Collective diffusion of a colloidal particles in a liquid crystal
aa r X i v : . [ c ond - m a t . s o f t ] D ec Collective diffusion of a colloidal particles in a liquid crystal.
B. I. Lev and A. G. Zagorodny
Bogolyubov Institute for Theoretical Physics National Academy ofScience of Ukraine, Metrologichna Str. 14-b, 03680 Kyiv, Ukraine (Dated: July 28, 2018)The collective diffusion effects in system of a colloidal particles in a liquid crystal has beenproposed. In this article described peculiarity of collective diffusion colloidal particles in a liquidcrystal, which can be observe experimentally. The diffusion coefficient should be crucial dependencefrom temperature and concentration of particles. In system colloidal particles arise from the elasticdistortion of the elastic director field the inter-particle interaction. These interaction can causenontrivial collective behavior, as the results the non-monotonic dependence collective diffusion. Arepredicted a nontrivial behavior collective diffusion of colloidal particles in a liquid crystal.
PACS numbers: 23.23.+x, 56.65.Dy
Colloidal particles in liquid crystals have attracted agreat research interest during the last years. Anisotropicproperties of the host fluid - liquid crystal give rise toa new class of a colloidal anisotropic interactions thatnever occurs in isotropic hosts. Liquid crystal colloidalsystem have much recent interest as models for diversephenomena in condense matter physics. The anisotropicinteractions are result an different structures of colloidalparticles such as linear chains in inverter nematic emul-sions [1, 4], 2D crystals [5] and 2D hexagonal structuresat nematic-air interface [6]. The authors of [7] observed3D crystal structures in the system of hard particles withdipole configuration deformation of the director field.Recently [8] was observed the anomalous diffusion sep-arate colloidal particle occurs at time scales that corre-spond to the relaxation times of director deformationsaround the particle. Once the nematic melts, the dif-fusion becomes normal and isotropic. Was shows thatthe deformations and fluctuations of the elastic directorfield profoundly influence diffusive regimes and was ob-tain as states are respectively termed sub-diffusion andsuper-diffusion.In this article will be investigate the collective behaviorof the diffusion that the results particles interaction withformation new spatially nonuniform distribution [9], [10],[11]. Will be study the collective behavior the system col-loidal particles. Nontrivial behavior collective diffusionof colloidal particles in liquid crystal has been predicted.Obtained results are in the temperature dependence ofdiffusion coefficient at various concentration, which havenon monotonic character and transparent the new phys-ical behavior the colloidal liquid crystal with taking intoaccount the interaction between particles.For colloidal particles in solution, collective also calledcooperative or mutual diffusion coefficient [12]- [15] is of-ten determined experimentally with the help of dynamiclight scattering. If one extrapolates this diffusion coeffi-cient to a vanishing concentration of particles, it reducesto the single-particle diffusion coefficient since the inter-actions between the particles are presumably negligiblethen. At various concentration, particle interactions in-fluence on the diffusion. At low enough concentrations, where many body interactions may be disregarded char-acter of concentration will be play crucial role in the de-termination of the diffusion coefficient.Complex investigation phase transition, collective dif-fusion, an auto-correlation function in this case allow thegive the new physical picture collective behavior colloidmedia which a based on the liquid crystal. The criticalnon-monotonic engagement of both the temperature de-pendence viscus-elastic response functions is found to farmore pronounces than for particle systems as a resultsof long-ranged interaction between the colloidal parti-cles by induce the deformation elastic director field. Thetemperature and concentration dependence viscous elas-tic response of a suspension of spherical colloid particlesin the vicinity of the isotropic- nematic is analyzed inthe men-field region. Explicit for the share rate temper-ature and concentration dependence of the static struc-ture factor are derived. Macroscopic expression for theanomalous part of the collective diffusion coefficient arederived, which are then expression as behavior the onethe particle with the static structure factor with the elas-tic deformation director field. The temperature depen-dence of the drag force in system colloidal particles aremotivation the nontrivial behavior collective diffusion ofcolloidal particles in liquid crystal.Colloidal liquid crystal are characteristically systemswith structure and time scales such that typical shearrate can bring them out of equilibrium and into some ex-otic states. Here we focus our attention on the diffusionmotion the particle which are format this state. The rhe-ological properties of liquid crystal and the structure ofphase-separating (size, shape of patterns and their spa-tial distribution ) strongly affect the physical propertiesof the resulting colloids. These aspects have been exten-sive studied recently, and there has been a considerableprogress in our understanding of the relation betweenstructures and rheological properties. It is expected thatthe breakdown of the of the rotation symmetry may in-duce optical and mechanical properties. The flow of aliquid crustal around a particle not only depends on itsshape and the viscosity coefficients but also on the direc-tion of the molecules [8]. The estimates of particle mo-bility in the liquid crystal, which were derived from thoseof viscous friction for a size Stokes sphere, give much notright values compared with relevant experimental date.One the possible reasons for such discrepancy may bean increase of the effective mass of an particles due for-mation of the deformation coat around it which movingtogether with the particle. This formation elastic defor-mation coat troth distribution director field around par-ticle increase the friction drag on particle moving in aliquid crystal. In other case the particle which is foreignin liquid crystal interacting with another similar particle[1]-[9]. The physical mechanism of this interaction is thatthe particle distorts the director distribution that can oc-cupy a region much greater than its dimensions and thusprovide an effective interaction with another similar par-ticle via mediation of the elastic field deformation. Therole this interaction between the colloidal particles is twoford. First of all, it enters the particle expression for theviscous-elastic response function. These macroscopic ex-pression are ensemble averages of phase function, amongwhich are the elastic interaction function. These long-range interaction function are responsible for the strongdivergence of the viscous elastic response function for col-loids as compared to states of the all systems. Second,the assemble average that represents the viscous elasticresponse function must be evaluated with respect to theshear rate distorted pair-correlation function. The shearrate dependence of this probability density function is theresult of an interplay between equilibrium restoring forcesand shear forces. The anomalous contribution is thatpart of the viscous elastic response function that divergesat the critical point due to the development of long-rangcorrelation induced by particle interaction truth field di-rector deformation. In this time, many investigationshave been carried out of the rheological properties of col-loidal liquid crystal suspension [16]-[20]. Therefore, acomparison between our results and those would only bepossible for the Newtonian viscosity, except for the essen-tial difference in the interaction between particles andliquid crystal, and particles interaction truth deforma-tion elastic field distribution director in the liquid crys-tal. A macroscopic evaluation of the viscous elastic con-sists of the steps : a) the first step realization formationmesophase with formation of a deformation coat ”solvateshell” around usual particle; b) the second step consist indescription influence in the particle interaction inducedby elastic deformation the director field; c) the last stepconsist the description the probability phase transitionin system particle wish accompanied the new structureformation, which observed experimentally. The approachproposed in this article makes it possible describe the col-lective diffusion in liquid crystal colloid by all area the de-termination temperature and concentration the matter.We use concept of the effective mass and friction dragof colloidal particle which moving in liquid crystal. Asshown [8], along with an particle traveling in liquid crys-tal, there also moves a director deformation field linkedwith extremely weak anchoring on the surface. The first one deals with spherical particle that have astrong anchoring strength on the surface [21], [24]. It cre-ates topological defect in the vicinity of the particle whichare necessary to satisfy the topological global boundaryconditions. A particle with strong planar anchoring cre-ates a pair of topological defects, known as boojums . Aparticle with strong homeotropic boundary conditions,on the other hand, are to create an equatorial disclina-tion ring or a hyperbolic hedgehog as a companion forthe radial hedgehog on the surface of the particle. Usingthe variational techniques and an electrostatic analogy,Lubensky et al. [21] obtained an approximate directordistribution near the particle with normal boundary con-ditions, as well as the long range pair interaction poten-tial between the particles.The second approach was proposed in [9], wherethe authors have examined the case of weak anchoringstrength for particles of a general shape. They have foundanalytically the pair interaction potential, taking into ac-count the different Frank constants and have expressedthe potential in terms of a tensor expressing of the shapeof the particle. In the article [11] was argue that thelong range interaction potential between particles in liq-uid crystal is determined by symmetry breaking of thedirector field in the vicinity of the particles. This sym-metry breaking is caused by two reasons: the shape of theparticle and the anchoring strength. In the case of weakanchoring it is determined primarily by the form of theparticle. In the case of the strong anchoring, on the otherhand, both factors are essential, because the director dis-tribution near the particle in this case is determined bytopological defects in its vicinity. In order to universallydescribe all these phenomena, was introduce the conceptof the deformation coat around the particle. The defor-mation coat embraces all the accompanying topologicaldefects, white it has the same symmetry as the resultantdirector field near the particle. The director distribu-tion outside the coat undergoes only smooth variationsand does not contain any topological defects. This col-loidal particles may also be regarded as micro particlesurrounded by a ”solvate shell” provided the interactionbetween such a particle and the molecules is much moreintense than the intermolecular interaction responsiblefor the liquid crystal formation. The solvable formationmay be regarded as a macro-particle, thus its interactionwith another similar formation can be described in termof the director field deformation. Taking into account thedirector distribution around separate particle, one canfind the change of orientation of the director induced bytwo particles, and determine the change of deformationenergy at the approach of those with separation of theenergy component corresponding to interaction betweenthose.In the case then the interaction of the individual par-ticle with the director field deformation produced by theother particles is determined by the anchoring on the sur-face of this particle is describe in article [9]. When thenumber particles is small and the week anchoring on thesurface, which realize in our situation the colloidal parti-cles in liquid crystal, the distribution the director field isdetermined in papers [17]. In next we briefly describe thisapproach. Strong anchoring directly implies that close tothe colloidal surfaces significant spatial variations of thedirector and even defects can appear, meaning that as-sumption of a roughly uniform director n = ( n x , n y ,
1) isnot valid anymore throughout the whole liquid crystallinevolume. In order to explain subsequent speculations weconsider first the case of homogeneous liquid crystal withuniform director n and one particle immersed into it.Anchoring of the liquid crystal with the surface of theparticle deformations of the director field around the par-ticle so that director n ( R ) varies from point to point. Inthe one-constant approximation the total free energy ofthe system: F = K Z dV (cid:2) ( div n ) + ( rot n ) (cid:3) + I dSW ( ν · n ) (1)where W is anchoring strength coefficient, ν is the unitnormal vector, integration H dS is carried out on thesurface of the particle, K is the Frank elastic constant.Far from the particle director field variations are small n ( R ) ≈ ( n x , n y , | n µ | ≪ µ = x, y ) and bulk freeenergy has the form: F b,linear = K Z dV (cid:8) ( ∇ n x ) + ( ∇ n y ) (cid:9) (2)which brings Euler-Lagrange equation of Laplace type:∆ n µ = 0 (3)At large distances R in general case it can be expandedin multiples, n µ ( R ) = q µ R + p µ u R + 3 u : ˆ Q µ : u R (4)with u α = R α /R ; p µ u = p αµ u α , u : ˆ Q µ : u = Q αβµ u α u β ,and µ = ( x, y ). This is the most general expression forthe director field. We note that the multiple expressiondoes not depend on the anchoring strength. It is valid onfar distances for any anchoring, weak and strong, withouttopological defects or with them. Of course in order tofind multiple coefficients we need to solve the problem inthe near nonlinear area either with computer simulationor Ansatz functions. Let’s imagine that we have foundall multiple coefficients for the particular particle (forinstance with computer simulation). After this presenta-tion we can use both approach to the description energyinteraction independence from value anchoring strengthon the surface particle. For particles with strong an-choring it is the far-region, because of strong directordeformations in the near-region. But for particles withweak anchoring distortions are small elsewhere and themultiple expansion is applicable in the near-region too.The symmetry of the coat is equivalent to the brokensymmetry of the director in the vicinity. A way how to avoid strong deformations of the directorfield and incorporate them into an analytical descriptionis to introduce a “coat region” [11]. This region acts asan effective colloid which incorporates all strong defor-mations of the director field around the real particle andhas the same symmetry as the director distribution. Inarticle [22] the general paradigma of the elastic interac-tion between colloidal particles in nematic liquid crys-tal was proposed according to each every particle withstrong anchoring and radius R has three zones arounditself. The rst zone for r < . R is the zone of topo-logical defects, and for determination of the distributiondirector field must use the nonlinear equation; the secondzone at the approximate distance range 1 . R < r < R isthe zone where crossover from topological defects to themain multiple moment takes place. The last third zone- is the zone of the main multiple moment, where higherorder terms can neglect. It is possible simple explanationof these presentation. In order to understanding and bydefinition size of this zones we can propose next way.The task of finding a director distribution around thespherical particle we must minimization of Frank energyand take into account the boundary condition. This taskwas solved in article [23]. In spherical coordinate we cantake ~n ( ~r ) = (sin β sin γ, sin β cos γ, cos β ), where β -polarand γ - azimuthal angle. In this representation existazimuthal symmetry relatively z . The difference equationin one constant approximation takes the form : ∇ β ( r ) − sin 2 β ( r )2 r sin θ = 0 (5)and the boundary condition on the surface the particle (cid:18) ∂β∂r + βr r = R = − W K sinθ (cid:19) (6)have the general solution β = P k C k r k +1 P k ( cosθ ) where P k ( cosθ ) associated with Legendre polynomial. Theboundary condition selects the solution , which in thecase week anchoring ( RW/K ) ≤ β ( r ) = ( RW/ K ) R sin 2 θ/r (7)where W - anchoring energy and R - radius of spheri-cal particle. In the case small deformation of directorfield ( RW/K ) <
4. This relation determine the con-dition of week anchoring. In the case strong anchor-ing ( R W/K ) ≥ a . This solution also wasobtain in the article [23] and take the form β ( r ) = θ −
12 arctan sin 2 θ cos 2 θ + ( ar ) (8)Inside the the disclination ring the solution of distribu-tion of director field take the form β = (cid:0) ar (cid:1) sin 2 θ where a radius of ring, which can be obtain from minimum offree energy. Can estimate the radius of ring in expres-sion a = R . In order to that disclination ring disappeararound the spherical particle is necessary such anchoringas this solution must be solution as in case week anchor-ing. Therefore was content ourselves with the quality es-timate with crossover value W ∗ ∼ Ka R = KR . In thecase the disclination ring not appear around the sphericalparticle. In this approach was obtained the free energywhich introduce the one particle in liquid crystal [24],[21]. In the case week anchoring F ≈ W R K and in thecase strong anchoring F ≈ KR . The uniform direc-tor distribution far from the particle has zero topologicalcharge and so there should be another topological de-fect near the particle to compensate the hedgehog in thecenter. Obviously, the director configurations have differ-ent symmetry. The non equatorial disclination ring andthe pair of radial and hyperbolic hedgehogs break mir-ror symmetry in the horizontal plane, while equatorialdisclination ring (Saturn-ring) retains it. Authors of [24]have shown by Monte-Carlo simulations, that the config-uration with hyperbolic hedgehog has lower energy, thanSaturn-ring. It has been confirmed in [21] with help ofthe dipole Ansatz , that though the equatorial ring hassome metastability, its energy is higher, than of the dipoleconfiguration.We can introduce the deformation coat which includeall strong deformation of director field. This deforma-tion area is new “immersed particle” and we can use onlyself-consistence approach in the case week anchoring andsmall deformation of director field inside this new inclu-sion. We can estimate the size this deformation coat if wetake into account the all energy which introduce in liq-uid crystal own particle. For the spherical particle withradius R in the case strong anchoring free energy of de-formation can present in the form [21] F strong ≈ πKR .We assume that around the spherical particle exist theSaturn-ring disclination. The size of deformation areadenote as R ∗ . As next step we calculate the free energywhich can introduce the deformation area inside. Thisenergy can obtain if suppose that the inside this defor-mation area we have the case week anchoring an can tomake well use the distribution director field as in caseweek anchoring. This energy was obtain in the article[24] F week = π W c R ∗ K , where W c critical value of energy an-choring for the case different size of particle when outsidethe particle not appear peculiarity in the distribution ofdirector field. This critical value was obtain also in thearticle [24] and is W c = Ka R where a radius of Saturn-rind disclination around spherical particle. Inside the de-formation area exist the disclination ring . Free energy ofthis Saturn-ring disclination we can present in the form[21] F disc ≈ πKa (cid:0) π + 1 (cid:1) +8 πK ( R ∗ − a ) , where size ofSaturn-ring can be present in the previous form. For es-timation of size of deformation coat we can compare thefree energy which real create inclusion in the case stronganchoring to sums free energy which create deformationcoat outside and free energy Saturn-ring disclination in- side this coat. From relation F week + F disc = F strong thesize of deformation area take the form R ∗ ≈ R . Thisestimation was made in the case Saturn-ring disclinationinside the deformation coat but this approach can be usein the case different defect which can appear inside thiscoat. In the case dipole configuration in distribution di-rector field inside this are coerced estimation is correctto, but the deformation coat will have the asymmetricform. The size of deformation coat in all case have thesame approximate size.When the determined the size area the elastic defor-mation director, we have possibility study friction dragand inertial characteristic particle. The friction drag onparticle moving in nematic liquid crystal is determined asresults computer calculation in the article [17], [19]. Wefocus our attention on the effective mass usual particlewhich moving in liquid crystal. As shown earlier, alongwith an particle traveling in liquid crystal, there alsomoves a director deformation field linked with the surfaceanchoring on this particle. Thus, an particle moving in-side nematic liquid crystal the kinetic energy T = mu / u ≤ ˜ Rt , where t is a time inter-val during which the director transition to steady statetakes place relevant estimates of a spherical size ˜ R ofthe deformation coat. In the framework of conception ofdeformation coat, the director distribution around an aparticle naturally determined as n = n ( t − R u ( t ′ ) dt ′ ) . Numerical estimation of the effective mass can be ob-tained if one known the director distribution around amoving particle. When take the distribution directorin the case week anchoring, which is description by [17]isotropic part effective mass the particle which movingin liquid crystal we arrive at the following as expression m eff = m + I ( W K ) R , according to results of the ar-ticles [18], where I is the density of the nematic liquidcrystal moment of the inertia.In the approach the Stokes-Einstein relation betweencoefficient diffusion end the mass Brownian particles D = kT / πµ ˜ R ( µ is the viscosity coefficient) we couldbe experimental date decrease the coefficient diffusion onthe second region decreasing temperature. In this regiontake the phase transition in the liquid crystal and for-mation the deformation coat around usual particle. Thisphenomena is possibility decreasing the mobility parti-cles which motivation the increasing friction drag. Thus,it becomes possible to attribute a decrease of the parti-cle mobility, observed under the transformation into theliquid crystal, to the deformation coating effect and, con-sequently, to the increasing effective mass of a particle.To describe the peculiarities of the particle system be-havior in the liquid crystal implies taking into accountinteraction via the director elastic field. We have alreadyshown that a foreign particle produces liquid crystal dis-tortion in a region much greater than the particle di-mensions and thus leads to an effective interaction withanother similar particle via the director field deforma-tion. In this sense, the interaction of the spherical parti-cle also is associated with the director elastic field defor-mation. The particle dispersed in the liquid crystal causelong ranged deformation of the director field. The self-consistent approach provides a possibility to avoid theabove difficulties, so we have managed to find the energyof the inter-particle interaction of particles introduced inthe liquid crystal. Having found the inter-particle inter-action energy, we can study the thermodynamic behaviorof an aggregate of such particles and describe the condi-tion for the creation of new structure [10, 11]. The char-acter and intensity of the inter-particle interaction in thesystem of foreign particles in liquid crystal can be suchthat a temperature and concentration phase transition inthe system and produce a spatially inhomogeneous distri-bution the particles [25]. The is first-order face transitionwhen the external field present itself the topological de-fect in nematic ordering. The type of interaction betweenthe colloidal particles is sufficiently long ranged to leadto a strong critical divergence of the shear viscosity. Itis therefore interesting to study the full temperature de-pendence of the shear viscosity of colloidal system nearthe critical point with transition in new phase and pe-culiarity when this transition accompanied the specialinhomogeneous distribution in system particles. The vis-cous elastic response function will turn out to be equalto two distinct additive contribution : an anomalous anda background contribution. The anomalous contribu-tion is that part of the viscous elastic response functionthat diverges at the critical point due to the developmentof long-range correlation. The background contributionmust be subtracted from experimental viscous elastic re-sponse function to obtain their observer anomalous con-tribution, which may then be compared to theoreticalprediction. Therefore a comparison between our resultsand those would only be possible for Newtonian viscos-ity, except for the essential difference in the inter-particlepotential. One of the pictorial structures is inadequateproperty to describe the substance. The distribution ofviscous elastic properties in the system interacting par-ticles is known in papers [12]-[15] sufficient detail. Wedesigner must fall back on methods of analysis the vis-cous elastic properties in colloidal suspension .As you are well aware, furthermore D eff ( ~k ) wave-vector-dependent effective diffusion coefficient equal to[12]-[15] D eff = DkT { dGd e ρ + q S } (9)where D is the usual diffusion coefficient spherical par-ticle which moving in condense matter. The first on theright-hand side of this equation describes shear flow dis-tortion the interacting Brownian particle, and have rep-resentation in the form G = e ρkT − π e ρ c (1 − c ) Z dr ′ r ′ dV ( r ′ ) dr ′ g eq ( r ′ ) (10)where e ρ is concentration of the media, c is friction pa-rameter particles which is foreign in liquid crystal, V ( r ′ )is the pair interaction energy between particles truth theelastic deformation director field, and g eq ( r ′ ) is the paircorrelation function. The last term in Eq. (8) describesthe diffusion limited tendency to restore the equilibriumstructure with S = 2 π e ρc Z dr ′ r ′ dV ( r ′ ) dr ′ { g eq ( r ′ ) + 18 e ρ (1 − c ) dg eq ( r ′ ) dr ′ } (11)Close to the critical point and also close to the of-criticalpart of the spinodal decomposition, which accompaniedthe first-order phase transition,, where DkT dGd e ρ is small,the effective diffusion coefficient is small for small wavevectors, a phenomenon that is commonly referred to ascritical slowing down.As we will be show, the general solution of diffusioncoefficient can be expressed in the term of explicit ex-pressions for structure factor for two more simple cases:the structure factor in the small inter-particle interac-tion truth the deformation director field of shear flowand under stationary shear flow in the present the inter-particle interaction with phase transition in spatial inho-mogeneous distribution this particles. The above equa-tion relate to the mean-field behavior of the structurefactor. This is due to transmutation the present relationto form; G = e ρkT + 2 π e ρ c (1 − c ) Z dr ′ r ′ V ( r ′ ) g eq ( r ′ ) − π e ρ c (1 − c ) Z dr ′ r ′ V ( r ′ ) dg ( r ′ ) dr ′ ) (12)The stiffness of the bond is lessened resulting in a lower-ing of dependence from temperature the effective diffu-sion coefficient in follows form: D eff = D { − T c − aT ) (13)where a = π e ρ c (1 − c ) R dr ′ r ′ V ( r ′ ) dg ( r ′ ) dr ′ , and T c = − π e ρ c (1 − c ) Z dr ′ r ′ V ( r ′ ) g eq ( r ′ ) (14)is the critical temperature first-order phase transi-tion,with accompaniment the formation spatial inhomo-geneous structure in the distribution in system particles.The relevance of the correlation length is that it measuresthe range over which colloidal particles in the unsortedsystem are correlated. Since dGd e ρ → ξ = q S/ dGd e ρ diverges. This means that at critical pointeach colloidal particle in system is correlated with allother colloidal particles. One may imagine that it willtake on finite force to break up these many correlation inorder to make the system flow, which means that the vis-cosity diverges on approach of the spinodal. The effectivecoefficient diffusion have the dependence of correlationlength in follow form D eff = Dξ − illustrate the decriesthe time relaxation from the needier the temperature thefirs-order phase transition.In order case, when the has to be taken into account allphysical processes, which motivation the structure tran-sition in our matter for this fact behavior the effectivediffusion coefficient maybe describe the follow physicalpicture: When the formation the liquid crustal, everyparticles dresses in to elastic deformation coat the inho-mogeneous distribution director field around the particle.This fact provide to increase effective mass every particlean increase the drag force motion usual particle and de-crease the effective coefficient diffusion. When the liquidcrystal are formation, generation the interaction betweenparticle by induced the director field deformation. Thisrepulsive interaction truth the elastic field deformationprovide to increase the collective effective diffusion coef-ficient. When the interaction in system particles moti-vate the firs-order phase transition which accompaniedthe inhomogeneous the distribution particle the fact arethe decrease the effective diffusion coefficient. This factare attributable to realization long range correlation insystem particle, which is foreign in liquid crystal. Basedon this physical picture and an experimental date thecomplete behavior the collective diffusion coefficient inour case is rendered possible by the addition of the ap-proximation formula: D eff = D (cid:2) − b ( T − T in ) (cid:3) { − T c − aT ) (15) where T in is temperature the phase transition our me-dia in liquid crystal state and b -coefficient which deter-mined dependence the effective mass(7) or friction dragthe moving particles which is foreign in liquid crustal.This relation describe the all region dependence the effec-tive collective diffusion coefficient from the temperatureand concentration particles in our media. The basic pat-tern of fracture can be produced at will by establishingappropriate experimental conditions. As conclusion, wecan note about the fact peculiarity behavior the effectivediffusion coefficient which take into account the dressesevery particles in to elastic deformation coat. This factprovide to increase effective mass every particle an in-crease the drag force motion usual particle and decreasethe effective coefficient diffusion. When the liquid crystalare formation, generation the interaction between parti-cle by induced the director field deformation. The inter-action truth the elastic field deformation provide to in-crease the collective effective diffusion coefficient. Whenthe interaction in system particles motivate the firs-orderphase transition the fact are the decrease the effectivediffusion coefficient. This fact are attributable to real-ization long range correlation in system particle. Basedon this physical picture is possible experimental observethe complete behavior the usual and collective diffusionprocess. Acknowledgement: this research was partiallyfunded by the State Fund for Fundamental Research ofUkraine (Project F 76/84). [1] P. Poulin, H. Stark, T. C. Lubensky and D. A. Weitz.
Science , 1770 (1997).[2] P. Poulin, V. Cabuil and D. A. Weitz.
Phys. Rev.Lett. , 4862 (1997).[3] P. Poulin, V. A. Raghunathan, P. Richetti and D. Roux. J.Phys. I France , 1557 (1994).[4] P. Poulin and D. A. Weitz. Phys.Rev. E , 626 (1998).[5] I. Muˇ s evic, M. ˇ S karabot, U. Tkalec, M. Ravnik and S.ˇ Z umer Science , 954, (2006).[6] V. Nazarenko, A. Nych and B. Lev Phys.Rev.Lett , ,13 August (2001).[7] A. Nych, U. Ognysta, M. Skarabot, M. Ravnik, S. Zumerand I. Musevic, Nature Communications , 1489 (2013).[8] T. Turiv, I. Lazo, A. Brodin, B. I. Lev, V. Reiffenrath,V. G. Nazarenko, O. D. Lavrentovich, Science , 1351(2013)[9] B. I. Lev and P. M. Tomchuk. Phys. Rev. E , 591(1999).[10] B. I. Lev, H. M. Aoki, H. Yokoyama Phys. Rev. E , Mol.Cryst. Liq. Cryst., 367, 537 (2001).[11] B. I. Lev, S. B. Chernyshuk, P. M. Tomchuk and H. Yokoyama, Phys. Rev E., ,(2002)[12] T. Ohtsuki and K. Okano, J. Chem. Rev. , , 1443 (1982)[13] Peter Prinsen and Theo Odijka, J. Chem. Rev. , ,115102 (2007)[14] J. K. Dhont and Gerhard Nagele, Phys. Rev. E
Phys. Rev. E
Phys. Rev.. E ,5561 (1997).[17] R. W. Ruhwandl and E. M. Terentjev. Phys. Rev. E ,5204 (1996)[18] E. D. Belockii, B. I. Lev and P. M. Tomchuk, , JETPLett. , 573 (1980); Mol. Crys. Liq. Crys. 213, 195(1993)[19] H. Stark, and D. Ventzki, Phys. Rev. E., 64, 031711(2001)[20] H. Stark, and D. Ventzki, Euro. Phys. Lett., 57, 60 (2003)[21] T. C. Lubensky, D. Pettey, N. Currier and H. Stark. Phys.Rev. E , 610 (1998).[22] S. B. Chernyshuk, Eur. Phys. J. E , 6 (2014). [23] O. V. Kuksenok, R. W. Ruhwandl, S. V. Shiyanovskiiand E. M. Terentjev. Phys. Rev. E , 5198 (1996).[24] E. M. Terentjev. Phys. Rev. E51