Exact Relations In The Theory Of Anisotropic Liquids

Exact relations which connect the order parameters, k = 0 Fourier transforms of the direct correlation function harmonics and parameters of the single-particle distribution function for the uniaxial uids are proposed. These relations are an algebraic represetation of the Lovett equation in anisotropic uids. It is shown that in the anisotropic (ori-entationally ordered) phase the transverse correlations of spins in magnetic uids and the correlations of the director transverse uctuations in nematics become of long-range character. Anisotropic uids have no long-range positional order, but they do exhibit orientational order. As a consequence, the single-particle distribution function (1) does depend only on the molecular orientation ! = (; ') and (1) = f(! 1), where is the density of molecules. These systems are polar liquids, magnetic uids and the simplest of the liquid crystals | nematics. Anisotropic uids without the presence of an external eld are the systems with spontaneously broken symmetry, because the direction of the orien-tational ordering is not predetermined. The spontaneous ordering breaks the continuous symmetry (rotational invariance that is intrinsic in isotropic systems) and leads to orientational excitations, or Goldstone modes, which rotate the direction of ordering without any energy cost. This results in the peculiar physical properties of anisotropic uids. 1. Stability analysis and the Lovett equation in anisotropic uids From the density functional theory the following stability condition of equilibrium systems is known 1] Z Z d(1)d(2) (1)(2) > 0; (1.1) where c(1; 2) is the direct pair correlation function, 1 (or 2) being spatial and orientational coordinates of particle 1 (or 2). The equation (1.1) follows directly from the condition of the free energy minimum with respect to arbitrary variations in the particle distribution (i): Z Z 2 F (1)(2) (1)(2)d(1)d(2) > 0: (1.2) But in the anisotropic uids without the presence of external elds some variations (connected to rotations of the director: (1) r !1 (! 1)) keep c T.


Stability analysis and the Lovett equation in anisotropic uids
From the density functional theory the following stability condition of equilibrium systems is known 1] Z Z d(1)d(2) (1; 2) (1) ?C(1; 2) (1) (2) > 0; (1.1) where c(1; 2) is the direct pair correlation function, 1 (or 2) being spatial and orientational coordinates of particle 1 (or 2).The equation (1.1) follows directly from the condition of the free energy minimum with respect to arbitrary variations in the particle distribution (i): Z Z 2 F (1) (2) (1) (2)d(1)d(2) > 0: in the case of zero external eld v(! 1 ).Thus, the Lovett equation is closely connected with the broken rotational invariance and correctly treats the symmetry of anisotropic uids.We will show now that equation (1.4) also provides natural treatment for Goldstone modes in systems with spontaneously broken symmetry.In such systems the susceptibilities to external eld v rotations (transverse susceptibilities) are in nite in the zero-eld limit.In anisotropic uids the orientation of the molecules breaks the continuous rotational symmetry (but not the translational invariance) and results in a spontaneous partial order.Therefore 3], the transverse susceptibility in anisotropic uid is in nite in the limit of a zero wave vector.As the total pair correlation function h(k; !; ! 0) is directly related to the susceptibility, then h(k; !; ! 0) also becomes in nite as k !0. In 3] it was shown from the Ornstein-Zernike (OZ) equation that h(k !0; !; ! 0) ! 1 corresponds to the existence of a unit eigenvalue of the integral operator C(k = 0; ! 1 ; ! 2 ) = 1=2 (! 1 ) R C(r; ! 1 ; ! 2 )dr 1=2 (! 2 ).One can see that the Lovett equation (1.4) for v(! 1 ) = const is an eigenequation for the operator C(k = 0; ! 1 ; ! 2 ) Z d! 1 C(k = 0; !; ! 1 ) i (k = 0; ! 1 ) = i (k = 0) i (k = 0; !) (1.5)   with eigenvector i (k = 0; !) = ?1=2(!)r !(!) and unit eigenvalue i (k = 0) = 1.Therefore, the direct correlation function and the single-particle distribution function complying with the Lovett and OZ equations treat correctly Goldstone modes in the system.But there are great di culties in the obtaining of a self-consistent solution to this system of equations.The most di cult confusion is connected with the treatment of the integrodi erential Lovett equation.In the next section we present the algebraic representation of the Lovett equation for an uniaxial uid in the form of exact relations.It will be shown that this representation correctly treats Goldstone modes.

Exact algebraic representation of the Lovett equation in uniaxial uids
In uniaxial uids the orientational distribution function f(!) is axially symmetric around a preferred direction n and depends only on the angle between the molecular orientation ! and n.It allows us to write f(!) in the form where the constant Z can be found from the normalization condition where P l (cos ) are the Legendre polynomials.
In this note we shall discuss the exact relations between S l , A l and where C mnl (r) are the coe cients of the orientational expansion of the direct correlation function C(r; ! 1 ; ! 2 ).
In the space-xed coordinate system with z-axis parallel to n the pair direct correlation function of linear molecules can be written in the form r is a separation-vector of molecules mass centres, !r being its orientation.
It should be noted, that for axially symmetric system + = .
For our purposes we use the Lovett-Mou-Bu -Wertheim equation which for anisotropic uids can be written in the form 5] where C(! 1 ; ! 2 ) = R C(r; ! 1 ; ! 2 )dr, r ! is the angular gradient operator for a linear molecule.The space-xed X, Y and Z components are given by r != il, where l is the angular momentum operator.Using the relations 4] (r ! ) y = l+ ?l? 2 ; (2.6) and expansions (2.1), (2.4) the y-component of (2.5) is obtained in the form: Taking into account that only independent of the azimuthal angle ' quantities yield nonzero average values and using the orthogonality of Y lm s, one gets the following matrix equation L = Ĉ Ŷ L; (2.9) where L is a column consisting of L M = p M(M + 1)A M , Ĉ and Ŷ are of order (N N) with matrix elements C 11 mn and Y 11 mn , where can be expressed via S l , N is a number of values of index m (or n) such that C 11 mn 6 = 0. Since the angular momentum operator l is hermitian we can write eq.
(2.5) in the equivalent form: (2.11) In the similar manner from (2.11) we can obtain a matrix equation for the coe cients of the function f(!) L = ĈP ; (2.12) where P contains order parameters of the system: P l = p l(l + 1) p 2l + 1 p 4 S l : (2.13) Thus, the symmetries of the angular gradient operator and the nematic system yield two matrix relations (2.9) and (2.12) which connect the system order parameters, zero Fourier transforms of the direct correlation function harmonics and coe cients A l of the single-particle distribution functions of the system.Joint use of (2.9) and (2.12) allows us to express f(!) via the order parameters only: L = Ŷ ?1 P (2.14)The obtained relations can be very useful for the calculation of the correlation functions of the anisotropic phase and serve to verify the tness of chosen models and approximations.For the illustration we suppose that the expansion (2.4) may contain at Y 00 (! R ) only Y m (! 1 ), Y n (! 2 ) with m; n = 0; 2. Then the eq.( 2.9) and ( 2 (2.17) where C 110(nem) 220 (r) is the corresponding harmonic of the direct correlation function for the nematic phase.It is worth to note that equations (2.15) determine an independent of A 2 relation between S 2 and S 4 since in this case A 2 is a single-valued function of S 2 (see (2.2)).The same situation is observed in the computer simulation for the system of thin hard platelets 8].The matrix Ĥ( ) is singular if det 1 ?pY ( ) Ĉ( ) pY ( ) = 0.The lat- ter condition is satis ed for = 1 in uniaxial uids due to the obtained relation (2.9).Thus, Goldstone modes are connected with the corresponding harmonics of the pair correlation function (h 110 mn0 (k = 0)).Since these harmonics are coupled with the transverse correlations of spins in magnetic uids and the correlations of the director transverse uctuations in nematics, the peculiarities h 110 mn0 (k = 0) ! 1 are responsible for a long-range character of the above-mentioned correlations in the ordered phases.

Z f(!)d! = 1 :
Spherical harmonics Y lm (!) satisfy the standard Condon and Shortley phase convention 4].The nematic ordering is de ned by the parameters S l = hP l (cos )i = Z d!f(!)P l (cos ); (2.2) and the rst of eq.(2.15) brings out the instability point of the isotropic phase (that is identical to the result of the density functional theory 6is the corresponding harmonic of the isotropic direct correlation function.As the single-particle distribution function in this case has the Mayer-Saupe form, we equate4 Y 11  22  with the rst non-trivial value of the self-consistency equation 4 Y 11 22 = 1:1142 with S 2 = 0:3236 7] and obtain the bifurcation point of the nematic solution for our model: