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Commons Attribution 3.0 licence. Living nematic is a realization of an active matter combining a nematic liquid crystal with swimming Any further distribution of bacteria. The material exhibits a remarkable tendency towards spatio-temporal self-organization this work must maintain attribution to the manifested in formation of dynamic textures of self-propelled half-integer topological defects author(s) and the title of (disclinations). Here we report on the study of such living nematic near normal inclusions, or tactoids, the work, journal citation and DOI. naturally realized in liquid crystals close to the isotropic-nematic (I–N) phase transition. On the basis of the computational analysis, we have established that tactoid’sI–N interface spontaneously acquire negative topological charge which is proportional to the tactoid’s size and depends on the concentration of bacteria. The observed negative charging is attributed to the drastic difference in the mobilities of+1/2 and−1/2 topological defects in active systems. The effect is described in the framework of a kinetic theory for point-like weakly-interacting defects with different mobilities. Our dedicated experiment fully conﬁrmed the theoretical prediction. The results hint into new strategies for control of active matter. 1. Introduction A class of non-equilibrium systems consisting of interacting agents transducing the energy stored in the environment into mechanical motion is often called active matter [1] . Examples range from bird ﬂocks [2], ﬁsh schools [3], bacteria suspensions [4, 5], microtubules–molecular motors assays [6–8] to colloidal rollers [9] and Janus ﬂoaters [10]. Active systems often exhibit properties not present at their dynamic counterparts, such as viscosity reduction [11, 12], enhanced mixing [13, 14], rectiﬁcation of chaotic motion [15, 16], active concentration depletion [17, 18] and many others. An important class of active matter termed active nematic is constituted by active agents with apolar (nematic) interactions. A variety of systems fall into this class, from the suspension of swimming bacteria in nematic liquid crystal [19–21], gliding assays of microtubules or actin ﬁlaments on a lawn of molecular motors [22, 23], migrating cells in the cell tissues [24–26]. The recent studies demonstrated an extraordinary role of the topological defects (±1/2 disclinations) in the spatio-temporal organization of these systems [21, 25, 26]. Up to date, most of the experimental and theoretical studies of active systems have focused on spatially- homogeneous systems where the role of the boundaries is not crucial. While active matter often coexists with an isotropic ﬂuid, the proximity effects of isotropic ﬂuid on active nematic are not well understood. Recent computational works [27, 28] studied the behavior of active nematic-isotropic mixtures. Active forces lead to undulation instability of the nematic/isotropic domains and ejection of+1/2 point topological defects in the bulk of the nematic phase. Consequently, the nematic-isotropic interface becomes negatively charged. Macroscopic inclusions (tactoids) spontaneously appear in liquid crystal as a result of isotropic/nematic (I–N) phase transition controlled by temperature and the concentration of mesogenic material. Although the phase coexistence occurs in a narrow temperature interval, it can be expanded by the addition of surfactants [29]. © 2018 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft New J. Phys. 20 (2018) 043027 M M Genkin et al It was experimentally observed, that bacteria in nematic phase swim around isotropic tactoids [19] and can be jammed in the tactoid’scusps [30]. Tactoids themselves play a profound role in many systems undergoing a phase transition. Besides liquid crystals, the examples include cosmological models of the early universe [31, 32], superﬂuidity [33, 34], solid state physics [35]. According to the Kibble–Zurek hypothesis of the early universe formation, after the symmetry-breaking phase transition, the domains with almost constant order parameter (OP) are created. Consequently, the domains with different OP orientation merge yielding topological defects. The shape and nematic structure of tactoids were the main focus of experimental and theoretical studies [36–40]. In these works, it is assumed that tactoid’s size is determined by the interplay of elastic forces, surface energy and anisotropic anchoring on the I–N interface. In this context, the energy minimization problem has to be solved to ﬁnd an optimum tactoid’s shape. However, the role of tactoids on the properties of active nematics is not understood. Here we perform computational and experimental studies on the role of tactoids in the organization of active collective motion. Tactoids are explicitly incorporated in an extended model of a living nematic [21]. The obtained nematic conﬁgurations are in good agreement with the experiments [36, 37]. We have found that in the regime of active turbulence the tactoid’s isotropic-nematic interface spontaneously acquires an average negative topological charge. These ﬁndings are rationalized in the framework of a simple kinetic model for two populations of point-like weakly-interacting defects with different mobilities. The dependence of a tactoid’s charge on the bacteria concentration and tactoid’s size is in a good agreement with computations. To support our theoretical predictions, we performed a dedicated experiment with swimming bacteria Bacillus subtilis suspended in a lyotropic chromonic disodium cromoglycate (DSCG) liquid crystal. Our experimental results agrees with computational modeling both qualitatively and quantitatively. 2. Description of computational model 2.1. Model The model is a signiﬁcant extension of our previous study [21]. A continuum approach is justiﬁed by the fact that characteristic length scale of the emergent patterns (for example, an average distance between topological defects) L∼200 μm signiﬁcantly exceeds average inter-bacterial distance l<10 μm. For the liquid crystal dynamics we use continuum 2D (depth-averaged) Edwards–Beris approach: (· ¶+ v )QS - -GHF + +F = 0, (1) t exter anch + ·(ss+s +s -pv I)- z= 0, (2) asact visc =·( v 0. 3) Here equation (1) describes the tensorial order parameter (OP) Q evolution, which is deﬁned as QI =- qn() n 2 ,where n is the unit vector of nematic orientation, q is the amplitude of the OP. Tensor S describes alignment with the ﬂuid ﬂow, H is the tensorial molecular ﬁeld and F is the external (wall) anchoring. exter Tensor F is a new term that describes anisotropic anchoring on the I–N interface. Equation (2) is the linear anch momentum balance, where we split the total stress tensor into isotropics , anisotropics ,viscouss and active s a visc s contributions. The last term in equation (2) is the contribution from the depth-averaging. Equation (3) is the act incompressibility condition. For the detailed deﬁnitions of the mentioned terms see appendix A. We couple equations (1)–(3) with the conservation laws for the bacteria concentration c that are cast in the form of advection-diffusion equations: ¶+cV ·(pw+vc)=Dc, t 0c 2w ¶+wV ·(pc+vw)=- +Dw.4 ( ) t 0c + − As in our previous work [21], we assume that at each point we have two concentrations of bacteria c and c swimming with a constant speed V along the directions p and-p correspondingly. Here p =(( cosff ), sin( )), where f is the local orientation angle of bacteria. Equation (4) describes evolution of the +- +- w=-cc total concentration ﬁeldcc=+c and the concentration difference (or polarization) ﬁeld . Here we allow bacteria reversal with the average rate τ. For additional details see [21]. The major difference with [21] is that the bacteria swimming direction p does not coincide with the nematic (director) orientation vector n. This generalization is needed because the nematic director n is not deﬁned in the isotropic phase. According to [21], angle f obeys the following relaxation equation ∣∣ Q ¶= f sin() 2qf- 2 + Df. (5) t t 2 New J. Phys. 20 (2018) 043027 M M Genkin et al Here the ﬁrst term describes the relaxation towards the nematic angle θ with the relaxation rate τ , and the second term describes the diffusion of orientation with the rate D . The rational for the choice of sin() 2qf - 2 term in equation (5) is the following. Since θ−f is a periodic function, the angle relaxation has to be governed by a periodic function as well, e.g. by a sine: a leading term in the expansion of a generic periodic function. Note that the relaxation towards the nematic director (due to elastic restoring torque) is taken to be proportional to the angle difference θ−f, see [41, 42]. This torque also has to satisfy several limiting conditions, e.g. at θ−f=π/2 it has to vanish. In equation (5) we used a different form (sine of the angle difference) that automatically satisﬁes the required limiting conditions. We observed that for typical relaxation time τ ≈1s, the bacterial orientation always coincides with the director in the nematic phase, which makes the difference θ−f very small. Since sinxx » for small x, our approach is equivalent to the conventional one [42]. In normal domains, where ∣Q∣ = 0, the alignment term ∣Q∣( sin 2qf - 2 ) disappears and bacterial orientation is governed purely by the diffusion term. For simplicity, we assume that the elastic effects in LC dominate the ﬂow alignment and neglect the advection and shear alignment terms (these terms will not change our results qualitatively, provided that the relaxation timet is sufﬁciently small). Equation (5) is difﬁcult to solve numerically due to angle discontinuity. Here, by analogy with the nematic tensor Q, we introduce an evolution equation for the traceless symmetric tensor P=- qp ( p I 2) for bacterial orientation: 3 2 ¶=PPac-4. P - +D P ()6 tp p t Here a and c are constants that control the amplitude q = ∣∣ P and that are chosen in such way that q =1in p p p the steady-state. F aligns P with the nematic orientation set by tensor Q with the alignment time τ (τ is of the Q 0 0 order 1 s and is much smaller than the bacteria reversal time τ≈60 s [21]). The alignment termFQ ~ is chosen in such way that it does not change the amplitude of tensor P. In the limit ∣P∣ = const we recover equation (5). For the deﬁnition of F see appendix A. Inside the tactoid, F vanishes, and bacterial orientation Q Q becomes independent of Q. In turn, the nematic tensor Q couples to the tensor P via the expression for active stress. Note, that the constants a and c are used to control the amplitude q of the tensor P and do not bear any p p p physical meaning (unlike similar terms a and c in the equation for the OP Q). Our model consists of equations (1)–(4), (6). The major differences of this study from our previous work [21] include: (i) introduction of an additional equation for the bacterial orientation (6), (ii) introduction of the isotropic phase in the form of a tactoid (see section 2.2) and (iii) anisotropic anchoring on the I–N interface. Note, that the anchoring on the I–N interface is typically described by the Rapini–Papoluar form [43].We implemented the I–N anchoring in a different way by introducing additional relaxation term F in anch equation (1). It can be shown that our approach is equivalent to Rapini–Papoular one, see appendix D. 2.2. Tactoid implementation In the biphasic domain, the tactoids can be introduced in the Landau–de Gennes free energy via additional term ~∣∣ Q . This term will make both isotropic and nematic states locally stable. However, since we are interested in ﬁxed isotropic domains (they can be produced, for example, by local heating), we make the coefﬁcient a in front of the quadratic term in the Landau–de Gennes free energy (see appendix A for the deﬁnition of free energy) spatially-dependent, so that the equilibrium OP’s amplitude q is some positive constant in nematic and zero in isotropic phase, see ﬁgure 1 where amplitude q = ∣Q∣is shown with color. Consider a nematic phase and suppose we want to introduce an isotropic tactoid. In order to do that, we set the coefﬁcient a to a negative value inside the tactoid, a=−a , and positive value outside of the tactoid, a=a 2 1 with a rapid, but continuous transition on the I–N boundary. To implement a tactoid deﬁned by the equation f (x, y)„0, with boundary f (x, y)=0, we set: ⎛ ⎞ ⎛ ⎞ aa + fx() , y ax() , y =- tanh⎜-+ ⎟ 1.+ a ()7 ⎜ ⎟ ⎝ ⎠ 2 ⎝ d ⎠ Here δ is an approximate width of the I–N interface (the actual width of the I–N interface also depends on the other parameters). In this work we used a =1, a =5. We considered isotropic tactoids of elliptical shape: 1 2 () yy - () xx - fx(),1 y = + -. ()8 2 2 R R 1 2 In order to create a nematic tactoid surrounded by an isotropic phase, we changed the sign of the coefﬁcient a to an opposite one in the entire computational domain. We considered nematic tactoids of circular shape: () yy - () xx - 0 0 fx(),1 y = + -. ()9 2 2 R R 0 0 Thus, we introduced tactoids in our system by spatial variation of the Landau–de Gennes coefﬁcient a. In our simpliﬁed approach, the tactoid shape is predeﬁned (although local shape adjustments can take place near the 3 New J. Phys. 20 (2018) 043027 M M Genkin et al Figure 1. Isotropic tactoids in active nematic medium. Nematic contour lines and the magnitude of the order parameter (A)–(C), ﬂuid velocity magnitude with streamlines (D)–(F), concentration ﬁelds with bacteria orientation contour lines (G)–(I) for three different regimes. Here the wall anchoring strength is ξ =0.02 and planar I–N anchoring strength is ξ =30. The thick black arrow on exter anch (A) indicates the prescribed director orientation set by the wall anchoring. Panels (A), (D), (G) depict an aligned state with mean bacteria concentration c = 0.2. Panels (B), (E), (H) display a stable periodic conﬁguration with c = 0.5, near the instability threshold, 0 0 see also Movie S1 available online at stacks.iop.org/NJP/20/043027/mmedia. Panels (C), (F), (I) display a chaotic regime with defects with c = 0.6, see also Movie S2. The red rectangle on (A) indicates the contour used to calculate the tactoid charge s . 0 t surface defects, or boojums, see ﬁgure 3(C). Although more complicated models [27] of the biphasic systems with the possibility of shape evolution can be considered, our experimental observations justify our assumption of nearly ﬁxed tactoid’s shape (see Movie 4). 2.3. Numerical implementation Nonlinear system of PDEs (1)–(4), (6) was integrated in a square domain with periodic boundary conditions, 1024×1024 mesh points were typically used. Other numbers of mesh points were also used to ensure that the chosen resolution is an optimal one and to exclude numerical artifacts. We used a combination of forward Euler method in time and pseudospectral method in space for the numerical integration, see appendix A. Initial conditions were either aligned along the x-axis with small noise or random from uniform distribution. 2.4. Additional implementation details There was an artifact related to the deﬁnition of the bacteria orientation angle f that was restricted to the interval + − fÎ-[] pp 2; 2 . We solved this problem by relabeling the concentrations c and c in the close proximity of the regions where f=±π/2. For additional implementation details see [21] (where the same thing was performed for the nematic angle θ). In addition, there was a discrepancy related to the fact, that nematic ﬁeld is not deﬁned in the defect core. We modiﬁed our scheme by signiﬁcantly reducing the reversal time τ (see equation (4)) in the close proximity of each defect core. This was performed by scaling τ by the factor ∣p∣ in the entire computational domain. This factor was approximately 1 nearly everywhere (including both nematic and isotropic phases), but in the cores of topological defects where it was very small. By using this technique, we signiﬁcantly reduced the concentration + − difference w in the cores of topological defects, which is equivalent to setting c ≈c (there is no difference + − between c and c at the singularity point). 4 New J. Phys. 20 (2018) 043027 M M Genkin et al 2.5. Experimental details The procedure of the living nematic sample preparation and data acquisition is similar to methods used in previous works [19, 21]. We used nearly crossed polarized microscopy for simultaneous visualization of bacteria and tactoids. The data was recorded with frame rate of 15 fps using 10× magniﬁcation. Bacterial concentration 9 −3 in the experiment was c ≈5 × 10 cells cm . The experiment was performed in a ﬂat glass/PDMS cell. A small (≈10 μl) drop of bacterial suspension in DSCG was sandwiched between a TiO coated glass slide and a polydimethylsiloxane (PDMS) ﬁlm. 20 μm spacers were used to ensure uniform thickness over the cell. The thickness of the PDMS ﬁlm was chosen to be approximately 1 mm, as a trade-off between the ﬁlm stiffness and the oxygen permeability. In order to keep the system in the biphasic state, a small electric current was transmitted through a TiO coated glass slide to heat the LC and create isotropic tactoids of various sizes. The temperature of the sample was in the range of T≈25 °C–30 °C. The nematic ﬁeld was reconstructed with a custom Matlab script. The orientation of each bacterium and correspondingly nematic ﬁeld in its vicinity was extracted from grayscale images by a gradient method. Assuming that an orientation of LC molecules changes slowly between the neighboring bacteria, we reconstruct the nematic ﬁeld by linear interpolation. The ﬁeld was smoothed out by a Gaussian low-pass ﬁlter with a standard deviation equals to the average bacterial length of 5 μm. To increase the accuracy of the method in areas where bacteria swim occasionally, we time-averaged the ﬁeld over 50 consecutive frames. 3. Results 3.1. Isotropic tactoids in nematic phase Isotropic tactoids in nematic phase with planar I–N anchoring are shown in ﬁgure 1. At sufﬁciently low bacterial concentrations and high wall anchoring (wall anchoring prescribes the overall nematic orientation, see appendix A and [21]), an aligned regime with the nematic orientation parallel to the anchoring direction was observed, ﬁgure 1(A). However, planar I–N anchoring forces the nematic orientation to deviate from the horizontal one in the tactoid proximity. The velocity for this case is very small, and attains the maximum value on the I–N interface near the tactoid’s cusps (due to non-zero gradients of the OP), ﬁgure 1(D). In this case, the bacteria concentrate outside of the tactoid near the cusps, whereas bacteria concentration is reduced inside the tactoid, ﬁgure 1(G). For moderate bacteria concentrations, the conﬁguration is characterized by periodic nematic undulations, ﬁgure 1(B). In this case, the hydrodynamic ﬂow forms lanes normal to the anchoring direction, ﬁgure 1(E). As in the previous case, the bacteria concentrate outside of the tactoid. However, the interplay of wall anchoring at the I–N interface and the bulk nematic orientation results in non-symmetric ﬂow structure around a tactoid, ﬁgure 1(H). For even higher bacteria concentrations, a chaotic regime with multiple topological defects was observed, ﬁgures 1(C), (F), (I). The defects enter and leave the tactoid and temporarily distort the planar anchoring at the I–N interface. Note that the instability threshold observed here and in our previous work is different from the isotropic case, where the system is always appears to be unstable [44]. This is because of an additional alignment introduced by the nematic ﬁeld. By comparing the perturbation torque generated by active stress, which is proportional to the bacterial concentration T =k cf with the restoring elastic (or external wall anchoring) act 1 torque T =k f, one can justify that the former exceeds the latter only for c>k /k . elastic 2 2 1 We measured time-averaged total topological charge of the tactoid s and averaged relative bacteria concentration inside the tactoid c :cc =á() -c c ñ, c is the space-averaged concentration inside the t ttact 0 0 tact tactoid, c is mean bacteria concentration,áñdenotes time-averaging. The charge was calculated from the phase difference in the counterclockwise direction over a region enclosing the tactoid (red rectangle in ﬁgure 1(A)), see appendixA. The dependences of topological charge s and concentration c on wall anchoring and bacteria concentration t t c are shown in ﬁgures 2(A), (D)(planar I–N anchoring). For small bacteria concentrations and strong wall anchoring, there are no topological defects and the tactoid’s topological charge is zero. For higher concentrations, in the regimes with defects, the average topological charge of the tactoid is always negative. The charge magnitude ﬁrst increases with the bacteria concentration but then saturates and falls down. The magnitude of the tactoid’s charge increases with the wall anchoring. There is also a minor suppression of the relative bacteria concentration inside the tactoid. The dependences of s and c versus tactoid’s size and mean t t bacteria concentration are shown in ﬁgures 2(B), (E). In this work, we ﬁxed the aspect ratio of isotropic tactoids at 1:3 and deﬁned tactoid’s size as the length of its major axis. Both magnitudes of s and c grow linearly with the t t tactoid size. Similar behaviors are also observed for the tactoids with homeotropic I–N anchoring, see appendix B. Note, that as long as the defects are persistently entering and leaving the tactoid, instant topological charge s˜ and relative bacteria concentration c ˜ signiﬁcantly ﬂuctuate with time, see ﬁgures 2(C), (F). t t 5 New J. Phys. 20 (2018) 043027 M M Genkin et al Figure 2. Topological charge and relative bacterial concentration difference for the isotropic tactoid in a nematic medium with planar I–N anchoring. Tactoid chargeáñ s (A) and relative concentration differenceác ñ (D) versus wall anchoring strength ξ and mean t t exter bacteria concentration c . Here the tactoid size is ﬁxed L =180. The tactoid chargeáñ s (B) and relative concentration difference 0 tact t ác ñ (E) versus tactoid size L and mean bacteria concentration c . Here the wall anchoring strength is ﬁxed to ξ =0.02. The t tact 0 exter dependence of s (C) and c (F) on time for c = 0.8, ξ =0.02 and c = 1.8, ξ =0.1. Both s and c signiﬁcantly ﬂuctuate in the t t 0 exter 0 exter t t chaotic regime with defects. Figure 3. Nematic tactoids in isotropic medium. Nematic ﬁeld conﬁgurations for ﬁxed c = 0.6 and different tactoid sizes: planar (A) and homeotropic (E) I–N anchoring for R=30, planar (B) and homeotropic (F) I–N anchoring for R=90, planar (C) and homeotropic (G) I–N anchoring for R=240, see also Movie S3. No wall anchoring ξ =0 was imposed, whereas planar I–N exter anchoring had a strength ξ =30. Red circle on (A) is the contour used to calculate the tactoid charge. The dependence of the anch tactoid chargeáñ s (D) and relative concentration differenceác ñ (H) on the tactoid size L and mean bacteria concentration c for t t tact 0 planar I–N anchoring. 3.2. Nematic tactoids in isotropic phase Nematic tactoids in isotropic phase can be produced by changing the sign of the Landau–de Gennes coefﬁcient a in the entire computational domain. In agreement with experiments [36], we obtained different nematic conﬁgurations versus bacteria concentrations and tactoid sizes. We studied the tactoids of circular shape with different radii R, and no wall anchoring. Conﬁgurations with mean bacteria concentration c = 0.6 for planar I–N anchoring and different tactoid sizes are shown in ﬁgures 3(A)–(C) and for homeotropic I–N anchoring are shown in ﬁgures 3(E)–(G). The obtained conﬁgurations are in good qualitative agreement with the experiment: for small tactoid size, the elastic forces dominate over surface anchoring, which corresponds to aligned (undisturbed) nematic conﬁguration (ﬁgures 3(A), (E)). For larger tactoid sizes, there were a few singularities on the tactoid’s surface (ﬁgures 3(B), (F)). For even larger tactoid sizes, conﬁgurations with many defects are observed, both inside the tactoid and on its surface. The singularities on the I–N interface are called boojums. Figure 3(B) depicts the so-calle ‘2c’ conﬁguration with 2 positive boojums that frequently arises in experiments 6 New J. Phys. 20 (2018) 043027 M M Genkin et al Figure 4. Experimental images of isotropic tactoids in nematic medium and comparison of the data from experiments and simulations. (A) An experimental image obtained by nearly-cross polarized microscopy. 17 visible tactoids are encircled, see also Movie S4. (B) Nematic ﬁeld (green lines) reconstructed from orientation of bacteria in 50 consecutive frames. (C) The dependence of tactoid’s topological charge on its size obtained from the experiment (crosses) and simulations (solid lines). Activity of bacteria A,a product of bacterial concentration c and dipole force strength, is extracted from linear ﬁt (dashed line) of the experimental data. The 9 −3 bacterial concentration in the experiment was 5 × 10 cells cm (D) Nematic conﬁguration around a single tactoid in the experiments. Positive defects are highlighted with red circles, negative with blue circles. Negative defects reside on the I–N interface, whereas positive ones in the bulk of nematic phase. (see e.g. [36, 38]). Note that even though the overall tactoid shape was predeﬁned (disk), there was a local shape adjustment near the boojums, ﬁgure 3(C), (F). We measured averaged charge s and relative bacteria concentration inside the tactoid c . To determine the t t tactoid charge, we calculated the phase difference over a circular contour with R=0.99 R , R is the radius of 0 0 the tactoid (red circle in ﬁgure 3(A)). Thus, we calculated the total topological bulk charge and did not take into account the charge produced by boojums on the I–N interface. Both s and c versus tactoid size and mean t t bacteria concentration are shown in ﬁgures 3(D), (H) for planar I–N anchoring (dependencies for homeotropic anchoring are shown in appendix B). For both planar and homeotropic anchorings, the averaged bulk tactoid’s charge is positive and grows with the bacteria concentration and tactoid’s size. Note, that due to topological charge conservation, it implies negative surface topological charge, so that I–N interface becomes negatively charged as in the case of isotropic tactoid, see section 3.1. The relative concentration difference inside the tactoid is always negative for planar anchoring, with its maximum (by absolute value) achieved for large tactoids and intermediate bacteria concentrations. For high bacterial concentrations, in highly chaotic regimes, the effect smears out. For the homeotropic case, relative concentration difference is positive, see appendix B. 3.3. Experimental veriﬁcation Experimental veriﬁcation was performed with strain 1085 of motile bacteria Bacillus subtilis suspended in lyotropic liquid crystal DSCG, see section 2.4. The microscopy images were acquired in nearly cross-polarized regime, where the angle between a polarizer and an analyzer was slightly less than π/2, ﬁgure 4(A). This regime allows to observe both swimming bacteria and tactoids seen as dark areas simultaneously. Images were processed with a custom Matlab script to reconstruct the nematic ﬁeld from the local orientation of bacteria, ﬁgure 4(B). Topological charges of 17 visible tactoids were determined by summing the topological charges of defects around each of tactoids, ﬁgure 4(D). The analysis of the experimental data reveals that the total topological charge is decreasing with the size of a tactoid, conﬁrming theoretical predictions, see ﬁgure 4(C). The accessible range of bacterial concentration in the experiments is limited by two factors. At low concentrations, the nematic ﬁeld can not be reconstructed in signiﬁcantly large areas due to the absence of 7 New J. Phys. 20 (2018) 043027 M M Genkin et al bacterial tracers. At high concentrations tracking of bacterial orientation is not reliable due to bacterial overlapping. These limitations may be overcome by adding passive ﬂuorescent orientational tracers into the system in future work. 4. Discussion The onset of tactoid spontaneous charge can be attributed to different defect mobilities in active systems:+1/2 are mobile, whereas−1/2 are effectively immobile and only entrained by the moving ones [25, 45, 46].In [21],we have found that the effective speed of+1/2 defects is approximately 5 times larger than of−1/2 defects. In order to enter or leave the tactoid, the defects need to overcome a barrier at the I–N interface. We veriﬁed that the barrier is practically independent of the anchoring strength and is due to sharp variation of the nematic OP across the interface. The barrier breaks the symmetry in±1/2 defect absorption and leads to a non-uniform defect distribution, and thus net charging. The charging increases with the bacteria concentration, since the defect concentration itself is proportional to the bacteria concentration [21]. However, at very large bacteria concentrations, the charging is overwhelmed by the inter-defect interactions. To support our hypothesis, we developed a stationary 1D Fokker–Plank description for positive and negative + − defects probability distribution functions P and P , see also appendix C and ﬁgure C1. We took into account different defects mobilities (manifested by different diffusion coefﬁcients) and pairwise interactions between the defects: this interaction is similar to the Coulomb interactions between charged particles (different signs attract, similar signs repel) but with the interaction force inversely proportional to the distance between defects F µ 1 r [47]. +- ⎛ ⎞ Px()¢ -P ()x¢ ++ 2 ++ DP¶=¶ ⎜m d, xP¢ - F¢() x P ⎟ ⎝ ⎠ xx - ¢ -+ ⎛ ⎞ Px() -P ()x ¢ ¢ -- -- ⎜ ⎟ DP¶=¶ m d. xP¢ - F¢() x P (10) ⎝ ⎠ xx - ¢ + - Here D are the defect diffusivities, D D , μ is the inter-defect interaction strength, F (x) describes a potential barrier at the I–N interface, and L is the total size of the system. Here we assume that the defect diffusivities are roughly proportional to the product of defect speed and its mean free path, which in turn depend on the concentration of bacteria. Note, that the description of defect’s motion with diffusion-like process is justiﬁed by the fact that at sufﬁciently high bacterial concentration defect statistics can be well-approximated by the Gaussian distribution [21] (in contrast to the stretched exponent distribution that is typical for ballistic defects). For equation (10), an asymptotic solution for the tactoid charge density can be obtained under the assumption of very large positive defect diffusivity D and weak defects interaction (see appendix C): ⎛ ⎞ D ⎛ ⎞ A mdP e L ⎜⎟ SP =- d() e1 - ⎜1- log + 1 ⎟. (11) ⎝ ⎠ ⎝ D d ⎠ Here S is the tactoid topological charge density, P is the mean defect density in the system (assumed to be proportional to the bacterial concentration c), δ is the width of I–N interface, A is the height of the potential barrier. This model explains spontaneous negative charging of the I–N interface if we assume A>0. It will immediately imply accumulation of negative charge for negative tactoids and vice versa. The charge magnitude depends on the mean defect (or bacteria) concentration asSC µ- P +CP with C >C , where C are 00 1 0 0 1 1,2 constants. Moreover, as long as we assumed that the defects exist only at the nematic medium or at the I–N interface, the charge density is independent of the tactoid’s size. Charge density decreases with the system size as SL µ-log d . The logarithmic divergence of S with the system size is an artifact of the assumption that the interaction strength between moving defects is 1/r, similar to the immobile ones. As was shown, see e.g. [48],a small velocity correction to the interaction force leads to regularization of the logarithmic divergence. The correspondence of the suggested simpliﬁed 1D and full 2D models becomes clear if one considers the case of the very strong external (wall) anchoring along the x-axis. In this case [21], all topological defects are moving predominantly along the direction normal to the wall anchoring (along the y-axis) so that the 2D system is equivalent to a quasi-1D system. In this one-dimensional representation, two defects, that are separated along the x-axis, can pass through each other without collision. Topological charge density, deﬁned as a ratio of the tactoid topological charge to its length s =s /L for L t different tactoid sizes and bacterial concentrations are shown in ﬁgures 5(A), (B). This data is extracted from ﬁgures 2, 3 and appendix B. One sees that for isotropic and nematic tactoids with planar and homeotropic I–N anchoring, the tactoid topological charge density versus concentration is practically independent of the tactoid’s 8 New J. Phys. 20 (2018) 043027 M M Genkin et al Figure 5. The magnitude of topological charge density of a tactoid versus mean bacteria concentration for three different tactoid sizes. (A) Isotropic tactoid with planar anchoring. Inset: with homeotropic anchoring. (B) Nematic tactoid with planar anchoring. Inset: with homeotropic anchoring. The topological charge density for all of the four cases is practically independent of the tactoid size and can be ﬁtted by a parabola. size and can be well approximated by a parabola, in agreement with equation (11). In addition, we calculated the tactoid charge for the isotropic rectangular tactoid that span along the entire horizontal direction (see appendix B) with a variable width h along the y-axis. We ﬁxed the total system size, thus with h increasing the size of the medium L was decreasing. We found out, that tactoid topological charge is practically independent of L and h for sufﬁciently small bacteria concentrations. However, there is a steep increase in the topological charge for large L that is probably associated with a very small tactoid width h, so that topological defects are able to traverse the tactoid. 5. Conclusions Here we investigated model of active nematic system in the biphasic regime. In our approach, the nematic and isotropic phases are created by spatial modulation of the Landau–de Gennes coefﬁcient a. Even though the overall tactoid shape was predeﬁned, local dynamical shape adjustments take place, e.g. near the negative boojums, ﬁgure 3(C). We implemented a nematic-dependent anisotropic anchoring and showed that our approach is equivalent to the conventional approach with additional contribution to the free energy in the Rapini–Papoular form, see appendix D. The main difference from our previous model [21] is an additional equation for the bacteria orientation required to prescribe bacterial motion inside the tactoid. We have found that the tactoids spontaneously acquire a non-zero topological charge. The charge density depends on the mean bacterial concentration and wall anchoring strength. This effect is illustrated and quantiﬁed with a simple 1D Fokker–Plank model that attributes spontaneous charge accumulation to different defect mobility. The model qualitatively captures the dependence of the tactoid topological charge on the mean bacterial concentration, tactoid size, and the system size. Our experiment fully conﬁrms the prediction and shows qualitatively and quantitatively similar dependence of the I–N charge for different tactoid sizes. The suggested mechanism is similar to that of spontaneous charging of colloids [49, 50] due to colloidal particles protonation and transport of mobile protons into the solution. Note that the tactoid charging is fundamentally different from topological defect pinning by material imperfections in equilibrium systems, e.g. pinning of vortices in type-II superconductors [51]. Whereas superconducting vortices are pinned by the imperfections irrespective of their sign, the topological defects show remarkable asymmetry in their pinning properties. Our work provides a better insight into the complex interactions between the topological defects and tactoids in non-equilibrium systems. Both topological defects and tactoids can be used as tools to control and manipulate active matter. Topological defects can be created and controlled by light [52, 53] or by atomic force surface-pattering [54]. Correspondingly, tactoids can be created by intense laser heating and controlled by electric or magnetic ﬁelds. It was reported that tactoids can be reversibly aligned and stretched by up to a factor of ﬁfteen in length using an electrical ﬁeld [55]. Our ﬁndings suggest that an array of tactoids created on demand can be used to capture negative defects and thus create a positively charged active nematic liquid. This can potentially introduce additional control capabilities over active matter as well as a new way of sorting and identifying bacteria in addition to other methods [56]. 9 New J. Phys. 20 (2018) 043027 M M Genkin et al Acknowledgments MMG, AS and ISA were supported by the US Department of Energy, Ofﬁce of Basic Energy Sciences, Division of Materials Science and Engineering (experimental work and simulations performed at ANL). ISA was also supported by the NSF grant PHY 1707900 (analysis of the Fokker–Plank system for the defects). Author Contributions ISA conceived the research. ISA and MMG developed the theory. MMG performed modeling and computing. And AS performed experiments and analyzed data. All authors wrote the manuscript. Competing Interests The authors declare that they have no competing ﬁnancial interests. Appendix A. Computational model A.1. Deﬁnitions Our nonlinear system consists of equations (1)–(4), (6). Both the OP and the bacterial orientation tensors are 2 × 2 traceless symmetric tensors that are deﬁned as follows: QI =- qn() n 2, P= q(pp-I 2). (A.1) Here q and q are the amplitudes of the corresponding tensors, n is nematic director, p is bacteria orientation, I is a 2 × 2 unit matrix. In equation (1) tensor S can be expressed thorough the symmetric A and antisymmetricΩ contributions of the velocity gradient tensor W=v : ⎛ ⎞ ⎛ ⎞ I I ⎜⎟ ⎜⎟ SA=+()xxWW Q+ + Q+ ()AQ + -x(2T+I)r(QW), (A.2) ⎝ ⎠ ⎝ ⎠ where ξ is an aspect ratio parameter. The molecular ﬁeld H can be deﬁned as a ﬁrst variation of Landau–de Gennes free energy F. Here we use one-constant elastic approximation: ⎡ ⎤ a b K Fr=- d QQ + () QQ + (¶ Q ),A(.3) ab ba ab ba g ab ⎢ ⎥ ⎣ ⎦ 24 2 HQ=-acQ TrQ+KQ. () A.4 The external wall anchoring F relaxes director towards a certain anchoring direction deﬁned by unit vector exter f . The form of this term is chosen such as it does not affect the amplitude q, see also [21]: ext FQ=- 4T x R r[(Q ff I2)R ], (A.5) exter pp 2 2 exter ext ext where ξ is the strength of external anchoring, R is the rotation matrix. In this work we always choose the exter π/ 2 external (wall) anchoring direction along the x-axis: f =() 0; 1 . ext The alignment of the LC director on the I–N interface is described by the term F : anch FQ=- 4T x R r((Q ff I2)R )(r). (A.6) anch pp22 anch ee Here ξ is the strength of I–N anchoring, f is a vector along the easy direction on the I–N interface (see [36]), anch () r is a step function that is non-zero only in a thin region that encloses I–N interface. The form of the term F is similar to the form of other relaxation terms, for example, F . anch exter In order to ﬁnd f , we rotated the normal vector to the I–N interface (which we set parallel to the OP’s amplitude gradient) by the easy angle: fR= (∣Q∣).A(.7) Here R is the rotation matrix by the easy angle θ . In this work, we only considered planar (θ =π/2) and q 0 0 homeotropic (θ =0) anchoring. Note, that we did not normalize easy direction vector, therefore the normalization factor ∣∣∣ Q ∣ will be absorbed in the amplitude ξ (see equation (D.10)). anch s s s Different stress contributionss, , , in equation (2) are the same as in our previous work, see [21]. s a act visc Equation (6) contains the term F that represents a relaxation of the bacterial orientation tensor P towards the nematic orientation Q. The form of this term is similar to the other relaxation terms: 10 New J. Phys. 20 (2018) 043027 M M Genkin et al FP = R Tr()PQR . (A.8) Q pp22 This term relaxes bacteria orientation deﬁned by the tensor P towards the nematic orientation deﬁned by OP Q and does not change the amplitude of the tensor P. In the isotropic phase ∣Q∣is practically zero, which makes F very small compared to the rotational diffusion term in equation (6). A.2. Reduction to a scalar system In order to convert our system to a set of scalar equations, we denote: ⎛pp ⎞ nn 12 12 QP = ,. = ⎜ ⎟ ()A.9 () nn - 21 pp - ⎝ ⎠ We transform our system into a set of scalar equations for two complex and three scalar variables: n=n +in , 1 2 p=p +ip , c, w,Ψ (equations (2) and (3) can be replaced with a single equation for the stream functionΨ 1 2 deﬁned as v =∂ Ψ, v =−∂ Ψ). For equations (1), (6) this transformation is performed by summing up (1, 1) x y y x component with its (1, 2) component multiplied by the complex unity i. As a result, the nondimensionalized version of equation (1) can be written in the following form (for details see SI to [21]): ⎛ ⎞ ¶n G 1 22 2 ⎜⎟ +Ynn - Y - -n ix ∣∣n - Y yx y x ⎝ ⎠ ¶t Er 2 ⎛ ⎞ 22 2 2 ⎜⎟ +na(∣++2i bn∣) Y+ixn Y ⎝ ⎠ Er -- inn ImF()** in(r)ImF(n )=0. (A.10) ext anch Here the following differential operators are used: =¶ +¶i, *=¶ -¶i, xy xy = · =¶ +¶ , xy 2 22 =¶ + 2i¶ ¶ - ¶ , xy x y 2 22 * =¶ - 2i¶ ¶ - ¶ . () A.11 xy x y Correspondingly, equations (2) and (3) are reduced to a single equation for the stream functionΨ. This can be performed by applying a curl operation to equation (4): ⎛ p ⎞ ⎜ ⎟ ´ º¶pp -¶.A().12 xy ⎝ ⎠ The nondimensionalized version of equation (2) is: 42 2 2 Er() Y- z Y = Im[(cc + Er)+(c)], (A.13) sact a where cn =-( )(n*) + x(14 - ∣n∣)(a +2c∣n∣)n 22 2 2 + x((22 nn *) + ( ∣n∣-1) n), cn= 2Im()n* , cc =Lp 2. () A.14 act Here the Ericksen number is Er = hVl K. In equation (4) bacteria orientation unit vector is deﬁned as , where is deﬁned as . equation (4) is already in p =(( cosff ), sin( )) fÎ-[] pp 2, 2 f = Arctan() pp , 2 scalar form. Equation (6) is reduced to a scalar equation for the complex variablepp=+ ip in a similar way 1 2 as equation (1). As a result, we get: ¶p 2 2 =- ap 4c ∣∣p p+ iI p m(np*)+ D p. (A.15) pp r ¶t t A.3. Calculation of tactoid’s topological charge In order to determine the topological charge of a particular region (for example, the region that encloses the tactoid), the sum of phase differences was calculated along the contour enclosing this region in the counterclockwise direction. Let us denote the nematic angle of those points in counterclockwise order as θ , i=1, ..., N. The topological charge of the region enclosed by the contour is: 11 New J. Phys. 20 (2018) 043027 M M Genkin et al Table A1. Model parameters. Parameter Value Description Parameters of LC K 10–15 Elastic constant (pN) Er 3.75 Ericksen number −1 Γ 1 Relaxation rate (ms kg ) −1 −1 η 0.5 Isotropic viscosity (kg m s ) −1 ξ 0–0.6 Wall anchoring strength(1s ) exter −1 ξ 2000–2500 I–N anchoring strength (mm s ) anch h 20–30 Thickness (μm) ξ 0.9 Aspect ratio parameter a −2–0.4 Landau–de Gennes coefﬁcient −2 (Nm ) c 0.8 Landau–de Gennes coefﬁcient −2 (Nm ) −2 a 0.4 LdG coefﬁcient in P equation (Nm ) −2 c 0.8 LdG coefﬁcient in P equation (Nm ) Parameters for bacterial transport l 5 Bacterial length (μm) −1 V 15 Bacterial speed (μms ) τ 50–100 Reversal time (s) τ 1 Bacteria relaxation time (s) 2 −1 D 200–300 Concentration diffusion (μm s ) 2 −1 D 100–200 Rotational diffusion (μm s ) Λ 187.5 Dipolar strength (pN μm) ⎡kN = ⎤ sN =- ⎢ mod(( 2qq )+p, 2p)-p⎥. (A.16) å mod()kk +1,4 4p ⎣ ⎦ k=1 For isotropic tactoid, the rectangular contour was used (see the red rectangle in ﬁgure 1(a)). For the nematic tactoid, the circular region was used with radius R=0.99R , where R is the tactoid’s radius (see ﬁgure 3(a)). 0 0 A.4. Scaling of model parameters All the lengths are normalized by the bacterial length l=5 μm. Velocities are normalized by the bacterial −1 −3 velocity V =15 μms . Time is normalized by l/V . Concentrations are normalized by l . Viscosities are 0 0 −1 −1 2 normalized by η=0.5 kg m s . The Landau–de Gennes coefﬁcients a and c are normalized by K/l , where K=10 pN is the average elasticity constant. Correspondingly, the diffusion coefﬁcient D is normalized by lV , c 0 2 2 viscous friction coefﬁcient ζ by η/l , the dipole strengthΛ by ηl V ,Γ by 1/η, wall anchoring strength ξ 0 exter normalized by V /l,I–N anchoring strength ξ by V l. The Ericksen number is Er=ηlV /K=3.75. The 0 anch 0 0 parameters used in our model are listed in table A1. Appendix B. Additional integration results The results for the isotropic tactoid in nematic medium with homeotropic I–N anchoring are shown in ﬁgure B1. Here we placed the tactoid along the y-axis. The dependencies of the tactoid topological charge and relative concentration difference on the wall anchoring, bacteria concentration and the size of the tactoid are shown in ﬁgures B1(C)–(F). The charge magnitude grows with the bacterial concentration and tactoid’s size in agreement with the Fokker–Plank model. However, the charge magnitude decreases with the wall anchoring strength ξ . exter This can be explained by the aligning effect of the wall anchoring: topological defects predominantly move in the direction normal to the wall anchoring or parallel to the y-axis (see ﬁgure B1(B)). As a result, the effective area of the tactoid that is exposed to the topological defects becomes smaller for strong wall anchoring, in contrast to the case with planar anchoring, where we placed the tactoid along the x-axis (see ﬁgure 1). The dependence of the tactoid’s topological charge and relative concentration difference for the nematic tactoids in isotropic medium with homeotropic I–N anchoring is shown in ﬁgure B2. Topological charge magnitude grows with bacterial concentration and tactoid’s size in agreement with the Fokker–Plank model. 12 New J. Phys. 20 (2018) 043027 M M Genkin et al Figure B1. Isotropic tactoid in a nematic medium with homeotropic I–N anchoring. Here we placed the tactoid parallel to the y-axis. Nematic ﬁelds for mean bacteria concentration c = 0.4 (A) and c = 0.6 (B). Tactoid size (the length of the major axis) is L =180, 0 0 tact wall anchoring strength is ξ =0.02. The tactoid topological chargeáñ s (C) and relative concentration differenceác ñ (E) versus exter t t wall anchoring strength ξ and mean bacteria concentration c . Here the tactoid size is ﬁxed L =180. The tactoid chargeáñ s (D) exter 0 tact t and relative concentration differenceác ñ (F) versus tactoid size L and mean bacteria concentration c . Here the wall anchoring t tact 0 strength is ﬁxed ξ =0.02. exter Figure B2. Nematic tactoids in isotropic medium with homeotropic I–N anchoring. The dependence of the tactoid topological charge (A) and relative concentration difference (B) on the tacoid size and mean bacteria concentration. 13 New J. Phys. 20 (2018) 043027 M M Genkin et al Figure B3. Isotropic tactoids in a nematic medium with planar I–N anchoring. The dependence of the tactoid topological charge (A) and relative concentration difference (B) on the I–N anchoring strength and mean bacteria concentration. Figure B4. Isotropic rectangular tactoids that span the entire horizontal direction. Illustration of a nematic ﬁeld (A) and the dependence of the tactoid topological charge on the vertical size of tactoid (B) for three different bacteria concentrations. Relative concentration difference is positive (since the I–N anchoring is homeotropic) with its maximum attained for small bacteria concentrations and large sizes. Next, we investigated the dependence of the tactoid topological charge and relative concentration difference on the strength of I–N anchoring ξ . The relevant dependencies are shown in ﬁgure B3 for the isotropic anch tactoids with planar anchoring. One sees that the tactoid topological charge slightly increases in magnitude with the I–N anchoring strength. Relative concentration difference also depends on the I–N anchoring strength, but the effect is relatively weak too. These dependencies indicate that the energy barrier on the I–N interface arises not exclusively due to the I–N anchoring, but also due to the gradients of the OP’s amplitude. In addition, we tested our prediction against rectangular tactoids that span along the entire horizontal direction (see ﬁgure B4(A)). We varied the tactoid width (vertical size), keeping the tactoid surface area constant, and gathered the statistics for three different bacterial concentrations. The dependence of the tactoid topological charge on the tactoid vertical size is shown in ﬁgure B4(B). One sees that for sufﬁciently large tactoid sizes, the topological charge practically does not depend on the tactoid width. Appendix C. Fokker–Plank model To support our hypothesis, we developed a stationary 1D Fokker–Plank description. Consider an 1D system xL Î [0, ], occupied by nematic phase. We place an isotropic tactoid to the left of the nematic place with a thin I–N interface of the width δ located at x Î[] 0, d (see ﬁgure C1). The system is populated by positive and negative defects that can reside either in the nematic phase or on the I–N interface. The total numbers of positive + − + − (N ) and negative (N ) defects are the same: N =N =N. We assume positive defects are more mobile (since+1/2 defects move approximately 5 times faster than−1/2 defects). We also introduce a potential barrier that prevents defects from escaping the tactoid, which is deﬁned by a step function: 14 New J. Phys. 20 (2018) 043027 M M Genkin et al Figure C1. Illustration of the 1D system described by the Fokker–Plank model. 0, for 0<< x d, Fx () = ⎨ (C.1) ⎩Ax ,for >L , where A>0 is the strength of the barrier. We assume that defects interact similarly to point electrical charges (those with the same topological charge repel, those with different attract) and the interaction strength decays with distance as 1/x. Thus, the drift consists of two contributions: one due to defect interaction and another due to a barrier on the tactoid’s surface. The corresponding Langevine equations for each individual defect can be written in the following form: ⎛ N N ⎞ ++ + ⎜ ⎟ ¶=xF¢()x +mx - + , ii åå ⎜ -+ ++ ⎟ xx - x -x ⎝ j=00 ji jj=¹, i ji ⎠ ⎛ ⎞ N N -- ⎜ ⎟ ¶=xF ()x +mx - +,C(.2) t åå ii ⎜ +- --⎟ xx - x -x ⎝ ji ji ⎠ j=00jj=¹, i ± ± where μ is the interaction strength, ξ are random forces with the magnitude D . We introduce the probability + − density distribution functions of positive and negative defects (P (x, t) and P (x, t) correspondingly). The sums in equation (C.2) can be written as integrals over P : BB = P()x dx. As a result, we obtain a coupled i ò Fokker–Plank equations for the probability density distribution functions of positive and negative defects (P (x, t) and P (x, t) correspondingly): L+- ⎡ ⎤ ⎛ ⎞ Px()¢-,, t P () x¢ t ++ + + ¶= Px() ,t D¶ P-¶ ⎜-F¢(x)+ m d, xP ¢⎟ () x t , ⎢ ⎥ txx x ⎝ ⎠ ⎣ 0 xx-¢ ⎦ ⎡ L-+ ⎤ ⎛ ⎞ Pxt ()¢-,, P () xt¢ -- - - ¶= Px() ,t D¶ P-¶ ⎜-F¢(x)+ m d, xP ¢⎟ () x t . (C.3) ⎢ ⎥ txx x ⎝ ⎠ ⎣ 0 xx-¢ ⎦ In order to simplify the equations, we assume D is very large, so that positive defects are distributed uniformly: P (x, t)=P . Our goal is to ﬁnd a stationary probability distribution function for negative defects: ⎡ L ⎤ ⎛ ⎞ Px()¢-P -- - ¶- ⎜ Fx¢()+ m d. xP¢= ⎟ ()x D¶ P ()x (C.4) x⎢ ⎥ xx ⎣⎝ ⎠ ⎦ 0 xx-¢ We integrate equation (C.4) over x to obtain: L - ⎛ ⎞ Px()¢-P -- - ⎜-¢Fx ()+ m d. xP¢= ⎟ ()x D¶ P ()x+C (C.5) ⎝ ⎠ 0 xx-¢ In order to ﬁnd the constant C, consider the limit L¥. For x¥, the ﬁrst three terms in equation (C.5) go to zero: the barrier is constant for x>δ, P() xL =P (since we always have the same amount of negative 1 0 and positive defects), so that¶ P = 0. The interaction term is also zero, since at large x′the numerator is zero, and at small x′the denominator is very large. Thus, C=0, and we obtain: L - ⎛ ⎞ Pxt ()¢- , P -- - ⎜-¢Fx ()+ m d, xP¢= ⎟ () x t D¶ P. (C.6) ⎝ ⎠ 0 xx-¢ Let us denote: Pxt ()¢- , P Ix () = d. x¢ () C.7 int 0 xx-¢ Then our equation can be written in a compact form: -- - (-¢Fx ( )+ mI (x))P ()x = D ¶ P ()x.C (.8) int x 15 New J. Phys. 20 (2018) 043027 M M Genkin et al This equation can be integrated again: -+ (( Fx) m I (x)dx) D - int Px () =Ce. (C.9) Here C is the arbitrary integration constant. We can interchange the integration order and integrate the interaction term I (x) over x: int ⎛ ⎞ ∣∣ xx-¢ Ix ()dl x=¢ (P (x )-P )n ⎜ ⎟dx¢. ()C.10 int 0 òò ⎝ ⎠ 0 L Note that we omit the integration constant, as long as we already have one in equation (C.9). We assume a small interaction strength: μ=òμ , where 1. Thus, in equation (C.9) we can write the exponent of a sum as a product of two exponents and expand the second one in Maclaurin series. We also expand the constant CC=+ C+ .... As a result, we obtain: ⎛ ⎞ m ⎛ ⎞ - ∣∣ xx-¢ --Fx () D - 2 Px ()=+ (C C )e1- (Px (¢)-P )ln ⎜ ⎟dxO ¢+ () . (C.11) ⎜ ⎟ 01 0 ⎝ ⎠ ⎝ D 0 L ⎠ - -- We solve equation (C.11) using the asymptotic expansion P ()xP=+ ()x P ()x+ .... Substituting this into equation (C.11), we immediately obtain the solution for the zeroth order: -Fx () D Px () =Ce. (C.12) This solution describes the defect distribution in the system without interaction. Constant C can be found from the normalization condition, if we know the mean defect density P . Using equation (C.1), we get: 1 1 - - -AD P== Px ()dx (CCdd+ (L- )e,) (C.13) 0 00 L L so that: AD PLe C = .C().14 AD L-+dde Thus, in the zeroth order: (( AF - x)) D PLe Px () = .C (.15) 0 - AD L-+dde From equation (C.11) we can obtain O(ò) equations: (( AF - x)) D (( AF-¢x)) D L ⎛ ⎞ m PLe ⎛ ⎞ - Le ∣∣ xx-¢ - 1 0 -Fx () D Px ()=- C e - 1ln ⎜ ⎟dx¢. () C.16 ⎜ ⎟ 1 1 - - - AD AD ⎝ ⎠ DL() -+dde 0 ⎝ L-+dde ⎠ L Let us consider the second term in equation (C.16). We can represent the integral as a sum of two integrals: one from 0 to δ, another from δ to L. Let us denote these two integrals as r (x) and r (x): 1 2 AD ⎛ ⎞ ⎛ ⎞ Le ∣∣ xx-¢ ⎜ ⎟ rx () º ⎜ -1l ⎟ n dx¢, ()C.17 AD ò ⎝ ⎠ ⎝ ⎠ 0 L-+dde L ⎛ ⎞ ⎛ ⎞ L ∣∣ xx-¢ ⎜ ⎟ rx () º ⎜ -1l ⎟ n dx¢. ()C.18 AD ò ⎝ ⎠ ⎝ ⎠ L-+dde L We can calculate the integrals, and as result, we obtain: x ∣∣ x-d AD () e1--(Lx dd)-+log +(d-x)log () L L rx () = ,C().19 1 - AD L-+dde AD () e1 - d ⎛ Lx - ∣∣ x - d ⎞ ⎜⎟ rx ()=- -+LLdd +() -x log +-(x )log . () C.20 AD ⎝ ⎠ L-+dde L L Substituting everything into equation (C.16), we obtain the P : m PL --- - 1 0 -Fx () D (A-Fx ( )) D A D PC=-ee ()e -1 1 1 x Lx--∣∣ x d xL()--dd ln (L- x) ln - L(x-d)ln L L L () C.21 AD 2 () L-+dde 16 New J. Phys. 20 (2018) 043027 M M Genkin et al Figure D1. Nematic-isotropic phase coexistence with thin I–N interface. In order to ﬁnd C , one has to use the normalization condition for the function P : 1 1 Px ()d0 x = . (C.22) Substituting P from (C.21) into (C.22), we derive the expression for C : m PL -- AD AD C=- ee()1 2D -- L-- d L d 22 AD AD dd((LL --e2 -d))ln -L(L--d e d)ln d L .C().23 AD 3 () L-+dde - -- - - Thus, our solution up to the ﬁrst order of ò is P =+PP , where P is given by (C.15) and P is given 01 0 1 by (C.21), (C.23). We can ﬁnd the mean tactoid charge density using the following formula: sP=-((Px))d.x (C.24) d 0 Substituting P , after some algebra, we obtain: AD PL()-- d(e1) s=- AD L-+dde LL-- d d ⎛ - ⎞ 2 AD dlog - Llog m LP e 1 d L ⎜ ⎟ ´- 1 .C().25 ⎜ AD 2 ⎟ D () L-+dd e ⎝ ⎠ Note that for ﬁnite tactoid in an inﬁnite medium L d, the interaction term diverges: AD ⎛ ⎞ m P e - L AD sP »-() e1 - 1- d log . (C.26) 0 ⎜ ⎟ ⎝ D d ⎠ Equation (C.26) still holds for large L provided that the interaction strength μ is sufﬁciently small. It predicts parabolic dependence of the tactoid’s charge density on defect density P . In addition, in this model we did not include the tactoid size explicitly (since topological defects can only exist in the nematic medium or on the I–N interface). Therefore, the charge density is independent of the tactoid size. Appendix D. Relation to Rapini–Papoular approach It is possible to show that our dynamical treatment of the I–N anchoring is equivalent to the conventional Rapini–Papoular approach, where the additional I–N energy is introduced into the expression for the elastic energy [43]. Meanwhile, we can justify that the normal to the I–N interface is parallel to the gradient of the OP’s amplitude ∣Q∣for smooth interfaces. Ignoring the isotropic tension (as long as a tactoid’s area is nearly constant), the anisotropic tension in the Rapini–Papoular approach is of the following form: sq=- w()q,D(.1) where w is the strength of the anisotropic anchoring, θ is an easy angle (the angle between the normal to the interface and the easy direction), and θ is the nematic angle. Consider a steady-state regime with zero ﬂuid velocity and without the wall anchoring. Under these assumptions, equation (A.10) can be written in the following form: 17 New J. Phys. 20 (2018) 043027 M M Genkin et al +na()x,, ync- n∣n∣ + x ()x yinIm(n*F)=0. (D.2) Here a(x, y) and c are Landau–de Gennes coefﬁcients, where the coefﬁcient a(x, y) is set according to (7). The factor Er/Γ is absorbed in the anchoring strength, where we deﬁnedx=G x Er . Consider the quasi 1D tanch problem with a nematic phase in the region x<0 and isotropic phase in the region x>0. Let us denote the interface shape with function ζ(y)(see ﬁgure D1). According to our deﬁnition of the coefﬁcient a, for the given interface shape: a() xy , =- a(x z(y)). Equation (D.2) can be written as two scalar equations for amplitude q iq and phase θ, if we substitute n = qe and separate the real and the imaginary parts: 23 2 2 +qa (x-zq (y))qb- q -q ((¶ ) + (¶q) )= 0, (D.3) xy 2 22 qq +qq20 () +qq +xq(q)(q -q)=. (D.4) y x 0 y x t In the last term of equation (D.4) there is an additional factor (q) that comes from the deﬁnition of the easy direction f . In addition, we omitted the step function () xy , since here we only consider the I–N interface and replaced with θ −θ assuming that this difference is small. sin() qq - 0 0 Here we make two main assumptions: (i) thin and smooth I–N interface:dzz ¢()yy (), where δ is the interface thickness, and (ii) spatial derivatives of the nematic angle are much smaller than those of the OP’s amplitude»qqdq . Assumption (i) is valid as long as we deﬁne tactoids with a smooth interface. Assumption (ii) is valid as long as the deviations of the nematic angle from the easy angle on the I–N interface are small. Assume that the solution for the unperturbed interface ζ(y)=0 is known: q(x)=G(x), θ(x)=0. The solution G(x) is constant in the nematic phase, zero in the isotropic phase and decays quickly in a narrow I–N interface region. Let us now consider a perturbation to the interface shape ζ(y). With the assumptions (i) and (ii), the solution of equation (D.3) is n(x, y)=G(x−ζ(y)). In this case normal to the interface vector - (xyzz ( ))= (1,- ¢(y)) is parallel to the gradient of OP’s amplitude- Gx ( zz (y))= G¢(1,-¢(y)). This justiﬁes that the normal to the interface is directed parallel to the OP’s amplitude gradient. Next, consider equation (D.4) and neglect all the y-derivatives (using assumption (i)): qqqq++20 xqq()q-q=. (D.5) xx x 0 x x Here we argue that the ﬁrst term governs the dynamics in nematic phase (elasticity), whether the second and third are the leading terms on the I–N interface since the second term contains q that is much greater than any θ derivatives and the third term counterbalances it. Thus, we neglect the ﬁrst term and integrate equation (D.5) over the width of the I–N interface: d d 2 d ((2d qq qx+-q qq))x=2qq∣-2 qqdx+x(qq-). (D.6) x 0 xx 0 x 0 x t t òò x 0 0 d Here integration by parts has been used. In the third term of the right-hand side, we neglected the numerical 2 2 factors when integrating q q and assumed q » q d under the integral sign. On the right-hand side of this x x expression, we can neglect the second term, as this integral is proportional to δ (note, that this term would have been generated by the ﬁrst term in equation (D.5), that we have already neglected). Thus we obtained that on the I–N interface: 20 qqx+-() qq=. (D.7) x 0 This equation is similar to the boundary conditions that one obtains when minimizing the nematic free energy with Rapini–Papoular contribution (see chapter 5 in [57]): ¶ff d FO s += 0, () D.8 ¶¢qq d where f and f are the elastic and I–N surface energy densities correspondingly. Plugging in f =¢ K() q 2, FO s FO f=- w() qq 2, we obtain: Kwqf ¢+ () -q = 0. (D.9) By comparing (D.7) and (D.9) and using ,we ﬁnd that: x=G x Er tanch 2G d x =.D().10 anch Erq L Here L =K/w is the anchoring extrapolation length. 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New Journal of Physics – IOP Publishing
Published: Apr 1, 2018
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