the same as that of theoriginal phase. In addition, the martensitic transition is generally a fastprocess and is often accompanied by an explosive crystal motion, such asjumping and suddencracking (the thermosalient effect)1–8,10–19. Therefore, toobservephase front propagation, ameasurement techniquewith ahigh timeresolution is required. For this reason, the phase front propagation has beensuccessfully observed only for non-thermosalient crystals so far13–15,21.Here, we developed a technique for acquiring an image of a crystalduring amartensitic phase transition, where the new and original phases areclearly distinguished. By combining this technique with an ultrahigh-speedcamera,weobserved thephase frontpropagation in single crystals of 1,2,4,5-tetrabromobenzene (TBB, Fig. 1a). A TBB crystal is a typical thermosalientcrystal, which exhibits martensitic transition accompanied by an explosivecrystal motion1,35. We observed ballistic phase front propagation from oneResearch Center for Electronic and Optical Materials, National Institute for Materials Science, Tsukuba, Japan. e-mail: takazawa.ken@nims.go.jpCommunications Chemistry |           (2026) 9:340 11234567890():,;1234567890():,;https://doi.org/10.1038/s42004-026-02119-yhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42004-026-02119-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42004-026-02119-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42004-026-02119-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42004-026-02119-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42004-026-02119-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s42004-026-02119-y&domain=pdfhttp://orcid.org/0000-0002-2169-5169http://orcid.org/0000-0002-2169-5169http://orcid.org/0000-0002-2169-5169http://orcid.org/0000-0002-2169-5169http://orcid.org/0000-0002-2169-5169mailto:takazawa.ken@nims.go.jpwww.nature.com/commschemedge to the other across a single crystal of TBB. The velocity of the phasefront reached~800m/s, which is 5–6 orders ofmagnitude higher than thosepreviously reported for martensitic transitions of non-thermosalient crys-tals. Themeasuredvelocitieswereup to~60%of the soundvelocity (phononvelocity) in theTBBcrystals, i.e., phase front propagationwas subsonic. Thisfinding indicates that the observed phenomena arenot adequately describedby linear elastic wave models. To account for the subsonic propagation, weemployed a one-dimensional nonlinear lattice model that explicitly takesinto account the fact that the transitionbetween the twophases isfirst-order.We demonstrate that ourmodel quantitatively explains the subsonic phase-front propagation in TBB crystals and reveal novel insights into themechanism of the organic martensitic transitions.ResultsTBB single crystalsA single crystal of TBB was reported to transform from β- to γ-phase nearroom temperature, around 40 °C upon heating1,35–37. The crystal structuresof both the β- and γ-phases belong to a monoclinic system with the spacegroup P21/n. The lattice parameters are a = 10.2355(5) Å, b = 10.7193(4) Å,c = 4.0233(2) Å,β = 100.184(5)°,V = 434.47(4) Å3, andZ = 2 for theβ-phaseat T = 30 °C, whereas they are a = 9.8719(5) Å, b = 11.0780(5) Å,c = 4.0549(3) Å,β = 100.585(5)°,V = 435.90(4) Å3, andZ = 2 for the γ-phaseat T = 40 °C37. As shown in Fig. 1b, c, the distortion of the unit cell throughthe β→ γ transition is anisotropic, i.e., it expands along the b- and c-axesand shrinks along the a-axis. The anisotropic cell distortion is a commonFig. 1 | Structure and morphology of TBB single crystals. a Chemical structure ofTBB. Crystal structures of b β-phase and c γ-phase. Dashed blue lines show the unitcell of β-phase. d Left panel: crossed-polarized microscope image of a TBB crystaloriented so that the c-axis is parallel to the analyzer. The arrows indicate directions ofthe polarizer and the analyzer. Right panel: image of the crystal at the extinctiondirection. e Crystal faces of a TBB single crystal. f Top view of a crystal. Theextinction direction (ED) is shown by the blue line.https://doi.org/10.1038/s42004-026-02119-y ArticleCommunications Chemistry |           (2026) 9:340 2www.nature.com/commschemcharacteristic of organic martensitic phase transitions33. The changes in thelattice constants are−3.55%,+3.35%, and+0.79% along the a-, b-, and c-axes, respectively, whereas that in the unit cell volume is only+0.33%owingto the anisotropy. A crystal in the γ-phase transforms back to the β-phaseupon cooling at ~35 °C. The temperature hysteresis of ~5 °C between theβ→ γ and γ→ β transitions indicates that the transition is first-order.TBB crystalswere grownvia slow evaporation froma saturated ethanolsolution. The crystals had an elongated plate-like shape with a length alongthe long axis of up to ~5mm and a thickness of a few hundredmicrometers(Fig. 1d). The crystal faces were identified based on previously reportedassignments38 and Bravais–Friedel–Donnay–Harke simulations (Fig. 1e)39.The primary crystal plane is (1 −1 0) and contains the crystallographic c-axis (Fig. 1f). Using a cross-polarized light microscope, the extinctiondirection was observed to decline by ~15° with respect to the c-axis, aspreviously reported (Fig. 1f and right panel of Fig. 1d)38.Effect of phase transitions on polarization propertiesThe β-crystals were heated from room temperature across the phase tran-sition temperature (~40 °C) under a polarized light microscope. Weobserved that the extinction direction slightly rotated upon the β→ γtransition. The rotation angle (Δθ) was measured for 15 crystals anddetermined to be Δθ = 1.55° (s = 0.43). In the following section, wedemonstrate that this rotation can be used to observe the spatio-temporalevolutions of the phase transition.The spectrum of light transmitted through a crystal under a cross-polarized microscope is given byI θ; λð Þ / sin2 2θð Þ � sin2 πRλ� �¼ A θð Þ � B λð Þ ð1Þwhere θ is the angle between the extinction direction of the crystal and thepolarization direction of the analyzer (Fig. 2a),R is the retardation expressedas R = dΔn, where d is the crystal thickness, and Δn is the birefringence40.Because |dA(θ)/dθ| is maximum at θ = π/8 and 3π/8, A(θ) most sensitivelychangeswithΔθ at these crystal orientations (Fig. 2b). For apositiveΔθ,A(θ)increases and decreases at θ = π/8 and 3π/8, respectively. Wemeasured I(θ,λ) of the TBB crystal at these orientations before and after the phase tran-sition. The blue curve in Fig. 2c shows I(θ = π/8, λ)measured at 23 °C,wherethe crystal is in the β-phase. The spectrum exhibits several fringes attributedto B(λ) in Eq. 1. The total intensity of the transmitted light was evaluated byaveraging I(θ = π/8, λ) over the spectral range and is indicated by the bluedashed line in the Fig. 2c. Subsequently, the crystal was heated above thetransition temperature (~40 °C), and I(θ = π/8, λ) was measured after ittransformed to the γ-phase (the red curve). The fringes shifted slightly withrespect to those of the β-phase, indicating that B(λ) was affected by thetransition. The average intensity of the spectrum is indicated by the reddashed line. It was determined that the intensity increased by ~11% uponthe β→ γ phase transition. A similar measurement was performed atθ = 3π/8 (Fig. 2d). At this orientation, the transmitted light intensitydecreased by ~12% through the β→ γ transitions. These changes inintensity are sufficiently large to distinguish between the β- and γ-phaseswhen the crystal is recorded using a video camera during the phasetransition.We further examined the effect of phase transition on both A(θ) andB(λ). We observed that the extinction direction rotated by Δθ ≈ 1.55° uponphase transition. Using Eq. 1, the change in A(θ) induced by this rotation([A(θ+Δθ)−A(θ)]/A(θ)) is calculated as 0.11 and−0.11 at θ = π/8 and3π/8, respectively. These values approximately agree with the observed changesin the intensities of the transmitted light, i.e., ~11% and ~−12% at θ = π/8and 3π/8, respectively. This agreement implies that the changesweremostlyattributable to those in A(θ). Because B(λ) provides multiple fringes in thespectra and the phase transition induces a small shift of them, the change inthe averaged B(θ) over the spectral range is small. The TBB crystals inves-tigated in this study all satisfied the conditions,where the change inB(λ)wasFig. 2 | Effect of phase transition on the polarization properties of a TBB crystal.aRotation of ED by the phase transition. EDβ and EDγ show the extinction directionof the β- and γ-crystal, respectively. b Curve of A(θ). Red arrows indicate increaseand decrease in A(θ) at θ = π/8 and 3π/8, respectively, through the β→ γ transition.c Transmission spectra of a TBB crystal oriented at θ = π/8 before (blue curve) andafter (red curve) the β→ γ transition. Averaged spectral intensities are indicated bydashed lines. d Same as (c) at θ = 3π/8.https://doi.org/10.1038/s42004-026-02119-y ArticleCommunications Chemistry |           (2026) 9:340 3www.nature.com/commschemnegligible for the transmitted light intensity. A detailed analysis of the effectof phase transition on B(λ) is provided in the Supplementary Information.Direct observation of phase front propagationWedemonstrated that theβ- andγ-phases canbedistinguishedbyorientingthe crystals at specific angles. Based on this result, we recorded videos of theTBB crystals during the phase transition using a high-speed camera. Fig-ure 3a shows snapshots of a TBB crystal with height and width of 0.6 and3.6mm, respectively, recorded by the camera operated at 5.4 × 105 fps withthe exposure time for each frame of 1/3,811,764 s (≈262 ns) during theγ→ β transition (see also SupplementaryVideo 1). The crystal was orientedat θ = π/8. The dark region, which corresponds to the β-phase, appeared atthe right edge of the crystal and extended toward the left edge of the crystalwith maintaining the clear boundary between the dark and bright regions(the phase front). The difference in intensity between the bright and darkregions was approximately 10%, which was consistent with the aboveanalysis. These images clearly indicate that the phase transition proceeds byphase front propagation.To measure the position of the phase front in each snapshot precisely,the image captured ~10 μs before the phase transition was subtracted fromthose recordedduring thephase transition (insets inFig. 3b).This procedureeliminates the inhomogeneity of the transmitted light that is unrelated to thephase transition and allows for a clear visualization of the phase front. Then,the intensityof the imagewas integrated along thevertical axis andplotted asa function of the horizontal position of the crystal (black curves in Fig. 3b).The positions of the phase fronts were determined by fitting a sigmoidfunction to the curves (red curves in Fig. 3b). The velocity of the phase front(vpf) was then evaluated from the difference of the phase front positionsbetween those in two adjacent frames as vpf = 5.4 × 105 (xn+ 1− xn) m/s,where xn is the position of the phase front in the nth frame (note that n = 1represents the frame, where the phase front was first captured). The eval-uated vpf values for this crystal were plotted against n in the right panel ofFig. 4 (red squares) and were 300–450m/s. The sigmoid curves in Fig. 3bshow that the width of the transition region varies for each snapshot. Thewidth can be affected by many factors, such as the tilt angle of the (0 0 1)plane with respect to the substrate, the movement of the phase front duringthe exposure time of the camera, and the slight tilt of the phase front withrespect to the long axis of the crystal. At present, it is not clear which factorsdetermine the width.We performed these measurements and analyses for ~30 crystals withlengths along the [0 0 1] direction ranging from 0.8 to 4.2 mm for bothβ→ γ and γ→ β transitions. The transition temperatures were 38 ± 1.5 and35 ± 1.5 °C for theβ→ γ andγ→ β transitions, respectively.Videos of someof them are provided in the Supplementary Video 2. The following featureswere observed for all the crystals: (1) thephase transition started at oneof theshort edges, and the phase front propagated along the [0 0 1] direction; (2)phase front was perpendicular to the [0 0 1] direction; and (3) when thephase transition started from one edge in the β→ γ transition induced byheating, it started from the other in the γ→ β transition induced by sub-sequent cooling.Fig. 3 | Phase front propagation in the phase transition of a TBB crystal recordedwith the high-speed camera. a Snapshots recorded during the phase transition.b Black curves: intensities of the transmitted light obtained by integrating the crystalimages along the vertical axis of the snapshots. Red curves: fitted curves using thesigmoid function. Insets: images obtained by subtracting those recorded 10 μs beforethe phase transition.https://doi.org/10.1038/s42004-026-02119-y ArticleCommunications Chemistry |           (2026) 9:340 4www.nature.com/commschemThe vpf values were plotted in the left (the β→ γ transitions) and right(the γ→ β transitions) panels of Fig. 4. For a large crystal, the phase frontwas captured over several frames before the entire crystal was transformed,providing several data points on the graph, whereas a small crystal providedfewdata points. Therefore, the number of data points is related to the crystalsize. For crystals smaller than~0.8mm, the phase frontwas capturedonly inone frameornot capturedat all. Thus, vpf for such small crystals couldnot beevaluated. The graph shows that vpf ranges 200–800m/s for both the β→ γand γ→ β transitions, and no clear correlation between vpf and thecrystal size was found. We do not have enough data on the correlationbetween vpf and a crystal thickness. However, we observed a tendencythat larger crystals have larger thicknesses. Therefore, the absence of cor-relation between vpf and the crystal size suggests that vpf does not correlatewith the thickness. Because the crystals were heated by thermal contactwith the substrate, a temperature gradient along the thickness of the crystalsmay exist and affect vpf. However, the absence of a correlation between vpfand the thickness further suggests that the effect of the temperature gradienton vpf is minimal.The fluctuations of vpf with n observed for some crystals may beattributed to the inhomogeneity of crystal quality. The vpf values and theirfluctuations are not reproducible. The phase front motions could beobserved repeatedly over several heating/cooling cycles for the same crystal.However, the repeated phase transitions caused gradual destruction(cracking and disintegration) of the crystal, leading to low vpf and largefluctuations. Therefore, the vfp values presented in Fig. 4 were all measuredfor the first heating/cooling cycles.The fact that the phase front is perpendicular to the [0 0 1] directionsuggests that the molecular motions in the (0 0 1) sheet are significantlyfaster than those in the [0 0 1] direction.Dye et al. studied the anharmonicityof molecular interactions in a TBB crystal in the β-phase and showed thatthe barrier formolecularmotions in the abplane is lower than that along thec-axis, supporting fastermotions in the (0 0 1) sheet41. This indicates that themolecular motions in the (0 0 1) sheet are cooperative and that the phasetransition proceeds in a sheet-by-sheet (layer-by-layer) manner (Fig. 5a), asobserved for other organic martensitic transitions21,22,42,43. On the otherhand, if the cooperativemotions occur in the (0 1 0) or (1 0 0) sheets, phase-frontpropagation should beobserved along thewidthof the crystal (Fig. 5b).This is clearly inconsistent with our observations. This layer-by-layer pro-gression of the phase transition allows us to model a TBB crystal as a one-dimensional chain of interacting masses, which represent the (0 0 1) sheets(Fig. 5c). A theoretical analysis of phase front propagation based on thismodel is provided in a later section.We observed that the phase transition always started not in themiddlebut at one of the short edges of the crystals. This is likely attributed to aninstability of the edges due to surface free energy. We also observed thatwhen the phase transition started at one of the short edges in the β→ γtransitionbyheating, it started at the other short edge in theβ→ γ transitionby subsequent cooling.This feature is alsounderstood in termsof the surfacefree energy. The surface free energy of both edges are, to a greater or lesserextent, different, and the β→ γ transition started from the edge with thehigher energy when the crystal is heated. Upon the phase transition, theenergies of both edges lowered. However, the edgewith the higher energy inthe β-phase should be more stabilized than the others. As a consequence,when the crystal is subsequently cooled, the γ→ β transition started at theother edge, which has the higher energy in the γ-phase.To investigate the mechanism behind the significantly fast phase frontpropagation, we compare the vpf values with the sound velocity (phononvelocity) in TBB crystals. The anisotropic sound velocities of a TBB singlecrystal in the β-phase were investigated using Brillouin light scattering41,44.Based on the elastic constants provided in ref. 41, the sound velocities in the[0 0 1] direction were calculated as vL = 2146m/s, vTA1 = 1283m/s, andvTA2 = 1469m/s, where vL and vTA1, 2 are the longitudinal and transversevelocities, respectively (Supplementary Information). Therefore, theobserved vpf value of 800m/s are approximately 40% and 60% of thelongitudinal and transverse soundvelocities, respectively, indicating that thephase front propagation in the TBB crystals is subsonic.The vpf values previously measured for martensitic, but non-thermosalient crystals, were on the order of μm/s to mm/s13,21, and there-fore, 5–6 orders of magnitude smaller than those observed for TBB crystals.This suggests that the fast phase front propagation is a key factor in theoccurrence of the thermosalient effect. The vpf values of these non-thermosalient crystals were analyzed based on the change rates of thevolume fractions of the two phases, which are governed by the free energycurve along the transition coordinate45. Yangui et al. successfully observedthe phase front motion in the martensitic phase transition of organic-inorganic perovskite crystals (vpf ≈ 1.6mm/s)46 and qualitatively analyzedthe phase front propagation based on the change rates of the volumefractions47.The vpf values as high as the sound velocity have been reported forinorganic crystals. Mariette et al. performed femtosecond time-resolved X-Fig. 4 | Velocity of the phase front (vpf). Left panel: the β→ γ transition. Rightpanel: the γ→ β transition. The vpf values obtained from the same crystal are shownby the same symbol and color in the left and right panels. The vpf values wereobtained using the relation vpf = 5.4 × 105 (xn+ 1− xn) m/s, where xn is the positionof the phase front in the nth frame. The xn values were determined by fitting sigmoidfunctions to the data points (see the text and Fig. 3b). The number of data points (N)is different for each crystal depending on its size (N = 118 (◄), 120 (▲), 85 (▽), 77(▼), 65 (◁), 97 (◆), 81 (►), 125 (●), 102 (○), 76 (△), 111 (◇), 104 (■)). The errorbars were evaluated from the standard errors of xn, which were obtained by thesigmoidal fitting.https://doi.org/10.1038/s42004-026-02119-y ArticleCommunications Chemistry |           (2026) 9:340 5www.nature.com/commschemray powder diffraction measurements on Ti3O5 nanoparticles during astructural phase transition induced by a short-pulsed laser irradiation48.Detailed analysis of the diffraction data showed that the phase transition istriggered by the precursor photo-induced electronic changes accompaniedby cell distortions and proceeds with the motion of the phase front withsound velocity (vpf ≈ 6.5 × 103 m/s). To explain the phase frontmotion, theyemployed a one-dimensionalmodel, where a strainwave (an acoustic wave)created by pulsed laser irradiation propagated across the crystal with thevelocity of sound, leaving behind the new phase. They demonstrated thatexperimental observations were quantitatively reproduced using the strainwave model49.The vpf values on the same order of magnitude as the sound velocityobserved for the TBB crystals strongly suggest that the phase front propa-gation is also associated with the acoustic wave traveling across the crystal.However, the strainwavemodel is not applicable toTBBcrystals because thevpf values are considerably less than the sound velocity. The solutions of alinearwave equation, including strainwaves, propagatewith an exact soundvelocity, in contrast to the subsonic vpf of TBB crystals.Moreover, vpf for theTBB crystals was observed to differ in the range 200–800m/s depending onthe crystal, suggesting that vpf is sensitively influencedby small differences inthe intrinsic and/or extrinsic properties of the individual crystals. Thisfeature is hardly accounted for by the linear models. These observations ledus to explore a suitable model for describing the phase front propagation inTBB crystals.Analysis using a nonlinear lattice modelHere, we employ a one-dimensional nonlinear lattice model in accordancewith Landau’s phase transition theory. Landau’s theory is based on freeenergy as a function of an order parameter. The free energy needs to beexpanded up to sixth order for a first-order phase transition, whereas thefourth order for a second-order phase transition50. As stated earlier, thetransition exhibited by the TBB crystals is of the first order, and the crystalcan be modeled as a one-dimensional chain of coupled point masses thatrepresent (0 0 1) sheets (Fig. 5c). Subsequently, a sixth-order on-sitepotential is incorporated into themass chainmodel. Theoretical analyses ofthis type of model system, according to Lagrangian mechanics, have beenconducted by several authors. We followed the work by Gooding et al.51.Using a dimensionless scalar order parameter ϕ(x, t) (x: position, t: time)that is detailed below, the on-site potential is written in the following form:Von�site ϕ� � ¼ 12ρ�ω02 ϕ2 � 1� �2ϕ2 þ Eg� �ð3Þwhere ρ* is a parameter linearly related to the density ρ,ω0 is the frequency,and Eg is a parameter that determines the potential profile (0 < Eg < 1/2)51.Our model, which consists of a harmonically coupled mass chain with theon-site potential, is schematically illustrated in Fig. 6a. The on-site potentialhas two stable equilibria at ϕ = ± 1, corresponding to the γ-phase, and ametastable equilibrium at ϕ = 0 (the β-phase). In this model, thepropagation of the phase front is viewed as a collective motion of themasses, where the displacement of a mass from ϕ = 0 to 1 occursconsecutively from one site to the neighboring site. From awavemechanicsperspective, this collectivemotion is described as the propagation of a kink-like solitary wave with velocity vpf.We sought such kink-like wave solutionsfor the model system.Previous studies demonstrated that this system has propagating kink-like solutions with a constant vpf only when the system is non-conservative51–54. Although vpf fluctuated during the propagation (Fig. 4),we assume vpf is constant throughout the transitions for simplicity. Subse-quently, the Rayleigh dissipation function defined as R = (1/2)γ(ϕt)2 wasintroduced (hereafter, the italic subscripts denote differentiation). The fieldequation is derived from the Lagrangian density of the system, that is,L = T−V, where T = (1/2)ρ*(ϕt)2 and V =Von-site+ (1/2)g(ϕx)2, and isgiven byϕtt ¼ �ω02 1� 2Eg� �ϕþ 2Eg � 4� �ϕ3 þ 3ϕ5h i� Γϕt þgρ�ϕxx ð4ÞFig. 5 | Layer-by-layer progression of the phase transition in TBB crystals.aUpper panel: the (0 0 1) sheet is indicated by the yellow plane. Lower panel: orangeand yellow planes represent those in the new and original phases, respectively. Thephase front propagates along the long axis of the crystal. b Upper panel: the (0 1 0)sheet is indicated by the yellow plane. Lower panel: orange and yellow planesrepresent those in the new and original phases, respectively. The phase front pro-pagates along the short axis of the crystal. c 1D chain model, where masses (blackcircles) representing the (0 0 1) sheets are connected with springs.https://doi.org/10.1038/s42004-026-02119-y ArticleCommunications Chemistry |           (2026) 9:340 6www.nature.com/commschemwhere Γ = γ/ρ* and g is the spring constant. The following dynamical kink-like solutions exactly satisfy the field equation:ϕ x; tð Þ ¼ 1þ expx � vpf t‘� ��  �12 ð5ÞwithΓvpf ¼ 4ω02Eg‘ ð6Þand‘2 ¼ gΓ2ρ� 4ω02 Γ2 þ 4ω02Eg2� �h i�1ð7Þwhere ‘ is the width of the kink51.Figure 6b shows the time evolution of ϕ(x, t), showing that the γ-phaseextends into the β-phase via propagation of their boundary with the con-stant velocity vpf. Using Eqs. 5–7, vpf is a monotonically decreasing functionof ‘Γ=Eg , asvpf ¼ � 12‘ΓEg !þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14‘ΓEg !2þ c02vuut < c0 ð8Þwhere c0 = (g/ρ*)1/2 is the sound velocity (Fig. 6c). The parameter Γ (=γ/ρ*)characterizes the dissipation of the system, which likely differs for eachcrystal depending on intrinsic factors, such as crystal quality and extrinsicfactors, such as the adhesion between the crystals and the substrates.Therefore, ‘Γ=Eg candiffer for each crystal, resulting in a different vpf. Theseresults indicate that the model qualitatively captures the characteristic fea-tures of the phase front propagation associated with the phase transition ofTBB crystals. It should benoted that in an actual crystal, themolecules in the(0 0 1) sheet at the phase front deflect both in and out of the (0 0 1) planeupon the phase transition (see Fig. 5a). Therefore, the phase front as apropagating wave has a mixed longitudinal-transverse character. Incontrast, in the one-dimensional model, the phase front propagates as apure longitudinal wave because the molecular motions are reduced to one-dimensional displacements of the point masses.Quantitative validation of the nonlinear lattice modelHaving confirmed the consistency of the model, next, we quantitativelyanalyzed the phase transition of the TBB crystal using the model. Forquantitative analysis, the dimensionless order parameter ϕ must be con-verted into a quantity with the dimension of length. In the ϕ-space, thedisplacement between themetastable and stable equilibria is unity (ϕ = 0–1).In an actual crystal, on the other hand, the displacement of the (0 0 1) sheetsinduced by the phase transition is given by the difference in the latticeconstant c between the β- and γ-phases (Δc). Considering this correspon-dence, it is reasonable to scale ϕ = 1 by Δc (=4.0549–4.0233 Å = 0.0316 Å).Using this conversion relation, and the previously reported thermal analysisdata, including V(0) = 330 J/mol (the enthalpy of the phase transition1,55),and ΔE = 2000 J/mol (the barrier height for the transformation37,56), Eg isuniquely determined as Eg = 2.2 × 10−2 fromEqs. 5 to 7 (see SupplementaryInformation). The obtained on-site potential curve is shown in Fig. 7a.Further, using this Eg value and the sound velocity c0 in Eq. 8, we canexplicitly evaluate all the other parameters (γ, ‘, andω0) fromEqs. 5 to 7. Asdescribed earlier, in the one-dimensional model, collective motions ofmolecules in all three directions (the [1 0 0], [0 1 0], and [0 0 1] directions),which give rise to the fast longitudinal and slow transverse waves, arereduced to the displacement of the masses along the propagation direction.Therefore, c0 does not need to agree with either the longitudinal andtransverse sound velocities and can take a value between them (vL ≥c0 ≥ vTL1, 2). Considering this point, we calculated the parameters for thehigh and low limiting c0 values, namely, c0 = vL and vT1. For both the c0values, the parameters were consistently and uniquely determined for agiven vpf value (Supplementary Information) and those for vpf = 200, 400,600, and 800m/s are listed in Table 1.To assess the validity of this analysis, we compared these results withthe experimental data. In our model, γ determines energy dissipation in theFig. 6 | Analysis of the phase front propagation using the nonlinear latticemodel.a 1Dmass chainmodel with the on-site potential with nondegenerateminima.V(0):energy difference between the metastable and stable minima. ΔE: barrier height.b Time evolution of the kink-like solution during the β→ γ transition induced byheating. c The phase front velocity (vpf/c0) as a function of ‘Γ=Eg .https://doi.org/10.1038/s42004-026-02119-y ArticleCommunications Chemistry |           (2026) 9:340 7www.nature.com/commschemsystem. The corresponding quantity for the actual crystals is the viscosity(damping), which is given by the imaginary part of the complex Young’smodulus, expressed as E* = E’+ iωE” (E’: Young’s modulus, ω: frequency).Danno et al. measured the E* values of several aromatic molecular crystals,such as triphenylene, 1,2,3,4-dibenzanthracene, and coronene, using forcedresonance vibration methods and showed that the ωE” values of thesecrystals are all on the order of 107–108 N/m257. The ω0γ values calculatedfromour analysis are listed in Table 1. For both the limiting values of c0 = vLand vTA1, they are on the order of 107 N/m2, showing good agreement withthe ωE” values of the aromatic crystals. This agreement strongly supportsthe validity of our analysis. The relationship between vpf and γ values inTable 1may suggest that variation in vpf is attributed to that of γ, which canbe influenced by the intrinsic and/or extrinsic properties of individualcrystals. However, experimental measurements of the elastic properties ofthe crystals are required to verify this assumption.Further support for the validity of the analysis is provided by exam-ining the kink width ‘. The evaluated ‘ is on the order of 10−12 m (Table 1).This is twoorders ofmagnitude smaller than the lattice constant c andon thesame order as the difference in the lattice constant c between the β- and γ-phases (Δc = 3.16 × 10−12 m), implying that the kink is always localized at asingle site, as schematically illustrated inFig. 7b.This indicates that thephasetransition proceeds in a layer-by-layer manner, as expected from ourexperimental observations and the martensitic phase transitions of otherorganic crystals20,33. If ‘ is sufficiently large to extend over multiple sites,those within the kink width have intermediate displacements between thestable and metastable equilibria (Fig. 7c). This corresponds to the situationwhere there extends a region with intermediate crystal structures at thephase boundary, and is inconsistent with the layer-by-layer progression ofthe phase transition.DiscussionIn this study, we investigated the martensitic phase transitions in singlecrystals of TBB and observed that the transitions proceeded with the sub-sonic phase front propagation across the crystals.We demonstrated that thenonlinear latticemodel successfully explained themotion of the phase front.The analysis reveals that the phase front is described as a kink-like solitarywave, which can be formed and propagate in a non-conservative system.This indicates that the viscosity of the crystals is the crucial factor in theformation and propagation of the phase front. A number of organic mar-tensitic crystals, including thermosalient crystals, have been discovered todate, and studies aimed at device applications of these crystals have beenextensively conducted. Our results serve as a promising guideline for ana-lyzing and understanding the phase transitions of these crystals and providevaluable information for their device applications.MethodsSample preparationTBB was obtained from Wako Pure Chemical Industries and purified byrecrystallization from ethanol. A saturated ethanol solution of TBB wasfiltered using a syringe filter with 200 nm pores and then placed in a con-tainer with a lid with holes. After the crystals were grown via slow eva-poration, they were washed with ethanol and dried under ambientconditions.Microscopic observations showed that approximately half of thecrystals were twinned, and only non-twinned single crystals were used forthe measurements.Optical measurementA TBB crystal was placed on a microscope cover glass (18mm× 18mm)and immobilized with a small amount of mineral oil to prevent movementof the crystal at the moment of phase transition. Although we attempted toperform themeasurements on crystals that were simply placed on the coverglass without immobilization, most of them hopped or broke upon thephase transitions. We could observe phase front propagation for only a fewcrystals that did not exhibit movement. The vpf values of these crystals wereobserved as ~600m/s, suggesting that the immobilization does not largelyaffect vpf. The sample was then placed on a transparent thermoplate (TokaiHit). The thermoplate was mounted on the rotational stage of a polarizedlight microscope (Olympus, BX51-P) equipped with a modified illumina-tion system.A 1.25 times objective lenswas used. Because the exposure timefor each frame in the high-speed camera recording was short (~262 ns),high-intensity illumination was required to capture images of the crystal.We constructed an illumination unit consisting of a 100W mercury lampand attached it to the microscope.High-speed videoswere recorded using ahigh-speed camera (Photron,FASTCAM) mounted on the microscope. A video of a crystal was loop-recorded while the sample temperature was varied at ~1 °C/min.When thecrystal was transformed, the loop recording was immediately stopped sothat the last ~3 s of the video was saved. Because the relative orientationbetween the stage and the camera is fixed, we can confirm that the crystalsdid not rotate upon the phase transitions by reviewing the videos. Thisensures that the movement of the boundary between the dark and brightregions of the crystals is solely attributed to the change in the crystalstructure.Table 1 | Calculated parameters (γ, ‘, andω0γ) for vpf = 200, 400,600, and 800m/svpf (m/s) γ (10−7 Ns/m2) l (10−12 m) ω0γ (107 N/m2)vL vTA1 vL vTA1 vL vTA1200 12.7 7.56 12.6 7.46 10.8 6.42400 6.29 3.64 12.4 7.17 5.34 3.09600 4.10 2.25 12.1 6.67 3.48 1.91800 2.97 1.50 11.7 5.90 2.52 1.27The sound velocities of vL = 2146m/s and vTA1 = 1283m/s were used.Fig. 7 | On-site potential and kink-like solutionsobtained by the theoretical analysis. a On-sitepotential. b Time evolution of the kink-like solution(from top to bottom) for ‘ < c. The color of a mass ateach site represents displacement. The masses at themetastable and stable minima are shown in red andblue, respectively, whereas those at intermediatebetween them are shown in purple (see the inset ina). c The same as (b) for ‘ > c.https://doi.org/10.1038/s42004-026-02119-y ArticleCommunications Chemistry |           (2026) 9:340 8www.nature.com/commschemReporting summaryFurther information on research design is available in the Nature PortfolioReporting Summary linked to this article.Data availabilityThe data that support the findings of this study are available from thecorresponding author upon reasonable request.Received: 17 November 2025; Accepted: 25 June 2026;References1. Lieberman, H. F., Davey, R. J. & Newsham, D. M. T. Br⋯Br and Br⋯Hinteractions in action: polymorphism, hopping, and twinning in 1,2,4,5tetrabromobenzene. Chem. Mater. 12, 490–494 (2000).2. Ding, J., Herbst, R., Praefcke, K., Kohne, B. & Saenger, W. 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Both authors contributed to writing the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containsSupplementary material available athttps://doi.org/10.1038/s42004-026-02119-y.Correspondence and requests for materials should be addressed toKen Takazawa.Peer review information Communications Chemistry thanks CélineMariette, Panče Naumov and the other anonymous reviewer(s) for theircontribution to the peer review of this work. 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Toview a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2026https://doi.org/10.1038/s42004-026-02119-y ArticleCommunications Chemistry |           (2026) 9:340 10https://doi.org/10.1038/s42004-026-02119-yhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/commschem Subsonic phase front propagation in martensitic transitions of 1,2,4,5-tetrabromobenzene single crystals Results TBB single crystals Effect of phase transitions on polarization properties Direct observation of phase front propagation Analysis using a nonlinear lattice model Quantitative validation of the nonlinear lattice model Discussion Methods Sample preparation Optical measurement Reporting summary Data availability References Author contributions Competing interests Additional information