Integral equation theory for nematic fluids

The traditional formalism in liquid state theory based on the calculation of the pair distribution function is generalized and reviewed for nematic fluids. The considered approach is based on the solution of orientationally inhomogeneous Ornstein-Zernike equation in combination with the Triezenberg-...

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Дата:	2010
Автор:	Holovko, M.F.
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Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2010
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/32105
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Цитувати:	Integral equation theory for nematic fluids / M.F. Holovko // Condensed Matter Physics. — 2010. — Т. 13, № 3. — С. 33002:1-29. — Бібліогр.: 56 назв. — англ.

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spelling	irk-123456789-321052012-04-09T12:23:28Z Integral equation theory for nematic fluids Holovko, M.F. The traditional formalism in liquid state theory based on the calculation of the pair distribution function is generalized and reviewed for nematic fluids. The considered approach is based on the solution of orientationally inhomogeneous Ornstein-Zernike equation in combination with the Triezenberg-Zwanzig-Lovett-Mou-Buff-Wertheim equation. It is shown that such an approach correctly describes the behavior of correlation functions of anisotropic fluids connected with the presence of Goldstone modes in the ordered phase in the zero- eld limit. We focus on the discussions of analytical results obtained in collaboration with T.G. Sokolovska in the framework of the mean spherical approximation for Maier-Saupe nematogenic model. The phase behavior of this model is presented. It is found that in the nematic state the harmonics of the pair distribution function connected with the correlations of the director transverse fluctuations become long-range in the zero- eld limit. It is shown that such a behavior of distribution function of nematic fluid leads to dipole-like and quadrupole-like long-range asymptotes for effective interaction between colloids solved in nematic fluids, predicted before by phenomenological theories. Традиційний формалізм у теорії рідин, що базується на розрахунку парної функції розподілу, узагальнено на нематичні плини. Розглянутий підхід базується на розв'язку орієнтаційно-неоднорідного рівняння Орнштейна - Церніке в поєднанні з рівнянням Трайцінберга - Цванціга - Ловета - Моу - Бафа - Вертгайма. Показано, що даний підхід коректно описує поведінку кореляційних функцій анізотропних флюїдів, обумовлену наявністю голдстоунівських мод у впорядкованій фазі за відсутності впорядковуючого зовнішнього поля. Розглянуто аналітичні результати, одержані у співпраці з Т. Г. Соколовською в межах середньо-сферичного наближення для нематогенної моделі Майєра - Заупе. Представлено фазову діаграму цієї моделі. Встановлено, що в нематичному стані гармоніки парної кореляційної функції, пов'язані з кореляціями флуктуацій поперечних до напрямку директора, стають далекосяжними за відсутності впорядковуючого поля. Показано, що така поведінка функції розподілу нематичного флюїду призводить до дипольно- та квадрупольно-подібних далекосяжних асимптотик ефективної міжколоїдної взаємодії в нематичних флюїдах, передбаченої раніше феноменологічними теоріями. 2010 Article Integral equation theory for nematic fluids / M.F. Holovko // Condensed Matter Physics. — 2010. — Т. 13, № 3. — С. 33002:1-29. — Бібліогр.: 56 назв. — англ. 1607-324X PACS: 05.20.Jj, 05.70.Np, 61.20.-p, 68.03.-g http://dspace.nbuv.gov.ua/handle/123456789/32105 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The traditional formalism in liquid state theory based on the calculation of the pair distribution function is generalized and reviewed for nematic fluids. The considered approach is based on the solution of orientationally inhomogeneous Ornstein-Zernike equation in combination with the Triezenberg-Zwanzig-Lovett-Mou-Buff-Wertheim equation. It is shown that such an approach correctly describes the behavior of correlation functions of anisotropic fluids connected with the presence of Goldstone modes in the ordered phase in the zero- eld limit. We focus on the discussions of analytical results obtained in collaboration with T.G. Sokolovska in the framework of the mean spherical approximation for Maier-Saupe nematogenic model. The phase behavior of this model is presented. It is found that in the nematic state the harmonics of the pair distribution function connected with the correlations of the director transverse fluctuations become long-range in the zero- eld limit. It is shown that such a behavior of distribution function of nematic fluid leads to dipole-like and quadrupole-like long-range asymptotes for effective interaction between colloids solved in nematic fluids, predicted before by phenomenological theories.
format	Article
author	Holovko, M.F.
spellingShingle	Holovko, M.F. Integral equation theory for nematic fluids Condensed Matter Physics
author_facet	Holovko, M.F.
author_sort	Holovko, M.F.
title	Integral equation theory for nematic fluids
title_short	Integral equation theory for nematic fluids
title_full	Integral equation theory for nematic fluids
title_fullStr	Integral equation theory for nematic fluids
title_full_unstemmed	Integral equation theory for nematic fluids
title_sort	integral equation theory for nematic fluids
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/32105
citation_txt	Integral equation theory for nematic fluids / M.F. Holovko // Condensed Matter Physics. — 2010. — Т. 13, № 3. — С. 33002:1-29. — Бібліогр.: 56 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT holovkomf integralequationtheoryfornematicfluids
first_indexed	2025-07-03T12:37:09Z
last_indexed	2025-07-03T12:37:09Z
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fulltext	Condensed Matter Physics 2010, Vol. 13, No 3, 33002: 1–29 http://www.icmp.lviv.ua/journal Integral equation theory for nematic fluids M.F. Holovko Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine Received August 13, 2010 The traditional formalism in liquid state theory based on the calculation of the pair distribution function is generalized and reviewed for nematic fluids. The considered approach is based on the solution of orientation- ally inhomogeneous Ornstein-Zernike equation in combination with the Triezenberg-Zwanzig-Lovett-Mou-Buff- Wertheim equation. It is shown that such an approach correctly describes the behavior of correlation functions of anisotropic fluids connected with the presence of Goldstone modes in the ordered phase in the zero-field limit. We focus on the discussions of analytical results obtained in collaboration with T.G. Sokolovska in the framework of the mean spherical approximation for Maier-Saupe nematogenic model. The phase behavior of this model is presented. It is found that in the nematic state the harmonics of the pair distribution function con- nected with the correlations of the director transverse fluctuations become long-range in the zero-field limit. It is shown that such a behavior of distribution function of nematic fluid leads to dipole-like and quadrupole-like long-range asymptotes for effective interaction between colloids solved in nematic fluids, predicted before by phenomenological theories. Key words: pair distribution function, integral equation theory, Maier-Saupe nematogenic model, Goldstone modes, colloid-nematic mixture PACS: 05.20.Jj, 05.70.Np, 61.20.-p, 68.03.-g Introduction The pair distribution function g(12) plays central role in the modern fluid theory. It establishes a bridge between microscopic properties modeled by interparticle interactions and the macroscopic ones such as structural, thermodynamic, dielectric and other properties. For homogeneous fluids the integral equation methods have been intensively used in fluid theory during the last decades [1, 2]. This technique is based on the analytical or numerical calculation of the pair distribution function by the solution of the Ornestein-Zernike (OZ) equation within different closures: Percus- Yevick (PY), hypernetted chain (HNC), mean spherical approximation (MSA) and its different modifications. In the presence of an external field the fluid becomes inhomogeneous and is de- scribed by the singlet distribution function ρ(1) that appears instead of the bulk density of the homogeneous liquid [3]. The external field determines the symmetry of the singlet distribution function and its dependence on coordinates of the fixed molecule 1. The transition from homoge- neous to inhomogeneous state leads to the broken symmetry of the system. As a result, the pair distribution function of inhomogeneous fluids loses the uniform invariance and does not have the symmetry of the pair potential. Besides external fields, the inhomogeneity can also be caused by a change of the system symmetry as a result of phase transition. Such a typical situation takes place in the case of crystallization, where at certain values of the density the periodic singlet distribution function branches off the uniform one. Thus, in the inhomogeneous case, the OZ equation includes the singlet distribution function ρ(1) and, besides the closure for the OZ equation, an additional relation between singlet and pair distribution functions is needed [1–3]. There are at least two exact relations that can be used for this aim. It could be the first member from the hierarchy of the Bogolubov-Born-Green- Kirkwood-Yvon (BBGKY) equation [1, 4] or the Triezenberg-Zwanzig-Lovett-Mou-Buff-Wertheim (TZLMBW) equation [1, 5–7]. We should note that in accordance with Bogolubov’s idea about the c© M.F. Holovko 33002-1 http://www.icmp.lviv.ua/journal M.F. Holovko quasiaverages [8] for a correct treatment of the system with spontaneously broken symmetries, an external field of infinitely small value should be introduced in order to stabilize the system. For molecular fluids, the inhomogeneity can be caused by the broken rotational invariance in addition to the break of the translational invariance. Such a situation appears at the phase tran- sition from isotropic to nematic liquid crystal phase, when at certain thermodynamical conditions the orientational-dependent singlet distribution function branches off the isotropic one. Similarly to the crystallization case, the pair distribution function loses the translational invariant form as a result of the broken symmetry. It loses its rotationally invariant form in the nematic case. The change of the fluid from isotropic to nematic state in the absence of external fields induces col- lective fluctuations, which develops orientational wave excitations, the so-called Goldstone modes. This leads to the divergence of the corresponding harmonics of the pair distribution function in the limit of zero wave vector k. Recently much effort has been devoted to generalization of the integral equation theory to orientationally ordered (anisotropic) fluids. In order to investigate the properties of molecular fluids in the nematic phase some Ansatzes based on the construction of an effective isotropic state are used [9, 10]. Another description of the isotropic-nematic phase transition is connected with the application of the TZLMBW equation, with the assumption that the direct correlation function in the nematic phase can be approximated to it by the form which is reduced in the isotropic case, i.e., by its rotationally invariant form. This procedure was used by Lipszyc and Kloczkowski [11] and Zhong and Petschek [12, 13]. They made an attempt to calculate the single-particle distribution function and the pair distribution functions in a self-consistent way on the basis of the OZ equation and the so-called Ward identity. The Ward identity relates the singlet distribution function to an integral of the pair direct correlation function. Later Holovko and Sokolovska [14] showed that this is nothing else than the TZLMBW equation in the functional differential form. Treating the direct correlation function in the PY approximation as the effective potential, Zhong and Petschek [12, 13] supposed that the direct correlation function should be rotationally invariant just like the initial potential. In order to remake the PY closure in a rotationally invariant form, they used a procedure named in [15] as an unoriented nematic approximation. It was shown that with the modified PY closure, the Ward identity is implemented and yields an infinite susceptibility in the limit of zero wave vector for the Goldstone modes. However, Holovko and Sokolovska [14] showed that the requirement of the rotational invariance for correlation functions leads to an incorrect conclusion about the divergence of the nematic structure factor in the limit of zero wave vector. In contrast to the TZLMBW, the BBGKY equation does not reproduce the correct zero-field divergence in the transverse susceptibility of nematic fluids [16]. Thus, it seems better to build a theory using the TZLMBW equation instead of the BBGKY one. The generalization of the integral equation theory for orientationally inhomogeneous molecular fluids was formulated by Holovko and Sokolovska [14, 17]. In this approach the self-consistent solution of the OZ and TZLMBW equations are used for the calculation of the pair and single- particle distribution functions in nematics. The developed method does not impose any additional approximations other than a closure for the OZ equation. A principal point of this approach is the use of exact relations obtained from TZLMBW equation for the nematic phase. In accordance with the Bogolubov idea [8] there was introduced an external field of infinitely small value which fixes the orientation of the nematic director. It was shown that the application of TZLMBW equation provides a correct description of the Goldstone modes in full accordance with the fluctuation theory of de Gennes [18]. Only harmonics of the distribution function connected with correlations of the director transverse fluctuations have to diverge at κ = 0, the others being finite. The developed integral equation approach was applied to the hard sphere Maier-Sauper nematic model. There was obtained an analytical solution for this model in MSA approximation [14, 19], which was used for the description of phase behaviour of the considered model [20]. The properties of hard sphere Maier-Saupe model were also studied by numerical solution of orientationally inho- mogeneous OZ equation in PY, MSA, HNC and reference HNC approximations [16, 21, 22]. The considered integral equation theory was also applied to the investigation of a planar nematic fluid in the presence of a disoriented field [23–26]. The obtained analytical results in MSA approximation 33002-2 Integral equation theory for nematic fluids for hard sphere Maier-Saupe nematic model were used in Henderson-Abraham-Barker approach [1, 27] for the investigation of a nematic fluid near hard wall in the presence of orienting field [28, 29]. These results were applied to the investigations of colloidal interactions in nematic-colloid dispersions [30–35]. This paper reviews the recent studies of nematic fluids within the framework of integral equation theory for orientationally inhomogeneous molecular fluids. This review is devoted to the memory of Tatjana Sokolovska who passed away a year ago. The remainder of the paper is organized as follows. The general formulation of integral equation, the orientational expansions of the pair correlation and the singlet distribution functions are presented in the first section. The solution of the MSA for hard sphere Maier-Saupe nematic model in the presense and in the absence of an external field is discussed in the second section. In that section, thermodynamic properties and phase behavior of this model are discussed. In the third section the integral equation approach is used for the description of a nematic fluid near hard wall that interacts with a uniform orienting field. Some aspects of intercolloidal interactions in a nematic fluid are studied by integral equation theory and are also discussed in this section. 1. Integral equations for orientationally inhomogeneous fluids: general relations In this paper we consider a fluid of spherical particles with diameter σ having an orientation specified by the unit vector ω. The fluid is subject to an external ordering field of the form v(1) = −W2 √ 5P2(cosϑ1) with W2 > 0 (1.1) which favors an order parallel to the direction n, P2(cos ϑ) = 3 2 (cos2 ϑ − 1) – is a Legendre poly- nomial of second order, ϑ is the angle between vectors ω and n, 1 indicates both position r1 and orientation ω1 of the molecule. We confine that in the interactions between the fluid particles, the orientational component is essentially a Maier-Saupe term [36] and assume that the intermolecular potential v(12) can be presented in the form v(1, 2) = vh(r12) + v0(r12) + v2(r12, ω1, ω2) (1.2) where vh(r12) is the hard sphere potential vh(r) = { ∞, for r < σ, 0, for r > σ. (1.3) The long-range attraction has an isotropic part v0(r) = −A0 exp(−z0r) r (1.4) and an anisotropic part v2(r, ω1, ω2) = −A2 exp(−z2r) r P2(cosϑ12) (1.5) where P2(cos ϑ12) is the second Legendre polynomial, ϑ12 is the angle between the axes of molecules 1 and 2. The parameters z0, z2 and A0, A2 determine the range and the strength of the coupling interactions. The molecular Ornstein-Zernike equation for orientational inhomogeneous fluids can be written in the form [1–3] h(1, 2) = C(1, 2) + ∫ d3ρ(3)C(1, 3)h(3, 2) (1.6) where d3 = dr3dω3, h(12) = g(12)− 1 and C(12) are, respectively, the total and direct correlation functions. 33002-3 M.F. Holovko A some closure relation which relates the correlation functions C(12) and h(12) to the pair potential v(12) should be added. In this paper we will use MSA closure, according to which h(1, 2) = −1 for r12 < σ, (1.7) C(1, 2) = −βv(1, 2) for r12 > σ. (1.8) Condition (1.7) is exact for the considered model since g12 = 0 for r12 < σ. The condition (1.8) assumes that the long-range asymptote of C(12) = −βv(12) is correct for the whole intermolecular distance r > σ. For orientational inhomogeneous fluid ρ(1) = ρf(ω), where ρ is the number density of the ordered phase, f(ω) is a single particle distribution function which can be written in the form [37] f(w) = 1 Z exp(−βv(1) + C(1)) (1.9) where the constant Z can be found from the normalization condition ∫ f(ω)dω = 1, (1.10) β = 1 kBT , kB is the Boltzmann constant, T is the temperature, C(1) is the singlet direct correlation function, which is the first in the hierarchy of direct correlation functions. By using the functional differentiation technique we can define the total and the direct pair correlation functions as [37] − 1 β δρ(1) δv(2) = ρ(1)δ(1, 2) + ρ(1)ρ(2)h(1, 2) , (1.11) −β δv(1) δρ(2) = δ(1, 2) ρ(1) − C(1, 2) . (1.12) where δ(12) is the Dirac δ-function of all coordinates of the molecules 1 and 2. In accordance with (1.9) the second of these relations can be written in the form of Ward identity introduced by Zhong and Petschek [12, 13] δC(ω1) δρ(ω2) = ∫ dr12C(r12, ω1, ω2). (1.13) It is important to note that after the inclusion of an external field v(1), the system instead of a rotational invariance possesses a rotational covariance [8]. This means that the Hamiltonian and the average values, like correlation functions, are rotationally invariant if the external field and the molecules are simultaneously rotated. As a result of this symmetry, one gets ∇ω1 ρ(ω1) = ∫ dr12dω2 δρ(ω1) δv(ω2) ∇ω2 v(ω2), (1.14) ∇ω1 v(ω1) = ∫ dr12dω2 δv(ω1) δρ(ω2) ∇ω2 ρ(ω2) (1.15) where for the considered case of linear molecules [38] ∇∇∇ω = [̂r×∇∇∇] = −eϑ 1 sin ϑ ∂ ∂φ + eφ ∂ ∂ϑ (1.16) is the angular gradient operator, eω and eϕ are two orthogonal unit vectors perpendicular to the unit vector r̂. Combination of the relations (1.11)–(1.12) and (1.14)–(1.15) yields integro-differential equations for the singlet distribution function – the TZLMBW equations for spatially homogeneous but 33002-4 Integral equation theory for nematic fluids orientationally non-uniform systems β∇ω1 v(ω1) + ∇ω1 ln ρ(ω1) = −β ∫ dr12dω2h(r12, ω1, ω2)ρ(ω2)∇ω2 v(ω2), (1.17) β∇ω1 v(ω1) + ∇ω1 ln ρ(ω1) = ∫ dr12dω2C(r12, ω1, ω2)∇ω2 ρ(ω2). (1.18) From equation (1.17) it follows directly that to have a non-trivial solution for ρ(ω1) the integral ∫ dr12h(r12, ω1, ω2) should diverge in the limit v(ω) → 0+. The divergence signals the appearance of the Goldstone modes. The equation (1.18) in the zero-field limit can be written in the form ∇ω1 ln ρ(ω1) = ∫ C(ω1, ω2)∇ω2 ρ(ω2)dω2 (1.19) which is an integro-differential form of the Ward identity (1.13), C(ω1, ω2) = ∫ C(r, ω1, ω2)dr. The next step of the integral equation theory for molecular fluids is usually connected with spherical harmonics expansions for orientational dependent functions g(12) or h(12), c(12) and f(ω). Due to orientational inhomogeneity of the fluid the traditional orientational invariance tech- nique [1, 38] should be slightly modified [14]. In uniaxial fluids, the orientational distribution function f(ω) is axially symmetric around a preferred direction n and depends only on the angle ϑ between the molecular orientation ω and n. It allows us to write the relation (1.9) for f(ω) in the form f(ω) = 1 Z exp { ∑ l>0 BlYl0(ω) } (1.20) where the spherical harmonics Ylm(ω) satisfy the standard Condon-Shortey phase convention [38]. The nematic ordering is defined by the parameters Sl = 〈Pl(cosϑ)〉 = ∫ dωf(ω)Pl(cosϑ), (1.21) where Pl(cosϑ) = √ 1 2l+1Yl,0(ω) are the Legendre polynomials. In the space-fixed coordinate system with z-axis parallel to n the direct and total pair correlation functions can be written in the form f(r, ω1, ω2) = ∑ m,n,l µ,ν,λ fµνλ mnl(r)Ymµ(ω1)Y ∗ nν(ω2)Ylλ(ωr) (1.22) where f(r, ω1, ω2) = h(12) or C(12), r is a separation vector of molecular mass center, ωr being its orientation. Due to invariance of a uniaxial system with respect to rotations around z-axis, µ+λ = ν. Since the pair potential (1.2) is independent of orientation of the intermolecular separation vector r, the harmonic coefficients that survive in the expansion (1.22) have only l = λ = 0 and µ + ν = 0. This permits to attain notational simplification from six indexes to three. In the MSA for the considered model, the expansion (1.22) reduces to f(r, ω1, ω2) = f000(r) + f200(r) [ Y20(ω1) + Y20(ω2) ] + ∑ \|µ\|≤ 2 f22µ(r)Y2µ(ω1)Y ∗ 2µ(ω2). (1.23) It should be noted that for isotropic case f200(r) = 0. Due to the uniaxial symmetry of a nematic the OZ equations for harmonics with different values of µ decouple. In the MSA for µ 6= 0 harmonics, we have h22µ(r12) = C22µ(r12) + ρ〈Y 2 2µ(ω)〉ω ∫ C22µ(r13)h22µ(r32)dr3 (1.24) 33002-5 M.F. Holovko with the closure h22µ(r12) = 0, r12 < σ, (1.25) C22µ(r12) = β 1 5 A2 1 r12 e−z2r12 , r12 > σ, where 〈. . . 〉ω = ∫ f(ω)(. . . )dω. The spherical harmonics Ymµ(ω) are normalized in such a way that ∫ Ymµ(ω)Y ∗ nν(ω)dω = δmnδµν . (1.26) For the case µ = 0, we obtained a more complex OZ equation. In Fourier space it may be presented in a matrix form Ĥ(k) = Ĉ(k) + Ĉ(k)ρ̂Ĥ(k), (1.27) where Ĥ(k) = ( h000(k) h020(k) h200(k) h220(k) ) , (1.28) Ĉ(k) = ( C000(k) C020(k) C200(k) C220(k), ) , (1.29) ρ̂ = ρ ( 1 〈Y20(ω)〉ω 〈Y20(ω)〉ω 〈Y 2 20(ω)〉ω ) , (1.30) hmnµ(k) = 4π ∞ ∫ 0 r2dr sin kr kr hmnµ(r) . (1.31) The closures of the equation (1.27) in the r-space are as follows for r < σ: h000(r) = −1, h020(r) = h200(r) = h220(r) = 0, (1.32) and for r > σ: C000(r) = βA0 e−z0r r , C020(r) = C200(r) = 0, (1.33) C220(r) = 1 5 βA2 e−z2r r . The space-fixed x, y, z-components of the angular gradient operator are given by∇∇∇ω = il, where l is the angular momentum operator. Using the relations [38] (∇ω)y = l+ − l− 2 , (1.34) l±Ymµ(ω) = [ m(m + 1) − µ(µ + 1) ] 1 2 Ym,µ±1(ω) (1.35) and expressions (1.20), (1.22) the y-component of (1.18) is obtained in the form ∑ l √ l(l + 1)(Bl + βW2 √ 30) [ Yl,1(ω1) − Yl,−1(ω1) ] = ∑ l′ ∑ mnµ ∫ Cµµ0 mn0(r)Ymµ(ω1) × Y ∗ nµ(ω2) √ l′(l′ + 1)Bl′ [ Yl′,1(ω2) − Yl′,1(ω2) ] ρ(ω2)dω2dr . (1.36) 33002-6 Integral equation theory for nematic fluids Taking into account that only quantities independent of the azimuthal angle ϕ yield non-zero average values and using the orthogonality of Ylm(ω) one gets the following equation L = ĈŶ L + βW2 √ 30l , (1.37) where L is a column with Ll = √ l(l + 1)Bl, Ĉ and Ŷ are matrices with elements Cmn = ∫ drC110 mn0(r), (1.38) Ymn = ρ ∫ dωf(ω)Ym1(ω)Y ∗ n1(ω), (1.39) l is a column with δl,2. After integration by parts the equation (1.18) can be written in the form β∇∇∇ω1 v(1) +∇∇∇ω1 ln ρ(ω1) = − ∫ ρ(ω2)dω2∇∇∇ω2 C2(ω1, ω2). (1.40) Hence, L = ĈP + βW2 √ 30l (1.41) where P is the column with Pl = ρ √ l(l + 1)(2l + 1)Sl . (1.42) Equations (1.37) and (1.41) connect the system order parameters Sl, zero Fourier transforms of the direct correlation function harmonics C110 mn0(r), the intensity of external field W2 and the coefficients Bl of the single particle distribution function f(ω). In the MSA approximation for the considered model (m, n) = (0, 2) and the equations (1.37) and (1.41) reduce to B2 = C22Y22B2 + βW2 √ 5 , (1.43) B2 = 5 ρC22S2 + βW2 √ 5 . (1.44) As a result, for a single particle distribution function f(ω) we will have f(ω) = √ 3 2 β W eff 2 D (√ 3 2 β W eff 2 ) exp [ β W eff 2 P2(cos ϑ) ] , (1.45) where β W eff 2 = β W2 √ 5 1 − C22Y22 , (1.46) D(x) is Dawson’s integral D(x) = e−x2 x ∫ 0 ey 2 dy. (1.47) In the absence of the external field W2 = 0 and in accordance with (1.44), (1.45) 1 = C22 Y22 , (1.48) B2 = 5 ρ C22 S2 . (1.49) Thus, in the absence of any field a the single distribution function f(ω), the problem results in the well-known Maier-Saupe equation [36] S2 = ∫ P2(cos ϑ) exp [ MS2P2(cosϑ) ] dω ∫ exp [ MS2P2(cosϑ) ] dω (1.50) 33002-7 M.F. Holovko where M = 5 〈\|Y21(ω)\|2〉ω . (1.51) After integration the equation (1.50) can be written in the form S2 = 3 4 [ 1 xD(x) − 1 x2 ] − 1 2 , (1.52) MS2 = 2 3 x2 . (1.53) The Maier-Saupe theory predicts a first-order phase transition from isotropic phase with S2 = 0 to nematic phase S2 6= 0. From the OZ equation for µ 6= 0 (1.24) using the equation (1.48) it is not difficult to prove that in the absence of external field the harmonics with µ = ±1 have a divergence. This divergence is connected with Goldstone modes. In the MSA approach for the considered model, this is the harmonic h221(r). 2. Hard sphere Maier-Saupe model: MSA description In this section we consider the analytical solution of OZ equation with MSA closure for the model considered in previous section. The obtained results will be used for the description of structure, thermodynamics, and phase behavior of this model. By the factorization method of Baxter and Wertheim [19, 39] the integral equation (1.24) for µ 6= 0 under conditions (1.25) can be reduced to a system of algebraic equations 12 5 η 〈\|Y2µ(ω)\|2〉ω βA2 σ = D ( 1 − Q̃2µ(z2) ) , (2.1) 2πg̃22µ(z2) [ 1 − Q̃2µ(z2) ] = D 2 exp [−2z2 σ] [1 − 2πg22µ(z2)], (2.2) −C = [1 − 2πg22µ(z2)] D (2.3) where η = 1 6 πρσ3, C and D are dimensionless coefficients of the Baxter factor correlation function Q2µ(r) = z ρ〈\|Y2µ(ω)\|2〉ω [ q0µ(r) + De−z2r ] (2.4) with the short-range part q0µ(r) = { C [ e−z2r − e−z2σ ] , r < σ, 0, r > σ, (2.5) Q̃2µ(z2) and g̃22µ(z2) are the dimensionless Laplace transforms of Q2µ(r) and h22µ(r) Q̃(z2) = ρ〈\|Y21(ω)\|2〉ω ∞ ∫ 0 e−z2tQ(t)dt, (2.6) g̃221(z2) = ρ〈\|Y21(ω)\|2〉ω z2 ∞ ∫ σ e−z2th221(t)tdt. (2.7) From the definition of the factor correlation function it follows that 1− ρ〈\|Y2µ(ω)\|2〉ω ∫ C22µ(r)dr = \|Q2µ(k = 0)\|2 (2.8) where Q2µ(k) = 1 − ρ〈\|Y2µ(ω)\|2〉ω ∞ ∫ 0 dreikrQ(r). (2.9) 33002-8 Integral equation theory for nematic fluids The joint use of (2.8) for µ = 1 and (1.43) gives us the additional equation to determine ρ〈\|Y21(ω)\|2〉ω Q21(k = 0) = √ βW2 B2 (2.10) and in the explicit form f = D + dc, f = 1− √ βW2 B2 (2.11) where d = e−z2σ∆1(z2σ). Here and below we use the symbols ∆n(x) = ex − n ∑ l=0 1 l ! xl. (2.12) Formulas (2.11), (2.2), and (2.3) yield the expression for D D = − 1 2a ( b + √ b2 − 4ac ) (2.13) where a = −d exp (−2z2σ) − (d − 1) [ d − f∆2 0(−z2σ) ] , (2.14) b = (d − 1) c f + f [ ∆2 0(−z2σ)f − d ] + df exp (−z2σ), (2.15) c = f [ 2d − ∆2 0(z2σ) ] . (2.16) Now from equation (2.1) for µ = 1 we can obtain the dependence between the ordering parameter ρ〈\|Y21(ω)\|2〉ω and the system parameters η, βA2 1 σ , W2σ A2 and z2σ βA2 σ η ρ 〈\|Y21(ω)\|2〉ω = 5 24 D [ 2 − f d ∆2 0(z2σ) − D ( 1− ∆2 0(−z2σ) d )] . (2.17) In the absence of external field (W2 = 0) f = 1. If we put in this case 〈\|Y21(ω)\|2〉ω = 1 we will obtain the instability condition of the isotropic phase with respect to the nematic phase formation [14, 40]. If we put 〈\|Y21(ω)\|2〉ω = 1.1142 we get the bifurcation of the nematic solution with the smallest value of the order parameter S2 = 0.3236. Since this is the first order phase transition these two conditions are not equivalent. In the presence of an orienting field (W2 > 0) the fluid can exhibit only uniaxial paranematic and nematic phases. When W2 < 0, the same fluid provides the phase transition into a biaxial nematic phase. At strong disorienting field (W2 → −∞) the molecules align perpendicularly to the field and the phase transition into a limiting biaxial phase takes place [23]. In [24] it was shown that the fluid becomes orientationally unstable with respect to spontaneous biaxial nematic ordering under the condition 1 − ρ〈\|Y22(ω)\|2〉ωρ ∫ C222(r)dr = 0. (2.18) This condition reduces to (2.17) after 〈\|Y21(ω)\|2〉ω changes to 〈\|Y22(ω)\|2〉ω . At the infinite field if we put 〈\|Y22(ω)\|2〉ω = 15 8 equation (2.18) gives us the instability condition with respect to the limiting biaxial phase [23]. Figure 1 shows dependence of the order parameter S2 on the product (βa2η)−1 calculated from (2.17) at different values of z2σ. Here and below we consider that An = anσ(znσ)2, where n = 0 and 2. This allows us to consider the mean field result as the Kac potential limit z2σ → 0 and/or z0σ → 0 [41]. For z2σ = 3 and βa2 = 1, η has a non-physical value. As we will see later, in this region the system goes to crystallization. 33002-9 M.F. Holovko Figure 1. The dependence of the order parameter S2 on density η and temperature (βa2) −1 at different values of z2σ calculated in the MSA approximation for nematogenic Maier-Saupe model. For µ = 0, using the factorization method of Wertheim-Baxter [18, 39], the integral equation (1.27) under the condition (1.32–1.33) can be reduced to the system of integral equations 2πr Cij(r) = −q ′ ij(r) + ∑ k,l ∫ dtq ′ ik(r + t)ρklqjl(t), (2.19) 2πr hij(r) = −q ′ ij(r) + 2π ∑ k,l ∫ dt(r − t)hik(\|r − t\|)ρklqlj(t). (2.20) It follows from the asymptotic behavior of the factor correlation functions that qij(r) has the form (i, j = 0, 2) qij(r) = q0 ij(r) + ∑ n=0,2 D (n) ij e−znr (2.21) where the short-range part q0 ij(r) =    1 2q ′′ ij(r − σ)2 + q ′ ij(r − σ) + ∑ n=0,2 C (n) ij ( e−znr − e−znσ ) , r < σ, 0, r > σ. (2.22) Below we shall use the following designations for dimensionless properties c (n) ij = ρ zn C (n) ij , d (n) ij = ρ zn D (n) ij , (2.23) g̃ij(zn) = ρ zn ∞ ∫ σ [hij(t) + δi0δj0] te −zntdt, (2.24) Q̃ij(zn) = ρ ∞ ∫ 0 qij(t)e −zntdt. (2.25) Finally, we obtain a system of algebraic equations for coefficients of factor correlation functions and Laplace transforms of a pair correlation function harmonics − c (n) ij = ∑ l [ δil − 2π ∑ k g̃ik(zn)Skl ] d (n) lj , (2.26) 12 2n + 1 βAn 1 σ η δinδjn = ∑ k d (n) ik [ δkj − ∑ l SklQ̃jl(zn) ] , (2.27) 33002-10 Integral equation theory for nematic fluids 2π ∑ k g̃ik(zn) [ δkj − ∑ l SklQ̃lj(zn) ] = 6 π ηe−znσ [ q ′′ ij (znσ)3 + q ′ ij (znσ)2 ] δi0 − ∑ m=0,2 zm zm + zn c (m) ij exp ( −(zm + zn)σ ) , (2.28) where Ŝ = 1 ρ ρ̂, ρ̂ is given by (1.30). Figure 2. The harmonics of the pair correlation functions in the Fourier space and the structure factor for nematogenic Maier-Saupe model in isotropic phase (z0σ = z2σ = 1, βa0 = 0.1, βa2 = 1, η = 0.28). In (2.27) we should expect that multiple solutions occur, of which only one is acceptable [42]. To choose the physical solution one can utilize the condition det [ 1 − ŜQ̂(s) ] 6= 0 for Re s > 0. (2.29) Using the obtained analytical solution of OZ equation for the considered model it is possible to calculate the structure factor and harmonics of the pair correlation functions. In figures 2 and 3 one can see the structure factor, and the Fourier-transforms of the pair correlation function harmonics hmnµ(k) for the isotropic and nematic phases correspondingly in the absence of the external field. 33002-11 M.F. Holovko Figure 3. The harmonics of the pair correlation functions in the Fourier space and the struc- ture factor for nematogenic Maier-Saupe model in the nematic phase (z0σ = z2σ = 1, βa0 = 0.1, βa2 = 1, η = 0.315). The structure factor of the system S(k) = 1 + ρ ∫ f(ω1)h(k, ω1, ω2)f(ω2)dω1dω2 = 1 + ρ [ h000(k) + 2h020(k)〈Y20(ω)〉ω + h220(k)〈Y20(ω)〉2ω ] . (2.30) We should note that in isotropic phase h220(k) = h221(k) = h222(k) and h200(k) = h020(k) = 0. 33002-12 Integral equation theory for nematic fluids In the nematic phase the contributions of these harmonics are very important. One can see in figure 3 that in the nematic phase the off-diagonal elements h200(k) = h020(k) at small k have comparable to h220(k) absolute value and opposite sign. Due to this the contributions of different harmonics into S(k) compensate at small k and S(k) in this region behaves similarly to isotropic case (figure 2). The small peak at small k in the nematic phase (figure 3) in S(k) is attributed to the appearance of additional interparticle effective attraction due to parallel alignment of molecules. It is important to note that h221(k) is the only harmonic that tends to infinity at k = 0 and this harmonic does not give any contribution to the structure factor which is finite at k = 0. Using equation (1.24) it is possible to show [14] that ρ 〈\|Y21(ω)\|2〉ω h221(k → 0) −→ (z2σ)2 (kσ)2 4 [(z2σ)2C exp (−z2σ) − 2] 2 (2.31) which implies the asymptotic behavior h221(r → ∞) −→ 1 6 (z2σ)2 [(z2σ)2C exp (−z2σ) − 2] 2 η 〈\|Y21(ω)\|2〉ω σ r . (2.32) It can be shown that this harmonic is connected with the correlations of the director fluctuations. This result confirms the prediction from the fluctuation theory of de Gennes [18]. Now we consider the thermodynamic properties. The structure factor at k = 0 gives us an isothermal compressibility 1 β ( ∂ρ ∂P ) T = S (k = 0) . (2.33) The average energy of interparticle interaction at the absence of external field is calculated by β∆E N = β2πρ ∞ ∫ 0 r2dr ∫ dω1f(ω1) ∫ dω2f(ω2) [v0(r) + v2(r, ω1, ω2)] × [ g000(r) + h200(r)Y20(ω1) + h020(r)Y20(ω2) + ∑ µ h22µ(r)Y2µ(ω1)Y ∗ 2µ(ω2) ] = 12η βA0 σ [ g000(z0σ) + 2 √ 5S2h200(z0σ) + 5S2 2h220(z0σ) ] + 12η βA0 σ × [ 5S2 2g000(z2σ) + 2 √ 5S2〈\|Y20(ω)\|2〉ωh200(z2σ) + ∑ µ ( 〈\|Y21(ω)\|2〉ω )2 h22µ(z2σ) ] . (2.34) Similarly, we can calculate the virial pressure β∆Pv ρ = −βρ 2 3 π ∞ ∫ 0 r3dr ∫ dω1f(ω1) ∫ dω2f(ω2) [ g000(r) + h200(r)Y20(ω1) + h020(r)Y20(ω2) + ∑ µ h22µ(r)Y2µ(ω1)Y ∗ 2µ(ω2) ] ∂ ∂r [vh(r) + v0(r) + v2(r, ω1ω2)] = 4η [ g000(σ+) + 2 √ 5S2h200(σ+) + 5S2 2h220(σ+) ] + 1 3 β∆E N + 4η βA0 σ z0 ∂ ∂z0 [ g000(z0σ) + 2 √ 5S2h200(z0σ) + 5S2 2h220(z0σ) ] + 4η βA2 σ z2 ∂ ∂z2 [ 5S2 2g000(z2σ) + 2 √ 5S2〈\|Y20(ω)\|2〉ωh200(z2σ) + ∑ µ ( 〈\|Y21(ω)\|2〉ω )2 h22µ(z2σ) ] , (2.35) 33002-13 M.F. Holovko where g000(r) = 1 + h000(r), g000(znσ) and hmnµ(znσ) are Laplace-transforms of corresponding functions at znσ. Figure 4. Some isotherms of equation of state for nematogenic Maier-Saupe model. 33002-14 Integral equation theory for nematic fluids Some isotherms calculated using the equation of state (2.35) are presented in figure 4 for three different regimes [43]: 1. The isotropic attraction is stronger than the anisotropic one ( a0 a2 = 2 ) ; 2. Isotropic attraction is absent (a0 = 0); 3. Strong anisotropic attraction and isotropic repulsion ( a0 a2 = −0.7 ) . For simplification we consider that z0σ = z2σ = 0.5. Nematic and isotropic branches are denoted by N and I correspondingly. At the first case when isotropic attraction is stronger than the anisotropic one at smaller densities and lower temperature (βa0 = 1) there is condensation between two isotropic phases which disappears at high temperature (βa0 = 0.9). In the second case when isotropic attraction is absent at high temperature (βa2 = 0.9) we observe the weak isotropic- nematic phase transition. At the lower temperature (βa2 = 1) we observe condensation in the nematic region. In the third case (a0 a2 = −0.7) at the lower temperature (βa2 = 4.25) we observe the liquid-gas phase transition between two nematic phases. The entire liquid-gas coexistence region including the critical point is within the nematic region. For the description of phase diagram we need to have the expression for the chemical potential of fluid which can be obtained by generalization of the Hoye-Stell scheme [44]. Unfortunately, this problem has nor been solved yet. We will consider it in a separate paper. Here we will instead use the density functional scheme developed by us in [20] for the chemical potential and the expression (2.35) for the pressure. For simplification we consider the case a0 = 0. Figure 5. Phase diagram of nematogenic Maier-Saupe model for different values of z2σ in the plane density-temperature. 33002-15 M.F. Holovko Figure 6. Phase diagram of the nematogenic Maier-Saupe model for different values of z2σ in the plane temperature-pressure. In figures 5 and 6 we present a phase diagram for the considered model at the planes η − kT a2 (density-temperature) and kT a2 − Pη ρa2 (temperature-pressure) at different z2σ. Dash-dotted line corresponds to the stability condition for the isotropic phase. As we can see, this condition overestimates the region of anisotropic phase. This overestimation increases with the increase of z2σ. But we should take into account that with the increase of z2σ the accuracy of MSA decreases. For z2σ = 0.5 at high temperature we observed a weak nematic transition of the first order. With decreasing temperature, the jump of density at the phase transition increases and at low temperature the orientational order is accompanied by condensation. The peculiarity of this condensation is that it occurs without a critical point. It means that there is no phase transition “nematic – condensed nematic”. In figure 7 the temperature dependence of the order parameter at the phase transition region is presented. In figure 5 the dotted line represents the crystallization Figure 7. Temperature dependence of the order parameter S2 at the phase transition region for the nematic phase. 33002-16 Integral equation theory for nematic fluids transition line. It was obtained using the Hansen-Verlet criterion [45]. According to this criterion the fluid becomes unstable when the height of the main peak in the structure factor S(k) becomes equal to 2.9 ± 0.1. Figure 6 gives evidence of the existence of temperature at which three phases coexist (isotropic, nematic, and solid). As we can see with the increase of z2σ, the triple point shifts to the region of higher pressure and lower temperature. Since with the increase of temperature the density of crystallization increases we can see that at high enough temperature the crystallization can forestall the nematic transition. We can note that for not so large value of z2σ the phase diagram presented in figure 5 agrees quite well with the results of [16] obtained in the framework reference HNC and from computer simulation. Figure 8. The dependence of reduced elastic constants on density and temperature. Let us consider the elastic properties of the considered model. Formal expressions for elastic constant in biaxial nematics in terms of direct correlation function have been given by Poniewiersky and Stecki [46]. It includes three elastic constants K1 (splay), K2 (twist) and K3 (bend) [47]. Since for the considered model correlation functions depend only on the angle ω12, the description of elastic properties reduces to one-constant approximation βK1 = βK2 = βK3 = βK = 1 6 ρ2 ∫ drdω1dω2r 2ḟ(ω1)ḟ(ω2)nx(ω1)nx(ω2)C2(r, ω1ω2) (2.36) 33002-17 M.F. Holovko where ḟ(ω1) = ∂f(ω) ∂ cosϑ . In the MSA approximation βK = 10πρ2S2 2 ∫ r4drC221(r). (2.37) Another way of calculating the elastic constants is connected with the application of the theory of hydrodynamic fluctuations [46]. In this way in one-constant approximation 1 βK = 1 3 lim k→0 k2h221(k) 〈\|Y21(ω)\|2〉ω 〈\|Y20(ω)\|〉2ω . (2.38) The results of our calculations are presented in figure 8. It is important to note that in our calculations both expressions (2.37) and (2.38) give the same results. The effect of a disorienting field on the phase diagram and on the elastic properties of the ordered fluids was studied by us in [23, 24]. It was shown that a disorienting field significantly increases the region of an ordered fluid. In the case of a strong disorienting field when the temperature decreases the orientational phase transition of the second order becomes a transition of the first order at a tricritical point. A disorienting field increases the ordering and the elastic properties of the model under consideration. 3. Application of the integral equation theory to colloid-nematic dispersi- ons In this section we review the results of recent publications of T. Sokolovska, R. Sokolovskii and G. Patey [28–35] about the generalization of the integral equation theory for colloid-nematic systems. The starting point of this generalization is the OZ equations for a two-component mixture of colloidal and nematic particles hCC(12) = CCC(12)+ ∫ d3ρC(3)CCC(13)hCC(32) + ∫ d3ρN(3)CCN(13)hCN(32), (3.1) hCN(12) = CCN(12)+ ∫ d3ρC(3)CCC(13)hCN(32) + ∫ d3ρN(3)CCN(13)hNN(32), (3.2) hNN(12) = CNN(12)+ ∫ d3ρC(3)CNC(13)hCN(32) + ∫ d3ρN(3)CNN(13)hNN(32) (3.3) in combination with TZLMBW equations for density distributions of colloidal and nematic parti- cles, respectively β∇vC(1) + ∇ ln ρC(1) = ∫ d2CCC(12)∇ρC(2) + ∫ d2CCN(12)∇ρN(2), (3.4) β∇vN(1) + ∇ ln ρN(1) = ∫ d2CNC(12)∇ρC(2) + ∫ d2CNN(12)∇ρN(2), (3.5) where the label 1 denotes the coordinates (r1, ω1) for nematogen and for spherical colloids 1 = (r1). Here we consider a dilute nematic colloids case for which OZ equations (3.1)–(3.3) reduce to hNN(12) = CNN(12) + ∫ d3ρN(3)CNN(13)hNN(32), (3.6) hCN(12) = CCN(12) + ∫ d3ρN(3)CCN(13)hNN(32), (3.7) hCC(12) = CCC(12) + ∫ d3ρN(3)CCN(13)hNC(32) (3.8) in combination with the usual TZLMBW equation (1.18) for a nematic subsystem β∇ω1 vN(1) + ∇ω1 ln ρN(1) ∫ d2CNN(12)∇ω2 ρn(2). (3.9) 33002-18 Integral equation theory for nematic fluids Equation (3.6) coincides with equation (1.6) for bulk nematic fluids. Equation (3.7) describes nematic fluids near colloidal particles. The function ρN(1) [1 + hNC(12)] gives distribution of a nematic fluid about a colloidal particle. This function takes into account all the changes at a given point r1 induced by a colloidal particle at the point r2. These include the changes in the local density and in the orientational distribution of the nematic fluid. Equation (3.8) describes the colloid-colloid correlations. It gives the colloid-colloid mean interaction force which at the HNC level is conveniently given by β wCC(12) = β vCC(12) + CCC(12) − hCC(12), (3.10) where vCC(12) is the direct pair interaction potential between colloidal particles. For nematic we consider the same model as in the previous sections. This is the model of hard spheres with an anisotropic interaction in the form (1.2). For simplification we put here A0 = 0. The nematogen interaction with the external field is given by (1.1). The model colloidal particles (C) are taken to be hard spheres of diameter R. Van der Waals or other direct colloid-colloid interactions could be included through the vCC(12) term in equation (3.10). We consider the size of a colloid to be much larger than the size of a nematic particle. The properties of a nematogenic fluid near the surface can then be described in the Henderson-Abraham-Barker (HAB) approach [27]. This approach reduces to equation (3.7) in the limit R → ∞. In this case we can switch from the colloid-nematic distance to the colloid-surface distance s12. The interaction of nematogens with the surface of a colloidal particle (anchoring) is modeled as was done for the flat wall case [28, 29] vNC(12) = { −AC exp [ −zC ( s12 − 1 2σ )] P2(ω, ŝ12) for s12 > 1 2σ, ∞, for s12 < 1 2σ, (3.11) where s12 is the vector connecting the nearest point of the surface of colloid 1 with the center of nematogen 2, and ŝ12 = s12 s12 . Note that the positive and the negative values of AC favor respectively the perpendicular and the parallel orientations of nematogen molecules with respect to the surface. The strength of the nematogen-colloid interaction is determined by AC and zC. We start with solving the OZ equation (3.7) for the wall-nematic correlation function with the MSA closure hWN(ŝ\|ω̂1, s1) = −1, s1 < 1 2 σ, CWN (ŝ\|ω̂1, s1) = −βvWN (ŝ\|ω̂1, s1) , s1 > 1 2 σ. (3.12) After application to correlation functions hWN and CWN orientational harmonics expansion fWN(ŝ\|ω̂1, s1) = fWN 000 (s1) + fWN 200 (s1)Y20(ŝ) + fWN 020 (s1)Y20(ω1) + ∑ µ fWN 22µ (s1)Y2µ(ŝ1)Y2µ(ω1). (3.13) OZ equation (3.7) reduces to the system of equations for harmonics coefficients fWN mnµ(s1). For µ 6= 0 hWN 22µ (s1) = CWN 22µ (s1) + ρ〈\|Y2µ(ω)\|2〉ω ∫ hWN 22µ (s2)C NN 22µ(r12)dr2 (3.14) and for µ = 0 hWN ij0 (s1) = CWN ij0 (s1) + ∫ dr2ρ ∑ i′j′ hWN ii′ ,0(s2)〈\|Yi′0(ω)Yj0(ω)\|〉ωCNN j′j0(r12) (3.15) where the indices can be 0 or 2. To solve equations (3.14) and (3.15) we can directly follow the Baxter-Wertheim factorization technique developed for calculating the wall-particle distribution functions [48]. As a result, for 33002-19 M.F. Holovko s > 1 2σ and for µ = 0, equations (3.14) and (3.15) can be written correspondingly in the form ĝWN(s) − ∞ ∫ 0 ĝWN(s − r)ρ̂q̂(r)dr = v̂WN [ 1 − q̂T (zC) ]−1 e−zCs + ( 1 0 0 0 ) [1− q̂(s = 0)] (3.16) and for µ 6= 0 hWN 22µ (s) = ρ 〈 \|Y2µ(ω)\|2 〉 ω ∞ ∫ 0 hWN 22µ (s − r)q22µ(r)dr + 1 5 βAC exp [ zC ( 1 2σ − s )] 1 − ρ 〈\|Y2µ(ω)\|2〉ω q22µ(zC) (3.17) where the matrix ĝWN(s) has the elements gWN ij (s) = δi0δj0 + hCN ij (s), v̂WN has the elements vWN ij = 1 5βACδi2δj2 exp [ 1 2σzC ] . The matrices ρ̂ and Baxter functions q̂(r) and q22µ(r) and its Laplace transforms q(zC) are discussed in previous sections. The right-hand sides of equations (3.16) and (3.17) determine the contact values of ĝWN ( 1 2σ ) and hWN 22µ ( 1 2σ ) for µ 6= 0. One can solve equations (3.16) and (3.17) at any distance s using the Perram method [47], for example. The wall-nematic distribution gWN(12) in the form (3.13) can be used to examine the structure near the wall. Using this distribution function it is reasonable to introduce the density profile which is defined as ρ(ŝ\|s) = ∫ gCN(ŝ1\|ω1s1)ρN(ω1)dω1 . (3.18) In contrast to the usual fluids, the nematic density profile depends not only on the distance from the wall s1, but also on the wall orientation ŝ. Noting the tensorial nature of orientational ordering near the wall, it is useful to define a generalized order parameter S(d̂) = ∫ P2(ω̂ · d̂)ρN(ω̂)dω̂ (3.19) where d̂ is an arbitrary unit vector. For the considered nematic model the unit vector d̂m that maximizes S(d̂) is chosen as a director. In the presence of an aligning field the bulk director is always parallel to this field and S(d̂m)/ρ = S2. The density-orientational profile of the generalized order parameter can be defined as S(ŝ1, s1, d̂) = ∫ P2(ω̂1d̂)ρN(ω1)g WN(ŝ1, s1, ω1)dω1 . (3.20) For a large distance from the wall s1 the distribution function gCN(ŝ1s1, ω1) tends to 1 and S(d̂) tends to the bulk value S(d̂) = ρs2P2(d̂ n̂). By analogy with the bulk definition, the unit vector d̂m that maximizes S(s, ŝ, d) at a given distance s1 can be taken to define a local director. The vector field dm(s1, ŝ) gives the director field configuration (the defect) around the wall. The maximum value Sm(s1) = S(s1, dm) gives the degree of local ordering at s1. From the investigation of [28, 29], for an orienting wall special long-range correlations were identified that are responsible for the reorientation of the bulk nematic at the zero external field. These correlations become stronger as the wall-nematic interaction is increased in range. They become longer ranged as the orienting field is weakened. The local director orientation can vary di- scontinuously with the distance from the wall when the orienting effect of the field and wall-nematic interaction are antagonistic. At high densities, when wall-nematic interaction favors orientations perpendicular to the surface, smectic-like structures were observed. An important property that can be calculated from (3.18) is the adsorption coefficient which describes the surface density excess. In contrast to usual fluids, for anisotropic fluids the adsorption coefficient depends on the surface orientation ŝ with respect to the external field and has the form Γ(ŝ) = ρ ∞ ∫ 1 2 σ ds1 ∫ dω1fN(ω1) [ gCN(s, ω1, s1) − 1 ] . (3.21) 33002-20 Integral equation theory for nematic fluids Above we introduced a generalized order parameter (3.20) related to a certain direction d̂. The adsorption excess per unit area associated with this order parameter is given by Γ(ŝ, d̂) = ρ ∞ ∫ σ 2 ds1 ∫ dω1P2(d̂ · ω̂) [ gWN(ŝ1ω̂, s1) − 1 ] f(ω1). (3.22) The maximum of this function gives information about the direction in which particles are mostly reoriented, and the minimum indicates the direction which they mostly abandon. There is a significant flaw in the HAB description of the MSA approach due to ignoring the non-direct interaction between colloid and nematic in the case of the inert hard wall (AC = 0). This problem has recently been solved in the framework of the inhomogeneous equation theory [50] (the so-called OZ2 approach [3]). Now we can return to colloid-nematic systems described by equations (3.7)–(3.8). To this end, we can use the results that had been obtained for a nematic near hard wall. We need to introduce the center-center colloid-molecule vector r12 which is parallel to ŝ1 and whose length is s1 + 1 2R, where R is the diameter of the colloidal particle. For example, the colloid-nematogen interaction vCN(12) can be expressed in terms of the vector connecting the nearest point on the surface of colloidal particle 1 with the center of nematogen 2. This transformation is the simplest for a spherical colloid vCN(12) = vCN(r12, ω̂2) = vWN(s = r12 − r̂12 2 R, ω̂ = ω̂2) = −AC exp [ −zC ( r12 − 1 2 (σ + R) )] P2(r̂12 · ω̂2). (3.23) For a sufficiently large colloidal particle, curvature effects are unimportant (for micron and submi- cron colloids), which suggests an ansatz that the direct correlation function CCN(12) can be taken from the wall-nematic solution CCN(r12, ω̂2) = CWN(s = r12 − r̂12 2 R, ω̂ = ω̂2). (3.24) This was found to be a good approximation [31, 32] because the wall-nematic direct correlation function is truly short-ranged outside the surface, while inside the core it rapidly tends to a function of ω̂2 that depends only on bulk properties. Thus, the direct correlation function is not very sensitive to the surface curvature. Now the nematic distribution around a colloidal particle and colloid-colloid mean force potential can be found from equations (3.7) and (3.9) by means of the Fourier transformation. As we already discussed in the previous section, the correlation functions CNN(12) and hNN(12) in the MSA approximation can be presented in the form (1.21) and the corresponding OZ equation reduces to equations (1.24) and (1.27) for harmonics h22µ(r) when µ 6= 0 and for harmonics hNN mn0(r) when µ = 0. The colloid-nematic correlation functions hCN(12) and CCN(12) can also be written as a spherical harmonic expansion fCN(r12, ω2) = fCN 000 (r12) + fCN 200 (r12)Y20(r̂12) + fCN 020 (r12)Y20(ω2) + ∑ \|µ\|62 fCN 22µ(r12)Y2µ(r̂12)Y2µ(ω2). (3.25) The axial symmetry of the bulk nematic allows equation (3.7) to be factorized into equations with different µ. These equations can be Fourier transformed to obtain k-space equations in terms of Hankel transforms of harmonics of the correlation functions fCN mnµ(k) = 4πim ∞ ∫ 0 r2drjm(kr)fCN mnµ(r), (3.26) 33002-21 M.F. Holovko where jm(x) is a spherical Bessel function. After Fourier transformation equation (3.7) for spherical harmonic coefficients can be written in the following form for µ = 0 ĤCN(k) [ 1 − ρ̂ ĈNN(k) ] = ĈCN(k), (3.27) where the hat denotes matrices with elements CCN mn(k) = 4πim ∞ ∫ 0 r2drjm(kr)CCN mn,0(r), (3.28) HCN mn(k) = 4πim ∞ ∫ 0 r2drjm(kr)hCN mn,0(r). (3.29) The matrix ĈNN(k) can be expressed in terms of the bulk Baxter functions (2.21) 1 − ρ̂NĈNN(k) = [ 1 − ŜNQ̂(−ik) ] [ 1 − ŜNQ̂T (ik) ] . (3.30) One can then immediately solve equation (3.27) for the Hankel transforms of harmonics hCN mn,0(r). Similarly for µ 6= 0 hCN 22µ(k) [ 1 − ρ 〈 \|Y2µ\|2 〉 ω CNN 22µ ] = CCN 22µ(k) (3.31) and in terms of Laplace transforms hCN 22µ(k) [1 − Q22µ(−ik)] [1 − Q22µ(ik)] = CCN 22µ(K). (3.32) Finally, using the inverse Hankel transformation hCN mnµ(r) = 4π (−i)m (2π)3 ∞ ∫ 0 k2dkjm(kr)hCN mnµ(k) (3.33) the total pair correlation function can be found in the r-space hCN(r12, ω2) = ∑ mn hCN mn0(r12)Ym0(r̂12)Yn0(ω2) + ∑ µ6=0 hCN 22µ(r12)Y2µ(r̂12)Y ∗ 2µ(ω2). (3.34) In this numerical calculation for CCN(r12, ω2) the ansatz (3.24) was used. The non-direct part of the potential of the mean force for a pair of colloidal particles can be written in the form − βwCC(12) = hCC(12) − CCC(12) = − ∑ l=0,2,4 βwCC l (r12)Yl0(r̂12). (3.35) After spherical harmonic expansions of the correlation functions using Fourier transforms of the coefficients of correlation functions in accordance with (3.8) we can write −βwCC(k) = ρ ∑ mn ∑ n′m′ ∑ µ CCN mnµ(k)hNC n′m′µ(k) 〈 Ynµ(ω)Y ∗ n′µ(ω) 〉 Ymµ(k̂)Y ∗ m′µ(k̂) = ρ ∑ mn ∑ µ [ hCC mn′µ(k) − Cmn′µ(k) ] Ymµ(k̂)Y ∗ m′µ(k̂), (3.36) where indexes m, m′, n, n′ are equal to 0 or 2. Note that [38] Ymµ(k̂)Ym′µ(k̂) = ∑ l [ (2m + 1)(2m′ + 1) (2l + 1) ] 1 2 ( m m′ l µ −µ 0 ) ( m m′ l 0 0 0 ) Yl0(k̂) (3.37) 33002-22 Integral equation theory for nematic fluids Figure 9. (a) Maps of the director dm(r) and the local ordering SN(r) in the isotropic regime (η = 0.2) and zero external field βW2 = 0. The colloidal particle is shown as a white circle of radius 1 2 R = 25. The axes denote the distance from the center of colloidal particle in units of σ. The director field is indicated by bars showing the director orientation. The local ordering 1 ρ SN(r) is shown by color, red regions are more ordered. Positions where 1 ρ SN(r) equals the bulk order parameter S2 are shown by thin black lines. (b) The potential of mean force βwCC(12) at η = 0.2 and βW2 = 0. One colloidal particle is shown as a white circle of radius 1 2 R and the grey stripe of width 1 2 R surrounding it denotes the region inaccessible to the center of the other colloidal particle due to the hard-core repulsion. The center-center distance in units of σ are indicated on both axes, and the color code is shown on the right. The blue regions are most attractive. The positions where the potential changes sign βwCC(12) are shown by solid black lines. and expression (3.36) can be rewritten in the form (3.35). In (3.37) l changes from 0 to m + m′. It means that l = 0, 2, 4. ( m m′ l µ−µ 0 ) and ( m m′ l 0 0 0 ) are the corresponding Clebsch-Gordon coefficients [38]. Due to the axial symmetry the contributions from different µ are separated again. Some results of such calculations taken from [30, 33] are presented in figures 9–12 for perpendi- cular anchoring (AC > 0). These illustrate the assorted structure and interactions that can occur in nematic colloids under different conditions. Pictures labelled (a) are maps of the local ordering and director field around single colloidal particles. Note that the local ordering in the bulk is ρS2, where ρ and S2 are the bulk density and the order parameter. Pictures labelled (b) present the re- sulting potentials of mean force between colloidal pairs. Equilibrium configurations of two colloidal particles are defined by absolute minima of the potential of mean force shown with blue. We plot the results for two densities η = 1 6πρσ3, which describe two different regimes. η = 0.2 corresponds to an isotropic phase at zero external field, whereas at η = 0.35 the fluid is a stable nematic. One can see that in the isotropic regime the external field promotes the chain formation of colloids along its direction (figure 10). In the nematic regime tilted chains of colloidal particles (figure 11) can be transformed by increasing the external field, which promotes colloidal aggregation in the phase perpendicular to the field direction (figure 12). In sum, a rich variety of equilibrium colloidal structures can be promoted by different fields without changing the composition of the system. Finally we consider the long-range behavior of colloid-colloid interactions [34]. These interac- tions result from colloid-induced distortions of nematic order and have been mainly described in the framework of phenomenological elastic theories [51, 52] which address the director distribu- tion around a single colloidal particle. In the integral equation theory in MSA approximation, the asymptotes connected with elastic behavior are determined by the OZ-relations among harmonics with µ = ±1. In the k-space these are 33002-23 M.F. Holovko Figure 10. As in figure 9, but at non-zero external field βW2 = 0.1. The field is directed along the vertical axis. Figure 11. As in figure 9, but in the nematic region, η = 0.35. Figure 12. As in figure 11, but at non-zero external field βW2 = 1. The field is directed along the vertical axis. hNN 221(k) = CNN 221 (k) + CNN 221(k)ρ 〈 \|Y21(ω)\|2 〉 ω hNN 221(k), (3.38) hCN 221(k) = CCN 221(k) + CCN 221(k)ρ 〈 \|Y21(ω)\|2 〉 ω hNN 221(k) (3.39) = CCN 221(k) + hCN 221(k)ρ 〈 \|Y21(ω)\|2 〉 ω CNN 221(k), (3.40) hCC 221(k) − CCC 221(k) = CCN 221(k)ρ 〈 \|Y21(ω)\|2 〉 ω hNN 221(k). (3.41) 33002-24 Integral equation theory for nematic fluids The equation (3.38) gives hNN 221(k) = CNN 221 (k) [ 1 − ρ 〈\|Y21(ω)\|2〉ω CNN 221(k) ] . (3.42) In the limit k → 0 in accordance with (1.43) 1 − ρ 〈 \|Y21(ω)\|2 〉 ω C221(k) = βW2 B2 + k2B2 + O(k4), (3.43) where B2 = 〈 \|Y21(ω)\|2 〉 ω βK [15ρS2 2 ] , (3.44) the elastic constant K is given by (2.37). Now if we put (3.43) into (3.42), after the inverse zeroth- order Hankel transformation, we will have hNN 221(r) r→∞−−−→ C exp (−r/ξ) r (3.45) where the decay length ξ = [ K/(W2ρS23 √ 5) ]1/2 (3.46) and the prefactor C = [ 4πρB2 〈 \|Y21(ω)\|2 〉 ω ]−1 = 3B2 2 4πβK . (3.47) In zero-field limit W2 = 0, ξ → ∞ and the result (3.45) coincides with our result (2.32) from the previous section. For a sufficiently large spherical colloidal particle, the ansatz (3.24) was suggested. Noting that j2(x) = x2 15 ( 1 − x2 14 + · · · ) at zero field CCN 221(k) k→0−−−→ −4π hWN 221 (s = 1 2σ) 30zc BR3k2 + O(k4) (3.48) where hWN(s = 1 2σ) is the contact value of hWN 221 (s). From (3.34) hCN 221(k) = CCN 221(k) [ 1 − ρ 〈 \|Y21(ω)\|2 〉 ω CNN 221 (k) ]−1 . (3.49) Now using (3.48) and (3.43) in the limit k → 0 we have hCN 221(k) = −4π 30 hWN 221 (s = 1 2σ) BzC R3 + O(k2). (3.50) Inverting the Hankel transform 4π (2π)3 i2 ∞ ∫ 0 k2dkj2(kr) = − 3 4π 1 r3 (3.51) one finds hCN(12) r→∞−−−→ 1 10 hWN 221 (s = 1 2σ) BzC R3 r3 12 [Y21(r̂12)Y ∗ 21(ω2) + c.c.] (3.52) where c.c. denotes the complex conjugate of the first term within the square brackets. For a pair of colloidal particles labelled by subscripts C and C ′ from equations (3.40) and (3.41) we have hCC′ 221 (k) − CCC′ 221 (k) = CCN 221(k)ρ 〈 \|Y21(ω)\|2 〉 ω CNC′ 221 (k) 1 − ρ 〈\|Y21(ω)\|2〉ω CNN 221 (k) . (3.53) 33002-25 M.F. Holovko At zero field and small k equation (3.53) takes the form hCC′ 221 (k) − CCC′ 221 (k) → ρ 〈 \|Y21(ω)\|2 〉 ω (4π)2 1 zC hWN 221 (s = 1 2 σ) 1 zC hW ′N 221 (s = 1 2 σ) R3R′3k2 302 . (3.54) The contribution of µ = ±1 terms to the Fourier transforms of colloid-colloid potential of mean force is [ hCC′ 221 (k) − CCC′ 221 (k) ] [ Y21(k̂)Y ∗ 21(k) + c.c. ] = [ hCC′ 221 (k) − CCC′ 221 (k) ] 2 [ 1 + 1 7 √ 5Y20(k̂) − 4 7 Y40(k̂) ] = β ∑ l=0,2,4 wCC′ l (k)Yl0(k̂). (3.55) Although in k-space three terms occur on the right-hand side of equation (3.55), at zero field the Y40(k̂) term alone determines the asymptotic behavior of the potential of mean force in r-space. Us- ing the inverse Hankel transformation for l = 4 and noting that k2Y40(k̂) becomes 105Y40(r̂)/(4πr5) in r-space we obtain βwCC′(r) r→∞−−−→ 8π 15 hWN 221 (s = 1 2σ) zC hW ′N 221 (s = 1 2σ) zC′ ρ 〈 \|Y21(ω)\|2 〉 ω R3R′3 r5 Y40(r̂). (3.56) This result was obtained taking into account only the “elastic harmonics” (µ = ±1) in expansion (3.36). It is assumed that elastic deformations of the director field are dominant at long distances. However, this assumption becomes unsatisfactory near phase boundaries where fluctuations in local ordering are large. These are results for the case when external field is absent and ξ → ∞. But the correlation length also influences the orientational behavior of the effective colloid-colloid interaction. The so-called quadrupole interaction (3.56) that determines the long-range behavior at infinite ξ transforms into a superposition of screened “multipoles” when ξ is finite [34] −βwCC′(r) r→∞−−−→ 4π ξ5 C(R, zC)C(R′, zC′)ρ 〈 \|Y21(ω)\|2 〉 ω (3.57) × [ −2K0( r ξ ) − 10 7 K2( r ξ )P2(r̂) + 24 7 K4( r ξ )P4(r̂) ] where C(R, zC) = hWN 221 (s = 1 2σ) 30zC [ R4 8ξ + R3 ] , (3.58) K0(x) = e−x x , K2(x) = 1 x3 ( 3 + 3x + x2 ) e−x, (3.59) K4(x) = 1 x5 ( 105 + 105x + 45x2 + 10x3 + x4 ) e−x . In the latest publication of T. Sokolovska [35] the problem of wall-colloid interaction in nematic solvents was discussed for “quadrupole” colloids. At weak field this interaction was obtained in the following form −βwWC(ξ) = π 2 ρ 〈 \|Y21(ω)\|2 〉 ω hWN 221 (s = 1 2σ)hCN 221(s = 1 2σ) zW zC × exp [ −1 ξ (s − 1 2 σ) ] sin2(2ϑs) 1 ξ2 [ R4 8ξ + R3 ] . (3.60) This is a new type of an effective force acting on colloidal particles in the presence of an external field. In contrast to the so-called “image” interaction [53] that is always repulsive at long distances, the force identified in [35] can be attractive or repulsive, depending on the type of anchoring at the wall and colloidal surface (AW 2 , AC 2 ). The effective force on a colloidal particle decreases with the distance s from the wall as exp(−s/ξ). 33002-26 Integral equation theory for nematic fluids 4. Conclusions The generalization and application of modern liquid state theory to the nematic and other liquid crystalline systems opens up new possibilities for the development of microscopic theory of liquid crystals. The leading role in this theory is played by the pair and singlet distribution functions, the knowledge of which makes it possible to describe the structure, thermodynamics, phase behavior, elastic and other properties depending on the nature of intermolecular interaction. A traditional way of calculating the pair distribution function is connected with the development of the integral equation theory which usually reduces to the solution of OZ equation with a corresponding closure relation. In this paper we present the review of the integral equation theory for orientationally ordered fluids. The considered approach is based on self-consistent solution of OZ equation for the pair distribution function together with the TZLMBW equation for the singlet distribution function. It is shown that such an approach correctly describes the behavior of correlation functions of anisotropic fluids connected with the presence of Goldstone modes in the ordered phase in the zero-field limit. Due to this peculiarity in the orientationally-ordered state, the harmonics of the pair distribution function connected with correlations of the director transverse fluctuations become long-range ones in the zero-field limit. It is important to note that these harmonics do not give a direct contribution into the structure factor of nematic fluids. This phenomenon ensures the finite value of the structure factor in the limit of zero wave vector. The presence of Goldstone modes in an ordered phase is responsible for some specific properties of anisotropic fluids such as its elastic properties, multipole-like long-range asymptotes for effective interaction between colloids solved in nematic fluids and so on. The capabilities of the formulated approach are illustrated through analytical results obtained in the framework of the mean spherical approximation for the Maier-Saupe nematogenic model. Out of the equation of state we select three types of phase diagrams depending on the ratio between isotropic and anisotropic interactions. For a strong isotropic attraction, we have the following phase transition between translational homogeneous phases: isotropic gas – isotropic liquid, isotropic gas – nematic and isotropic liquid – nematic. For a strong anisotropic interaction we observed a phase transition only between phases with different symmetries. In the isotropic repulsion case we also observed the nematic gas – nematic liquid phase transition. Using the Hansen-Verlet criterion [45] for crystallization, the point of coexistence of isotropic, nematic and crystalline phases was found. The effect of the disorienting field can significantly increase the region of the ordered fluid [23, 24]. The integral equation approach was also extended to a description of nematic fluid near a planar wall and a colloidal surface, as well as to colloidal-colloidal interaction in the presence of a uniform orienting field. The function ρNC(ω, r12) = ρf(ω) [1 + hNC(ω, r12 = r1 − r2)] provides the distribution of nematic fluid about the colloidal particle. This function takes into account all the changes at a given point r1 induced by the colloidal particle at r2. They include the changes in the local density and in the orientational distribution of the nematic fluid. The function ρNC(ω, r) defines the density-orientational profile of the generalized order parameter SC(r, d̂) = ∫ P2(ωωωd)ρNC(ω, r)dω (4.1) which is connected with the director field configuration around the colloid d̂m that maximizes SC(r,d) at a given point r. The application of anisotropic integral equation theory opens up new possibilities for the de- scription of intercolloidal interactions in nematic solvents. Contrary to elastic theories [51, 52] which describe intercolloidal interactions only for asymptotically large distances, when correlation lengths are much larger than the particle size, the integral equation theory can describe the in- tercolloidal interactions at small and intermediate distances in the presence of an external field. These interactions are important for the description of colloidal phase diagrams and structure as well as other colloidal properties in order to be controlled with external fields. In contrast to phe- nomenological elastic theories, the integral equation method does not assume boundary conditions at colloidal surfaces but instead calculates them. From investigations of potentials of the mean force 33002-27 M.F. Holovko for pairs of identical colloidal particles with perpendicular anchoring [33] it was concluded that effective colloid-colloid interactions are determined by three main factors, namely the phase tran- sition in confined geometry, depletion effects and elastic interactions between the nematic coating surrounding the colloidal particles. Varying the external field shifts the relative importance of these factors and significantly alters the effective interactions. In the framework of the integral equation theory it is also possible to involve colloidal particles of different size and form, ranging up to wall-colloid interactions. Effective potentials for colloidal pairs with asymmetric anchoring (e. g. perpendicular and parallel) are of interest as well. This can be also attributed to the effect of the presence of a third species in nematic colloids. 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Sokolovska T.G., Sokolovskii R.O., Phys. Rev. E59, R3819, 1999. Теорiя iнтегральних рiвнянь для нематичних флюїдiв М.Ф.Головко Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, Львiв, 79011, Україна Традицiйний формалiзм у теорiї рiдин, що базується на розрахунку парної функцiї розподiлу, узагальнений на нематичнi плини. Розглядуваний пiдхiд базується на розв’язку орiєнтацiйно- неоднорiдного рiвняння Орнштейна-Цернiке в поєднаннi з рiвнянням Трайцiнберга-Цванцiга- Ловета-Моу-Бафа-Вертгайма. Показано, що даний пiдхiд коректно описує поведiнку кореляцiйних функцiй анiзотропних флюїдiв, обумовлену наявнiстю голдстоунiвських мод у впорядкованiй фазi при вiдсутностi упорядковуючого зовнiшнього поля. Ми зосереджуємось на обговореннi аналiти- чних результатiв отриманих у спiвпрацi з Т.Г. Соколовською в рамках середньо-сферичного набли- ження для нематогенної моделi Майєра-Заупе. Представлена фазова дiаграма цiєї моделi. Вста- новлено, що в нематичному станi гармонiки парної кореляцiйної функцiї, пов’язанi з кореляцiями флуктуацiй поперечних до напрямку директора, стають далекосяжними при вiдсутностi впорядко- вуючого поля. Показано, що така поведiнка функцiї розподiлу нематичного флюїду приводить до дипольно- та квадрупольно-подiбних далекосяжних асимптотик ефективної мiжколоїдної взаємодiї в нематичних флюїдах, передбаченої ранiше феноменологiчними теорiями. Ключовi слова: парна функцiя розподiлу, теорiя iнтегральних рiвнянь, нематогенна модель Майєра-Заупе, моди Голдстоуна, колоїдно-нематична сумiш 33002-29 Integral equations for orientationally inhomogeneous fluids: general relations Hard sphere Maier-Saupe model: MSA description Application of the integral equation theory to colloid-nematic dispersions Conclusions

Integral equation theory for nematic fluids

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