Impact of surface roughness on liquid-liquid transition

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Science Advances  17 Feb 2017:
Vol. 3, no. 2, e1602209
DOI: 10.1126/sciadv.1602209


Liquid-liquid transition (LLT) in single-component liquids is one of the most mysterious phenomena in condensed matter. So far, this problem has attracted attention mainly from the fundamental viewpoint. We report the first experimental study on an impact of surface nanostructuring on LLT by using a surface treatment called rubbing, which is the key technology for the production of liquid crystal displays. We find that this rubbing treatment has a significant impact on the kinetics of LLT of an isotropic molecular liquid, triphenyl phosphite. For a liquid confined between rubbed surfaces, surface-induced barrierless formation of the liquid II phase is observed even in a metastable state, where there should be a barrier for nucleation of the liquid II phase in bulk. Thus, surface rubbing of substrates not only changes the ordering behavior but also significantly accelerates the kinetics. This spatiotemporal pattern modulation of LLT can be explained by a wedge-filling transition and the resulting drastic reduction of the nucleation barrier. However, this effect completely disappears in the unstable (spinodal) regime, indicating the absence of the activation barrier even for bulk LLT. This confirms the presence of nucleation-growth– and spinodal decomposition–type LLT, supporting the conclusion that LLT is truly a first-order transition with criticality. Our finding also opens up a new way to control the kinetics of LLT of a liquid confined in a solid cell by structuring its surface on a mesoscopic length scale, which may contribute to making LLT useful for microfluidics and other industrial applications.

  • liquid-liquid transition
  • surface effects
  • rubbing
  • wedge filling
  • pattern evolution


Generally, surface wettability and geometry seriously affect a phase transition near a surface through a coupling of the surface field to the order-parameter field. This leads to unique phenomena absent in bulk (14). Such examples can be seen for crystallization, vapor-liquid phase transition (5), and phase separation of a binary mixture (6) in contact with a substrate. Heterogeneous nucleation (that is, preferential nucleation on a wettable surface) is one of such classical examples widely seen in our daily lives, ranging from cloud and frost formation in nature to the production of high-quality single crystals in industry. These phenomena should not be regarded as a special exception to bulk phase transition because it is almost impossible to avoid the presence of surfaces in any real systems, particularly, in small-size systems relevant to nanotechnology applications. Recently, there have been growing interests in controlling self-organized pattern evolution by nanostructured functional surfaces using an interplay between phase ordering and wetting (49).

Here, we focus our attention on how substrate surface geometries affect liquid-liquid transition (LLT) of a single-component liquid, which is the first-order phase transition between two liquid states, liquids I and II. Recently, LLT has attracted considerable attention because of its counterintuitive nature (1013). LLT accompanies distinct changes of bulk physical and chemical properties such as density (14), refractive index (14, 15), dielectric constant (16, 17), polarity (18), viscosity (15), fragility (19), and miscibility (20). Controlling these properties while maintaining fluidity can be important for practical applications of LLT because the state of a liquid is key to various transport and chemical reaction processes. LLT offers an intriguing possibility to change the physical and chemical properties of a liquid without modifying its molecular structure. In addition to the change in these bulk properties, we recently found a difference in the wettability between liquids I and II of triphenyl phosphite to a solid substrate (21), suggesting a possibility of controlling LLT by a surface field. There has been a long-standing debate on the physical nature of the phase transition in triphenyl phosphite (16, 2234) since its first observation by Cohen et al. (14) and Ha et al. (14, 35). The difficulty comes from the fact that the transition takes place below the melting point, that is, in a state intrinsically metastable against crystallization. However, recent experimental studies (18, 3638) have shown that the transition is LLT and not merely due to nanocrystal formation.

LLT looks counterintuitive if we assume that a liquid has a completely random structure. However, LLT can be naturally explained by introducing a new additional scalar order parameter (3942) in addition to density ρ, a standard order parameter describing the liquid state. We argued that this new order parameter should be the fraction of locally favored structures, S. Although ρ is a conserved order parameter, S is a nonconserved one because locally favored structures can be created and independently annihilated, as spin flipping in magnets. This explains why liquid I completely transforms into liquid II without coexistence, unlike phase separation of a mixture. Then, LLT is regarded as a gas-liquid–like cooperative ordering of locally favored structures: Liquids I and II are a gas (dilute) and a liquid (dense) state of locally favored structures, respectively. Thus, the transformation from liquid I to II is the process of the increase in the fraction of locally favored structures (37). In this scenario, a gas-liquid transition is dominated by ρ, whereas an LLT is dominated by S.

Here, we report the first experimental study on an impact of surface nanostructuring on LLT. By comparing the ordering process of LLT of triphenyl phosphite in contact with rubbed substrates to that with unrubbed smooth ones for both nucleation-growth (NG)– and spinodal decomposition (SD)–type LLT, we find that surface structuring by rubbing, which is widely used in liquid-crystal industries, strongly accelerates the kinetics of NG-type LLT. We show that rubbing significantly affects the pattern formation and accelerates the kinetics for NG-type LLT but has no effects on SD-type LLT. This sheds new light on the very nature of LLT. Furthermore, this controllability of the kinetics of LLT by wedge-wetting effects will open a new avenue of LLT research toward microfluidics and other industrial applications.


Rubbing effects on NG-type LLT

First, we show (in Fig. 1A) a typical pattern evolution process of LLT in a cell made of unrubbed planar surfaces, which was observed with phase-contrast microscopy at the annealing temperature Ta = 218 K (see movie S1). This temperature is located in the metastable region of LLT or below the binodal temperature TBN but above the spinodal temperature TSD = 215.5 K. In an unrubbed cell, spherical droplets of liquid II are nucleated, increase their size linearly with time in the matrix of liquid I, and, finally, fill up an entire space. The schematic explanation of NG-type pattern evolution in a system confined between two solid substrates is also shown in Fig. 1A. On the other hand, in a rubbed cell, whose surfaces have plenty of submicrometer-scale grooves formed along the rubbing direction (white arrow), barrierless formation of liquid II domains is observed (see Fig. 1B and movie S2). In this case, LLT is initiated by the barrierless formation of liquid II domains on the rubbed substrates, which is followed by the growth toward the middle of the cell, resulting in the gradual increase of the intensity of the image with time. This is unlike in the above case of typical NG-type pattern evolution, where dark droplets of liquid II are directly formed in the middle of the matrix of liquid I (see schematic pictures in Fig. 1). As shown below, this leads to a difference in the temporal change in the intensity distribution (see Fig. 2). This behavior is fundamentally different from the ordinary NG-type transformation shown in Fig. 1A, despite the fact that both processes take place at thermodynamically identical conditions. However, even in Fig. 1B, we can see the nucleation of dark droplets (see, for example, encircled area) in the intermediate stage of the process in addition to the pattern formed near the surfaces. We found that these dark droplets of liquid II do not appear for a cell whose thickness is less than 5 μm. Together with microscopy observations with different focal depths, we conclude that nucleation of dark liquid II droplets overcoming a barrier takes place in the middle (bulk) part of the cell after the surface-assisted barrierless formation of the liquid II domains takes place (see schematic in Fig. 1A). We note that in the later stage, these droplets are merged with the liquid II wetting layers formed on the walls, which propagate from the surfaces toward the middle of the cell and thus eventually merge with the droplets. We confirm that the surface effects are significant, at least for a cell thickness less than a few tens of micrometers.

Fig. 1 Comparison of pattern evolution between LLT on smooth unrubbed and rubbed surfaces.

Pattern evolution observed at Ta = 218 K in an unrubbed smooth (A) and a rubbed (B) cell with phase-contrast microscopy. The pattern formation process can also be seen in movies S1 and S2 for (A) and (B), respectively. The cell thickness is 10 μm for both cases. Scale bars, 20 μm. Schematics illustrate cross-sectional views of the ordering processes.

Fig. 2 Dynamic acceleration of LLT on rubbed surfaces relative to that on unrubbed ones.

Temporal change of P(I) during LLT at Ta = 218 K for a smooth (A) and a rubbed (B) cell. a.u., arbitrary units. (C) Time evolution of the normalized dielectric relaxation strength of liquid II, ΔεII(t)/ΔεII, at Ta = 219 K (squares) and 220 K (circles) for rubbed (blue) and unrubbed (red) cells. It is found that, for both temperatures, the evolution of ΔεII becomes two or three times faster in rubbed cells than in usual cells.

We also measured the temporal change in the intensity distribution P(I) of a microscopy image, which is proportional to the density distribution P(ρ). Note that phase-contrast microscopy detects the difference in the refractive index, which is proportional to the density difference: A darker contrast means a higher density in our measurements (43). Because the locally favored structure has a higher density than the normal liquid structure, P(ρ) reflects P(S). As shown in Fig. 2A, the temporal change of P(I) for the unrubbed cell (see Fig. 1A) is identical to that of our previous study (44): Right large and left small peaks corresponding to liquid I (bright matrix region) and liquid II (dark droplets), respectively, coexist in the intermediate stage of LLT, which is characteristic of NG-type transformation. It is well known that, in NG-type ordering, nuclei with the final order-parameter value (forming the peak of low intensity in this case) appear in the majority matrix (forming the peak of high intensity), and the peak of nuclei grows without a big change in the peak position (or the order-parameter value). Here, we note that the left small peak is very broad due to a focal depth problem (see the Supplementary Materials for details). In contrast, P(I) for the rubbed cell (see Fig. 1B) does not have a bimodal-like shape but rather broadens while continuously changing its peak position from the right peak of liquid I to the left peak of liquid II without creating another peak (see Fig. 2B). In the rubbed cell, the growth of the liquid II phase is not limited by the nucleation process, because there is no barrier for its formation, but mainly by the growth process toward the middle of the cell. Thus, the continuous increase of the wetting layer thickness is the origin of the continuous shift of the peak intensity (see the schematic in Fig. 1B). This temporal change of P(I) is reminiscent of the behavior of SD-type LLT (44). This similarity comes from the barrierless formation of the liquid II phase, but, at the same time, there is a crucial difference in the origin of the continuous change in the peak intensity: For ordinary SD-type LLT, the order parameter itself changes continuously with time, whereas in the above case, the continuous nature comes from the fact that the intensity is proportional to the thickness of the wetting layer, which increases continuously with time. Note that the intensity of the image is the integration along the thickness direction. Furthermore, by comparing Fig. 1 (A and B), we can clearly see that not only the spatial pattern but also the kinetics of LLT are significantly accelerated by surface rubbing due to the barrierless ordering.

To see this more quantitatively, we performed dielectric spectroscopy measurements and evaluated the time evolution of the normalized dielectric strength of liquid II, ΔεII(t)/ΔεII, which is a measure of the degree of the transformation from liquid I to liquid II. Figure 2C shows the time evolution of ΔεII(t)/ΔεII at Ta = 219 and 220 K for a sample sandwiched between rubbed substrates together with that between unrubbed smooth ones. We found that, for both temperatures, the growth of ΔεII(t)/ΔεII in the rubbed cell is accelerated two- or threefold compared to that in the unrubbed one. As will be discussed below, LLT in a rubbed cell proceeds as if there is no activation barrier even in a metastable state above TSD. Because the thermodynamic condition is identical and, accordingly, the structural relaxation time is the same between the two cases, the growth kinetics of the order parameter should also be the same. Thus, we conclude that the dynamical acceleration of LLT is a consequence of the absence of the activation barrier for nucleation of liquid II droplets on rubbed surfaces: Surface rubbing induces spontaneous barrierless growth of the order parameter even in a metastable state of the bulk liquid.

Origin of rubbing-induced acceleration of NG-type LLT

Next, we consider the physical origin of the spatiotemporal modulation of LLT induced by rubbing. Figure 3A shows a confocal microscopy image of a rubbed substrate. We confirmed a large number of microgrooves aligned along the rubbing direction, with typical intervals of a few micrometers. Figure 3B shows a transient spatial pattern of LLT at Ta = 220 K. There, we can see complete filling of the microgrooves by liquid II as well as partial filling by liquid II droplets on aligned microgrooves, which is the origin of the anisotropy in the two-dimensional (2D) power spectrum pattern [identical to the light-scattering pattern (45)], as shown in Fig. 3C. We can see a bright line at qx = 0 in addition to the ordinary isotropic spinodal pattern, indicating the presence of nearly 1D liquid II domains along the rubbed direction (or, on the microgrooves).

Fig. 3 Anisotropic wetting morphologies induced by rubbing.

(A) A confocal micrograph of the 10× rubbed substrate. The color in the image corresponds to the height level: The planar area appears yellow. Black indicates height lower than the planar surface, corresponding to scratches (grooves) formed by rubbing, whereas blue indicates height higher than the planar surface, corresponding to ridges or impurities on the substrate. Note that the depth of the microgrooves is less than the resolution of optical and phase-contrast microscopy. Because of the random nature of grooves, their precise characterization by AFM was difficult. (B) A transient pattern observed with phase-contrast microscopy during LLT on the 10× rubbed substrate at Ta = 220 K. The white arrow indicates the rubbing direction (that is, the x direction). The complete filling of a wedge by liquid II appears as a filament-like shape (regions enclosed by the blue rectangles), whose situation is depicted in the upper right schematic. In addition, the partial filling by liquid II, whose situation is depicted in the lower right schematic, is also observed. Such coexistence of various wetting patterns may be a consequence of distributions of the wedge angle, depth, and length of grooves formed by rubbing. (C) A 2D power spectrum pattern of the image in (B) [see Tanaka et al. (45) for the calculation method]. Scale bars, 50 μm (A and B).

The ordering behavior of liquid II on a rubbed substrate is reminiscent of the so-called wedge filling of a liquid (46), provided that a microgroove can be regarded as a wedge. The wetting morphology for a wedge is classified into three types in terms of the relation between the contact angle to the surface (θ) and the wedge angle (α) (47): For θ < 90° − α/2, a liquid completely fills the wedge with a negative Laplace pressure. For 90° − α/2 < θ < 90° + α/2, a liquid partially fills the wedge in a droplet-like form with a positive Laplace pressure. For θ > 90° + α/2, no wedge filling takes place. The last condition is satisfied for a water/glycerol mixture, which also exhibits LLT (17). For this system, we did not see any indication of surface-assisted modulation of LLT but only observed a usual NG-type process similar to that in Fig. 1A. This difference in the behavior between triphenyl phosphite and a water/glycerol mixture is a consequence of the rather low wettability of liquid II of the latter to the glass substrate compared to that of the former.

Our two-order–parameter model of liquids (3941) suggests that there should be a similarity in the pattern evolution between LLT influenced by a rubbed surface and a gas-liquid phase transition under the influence of wedge filling (see Introduction) (5, 9, 47). We recently demonstrated that, in the process of LLT of triphenyl phosphite in contact with a solid wall, a transition from partial to complete wetting of liquid II takes place on an unrubbed planar substrate when approaching TSD (21), which is reminiscent of “critical point wetting” near a gas-liquid critical point (1, 3). As schematically shown in Fig. 4, the decrease of θ near TSD and TW should also lead to a transition from partial to complete filling (wedge-filling transition) at a filling temperature TF, which satisfies θ(TF) = 90° − α/2. We confirmed that this transition takes place around 220 K: TF ≈ 220 K (see movie S3). Note that the distributions (or the randomness) of both the wedge angle and the length of the microgrooves produced by rubbing may make the transition rather broad.

Fig. 4 Relationship between the wedge-filling transition, wetting transition, and spinodal point of LLT.

In the top row, we show schematics of the free energy F for two situations, where liquid I is unstable (left) and metastable (right) against liquid II. In our system, the contact angle of liquid II decreases toward zero when approaching TSD because the interfacial tension vanishes toward TSD = 215.5 K (21). Thus, a transition from partial to complete wetting (wetting transition) takes place at the wetting temperature TW above TSD. For a wedge whose angle is α, the decrease of θ near TSD and TW also induces a transition from partial to complete filling (wedge-filling transition) at the filling temperature TF, where the condition θ(TF) = 90° − α is satisfied. The two phase-contrast microscopy images indicate the formation of the liquid II phase on a well-developed wedge at 218 and 220 K, respectively. Scale bars, 10 μm. Partially filled liquid II begins to appear in the form of droplets above 220 K. The filling temperature TF of our system is thus considered to be located around 220 K. Note that, in the rubbed cell we used, the surface of the substrate was coated by polyimide, whose contact angle was estimated as 81° at 220 K from our previous study (21). The wedge angle of the microgrooves is thus estimated as 9°. Because of the high contact angle, TW is expected to be located near the spinodal temperature TSD, that is, TWTSD, which makes it very difficult to experimentally determine the wetting transition TW on a planar polyimide surface.

Nucleation of droplets basically proceeds with the formation of a new interface between liquids I and II, whose profile is determined to locally minimize the free energy (48). For a system satisfying the filling condition (T < TF or θ < 90° − α/2), liquid II droplets nucleate and grow while filling a wedge in liquid I. On the other hand, in the critical regime (TTF or θ ≈ 90° − α/2), nucleation of liquid II on a wedge is expected to accompany contact-line fluctuations induced by critical wedge filling. Furthermore, the kinetics of LLT on rubbed substrates are distinct from the usual NG-type transformation, which is the activation process overcoming a free-energy barrier associated with the formation of a new interface. For α/2 ≤ 90° − θ, such a barrier for the formation of a new phase vanishes selectively on wedges, leading to a situation similar to the formation of a new phase on a complete wettable surface. Then, the formation of liquid II on a rubbed substrate should proceed spontaneously without an activation barrier, despite the fact that the system is still in a thermodynamically metastable state and not in an unstable state (above TSD). We argue that this is the origin for the kinetic acceleration of LLT in a rubbed cell (see Fig. 2C). We tried to obtain the detailed geometrical characteristics of the surface pattern produced by rubbing with atomic force microscopy (AFM). However, it was difficult to characterize the angle α because of the random nature of rubbed surfaces and artifacts coming from the shape of the probe of AFM (see the Supplementary Materials). To make a quantitative check, we would need to use well-controlled surface structures as done by Seemann et al. (49), which remains a potential area of future research.

Rubbing effects on SD-type LLT

Finally, we consider wedge-wetting effects on SD-type LLT. Figure 5A shows a spatial pattern evolution process of SD-type transformation on a rubbed surface at 214 K. We also show a direct comparison of the evolution of ΔεII between a rubbed and an unrubbed cell in Fig. 5B. Unlike the NG case, there are no effects of rubbing on pattern evolution. The patterns are isotropic (see also the isotropic 2D power spectrum images shown in the insets of Fig. 5A), and there is no kinetic acceleration. This behavior is fully consistent with our previous observation that surface-wetting effects have no impact on SD-type LLT even for a complete wetting surface (21). The absence of wedge- and surface-wetting effects is a direct consequence of (i) the short-range nature of the surface field (21) and (ii) the nonconserved nature of the order parameter (40). The former leads to an extremely slow growth of wetting (or filling) layer (~ ln t) for an unstable regime (50) compared to bulk coarsening. Thus, fast-growing fluctuations in bulk overwhelm slowly growing wetting layers. This is because the latter allows only a local change of the order parameter near the surface and thus cannot be delocalized because of the irrelevance of diffusion and/or flow unlike in a system of a conserved order parameter (51). This is a consequence of the nonconserved nature of the order parameter, which can change locally without accompanying diffusion. On the other hand, for a system of a conserved order parameter (for example, phase separation of a binary mixture and a vapor-liquid transition), wedge-wetting effects are expected to have a serious influence on the kinetics of SD-type transformation via surface-induced composition waves and interfacial tension–driven flow, reflecting its nonlocal nature (51).

Fig. 5 Pattern evolution during SD-type LLT on a rubbed substrate.

(A) Pattern evolution observed with phase-contrast microscopy at Ta = 214 K. We observed the initial growth of the amplitude of the order-parameter (S) fluctuations through its coupling to the density (at 200 min) and its coarsening in the later stage (at 300 and 400 min) during LLT. This pattern evolution is identical to that of typical SD-type LLT in triphenyl phosphite (15, 44). The white arrow indicates the rubbing direction. Scale bar, 50 μm. Insets (at 200 and 300 min) are 2D power spectrum images obtained by digital image analysis (40), which is isotropic, unlike in the case of NG-type LLT (see Fig. 3C). (B) A direct comparison of ΔεII(t)/ΔεII at Ta = 214 K between the rubbed and smooth unrubbed surface case. (C) Temperature dependence of the ratio of the incubation time of LLT between the rubbed surface case (τR) and the smooth unrubbed surface case (τ0). We define τR and τ0 as the time when an SD-like pattern appears on the surface and when the area fraction of droplets of liquid II in liquid I matrix reaches 0.02, respectively. LLT on the rubbed surface proceeds spontaneously without an activation energy below TF, whereas LLT in bulk does so only below TSD. Above TSD, LLT in the unrubbed surface case (or in bulk) proceeds by overcoming an activation barrier required for nucleation of liquid I droplets.

We have shown that the effects of rubbing on LLT are significantly different between below TSD and above TSD. Figure 5C shows the ratio of the incubation time of LLT in an unrubbed cell (τ0) to that in a rubbed cell (τR) below TF. Here, we independently determine the spinodal point TSD by optical microscopy measurements (vertical dashed lines) (15). Although the accuracy of the determination of the ratio τ0R is not remarkably high (see error bars in Fig. 5C), we can see that, by decreasing the temperature, τ0R drastically decreases toward TSD in the metastable (NG) region and then becomes a constant (τ0R = 1) in the unstable (SD) region, which means that there is no kinetic acceleration below TSD. This strongly supports the scenario described above. The relation τ0R = 1 suggests that the bulk growth is dominant even in a rubbed cell below TSD (see above). Note that as the system approaches TSD, the nucleation barrier for liquid II droplets decreases, and thus, rubbing effects gradually become less important. Accordingly, the difference in the kinetics between the rubbed and unrubbed cases gradually becomes smaller toward TSD and finally disappears completely at TSD.


In conclusion, we have demonstrated that the spatiotemporal pattern evolution of NG-type LLT in triphenyl phosphite is seriously influenced by surface rubbing, which induces a wedge-filling transition. Even in a metastable region, rubbing induces spontaneous barrierless transformation from liquid I to liquid II on the substrates and thus significantly accelerates the kinetics of LLT. We also show that the absence of wedge-wetting effects on the kinetics of SD-type LLT is unique to the ordering governed by a nonconserved order parameter. We note that this is markedly different from a system of a conserved order parameter (for example, a binary liquid mixture), where surface-wetting effects seriously affect the pattern evolution of SD and accelerate the kinetics via diffusion or hydrodynamic flow (51).

Furthermore, the disappearance of wedge-wetting effects below the spinodal temperature TSD strongly suggests that there is no activation barrier for the transformation of liquid I to liquid II below TSD. That is, the transition mode in fact changes from NG-type (metastable) to SD-type (unstable) LLT at TSD. The presence of SD-type ordering further supports that the transition is primarily LLT and “not” nanocrystal formation [see also recent research (18, 3638, 41, 42) and references therein]. As mentioned above, the behavior also indicates that the order parameter governing the LLT is of a nonconserved nature. These findings shed new light on the physical nature of LLT, including the characterization of the nature of the order parameter governing it.

Finally, from an applications viewpoint, our finding may open up a previously unreported way to control the kinetics of LLT in a metastable state, which leads to a change in liquid properties (such as density, refractive index, chemical properties, and transport properties), by using surface structuring of a container. Such surface effects should be particularly important for nano- and microfluidic applications. Furthermore, the development of surface fabrication techniques capable of providing designed nanostructures would offer more accurate spatiotemporal control of LLT.


Experimental details

The sample used in this study is triphenyl phosphite, which was purchased from Acros Organics. We used it after extracting only a crystallizable part to remove impurities. We carefully avoided moisture by using dried nitrogen gas in order to prevent chemical decomposition of triphenyl phosphite because the molecules are known to decompose under the presence of water due to ester hydrolysis. The temperature was controlled within ±0.1 K by a computer-controlled hot stage (LK-600PH; Linkam Scientific) with a cooling unit (L-600A; Linkam Scientific). Sample cells used in this measurement, which were composed of two parallel glass substrates coated with 3× or 10× rubbed polyimide (rubbed cells) and with an unrubbed polyimide (usual cells), were purchased from E.H.C. Co. Ltd. For the rubbed cells, both top and bottom surfaces were rubbed along the same direction. We used a Leica SP5 confocal microscope to measure a surface roughness pattern of the rubbed substrate. An indium tin oxide transparent electrode was also deposited on the glass substrate, which allowed us to simultaneously perform dielectric spectroscopy measurements and microscopy observation. Here, the transparent electrode had no effect on the wetting dynamics of LLT because polyimide was further coated on the electrodes. The cell thickness was fixed to 5 or 10 μm, with an error of ±1 μm. We observed a transformation process from liquid I to liquid II with phase-contrast microscopy (BH2-UMA; Olympus). Broadband dielectric measurements were performed in a frequency range of 10 mHz to 1 MHz with an Impedance/Gain-Phase Analyzer (Solartron SI 1260).


Supplementary material for this article is available at

section S1. The shape of the intensity distribution function P(I)

section S2. The AFM observation of the rubbed substrate

fig. S1. Relation between a phase-contrast image and the intensity distribution function.

fig. S2. AFM image of the rubbed surface.

movie S1. LLT in the smooth unrubbed cell observed at 218 K (the same process as Fig. 1A).

movie S2. LLT in the rubbed cell observed at 218 K (the same process as Fig. 1B).

movie S3. LLT in the rubbed cell at four different temperatures.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.


Acknowledgments: We thank J. Russo for critical reading of the manuscript and K. Nagashima for technical support of the AFM measurements. Funding: This work was partially supported by the Grants-in-Aid for Scientific Research (S) (grant no. 21224011) and Specially Promoted Research (grant no. 25000002) from the Japan Society for the Promotion of Science (JSPS). Author contributions: H.T. conceived and supervised the project. K.M. performed the experiments and analyzed the data. K.M. and H.T. wrote the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors
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