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On the validity of the elastic expansion of the free energy of nematic liquid crystals
IO May 1999 PHYSICS LETTERS A
Physics Letters A 255 (1999) 165-172
On the validity of the elastic expansion of the free energy of nematic liquid crystals S. Faetti INFM and Dipartimento di Fisica, Universita di Pisa, via Buonarroti 2, 56127 Piss, Italy
Received 11 January 1999; revised manuscript received 21 February 1999; accepted for publication 22 February 1999 Communicated by V.M. Agranovich
Abstract The elastic free energy Fe1of nematic liquid crystals is a truncated power expansion of the free energy F in the gradients of the director field n(r). In principle, the cut-off distortion length for &I could be estimated from the direct comparison between the elastic terms and the higher order ones. However, in the case of dispersion interactions, the higher order elastic constants diverge and this comparison is not possible. The divergence of the higher order constants poses some doubt about the internal consistency of the elastic expansion itself. In this paper, we use a molecular model of a nematic liquid crystal to calculate the total free energy and its elastic approximation. The two free energies are calculated numerically for some director configurations as a function of the characteristic distortion length Ldis. A good agreement between the two free energies is found for slow director fields, and large deviations are observed only if ,!& M a, where a is the molecular length. This result demonstrates the validity of the elastic expansion and shows that the higher order terms affect only the behavior at very short length scales. @ 1999 Elsevier Science B.V. PACS: 61.30.-v;
61.30.Gd; 61.3O.C~
1. Introduction
The macroscopic behavior of nematic liquid crystals is described by the director field n(r) which represents the average orientation of the long molecular axes [ 11. The equilibrium configuration of the director field corresponds to a minimum of the free energy F. For slowly and smoothly varying director fields, the bulk free energy is approximated by a truncated expansion in the first and second director gradients which is known as the elastic free energy [ 2-41. Here we restrict our attention to planar director distortions with the director field which lies everywhere in the Xz plane and depends only on the z-coordinate. Then, the director field has the form
R Z [sinf3(z),O,cos8(z)],
(1)
where 8 ( z ) is the local angle between the director and the z-axis. In such a planar case, the bulk elastic free energy density is
d
sin2e8
413~
--y[
1 9
(2)
where fb is an unexential constant term, KII, K33 and K13 are the splay, the bend and the splay-bend elastic constants, respectively and dots stand for a zderivative. The elastic free energy can be obtained with a phenomenological approach by looking at the more general expansion in the director gradients which sat-
0375-9601/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved. PII SO375-9601(99)00143-7
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Letters A 255 (1999) 165-I 72
isfies the symmetry properties of nematic liquid crystals. From a molecular point of view, it is possible to connect the elastic constants with the interparticles interaction [4-61. In this microscopic approach, the elastic form is obtained by the direct expansion of a molecular free energy. The elastic free energy F,t per unit surface area of a nematic slab is obtained by integration of fe, and is given by F-1 = 6 +
sin28, K13-
2
. 01 -
sin2e2 K13202
d/2
+
I
~[Kiisin2B+Ksscos281~2dz,
(3)
-d/2
where 01 and t$ are the director angles at the two surfaces z = -d/2 ( 1) and z = d/2 (2) of the nematic layer. Fb is the z-integral of the constant bulk homogeneous term fb in Eq. (2). According to Eq. (3)) the splay-bend elastic term affects only the surface free energy. For this reason, K13 is usually called the surface-like elastic constant. This constant leads to some paradoxical consequences and has been the object of some debate in the literature [ 7-211. Some years ago, Somoza and Tarazona [ 181 questioned the actual existence of the splay-bend elastic constant. They showed that the procedure to pass from the non-local molecular energy density to the local elastic energy density is not univocal but requires always the preliminary and arbitrary choice of a mapping. Of course, a physically well-defined elastic constant should be independent of this arbitrary choice. This is just what happens as far as the Frank-Oseen elastic constants K11, K22, Ks3 and K24 are concerned, while the splay-bend elastic constant Ki3 is mappingdependent [ 181. In particular, there exists a special mapping which gives KIT = 0. Then, Somoza and Tarazona concluded that the bulk elastic constant Krs is not a well-defined physical parameter. The argument above was criticized by Teixera et al. [ 61 and defended by Yokoyama [ 191. Using the functional density theory, and exploiting the symmetry properties of the interactions of nematic liquid crystals, Yokoyama reached the conclusion that the bulk splay-bend elastic constant must be zero. However, he also showed that a surface term, which is formally similar to the classical splay-bend term, is present in the surface free energy due to the interfacial effects. Specific contribu-
tions of the interfacial interactions to the surface elastic free energy were already introduced in Refs. [ 13,21231. In a recent paper, Faetti and Faetti [ 201 showed that the Somoza-Tarazona paradox does not occur if the interfacial interactions are taken into account in the elastic expansion. These interactions produce two contributions to the free energy: an anchoring energy W(t?,) [24] which depends only on the surface director angle 8, and an elastic interfacial term which has the same functional form as the Kis term. Then, the surface elastic constant Ki3 in Eq. (3) is the sum of a bulk and of an interfacial contribution. Singularly, these terms are mapping-dependent, but the total surface elastic constant does not depend on the mapping [ 201. This means that the separation of the surface elastic term into a bulk and a surface contribution is an arbitrary and subjective choice, while the total surface free energy is univocally defined. Then, it is always possible to make the Yokoyama choice of a vanishing bulk splay-bend constant, provided a suitable surface contribution is introduced. For simplicity of notation, we will still denote by Ki3 the total renormalized surface constant. Then, the total elastic energy per unit surface area of the nematic slab is: F,‘, = &“,-r + Fbulk = sin20t . + K13el 2
Fb +
w(h)
+
w(f92>
sin 2e2 . K13e2 2
d/2
+
I
i [K,, sin2 e + K33 cos2
01 tj2
dz.
(4)
-d/2
The expressions of K13 and W( 0,) in terms of the parameters of the intermolecular interaction were given in Refs. [20,23] for a generic two-body interaction law. The elastic energy in Eq. (4) was obtained by making a systematic expansion of the molecular energy in the derivatives of the director angle 0 and disregarding the terms of order higher than the square of the first derivative. In principle, the range of validity of this truncated expansion could be estimated by comparing the higher order elastic terms which are disregarded in the expansion with the classical elastic ones. This requires the calculation of the higher order elastic constants. However, in the important case of long-range dispersion forces, this calculation is not possible because the higher order elastic constants di-
S. Faetti/Physic.s
Letters A 255 (1999)
verge for an infinite nematic medium. This fact generates some doubt about the validity of the truncated elastic expansion of the mean field molecular energy and, in particular, on the effective presence of a nonvanishing Kis-term. Furthermore, it is not possible to estimate the cut-off distortion length L,, for the elastic energy using this procedure. This is, perhaps, the main reason for the presence in the literature of very different points of view on the role of the higher order terms. Some authors [ 11,12,14,15] retain that the higher order elastic terms become important only at very short scale lengths and, thus, the cut-off length is L,, M a, where a is the molecular length. Other authors [ 131 retain that L,, is a macroscopic length. We emphasize here that the knowledge of the actual effect of the higher order elastic terms is important as far as the equilibrium director field is concerned. Indeed, the Kis-constant in Eq. (4) makes the free energy unbounded from below [7,8] and no continuous director field can minimize the elastic free energy. This is known in the literature as the OldanoBarber0 paradox. The higher order elastic terms play a basic role because they bound the energy from below so that a minimizing director field actually exists. However, different assumptions on the effects of the higher order contributions lead to substantially different theoretical predictions about the equilibrium director field [ 11,13-15,171. In this paper we investigate the validity of the elastic expansion of the molecular energy using a numerical approach. We consider nematic molecules interacting through induced dipole-induced dipole interactions and we calculate the energy F and its elastic approximation Fe! for a nematic slab of thickness d. In this calculation, both the bulk and the interfacial contributions are taken into account. The energies F and Fe, which correspond to some director distortions are calculated numerically as a function of the characteristic distortion length L&s. The comparison between these two energies allows us to investigate the validity of the elastic expression proposed in Ref. [20] and the role of the higher order terms. A good agreement between the energy F and the elastic one is obtained for L&s >> a independently on the kind of director distortion. This result gives a direct proof for the validicy of the theoretical approach followed in Ref. [ 201 to obtain the elastic energy and the surface K,j-term. In particular, it is entirely confirmed that the free en-
167
165-172
ergy F is correctly approximated only constant Ki3 is different from zero and given in Ref. [ 201. Large deviations of ergy from the molecular one are only
if the surface has the value the elastic enobserved for
The molecular model and the elastic free energy Here we consider the case of a semi-infinite NLC lying in the semispace z < 0 and in contact with its vapor phase at z = 0. The NLC is assumed to have a perfect order (S = 1) so that the director at point t coincides with the long axis of the molecules at the same point. With this assumption, there is no microscopic orientational disorder and the free energy is reduced to the energy alone. Furthermore, the molecular density p is assumed to be uniform everywhere in the NLC. Finally we assume that the molecules have a spherical shape with diameter a. In this context, the anisotropy of the interactions which is responsible for the nematic order is only ascribed to the long-range attractive dispersion forces. All these assumptions have been extensively used in the literature to study the elastic properties of nematic liquid crystals [ 4,5,25-29 1. Let U(0(ri).ti(r2),ri - r2) be the interaction energy between two nematic molecules at points rl and r2, respectively. In the case of planar director fields, the free energy per unit surface area is a functional of /3(t) given by 0
0
F=
=
J
dz
J
-co
--m
0
00
J
--M
dz
J
‘%~(z),~(z’),z
- z’) dz’
G(B(z),B(z’),s)ds,
(5)
:
wheres=z-z’.FunctionG(B(z),B(z’),z-z’) defined as
is
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S. Faetti/Physics Letters A 255 (1999) 165-172
G(e(z),@(z’),z -z’) cc =-
:
I I dx’
dy’p*WB(z)Az’),r,
%L(i + j,O,),
(13)
i,j,k=O
- r2)
where iijk iS given by
-ca
xH(T-
2
w(e,) =
co
a),
(6) Cc
where ri = (x,y,z), r2 = (x’,y’,z’) and r = ]ri ral. H(r - a) is the step function: H = 0 for r < a and H = 1 for I > a. The step function H takes into account the impenetrability of molecules. The most general expression for the two-body interaction energy is [25]:
iijk = 47~
loo Jijk(T)(II'u)'(n'.u)j(n'n')',
c
L in Eq. (13) is a function of 8, which depends on the even number i + j. Typical values are: L(o,e,)
= 1,
L(2,e,)
=
w,e,)
=
requires i + j = forces, functions
where Aijk is a constant and t satisfies the condition t > 5. The bulk constants Kii, K33 and KF3 (the suffix b indicates the bulk elastic contribution to Ki3 calculated using the standard mapping) were obtained in Ref. [ 251. According to Ref. [ 201, for a semi-infinite nematic sample in contact with its vapor phase, the total of the surface constant Ki3 coincides with Kp3. The explicit expressions of these constants are: * 2 lijk ijk=O (i+j+l):i+j+3)’ I1
(9)
[i+y-l+‘]’
(10)
00
Jijk (r) r4 dr =
(17)
24
It.
u)(n’ . u) - (n. n’)]? (18)
This energy corresponds with A,l, = -12A,
to the case t = 6 in Eq. (8)
AW2 = 2A,
A220 = 18A.
(19)
Then, from Eqs. (9)-( 13), we get K1,=K33=K=-
8?rA 21u ’
K,3 = --,
167TA 35u
(20)
and = 2
[5+6C0s2es-3C0s4e,].
(21)
W( 0,) represents the excess of surface free energy of a sample aligned uniformly at the angle 8, and reaches its minimum value for 8, = n-/2, i.e. for a planar alignment of the director on the surface. The expansion procedure of Ref. [25] can also be used to obtain the bulk higher order elastic constants. All these constants can be expressed in terms of integrals of the kind
(11)
Iijk
K33 = ; 2 i,j,k=0(i+j+3)
4TTAijk (t - 5)a’-5 *
a
The anchoring
(16)
Consider, now, the Nehring-Saupe induced dipoleinduced dipole interaction energy [ 4 3:
w(fl,) 2 i.,k,(i+j+lyt!i+j+3)
I
’ +~*‘s,
U( n, n’, r) = - -+3(
Jijk(T) = %,
Iijk= 4?T
(15)
3 + 6 ~0s~ es -code,
where u = (rt - rz)/r. Symmetry even [ 251. In the case of dispersion J;jk( r) are given by
K,,=;
(14)
(7)
‘* ijk=O ..
K,3 = -4
4?TAijk (t - 4)&4’
a
U(n, n’, rl - r2) =--
Jijk (r) r3 dr =
I
(12)
M
Jijk ( r) rs dr,
Bijk = energy W( 0,) is [ 231:
I (1
(22)
169
S. Faetti/Physics Letters A 255 (1999) 165-i-172
with s 2 5. The parameters Bijk defined in Eq. (22) diverge because Jijk ( r) decays as 1/r6. This divergence leads to an analog divergence of the higher order elastic constants. Then, the truncation of the elastic free energy at the first quadratic contributions should be not correct. The divergence of the higher order elastic constants is the direct consequence of the intrinsically non-local character of the long-range dispersion interactions. However, the divergence of the higher order elastic constants holds for an infinite nematic sample. Only with this assumption, the upper extreme in the integral in Eq. (22) is 00. For a finite nematic slab, the higher order elastic constants in a point z are given by similar integrals where 00 is replaced by the distances from the two interfaces. Then, these higher order elastic constants are always finite and depend greatly on the z-coordinate. In such a case, the truncation of the elastic series is formally correct and the elastic energy is expected to approximate reasonably the molecular energy for sufficiently slow director fields. It is not evident, however, what is the cut-off length L,, for the elastic energy. 3. Numerical comparison
between the molecular
and the elastic energy
I+
_K,3sin28
s
-ev(-d/5)1 611 +eM-d/O1
(es-eb)u
’
(24)
Fig. la shows the energy per unit surface F = F ( 0 ( z ) ) - F ( 0,) and the elastic energy F,I versus the
Here we consider a nematic slab of thickness d with two nematic-vapor interfaces at z = -d/2 and z = d/2, respectively. We are interested to investigate the effect of the higher order elastic terms by a direct comparison between the molecular energy F and its elastic approximation. Consider the trial director distortion e(z) = & + (0, - &) x exp((z - d/2)/5)
much greater than that due to the director distortion. In such a case, the accuracy of the calculation of the distortion energy becomes insufficient. The accuracy is increased by many orders of magnitude, if the integralofG(e(z),e(z’),z -z’) -c(e,,e,,z -z’) is calculatedinplaceofthatofG(8(z),8(z’),z-z’). This corresponds to subtracting the energy of a nematic layer aligned uniformly at the angle BSwhich is given by F( 0,) = F,,+2W( 0,). We denote by F the resulting energy (F = F (8( z ) ) - F (0,) ) . Calculations are made for a macroscopic nematic slab having the thickness d = 104a, which corresponds to d M 10 pm ifa% 1 nm. The elastic energy F,I = Fi, - F(8,) is given by Eq. (3) with Fb = 0 and with Ktt = K33 = K and Ki3 given in Eq. (20). Substituting e(z) of Eq. (23) in Eq. (3), we find:
+ exp(-(t
+ d/2)/5)
exp( -d/O
(23) This is an even function which has the value BSat the two surfaces z = fd/2 and, for 5 < d, approaches eb in the bulk with a characteristic decay length 5. The energy F which corresponds to this director distortion can be calculated numerically making the double integration in Eq. (5). This calculation requires the substitution in Eq. (5) of the analytical expression of function G(B(z),S(z’),z - z’) which is given in Ref. [ 261. For slow director distortions (6 >> d) , the homogeneous part (& in Eq. (4) ) of the energy is
adimensional distortion rate k = a/[ when the surface angle 8, is equal to the easy angle Be = 7r/2 and the bulk angle is et, = 7r/4. In such a condition, sin 28, = 0 and the Ktj-term in Eq. (24) vanishes. Then, the elastic energy contains only the quadratic terms and increases as the distortion rate increases. The energies are expressed in adimensional units (F -+ Fp2a2/A). The estimated numerical accuracy on F is much better than lo-“. The full line corresponds to F and the broken line corresponds to Fe]. Fig. lb shows the relative difference A = (F - Fe]) /F versus k in a logarithmic scale. Figs. 2a,b show the analogous results obtained when the surface angle is different from the easy angle 0, = 7r/2. In particular, 8, = 7r/4 and 8b = 7r/2. In such a case, the linear elastic term (K13) is different from zero and the elastic energy is a linear decreasing function of k. The elastic constant Kt3 is a parameter which describes macroscopically this decreasing of the energy with k. The dotted line in Fig. 2a denotes the energy which is obtained using the elastic form, but assuming K13 = 0. In such a case, the elastic energy
170
S. Faetti/Physics
Letters A 255 (1999) 165-l 72
--
-1.0 0.0
0.2
0.4
0.6
lo-5 1o-4 10”
0.8
1.0
1o-2 10-l loo k
Fig. I. (a) Comparison between the molecular energy and the elastic energy of the director distortion in Eq. (24) versus the dimensional distortion rate k = a/,$. The surface and bulk angles are 6, = x/2 and l)h = 7r/4, respectively. The full line is the molecular energy 7 = F (8( z ) ) - F( 8, ), while the broken line corresponds to the elastic form. Adimensional units are used on the vertical axis(F -+ Fp*a’/A). (b) Relative difference d = (F - F,,/F) versus k. Logarithmic scales are used on both the axes. Some oscillations of A close to k x 10e5 ate produced by numerical noise which is estimated to be z 10-12. Indeed, the elastic free energy is a function which decreases greatly by decreasing k if k < u/d. For instance, F = 3 x lo-lo for k = 10e5, and 7 = 2 x 10e7 for k = 5 x lO-s.
is an increasing function of k. The difference between the broken and dotted lines represents the contribution of constant Kr3 to the elastic energy. Calculations with other values of 8, and @b lead to qualitatively similar results. A in Figs. lb and 2b does not go to zero for k -+ 0, but approaches a virtual constant minimum value Ain. This behavior can be understood if we recall that the expressions of Krr and K33 given in Eq. (20) were obtained for an infinite nematic sample, while here we consider a nematic slab of thickness d. For a finite
T
0.0
2a
I I
0.2
I I
I I
0.6
0.4
, I
0.8
1.0
k Fig. 2. (a) The same as in Fig. la but for & = 7r/4 and 6, = ?r/2. The dotted line represents the elastic energy obtained using the value of K = Ktl = K33 given in Eq. (20) but putting Kt3 = 0. (b) Relative difference A = (P - F,,/P) versus k for & = 7r/4 and 8b = 7r/2. Logarithmic scales are used on both the axes.
nematic slab, the local bulk elastic constants depend on the distance from the interfaces although this dependence becomes very small far from the interfaces. According to the theoretical analysis in Refs. [ 23,261, the local correction to the elastic constant is lAK[ x Ku/L, where L > a is the distance from the interface. Similar size-effects concern the surface elastic constant K,s. By assuming L z d/4 as the average distance of a bulk point from one of the interfaces, we estimate the relative variation of the elastic energy [Al = 4a/d M 4 x lo- 4 which is of the order of magnitude of Atin in Figs. Ib and 2b. This interpretation seems to be confirmed by the study of the dependence of Ak,, on the thickness d of the nematic layer. Fig. 3 shows such a dependence for 8, = 7r/4 and & = g/2. We see that Atin is proportional to a/d, for sufficiently small values of u/d. The higher order elastic terms should not be responsible for Akn because their contribution would be ex-
S. Faetri/Physics Lelters A 255 0999)
0 2
4 6 a/d
10~10‘~
Fig. 3. A,i, versus a/d in the case @, = 7714 and @h = r/2.
Points
represent the numerical results.
petted to increase by increasing the layer thickness (the higher order constants diverge for d + co), in contrast with the results shown in Fig. 3. For very small adimensional distortion rates k < 10e5, A shows a great increase by decreasing k. This effect could be due to the higher order elastic contributions. However, in this region, the energy 7 is smaller than 10-‘“, and the numerical uncertainties play an important role. Therefore, the observed effect is probably due to these numerical uncertainties. For this reason, the numerical results obtained for k < lop5 have not been reported in Figs. 1 and 2. We remind that k < lop5 corresponds here to a distortion length greater than 1Od z 100 pm. Figs. 1 and 2 have some important consequences. First of all, they demonstrate that the elastic energy F,t in Eq. (3) becomes very close to the free energy F for k + 0. This means that the elastic expression which was obtained in Ref. [20] and, in particular, the total surface elastic term (KIT) describes satisfactorily the behavior of the energy for slow and smooth director fields. Then, the truncation of the elastic series at the quadratic terms is essentially correct for a finite nematic slab, although the higher order elastic constants diverge for an infinite nematic sample. In particular, it is confirmed that the total contribution of the interfacial and bulk interactions leads to a nonvanishing value of the resulting surface elastic constant K13. If the surface splay-bend term would not be introduced, the elastic energy would differ greatly from the actual free energy also for k + 0 (look at the dotted and broken lines in Fig. 2a). Then, the surface constant KIT is not due to some artifact of the
165-I 72
171
truncated elastic expansion but reflects an actual and important energy contribution which characterizes the energy behavior for slow director distortions. According to Figs. lb and 2b, the elastic energy reproduces the molecular energy better than 3% if k < 0.01. The difference AF = F - Fe, increases slowly by increasing the distortion rate k and becomes comparable with Fe1 only for k M 1, that is for a distortion length of the order of magnitude of a molecular length. This means that the higher order elastic contributions play an important role only for director distortions which have a very short characteristic length, in agreement with the qualitative predictions of simple dimensional arguments. Similar results are obtained if the energies F and Fe1 are calculated for director distortions which are still characterized by a distortion length 5 but have a different functional form. For instance, we have studied the cases of sinusoidal distortions and Gaussian distortions. Also in these cases, a satisfactory agreement is found for k -+ 0, while significant differences are observed only if the distortion length becomes comparable to the molecular length. Then, the results above are general enough and are not a consequence of the special choice of a given trial function.
4. Conclusive
remarks
The numerical results in Section 2 show that the elastic energy which is obtained as a truncated expansion of a molecular energy represents a satisfactory approximation. Then, the divergence of the higher order elastic constants does not represent a reason for the failure of the truncated elastic expansion, at least in the case of a nematic layer and of planar director distortions. The elastic energy is very close to the molecular energy for slow director distortions and appreciable deviations are observed only at molecular length scales. In particular, the numerical results demonstrate that the KIT constant which was obtained in Ref. [ 201 taking into account the interfacial effects reproduces correctly the actual behavior of the molecular energy. Of course, the continuous molecular model used here is very crude due to the simplifying assumption of spherical molecules, of perfect nematic order and of uniform molecular density. More refined molecular models [ 61 or Molecular dynamics calculations [ 301
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S. Faetti/Physics
Letters A 255 (1999) 165-I72
should be needed to obtain reliable values of the bulk and surface elastic constants of nematic liquid crystals. We recall here that the surface elastic energy which is obtained by the Taylor expansion of a molecular energy should not be confused with the thermodynamic free energy which enters the continuum theory of nematic liquid crystals. This important and subtle theoretical aspect has been analyzed in some previous papers and will not be discussed here. For such a discussion and for the analysis of the physical meaning and consequences of the Kls-term we refer the reader to Section IV in Ref. [20] and references therein.
Acknowledgement
Many thanks are due to G. Barber0 for stimulating discussions and a critical reading of the manuscript. References
[Zl [31 [41 [51 [61
PG. de Gennes, The Physics of Liquid Crystals (Clamndon, Oxford, 1974). C.W. Oseen, Trans. Faraday Sot. 29 (1933) 833. EC. Frank, Discuss. Faraday Sot. 25 (1958) 19. J. Nehring, A. Saupe, J. Chem. Phys. 54 (1971) 337; 55 (1972) 5527. G. Vertogen, Physica A 117 (1983) 227. P1.C. Texeira, V.M. Pergamenshchick, T.J. Sluckin, Mol. Phys. 80 (1993) 1339.
I71 G. Barbero, C. Oldano, Mol. Cryst. Liq. Cryst. 170 (1989) 99. [8] C. Oldano, G. Barbero, Phys. L&t. A 110 (1985) 213. [9] HP Hinov, Mol. Cryst. Liq. Cryst. 148 (1987) 197. [lo] HP Hinov, Mol. Cryst. Liq. Cryst. 178 (1990) 53. [ 111 G. Barbero, A. Strigaui, Liq. Cryst. 5 (1989) 693. [ 121 G. Barbero, A. Sparavigna, A. Strigazzi, Nuovo Cim. D 12 (1990) 1259. [ 131 V.M. Pergamenshchick, Phys. Rev. E 48 (1993) 1254. [ 141 S. Faetti, Phys. Rev. E 49 (1994) 4192. [ 151 S. Faetti, Phys. Rev. E 49 (1994) 5332. [ 161 S. Stallinga, G. Vertogen, Phys. Rev. E 53 (1996) 1692. [ 171 V.M. Pergamenshchick, Phys. Rev. E 58 (1998) R16; Phys. Lea. A 243 (1998) 167. [18] A.M. Somoza, P. Tarazona, Mol. Phys. 72 (1991) 991. [ 191 H. Yokoyama, Phys. Rev. E 55 ( 1997) 2938. [20] M. Faetti, S. Faetti, Phys. Rev. E 57 (1998) 6741. [Zl ] G. Barhero, 2. Gabbasova, Yu.A. Kosevich, J. Phys. II (France) 1 (1991) 1505. [22] G. Barbero, S. Ponti, Phys. Len. A 239 (1998) 267. [23] S. Faetti, M. Riccardi, J. Phys. II (France) 5 (1995) 1165. [24] J. Cognard, Mol. Cryst. Liq. Cryst. sup. 1 ( 1982) 1. [25] G. Barbero, R. Barberi, in: The Physics of Liquid Crystalline Materials, eds. I.C. Khoo, F. Simoni (Gordon and Breach, Philadelphia, 1991) [26] S. Faetti, M. Nobili, J. Phys. II (France) 4 (1994) 1617. [27] P Galatola, C. Oldano, M. Rajteri, G. Barbero, Phys. Lett. A 210 (1996) 101. (281 M. Rajteri, G. Barbero, P. Galatola, C. Oldano, S. Faetti, Phys. rev. E 53 (1996) 6093. [29] G. Skacey, V.M. Pergamenshchik, A.L. Alexe-Ionescu, G. Barbero, S. Zumer, Phys. Rev. E 56 (1997) 571. [ 301 J. Stelzer, L. Longa, H.R. Tmbin, Mol. Cryst. Liq. Cryst. 262 (1995) 455.
Physics Letters A 255 (1999) 165-172
On the validity of the elastic expansion of the free energy of nematic liquid crystals S. Faetti INFM and Dipartimento di Fisica, Universita di Pisa, via Buonarroti 2, 56127 Piss, Italy
Received 11 January 1999; revised manuscript received 21 February 1999; accepted for publication 22 February 1999 Communicated by V.M. Agranovich
Abstract The elastic free energy Fe1of nematic liquid crystals is a truncated power expansion of the free energy F in the gradients of the director field n(r). In principle, the cut-off distortion length for &I could be estimated from the direct comparison between the elastic terms and the higher order ones. However, in the case of dispersion interactions, the higher order elastic constants diverge and this comparison is not possible. The divergence of the higher order constants poses some doubt about the internal consistency of the elastic expansion itself. In this paper, we use a molecular model of a nematic liquid crystal to calculate the total free energy and its elastic approximation. The two free energies are calculated numerically for some director configurations as a function of the characteristic distortion length Ldis. A good agreement between the two free energies is found for slow director fields, and large deviations are observed only if ,!& M a, where a is the molecular length. This result demonstrates the validity of the elastic expansion and shows that the higher order terms affect only the behavior at very short length scales. @ 1999 Elsevier Science B.V. PACS: 61.30.-v;
61.30.Gd; 61.3O.C~
1. Introduction
The macroscopic behavior of nematic liquid crystals is described by the director field n(r) which represents the average orientation of the long molecular axes [ 11. The equilibrium configuration of the director field corresponds to a minimum of the free energy F. For slowly and smoothly varying director fields, the bulk free energy is approximated by a truncated expansion in the first and second director gradients which is known as the elastic free energy [ 2-41. Here we restrict our attention to planar director distortions with the director field which lies everywhere in the Xz plane and depends only on the z-coordinate. Then, the director field has the form
R Z [sinf3(z),O,cos8(z)],
(1)
where 8 ( z ) is the local angle between the director and the z-axis. In such a planar case, the bulk elastic free energy density is
d
sin2e8
413~
--y[
1 9
(2)
where fb is an unexential constant term, KII, K33 and K13 are the splay, the bend and the splay-bend elastic constants, respectively and dots stand for a zderivative. The elastic free energy can be obtained with a phenomenological approach by looking at the more general expansion in the director gradients which sat-
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Letters A 255 (1999) 165-I 72
isfies the symmetry properties of nematic liquid crystals. From a molecular point of view, it is possible to connect the elastic constants with the interparticles interaction [4-61. In this microscopic approach, the elastic form is obtained by the direct expansion of a molecular free energy. The elastic free energy F,t per unit surface area of a nematic slab is obtained by integration of fe, and is given by F-1 = 6 +
sin28, K13-
2
. 01 -
sin2e2 K13202
d/2
+
I
~[Kiisin2B+Ksscos281~2dz,
(3)
-d/2
where 01 and t$ are the director angles at the two surfaces z = -d/2 ( 1) and z = d/2 (2) of the nematic layer. Fb is the z-integral of the constant bulk homogeneous term fb in Eq. (2). According to Eq. (3)) the splay-bend elastic term affects only the surface free energy. For this reason, K13 is usually called the surface-like elastic constant. This constant leads to some paradoxical consequences and has been the object of some debate in the literature [ 7-211. Some years ago, Somoza and Tarazona [ 181 questioned the actual existence of the splay-bend elastic constant. They showed that the procedure to pass from the non-local molecular energy density to the local elastic energy density is not univocal but requires always the preliminary and arbitrary choice of a mapping. Of course, a physically well-defined elastic constant should be independent of this arbitrary choice. This is just what happens as far as the Frank-Oseen elastic constants K11, K22, Ks3 and K24 are concerned, while the splay-bend elastic constant Ki3 is mappingdependent [ 181. In particular, there exists a special mapping which gives KIT = 0. Then, Somoza and Tarazona concluded that the bulk elastic constant Krs is not a well-defined physical parameter. The argument above was criticized by Teixera et al. [ 61 and defended by Yokoyama [ 191. Using the functional density theory, and exploiting the symmetry properties of the interactions of nematic liquid crystals, Yokoyama reached the conclusion that the bulk splay-bend elastic constant must be zero. However, he also showed that a surface term, which is formally similar to the classical splay-bend term, is present in the surface free energy due to the interfacial effects. Specific contribu-
tions of the interfacial interactions to the surface elastic free energy were already introduced in Refs. [ 13,21231. In a recent paper, Faetti and Faetti [ 201 showed that the Somoza-Tarazona paradox does not occur if the interfacial interactions are taken into account in the elastic expansion. These interactions produce two contributions to the free energy: an anchoring energy W(t?,) [24] which depends only on the surface director angle 8, and an elastic interfacial term which has the same functional form as the Kis term. Then, the surface elastic constant Ki3 in Eq. (3) is the sum of a bulk and of an interfacial contribution. Singularly, these terms are mapping-dependent, but the total surface elastic constant does not depend on the mapping [ 201. This means that the separation of the surface elastic term into a bulk and a surface contribution is an arbitrary and subjective choice, while the total surface free energy is univocally defined. Then, it is always possible to make the Yokoyama choice of a vanishing bulk splay-bend constant, provided a suitable surface contribution is introduced. For simplicity of notation, we will still denote by Ki3 the total renormalized surface constant. Then, the total elastic energy per unit surface area of the nematic slab is: F,‘, = &“,-r + Fbulk = sin20t . + K13el 2
Fb +
w(h)
+
w(f92>
sin 2e2 . K13e2 2
d/2
+
I
i [K,, sin2 e + K33 cos2
01 tj2
dz.
(4)
-d/2
The expressions of K13 and W( 0,) in terms of the parameters of the intermolecular interaction were given in Refs. [20,23] for a generic two-body interaction law. The elastic energy in Eq. (4) was obtained by making a systematic expansion of the molecular energy in the derivatives of the director angle 0 and disregarding the terms of order higher than the square of the first derivative. In principle, the range of validity of this truncated expansion could be estimated by comparing the higher order elastic terms which are disregarded in the expansion with the classical elastic ones. This requires the calculation of the higher order elastic constants. However, in the important case of long-range dispersion forces, this calculation is not possible because the higher order elastic constants di-
S. Faetti/Physic.s
Letters A 255 (1999)
verge for an infinite nematic medium. This fact generates some doubt about the validity of the truncated elastic expansion of the mean field molecular energy and, in particular, on the effective presence of a nonvanishing Kis-term. Furthermore, it is not possible to estimate the cut-off distortion length L,, for the elastic energy using this procedure. This is, perhaps, the main reason for the presence in the literature of very different points of view on the role of the higher order terms. Some authors [ 11,12,14,15] retain that the higher order elastic terms become important only at very short scale lengths and, thus, the cut-off length is L,, M a, where a is the molecular length. Other authors [ 131 retain that L,, is a macroscopic length. We emphasize here that the knowledge of the actual effect of the higher order elastic terms is important as far as the equilibrium director field is concerned. Indeed, the Kis-constant in Eq. (4) makes the free energy unbounded from below [7,8] and no continuous director field can minimize the elastic free energy. This is known in the literature as the OldanoBarber0 paradox. The higher order elastic terms play a basic role because they bound the energy from below so that a minimizing director field actually exists. However, different assumptions on the effects of the higher order contributions lead to substantially different theoretical predictions about the equilibrium director field [ 11,13-15,171. In this paper we investigate the validity of the elastic expansion of the molecular energy using a numerical approach. We consider nematic molecules interacting through induced dipole-induced dipole interactions and we calculate the energy F and its elastic approximation Fe! for a nematic slab of thickness d. In this calculation, both the bulk and the interfacial contributions are taken into account. The energies F and Fe, which correspond to some director distortions are calculated numerically as a function of the characteristic distortion length L&s. The comparison between these two energies allows us to investigate the validity of the elastic expression proposed in Ref. [20] and the role of the higher order terms. A good agreement between the energy F and the elastic one is obtained for L&s >> a independently on the kind of director distortion. This result gives a direct proof for the validicy of the theoretical approach followed in Ref. [ 201 to obtain the elastic energy and the surface K,j-term. In particular, it is entirely confirmed that the free en-
167
165-172
ergy F is correctly approximated only constant Ki3 is different from zero and given in Ref. [ 201. Large deviations of ergy from the molecular one are only
if the surface has the value the elastic enobserved for
The molecular model and the elastic free energy Here we consider the case of a semi-infinite NLC lying in the semispace z < 0 and in contact with its vapor phase at z = 0. The NLC is assumed to have a perfect order (S = 1) so that the director at point t coincides with the long axis of the molecules at the same point. With this assumption, there is no microscopic orientational disorder and the free energy is reduced to the energy alone. Furthermore, the molecular density p is assumed to be uniform everywhere in the NLC. Finally we assume that the molecules have a spherical shape with diameter a. In this context, the anisotropy of the interactions which is responsible for the nematic order is only ascribed to the long-range attractive dispersion forces. All these assumptions have been extensively used in the literature to study the elastic properties of nematic liquid crystals [ 4,5,25-29 1. Let U(0(ri).ti(r2),ri - r2) be the interaction energy between two nematic molecules at points rl and r2, respectively. In the case of planar director fields, the free energy per unit surface area is a functional of /3(t) given by 0
0
F=
=
J
dz
J
-co
--m
0
00
J
--M
dz
J
‘%~(z),~(z’),z
- z’) dz’
G(B(z),B(z’),s)ds,
(5)
:
wheres=z-z’.FunctionG(B(z),B(z’),z-z’) defined as
is
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S. Faetti/Physics Letters A 255 (1999) 165-172
G(e(z),@(z’),z -z’) cc =-
:
I I dx’
dy’p*WB(z)Az’),r,
%L(i + j,O,),
(13)
i,j,k=O
- r2)
where iijk iS given by
-ca
xH(T-
2
w(e,) =
co
a),
(6) Cc
where ri = (x,y,z), r2 = (x’,y’,z’) and r = ]ri ral. H(r - a) is the step function: H = 0 for r < a and H = 1 for I > a. The step function H takes into account the impenetrability of molecules. The most general expression for the two-body interaction energy is [25]:
iijk = 47~
loo Jijk(T)(II'u)'(n'.u)j(n'n')',
c
L in Eq. (13) is a function of 8, which depends on the even number i + j. Typical values are: L(o,e,)
= 1,
L(2,e,)
=
w,e,)
=
requires i + j = forces, functions
where Aijk is a constant and t satisfies the condition t > 5. The bulk constants Kii, K33 and KF3 (the suffix b indicates the bulk elastic contribution to Ki3 calculated using the standard mapping) were obtained in Ref. [ 251. According to Ref. [ 201, for a semi-infinite nematic sample in contact with its vapor phase, the total of the surface constant Ki3 coincides with Kp3. The explicit expressions of these constants are: * 2 lijk ijk=O (i+j+l):i+j+3)’ I1
(9)
[i+y-l+‘]’
(10)
00
Jijk (r) r4 dr =
(17)
24
It.
u)(n’ . u) - (n. n’)]? (18)
This energy corresponds with A,l, = -12A,
to the case t = 6 in Eq. (8)
AW2 = 2A,
A220 = 18A.
(19)
Then, from Eqs. (9)-( 13), we get K1,=K33=K=-
8?rA 21u ’
K,3 = --,
167TA 35u
(20)
and = 2
[5+6C0s2es-3C0s4e,].
(21)
W( 0,) represents the excess of surface free energy of a sample aligned uniformly at the angle 8, and reaches its minimum value for 8, = n-/2, i.e. for a planar alignment of the director on the surface. The expansion procedure of Ref. [25] can also be used to obtain the bulk higher order elastic constants. All these constants can be expressed in terms of integrals of the kind
(11)
Iijk
K33 = ; 2 i,j,k=0(i+j+3)
4TTAijk (t - 5)a’-5 *
a
The anchoring
(16)
Consider, now, the Nehring-Saupe induced dipoleinduced dipole interaction energy [ 4 3:
w(fl,) 2 i.,k,(i+j+lyt!i+j+3)
I
’ +~*‘s,
U( n, n’, r) = - -+3(
Jijk(T) = %,
Iijk= 4?T
(15)
3 + 6 ~0s~ es -code,
where u = (rt - rz)/r. Symmetry even [ 251. In the case of dispersion J;jk( r) are given by
K,,=;
(14)
(7)
‘* ijk=O ..
K,3 = -4
4?TAijk (t - 4)&4’
a
U(n, n’, rl - r2) =--
Jijk (r) r3 dr =
I
(12)
M
Jijk ( r) rs dr,
Bijk = energy W( 0,) is [ 231:
I (1
(22)
169
S. Faetti/Physics Letters A 255 (1999) 165-i-172
with s 2 5. The parameters Bijk defined in Eq. (22) diverge because Jijk ( r) decays as 1/r6. This divergence leads to an analog divergence of the higher order elastic constants. Then, the truncation of the elastic free energy at the first quadratic contributions should be not correct. The divergence of the higher order elastic constants is the direct consequence of the intrinsically non-local character of the long-range dispersion interactions. However, the divergence of the higher order elastic constants holds for an infinite nematic sample. Only with this assumption, the upper extreme in the integral in Eq. (22) is 00. For a finite nematic slab, the higher order elastic constants in a point z are given by similar integrals where 00 is replaced by the distances from the two interfaces. Then, these higher order elastic constants are always finite and depend greatly on the z-coordinate. In such a case, the truncation of the elastic series is formally correct and the elastic energy is expected to approximate reasonably the molecular energy for sufficiently slow director fields. It is not evident, however, what is the cut-off length L,, for the elastic energy. 3. Numerical comparison
between the molecular
and the elastic energy
I+
_K,3sin28
s
-ev(-d/5)1 611 +eM-d/O1
(es-eb)u
’
(24)
Fig. la shows the energy per unit surface F = F ( 0 ( z ) ) - F ( 0,) and the elastic energy F,I versus the
Here we consider a nematic slab of thickness d with two nematic-vapor interfaces at z = -d/2 and z = d/2, respectively. We are interested to investigate the effect of the higher order elastic terms by a direct comparison between the molecular energy F and its elastic approximation. Consider the trial director distortion e(z) = & + (0, - &) x exp((z - d/2)/5)
much greater than that due to the director distortion. In such a case, the accuracy of the calculation of the distortion energy becomes insufficient. The accuracy is increased by many orders of magnitude, if the integralofG(e(z),e(z’),z -z’) -c(e,,e,,z -z’) is calculatedinplaceofthatofG(8(z),8(z’),z-z’). This corresponds to subtracting the energy of a nematic layer aligned uniformly at the angle BSwhich is given by F( 0,) = F,,+2W( 0,). We denote by F the resulting energy (F = F (8( z ) ) - F (0,) ) . Calculations are made for a macroscopic nematic slab having the thickness d = 104a, which corresponds to d M 10 pm ifa% 1 nm. The elastic energy F,I = Fi, - F(8,) is given by Eq. (3) with Fb = 0 and with Ktt = K33 = K and Ki3 given in Eq. (20). Substituting e(z) of Eq. (23) in Eq. (3), we find:
+ exp(-(t
+ d/2)/5)
exp( -d/O
(23) This is an even function which has the value BSat the two surfaces z = fd/2 and, for 5 < d, approaches eb in the bulk with a characteristic decay length 5. The energy F which corresponds to this director distortion can be calculated numerically making the double integration in Eq. (5). This calculation requires the substitution in Eq. (5) of the analytical expression of function G(B(z),S(z’),z - z’) which is given in Ref. [ 261. For slow director distortions (6 >> d) , the homogeneous part (& in Eq. (4) ) of the energy is
adimensional distortion rate k = a/[ when the surface angle 8, is equal to the easy angle Be = 7r/2 and the bulk angle is et, = 7r/4. In such a condition, sin 28, = 0 and the Ktj-term in Eq. (24) vanishes. Then, the elastic energy contains only the quadratic terms and increases as the distortion rate increases. The energies are expressed in adimensional units (F -+ Fp2a2/A). The estimated numerical accuracy on F is much better than lo-“. The full line corresponds to F and the broken line corresponds to Fe]. Fig. lb shows the relative difference A = (F - Fe]) /F versus k in a logarithmic scale. Figs. 2a,b show the analogous results obtained when the surface angle is different from the easy angle 0, = 7r/2. In particular, 8, = 7r/4 and 8b = 7r/2. In such a case, the linear elastic term (K13) is different from zero and the elastic energy is a linear decreasing function of k. The elastic constant Kt3 is a parameter which describes macroscopically this decreasing of the energy with k. The dotted line in Fig. 2a denotes the energy which is obtained using the elastic form, but assuming K13 = 0. In such a case, the elastic energy
170
S. Faetti/Physics
Letters A 255 (1999) 165-l 72
--
-1.0 0.0
0.2
0.4
0.6
lo-5 1o-4 10”
0.8
1.0
1o-2 10-l loo k
Fig. I. (a) Comparison between the molecular energy and the elastic energy of the director distortion in Eq. (24) versus the dimensional distortion rate k = a/,$. The surface and bulk angles are 6, = x/2 and l)h = 7r/4, respectively. The full line is the molecular energy 7 = F (8( z ) ) - F( 8, ), while the broken line corresponds to the elastic form. Adimensional units are used on the vertical axis(F -+ Fp*a’/A). (b) Relative difference d = (F - F,,/F) versus k. Logarithmic scales are used on both the axes. Some oscillations of A close to k x 10e5 ate produced by numerical noise which is estimated to be z 10-12. Indeed, the elastic free energy is a function which decreases greatly by decreasing k if k < u/d. For instance, F = 3 x lo-lo for k = 10e5, and 7 = 2 x 10e7 for k = 5 x lO-s.
is an increasing function of k. The difference between the broken and dotted lines represents the contribution of constant Kr3 to the elastic energy. Calculations with other values of 8, and @b lead to qualitatively similar results. A in Figs. lb and 2b does not go to zero for k -+ 0, but approaches a virtual constant minimum value Ain. This behavior can be understood if we recall that the expressions of Krr and K33 given in Eq. (20) were obtained for an infinite nematic sample, while here we consider a nematic slab of thickness d. For a finite
T
0.0
2a
I I
0.2
I I
I I
0.6
0.4
, I
0.8
1.0
k Fig. 2. (a) The same as in Fig. la but for & = 7r/4 and 6, = ?r/2. The dotted line represents the elastic energy obtained using the value of K = Ktl = K33 given in Eq. (20) but putting Kt3 = 0. (b) Relative difference A = (P - F,,/P) versus k for & = 7r/4 and 8b = 7r/2. Logarithmic scales are used on both the axes.
nematic slab, the local bulk elastic constants depend on the distance from the interfaces although this dependence becomes very small far from the interfaces. According to the theoretical analysis in Refs. [ 23,261, the local correction to the elastic constant is lAK[ x Ku/L, where L > a is the distance from the interface. Similar size-effects concern the surface elastic constant K,s. By assuming L z d/4 as the average distance of a bulk point from one of the interfaces, we estimate the relative variation of the elastic energy [Al = 4a/d M 4 x lo- 4 which is of the order of magnitude of Atin in Figs. Ib and 2b. This interpretation seems to be confirmed by the study of the dependence of Ak,, on the thickness d of the nematic layer. Fig. 3 shows such a dependence for 8, = 7r/4 and & = g/2. We see that Atin is proportional to a/d, for sufficiently small values of u/d. The higher order elastic terms should not be responsible for Akn because their contribution would be ex-
S. Faetri/Physics Lelters A 255 0999)
0 2
4 6 a/d
10~10‘~
Fig. 3. A,i, versus a/d in the case @, = 7714 and @h = r/2.
Points
represent the numerical results.
petted to increase by increasing the layer thickness (the higher order constants diverge for d + co), in contrast with the results shown in Fig. 3. For very small adimensional distortion rates k < 10e5, A shows a great increase by decreasing k. This effect could be due to the higher order elastic contributions. However, in this region, the energy 7 is smaller than 10-‘“, and the numerical uncertainties play an important role. Therefore, the observed effect is probably due to these numerical uncertainties. For this reason, the numerical results obtained for k < lop5 have not been reported in Figs. 1 and 2. We remind that k < lop5 corresponds here to a distortion length greater than 1Od z 100 pm. Figs. 1 and 2 have some important consequences. First of all, they demonstrate that the elastic energy F,t in Eq. (3) becomes very close to the free energy F for k + 0. This means that the elastic expression which was obtained in Ref. [20] and, in particular, the total surface elastic term (KIT) describes satisfactorily the behavior of the energy for slow and smooth director fields. Then, the truncation of the elastic series at the quadratic terms is essentially correct for a finite nematic slab, although the higher order elastic constants diverge for an infinite nematic sample. In particular, it is confirmed that the total contribution of the interfacial and bulk interactions leads to a nonvanishing value of the resulting surface elastic constant K13. If the surface splay-bend term would not be introduced, the elastic energy would differ greatly from the actual free energy also for k + 0 (look at the dotted and broken lines in Fig. 2a). Then, the surface constant KIT is not due to some artifact of the
165-I 72
171
truncated elastic expansion but reflects an actual and important energy contribution which characterizes the energy behavior for slow director distortions. According to Figs. lb and 2b, the elastic energy reproduces the molecular energy better than 3% if k < 0.01. The difference AF = F - Fe, increases slowly by increasing the distortion rate k and becomes comparable with Fe1 only for k M 1, that is for a distortion length of the order of magnitude of a molecular length. This means that the higher order elastic contributions play an important role only for director distortions which have a very short characteristic length, in agreement with the qualitative predictions of simple dimensional arguments. Similar results are obtained if the energies F and Fe1 are calculated for director distortions which are still characterized by a distortion length 5 but have a different functional form. For instance, we have studied the cases of sinusoidal distortions and Gaussian distortions. Also in these cases, a satisfactory agreement is found for k -+ 0, while significant differences are observed only if the distortion length becomes comparable to the molecular length. Then, the results above are general enough and are not a consequence of the special choice of a given trial function.
4. Conclusive
remarks
The numerical results in Section 2 show that the elastic energy which is obtained as a truncated expansion of a molecular energy represents a satisfactory approximation. Then, the divergence of the higher order elastic constants does not represent a reason for the failure of the truncated elastic expansion, at least in the case of a nematic layer and of planar director distortions. The elastic energy is very close to the molecular energy for slow director distortions and appreciable deviations are observed only at molecular length scales. In particular, the numerical results demonstrate that the KIT constant which was obtained in Ref. [ 201 taking into account the interfacial effects reproduces correctly the actual behavior of the molecular energy. Of course, the continuous molecular model used here is very crude due to the simplifying assumption of spherical molecules, of perfect nematic order and of uniform molecular density. More refined molecular models [ 61 or Molecular dynamics calculations [ 301
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Letters A 255 (1999) 165-I72
should be needed to obtain reliable values of the bulk and surface elastic constants of nematic liquid crystals. We recall here that the surface elastic energy which is obtained by the Taylor expansion of a molecular energy should not be confused with the thermodynamic free energy which enters the continuum theory of nematic liquid crystals. This important and subtle theoretical aspect has been analyzed in some previous papers and will not be discussed here. For such a discussion and for the analysis of the physical meaning and consequences of the Kls-term we refer the reader to Section IV in Ref. [20] and references therein.
Acknowledgement
Many thanks are due to G. Barber0 for stimulating discussions and a critical reading of the manuscript. References
[Zl [31 [41 [51 [61
PG. de Gennes, The Physics of Liquid Crystals (Clamndon, Oxford, 1974). C.W. Oseen, Trans. Faraday Sot. 29 (1933) 833. EC. Frank, Discuss. Faraday Sot. 25 (1958) 19. J. Nehring, A. Saupe, J. Chem. Phys. 54 (1971) 337; 55 (1972) 5527. G. Vertogen, Physica A 117 (1983) 227. P1.C. Texeira, V.M. Pergamenshchick, T.J. Sluckin, Mol. Phys. 80 (1993) 1339.
I71 G. Barbero, C. Oldano, Mol. Cryst. Liq. Cryst. 170 (1989) 99. [8] C. Oldano, G. Barbero, Phys. L&t. A 110 (1985) 213. [9] HP Hinov, Mol. Cryst. Liq. Cryst. 148 (1987) 197. [lo] HP Hinov, Mol. Cryst. Liq. Cryst. 178 (1990) 53. [ 111 G. Barbero, A. Strigaui, Liq. Cryst. 5 (1989) 693. [ 121 G. Barbero, A. Sparavigna, A. Strigazzi, Nuovo Cim. D 12 (1990) 1259. [ 131 V.M. Pergamenshchick, Phys. Rev. E 48 (1993) 1254. [ 141 S. Faetti, Phys. Rev. E 49 (1994) 4192. [ 151 S. Faetti, Phys. Rev. E 49 (1994) 5332. [ 161 S. Stallinga, G. Vertogen, Phys. Rev. E 53 (1996) 1692. [ 171 V.M. Pergamenshchick, Phys. Rev. E 58 (1998) R16; Phys. Lea. A 243 (1998) 167. [18] A.M. Somoza, P. Tarazona, Mol. Phys. 72 (1991) 991. [ 191 H. Yokoyama, Phys. Rev. E 55 ( 1997) 2938. [20] M. Faetti, S. Faetti, Phys. Rev. E 57 (1998) 6741. [Zl ] G. Barhero, 2. Gabbasova, Yu.A. Kosevich, J. Phys. II (France) 1 (1991) 1505. [22] G. Barbero, S. Ponti, Phys. Len. A 239 (1998) 267. [23] S. Faetti, M. Riccardi, J. Phys. II (France) 5 (1995) 1165. [24] J. Cognard, Mol. Cryst. Liq. Cryst. sup. 1 ( 1982) 1. [25] G. Barbero, R. Barberi, in: The Physics of Liquid Crystalline Materials, eds. I.C. Khoo, F. Simoni (Gordon and Breach, Philadelphia, 1991) [26] S. Faetti, M. Nobili, J. Phys. II (France) 4 (1994) 1617. [27] P Galatola, C. Oldano, M. Rajteri, G. Barbero, Phys. Lett. A 210 (1996) 101. (281 M. Rajteri, G. Barbero, P. Galatola, C. Oldano, S. Faetti, Phys. rev. E 53 (1996) 6093. [29] G. Skacey, V.M. Pergamenshchik, A.L. Alexe-Ionescu, G. Barbero, S. Zumer, Phys. Rev. E 56 (1997) 571. [ 301 J. Stelzer, L. Longa, H.R. Tmbin, Mol. Cryst. Liq. Cryst. 262 (1995) 455.