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Spontaneous surface polarization at the liquid crystal-substrate interface
PhysicsLettersAl7O(1992)4l—44 North-Holland
PHYSICS LETTERS A
Spontaneous surface polarization at the liquid crystal—substrate interface G. Barbero Dipartimento di Fjsica delPolitecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
and Yu.A. Kosevich All-Union Surface and Vacuum Research Centre,Andreyevskaya Nab. 2, 117334 Moscow, Russian Federation Received 1 June 1992; revised manuscript received 31 July 1992; accepted for publication 19 August 1992 Communicated by J. Flouquet
The spontaneous surface polarization at the surface between a nematic liquid crystal and a solid substrate is predicted. It is shown that it can be due to the second order elastic constant of the K 13 type, weak surface anchoring on the solid substrate and flexoelectric properties of the considered liquid crystal. Moreover it is shown that this surface polarization is proportional to the surface scalar order parameter, and that it vanishes for homeotropic or planar alignment. The temperature dependence of the surface polarization is obtained by using a simple form of the nematic-.substrate surface energy, responsible for the orientation induced on the nematic by the solidsubstrate. Some special cases are discussed. In particular it is shown that in a special case the surface polarization can be nearly independent ofthe temperature.
The well-known origin of the bulk flexoelectricity in nematic liquid crystals (LC) is a nonuniformity of the spatial distribution of the nematic director ii. It gives a bulk polarization P proportional to the spatial derivatives of n [1,21. In a recent paper [3] it was predicted that a near-surface nonhomogeneous distribution oforder the nematic director can be Kinduced by the second elastic constant of the and a weak surface anchoring on the substrate.13Intype this Letter we show that in LC such a near-surface distortion leads to a spontaneous normal surface polarization at a solid—LC interface. This spontaneous surface polarization is proportional to the surface scalar order parameter, it depends on the interfacial interaction and goes to zero at the free surface of the LC. The considered phenomenon can explain (at least qualitatively) the origin of the observed [4] temperature independent polar ordering of the LC molecules in a monolayer at the LC—glass interface, which is absent at the free surface of LC. Moreover two boundaries of the LC layer with asymmetric in-
terfaces can produce a nonzero total polarization of the layer even in the absence of an external electric field, in contrast to the consideration reported in ref.
[51. For the macroscopic description of the surface polarization of LC, let us assume3Ra d3R’, two-body interacbetween the tion of the kind f( n, n’, r) d 3R, d3R’ characterized by is = is (R) volume and ii’ =elements is (R’ = Rd+ r). In this frame, in the case in which is changes slowly over a distance of the order of the range of the molecular forces giving rise to the nematic phase,f(n, is’, r) can be expanded in a power series of the spatial derivatives of is. Standard calculations allow one to evaluate the elastic bulk energy density j~,at a given point of the sample by integrating jfover r. In this way fb is deduced in terms of the first and second order spatial derivatives of is, known as the Frank energy density for LC. In the Frank expression for the bulk elastic energy density appear five elastic constants: K 11 (i = 1, 2, 3) and K24, K1 These elastic constants can be expressed in terms ~.
0375-9601/92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
41
Volume 170, number 1
PHYSICS LETTERSA
of the integrals off(n, is, r), (af/8n~)~~=~ and (02f/ ôn~)~= “,M~ ~ over d 3r. It follows that the elastic constants are independent of is only in the bulk, whereas an explicit dependence on the boundary surface and on is is expected in a boundary layer of thickness d. In the case in which the limiting surface is flat and is remains in a plane, and consequently it can be characterized by a tilt angle 0, n = (sin 0, 0, cos 0), it is possible to introduce an effective surface energy f 5=f~N.N) +f~’~.In this expression f~N.N)=f~fb dz is due to the incomplete nematic— nematic interaction, and f ~N,s) is the true surface energy due to the direct nematic—substrate interaction. In the hypothesis in which the bulk first spatial derivative is negligible with respect to the surface gradient of 0 we obtain [3] a generalized version of the Rapini—Papoular expression of the surface free energy f~for nematic LC: f,=~W(O
2 0—6’)
+ (/3+Ee)00(O. —0~)+ ~y(O, _0~)2.
(1)
In expression (1) the surface energy is characterized by two angles: 0~= 0(0) is the tilted angle at the geometrical interface z = 0 and 0. = 0 ( z = d) is that at the end of the “diffuse” near-surface layer (of finite microscopic thickness d) of the LC. The parameter w and the angle 9 describe the anchoring energy and the easy axis orientation at the interface in the limit of weak anchoring. The parameters y and Ji describe, respectively, the first and the second order surface— bulkinteractions and are proportional to the K33 and K~3,K1 elastic moduli, respectively [3]. They are given by 13=—+/d,
y=/d,
where ( > denotes the average evaluated over a layer of thickness d of the order of the interaction range of the molecular forces giving rise to the nematic phase, Let us consider now the validity of the macroscopic approach to the surface problems presented above. As is well known various intermolecular forces are responsible for the nematic phase. Among others, for instance, forces connected to steric interactions and forces coming from induced dipole— induced dipole interaction. In general some forces, 42
19 October 1992
like the ones due to steric interactions, are short range, whereas others, like the induced dipole—induced dipole interaction, are long range and depend on r as r The parameter d introduced before is the largest interaction range among the forces responsible for the elastic properties of the LC. Consequently the validity of the macroscopic approach has to be tested by considering the interaction range of the dispersion forces, which is of the order of hundred ~.
Angstrom, i.e. quasi macroscopic. From this observation it follows that our macroscopic approach, by means of which the effective surface energy has been introduced, works well. A few words about the physical meaning of 00 0. can clarify the role of the nematic—nematic interaction in the nematic surface orientation. To do this let us reconsider the above. two-body interaction f( n, (a) is’, 3Rd3R’ discussed In the case in which r) d f(n, n’, r) =g(r)h(n~n’),i.e. f(n, is’, r), is independent of the direction of r, K 1 and K13 are identically zero. Consequently f3~ 0, and 0~= O~as shown in ref. [3]. On the contrary in the case in which (b) f(n, is’, r)=f(n.n’, is~r,n’~r),K13~’0and K1#0. giving 00 ~ O~.These analytical results contained in our theory can be easily understood by the following simple argument. Let us consider a molecule at the point R, oriented along n, interacting with the molecules within the interaction sphere, which are supposed perfectly aligned along a unique direction n’. In order to obtain nematic order the minimum energy of the considered molecule must occur for n = is’. If the interaction energy is of type (a) the minimum is obtained for n’ =n. However in the case in which f is of kind (b) this minimum is obtained by averaging all two-body energies, which minimum occurs, in general, for is’ obliquely oriented with respect to n. If now some surrounding molecules are absent, because the considered molecule is near the boundary, the minimum ofthe interaction energy generally occurs for is = is’ for an interaction energy of type (a), whereas for one of kind (b) it can occur for n ~ is’. In the first case 0~= 0~,and in the second case a boundary “discontinuity” of 0, localized over d is expected. In expression (1), in addition to the one proposed in ref. [31,we took into consideration also an external electric field E directed along the normal to the interface. The parameter e is the effective surface
Volume 170, number 1
PHYSICS LETTERS A
flexoelectric coefficient (which is of the order of the bulkone). In the surface energyj given by (1) we neglect the electrostatic energy connected with the surface polarization. This approximation works well because, as will be shown later, the surface polarization is proportional to the square of the surface tilt angle. Consequently the electrostatic energy, proportional to the square of the polarization, is found to be proportional to the fourth power of the surface tilt angle, and hence negligible in our second order approximation. The values of 0~and O~,considered as independent surface parameters, are determined by minimizing the eq. total(1)) (thefree Frank bulkfb [1,21 by energy at z= 0: and surfacef. given
19 October 1992
(5)
pO=_et~92
Another feature of p~is its linear dependence on the modulus ofthe order parameter and thus a weak dependence on the temperature, unlike the surface order-electric polarization which is of different nature [6,71. This linear dependence of p. on the scalar order parameter follows directly from eq. (5), valid in the strong anchoring case. In fact, the ratio fl/y is nearly independent of 5, as is clear from the definition of the surface parameters reported above. Hence, by taking into account that e= e0S, where S is the ne0°—~Pso5, matic Pso scalar order parameter, we obtain part p5 of the where is the temperature independent
of~,’ao,—of~/oo’=0, af~/oo
0=0,
(2)
where 0’ = ô0/8z. Then the considered spontaneous surface polarization p~(per unit area) is determined as the derivative of the surface energyJ with respect to the electric field E (in the limit of vanishingly small E). In this limit we can neglect in (2) the bulk distortions of the nematic director, and easily obtain for v~the expression 2 Wy)]2. (3) p5 = 8f5/t9E= e/Jy [ W8/ (p —
—
—
Expression (3) can be obtained also in another way. The flexoelectric polarization is P= e1 1n div ,, e33isxrotn. our unidimensional caseinthe z cornponent of P In is found to be P~ = e00’, second order of 0, where e=e 11 +e33. In the surface layer the variation of 0 is 1.~0=0~—0~ = (fly) 0~,as shown in ref. [3]. Hence 0’ —~Li0/d,giving P~—~ The surface polarization p~is obtained by—e(E~~0/d)0. integrating P~over d, and it is found to be p <0> ~(0~+ 0~) 0~.It follows5that et~0<0>, where —
—
—
-~
~
_e~0o2.
—~
‘~ —
(4)
As shown in ref. [31, Oo = (1 —fl2/y W) _19~ By substituting this expression for 0~in (4) we reobtain (3). As is seen from (3), the spontaneous surface polarization p 5 is considerably the surface anchoring energy Wanddetermined in the limitbyW=0, p~
surface polarization. It is important to underline that the scalar order parameter we are considering is the surface order parameter, whose temperature dependence is different from that in the bulk, as discussed in ref. [81. The conclusion about the dependence of p5 on T holds in the case in which the easy axis orientation at the interface 9 can be considered ternperature independent. In the opposite case where 99(T) the However, temperature dependence of p5ofisstrong more complicated. at least in the case anchoring the hypothesis that 9 is independent of T is reasonable. In the event of weak anchoring the p 5 versus T function can be deduced by supposing that 2(T), y(T)=G52(T), as is well known, /J(T)=B5 and furthermore W( T) =AS( T) + CS2 ( T), where B, G, A and C are temperature independent. In the expression fori.e.W(from T) the term, AS( T), comes 1’1’~, the first direct interaction of the from f~ nernatic with the substrate [8], whereas the second 2 ( T), is due partially to the direct nematic— term, CS interaction and to the incomplete nernatic— substrate nematic interaction [9]. In the present case we can still imagine 9 temperature independent. In fact the actual easy direction depends on the value of W, which we are considering T-dependent. By substituting the expressions for fl( T), y( T) and W( T) into eq. (4) one obtains 1
p 5(T)=p50
goes to zero. Furthermore in the limit W—. c (strong anchoring case) p5 is
—
2 s B GCA/C+S)
where p5~coincides with that introduced before. 43
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PHYSICS LETTERS A
Note that in case A =0 (no linear term in S is present in W( T)) the temperature behaviour of p5 ( T) is similar to that deduced in the strong anchoring case. Furthermore in the case in which A and C have opposite sign, p5 could be nearly independent of T. Note that the signs of A and C strongly depend on the LC—substrate interface [8,91. In the thin layer of LC with symmetric interfaces the spontaneous surface polarizations at the upper and lower sides exactly cancel, while for asymmetric interfaces a net polarization exists (dueoftoanthe dif2) even in the absence exterference of (W9) nal electric field. The near-surface gradients and the thickness of the surface layer in which p 5 exists do not directly enter in the proposed theory and therefore it is applicable even for very thin polarized layers [4]. The theory of electrodynamic properties of two-dimensional transient layers can be used for the calculation of the phase shift of the light transmitted across the thin bounded LC layer [10]. In conclusion we state that in the case in which a surface distortion of an average nematic molecular orientation is present, a surface electnc polarization may appear. The considered surface distortion is due to the surface second order elastic modulus of the K13 type, which is hence the source of the surface polarization. As shown this surface polarization exists only if the actual orientation imposed by the surface is tilted (neither homeotropic, nor planar). Of course other mechanisms could give rise to surface polarization, as discussed in ref. [11]. More precisely, surface polarization may be due to: (i) ferroelectric order at the surface, connected to the different chemical affinities of the two ends of the nematic molecules with the solid substrate, (ii) order electric polarization, coming from the spatial variation of the scalar order parameter imposed by the surface, (iii) selective ion adsorption. However, these types of surface polarizations are
44
19 October 1992
different from the one considered in our paper, because they can exist also in uniform samples. The surface polarization considered in the present Letter is similar to the one discussed by Tripathi et al. [12]. In a recent paper [12] an experimental observation of a spontaneous electric polarization very near a chiral nematic substrate interface with a non-zero surface tilt has been reported. From a phenomenological point of view this interesting observation is closely related to our prediction (see eq. (3)). However, far asofthe is oriented in thesoplane theobserved interfacepolarization and a net polarization exists in a cell with symmetric interfaces, this spontaneous two-dimensional electric polarization has a (symmetry breaking) nature different from that predicted in our paper.
References [1] PG.
de Gennes, The physics of liquid crystals (Clarendon, Oxford, 1974). 121 S. Chandrasekhar, Liquid crystals (Cambridge Univ. Press, Cambridge 1976). 131 G. Barbero, Z. Gabbasova and Yu.A. Kosevich, J. Phys. II (Paris) 1(1991)1505. [4] P. Guyot-Sionnest, H. Hsiung and YR. Shen, Phys. Rev. Lett. 57 (1986) 2963. [5] S.D. Lee and J.S. Pastel Phys. Rev. Lett. 65 (1990) 56. 161 G. Barbero, I. Dozov, J.F. Palierne and G. Durand, Phys. Rev. Lett. 56 (1986) 2056.
[71BE. Vugmeister and Yu.A.
Kosevich, Soy. Phys. Solid State 31(1989)1871. [8] Ping Sheng, Phys. Rev. A 26 (1982) 1610. [91A. Goossens, Mol. Cryst. Liq. Cryst. 124 (1985) 305; T.J. Sluckin and A. Poniewierski, in: Fluid interfacial phenomena, ed. C.A. Croxton (Wiley, New York, 1986); G. Barbero, Z. Gabbasova and M.A. Osipov, J. Phys. II (Paris) 1(1991)691. [10] Yu.A. Kosevich, Soy. Phys. JETP 69 (1989) 200. [11] G. Barbero, A.N. Chuvyrov, A.P. Krekhov andO.A. Scaldin, J. AppI. Phys. 69 (1991) 6343. [12] S. Tripathi, Mm Hua Lu, E.M. Terentjev, R.G. Petschek and C. Rosenblatt, Phys. Rev. Lett. 67 (1991) 3400.
PHYSICS LETTERS A
Spontaneous surface polarization at the liquid crystal—substrate interface G. Barbero Dipartimento di Fjsica delPolitecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
and Yu.A. Kosevich All-Union Surface and Vacuum Research Centre,Andreyevskaya Nab. 2, 117334 Moscow, Russian Federation Received 1 June 1992; revised manuscript received 31 July 1992; accepted for publication 19 August 1992 Communicated by J. Flouquet
The spontaneous surface polarization at the surface between a nematic liquid crystal and a solid substrate is predicted. It is shown that it can be due to the second order elastic constant of the K 13 type, weak surface anchoring on the solid substrate and flexoelectric properties of the considered liquid crystal. Moreover it is shown that this surface polarization is proportional to the surface scalar order parameter, and that it vanishes for homeotropic or planar alignment. The temperature dependence of the surface polarization is obtained by using a simple form of the nematic-.substrate surface energy, responsible for the orientation induced on the nematic by the solidsubstrate. Some special cases are discussed. In particular it is shown that in a special case the surface polarization can be nearly independent ofthe temperature.
The well-known origin of the bulk flexoelectricity in nematic liquid crystals (LC) is a nonuniformity of the spatial distribution of the nematic director ii. It gives a bulk polarization P proportional to the spatial derivatives of n [1,21. In a recent paper [3] it was predicted that a near-surface nonhomogeneous distribution oforder the nematic director can be Kinduced by the second elastic constant of the and a weak surface anchoring on the substrate.13Intype this Letter we show that in LC such a near-surface distortion leads to a spontaneous normal surface polarization at a solid—LC interface. This spontaneous surface polarization is proportional to the surface scalar order parameter, it depends on the interfacial interaction and goes to zero at the free surface of the LC. The considered phenomenon can explain (at least qualitatively) the origin of the observed [4] temperature independent polar ordering of the LC molecules in a monolayer at the LC—glass interface, which is absent at the free surface of LC. Moreover two boundaries of the LC layer with asymmetric in-
terfaces can produce a nonzero total polarization of the layer even in the absence of an external electric field, in contrast to the consideration reported in ref.
[51. For the macroscopic description of the surface polarization of LC, let us assume3Ra d3R’, two-body interacbetween the tion of the kind f( n, n’, r) d 3R, d3R’ characterized by is = is (R) volume and ii’ =elements is (R’ = Rd+ r). In this frame, in the case in which is changes slowly over a distance of the order of the range of the molecular forces giving rise to the nematic phase,f(n, is’, r) can be expanded in a power series of the spatial derivatives of is. Standard calculations allow one to evaluate the elastic bulk energy density j~,at a given point of the sample by integrating jfover r. In this way fb is deduced in terms of the first and second order spatial derivatives of is, known as the Frank energy density for LC. In the Frank expression for the bulk elastic energy density appear five elastic constants: K 11 (i = 1, 2, 3) and K24, K1 These elastic constants can be expressed in terms ~.
0375-9601/92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
41
Volume 170, number 1
PHYSICS LETTERSA
of the integrals off(n, is, r), (af/8n~)~~=~ and (02f/ ôn~)~= “,M~ ~ over d 3r. It follows that the elastic constants are independent of is only in the bulk, whereas an explicit dependence on the boundary surface and on is is expected in a boundary layer of thickness d. In the case in which the limiting surface is flat and is remains in a plane, and consequently it can be characterized by a tilt angle 0, n = (sin 0, 0, cos 0), it is possible to introduce an effective surface energy f 5=f~N.N) +f~’~.In this expression f~N.N)=f~fb dz is due to the incomplete nematic— nematic interaction, and f ~N,s) is the true surface energy due to the direct nematic—substrate interaction. In the hypothesis in which the bulk first spatial derivative is negligible with respect to the surface gradient of 0 we obtain [3] a generalized version of the Rapini—Papoular expression of the surface free energy f~for nematic LC: f,=~W(O
2 0—6’)
+ (/3+Ee)00(O. —0~)+ ~y(O, _0~)2.
(1)
In expression (1) the surface energy is characterized by two angles: 0~= 0(0) is the tilted angle at the geometrical interface z = 0 and 0. = 0 ( z = d) is that at the end of the “diffuse” near-surface layer (of finite microscopic thickness d) of the LC. The parameter w and the angle 9 describe the anchoring energy and the easy axis orientation at the interface in the limit of weak anchoring. The parameters y and Ji describe, respectively, the first and the second order surface— bulkinteractions and are proportional to the K33 and K~3,K1 elastic moduli, respectively [3]. They are given by 13=—
y=
where ( > denotes the average evaluated over a layer of thickness d of the order of the interaction range of the molecular forces giving rise to the nematic phase, Let us consider now the validity of the macroscopic approach to the surface problems presented above. As is well known various intermolecular forces are responsible for the nematic phase. Among others, for instance, forces connected to steric interactions and forces coming from induced dipole— induced dipole interaction. In general some forces, 42
19 October 1992
like the ones due to steric interactions, are short range, whereas others, like the induced dipole—induced dipole interaction, are long range and depend on r as r The parameter d introduced before is the largest interaction range among the forces responsible for the elastic properties of the LC. Consequently the validity of the macroscopic approach has to be tested by considering the interaction range of the dispersion forces, which is of the order of hundred ~.
Angstrom, i.e. quasi macroscopic. From this observation it follows that our macroscopic approach, by means of which the effective surface energy has been introduced, works well. A few words about the physical meaning of 00 0. can clarify the role of the nematic—nematic interaction in the nematic surface orientation. To do this let us reconsider the above. two-body interaction f( n, (a) is’, 3Rd3R’ discussed In the case in which r) d f(n, n’, r) =g(r)h(n~n’),i.e. f(n, is’, r), is independent of the direction of r, K 1 and K13 are identically zero. Consequently f3~ 0, and 0~= O~as shown in ref. [3]. On the contrary in the case in which (b) f(n, is’, r)=f(n.n’, is~r,n’~r),K13~’0and K1#0. giving 00 ~ O~.These analytical results contained in our theory can be easily understood by the following simple argument. Let us consider a molecule at the point R, oriented along n, interacting with the molecules within the interaction sphere, which are supposed perfectly aligned along a unique direction n’. In order to obtain nematic order the minimum energy of the considered molecule must occur for n = is’. If the interaction energy is of type (a) the minimum is obtained for n’ =n. However in the case in which f is of kind (b) this minimum is obtained by averaging all two-body energies, which minimum occurs, in general, for is’ obliquely oriented with respect to n. If now some surrounding molecules are absent, because the considered molecule is near the boundary, the minimum ofthe interaction energy generally occurs for is = is’ for an interaction energy of type (a), whereas for one of kind (b) it can occur for n ~ is’. In the first case 0~= 0~,and in the second case a boundary “discontinuity” of 0, localized over d is expected. In expression (1), in addition to the one proposed in ref. [31,we took into consideration also an external electric field E directed along the normal to the interface. The parameter e is the effective surface
Volume 170, number 1
PHYSICS LETTERS A
flexoelectric coefficient (which is of the order of the bulkone). In the surface energyj given by (1) we neglect the electrostatic energy connected with the surface polarization. This approximation works well because, as will be shown later, the surface polarization is proportional to the square of the surface tilt angle. Consequently the electrostatic energy, proportional to the square of the polarization, is found to be proportional to the fourth power of the surface tilt angle, and hence negligible in our second order approximation. The values of 0~and O~,considered as independent surface parameters, are determined by minimizing the eq. total(1)) (thefree Frank bulkfb [1,21 by energy at z= 0: and surfacef. given
19 October 1992
(5)
pO=_et~92
Another feature of p~is its linear dependence on the modulus ofthe order parameter and thus a weak dependence on the temperature, unlike the surface order-electric polarization which is of different nature [6,71. This linear dependence of p. on the scalar order parameter follows directly from eq. (5), valid in the strong anchoring case. In fact, the ratio fl/y is nearly independent of 5, as is clear from the definition of the surface parameters reported above. Hence, by taking into account that e= e0S, where S is the ne0°—~Pso5, matic Pso scalar order parameter, we obtain part p5 of the where is the temperature independent
of~,’ao,—of~/oo’=0, af~/oo
0=0,
(2)
where 0’ = ô0/8z. Then the considered spontaneous surface polarization p~(per unit area) is determined as the derivative of the surface energyJ with respect to the electric field E (in the limit of vanishingly small E). In this limit we can neglect in (2) the bulk distortions of the nematic director, and easily obtain for v~the expression 2 Wy)]2. (3) p5 = 8f5/t9E= e/Jy [ W8/ (p —
—
—
Expression (3) can be obtained also in another way. The flexoelectric polarization is P= e1 1n div ,, e33isxrotn. our unidimensional caseinthe z cornponent of P In is found to be P~ = e00’, second order of 0, where e=e 11 +e33. In the surface layer the variation of 0 is 1.~0=0~—0~ = (fly) 0~,as shown in ref. [3]. Hence 0’ —~Li0/d,giving P~—~ The surface polarization p~is obtained by—e(E~~0/d)0. integrating P~over d, and it is found to be p <0> ~(0~+ 0~) 0~.It follows5that et~0<0>, where —
—
—
-~
~
_e~0o2.
—~
‘~ —
(4)
As shown in ref. [31, Oo = (1 —fl2/y W) _19~ By substituting this expression for 0~in (4) we reobtain (3). As is seen from (3), the spontaneous surface polarization p 5 is considerably the surface anchoring energy Wanddetermined in the limitbyW=0, p~
surface polarization. It is important to underline that the scalar order parameter we are considering is the surface order parameter, whose temperature dependence is different from that in the bulk, as discussed in ref. [81. The conclusion about the dependence of p5 on T holds in the case in which the easy axis orientation at the interface 9 can be considered ternperature independent. In the opposite case where 99(T) the However, temperature dependence of p5ofisstrong more complicated. at least in the case anchoring the hypothesis that 9 is independent of T is reasonable. In the event of weak anchoring the p 5 versus T function can be deduced by supposing that 2(T), y(T)=G52(T), as is well known, /J(T)=B5 and furthermore W( T) =AS( T) + CS2 ( T), where B, G, A and C are temperature independent. In the expression fori.e.W(from T) the term, AS( T), comes 1’1’~, the first direct interaction of the from f~ nernatic with the substrate [8], whereas the second 2 ( T), is due partially to the direct nematic— term, CS interaction and to the incomplete nernatic— substrate nematic interaction [9]. In the present case we can still imagine 9 temperature independent. In fact the actual easy direction depends on the value of W, which we are considering T-dependent. By substituting the expressions for fl( T), y( T) and W( T) into eq. (4) one obtains 1
p 5(T)=p50
goes to zero. Furthermore in the limit W—. c (strong anchoring case) p5 is
—
2 s B GCA/C+S)
where p5~coincides with that introduced before. 43
Volume 170, number I
PHYSICS LETTERS A
Note that in case A =0 (no linear term in S is present in W( T)) the temperature behaviour of p5 ( T) is similar to that deduced in the strong anchoring case. Furthermore in the case in which A and C have opposite sign, p5 could be nearly independent of T. Note that the signs of A and C strongly depend on the LC—substrate interface [8,91. In the thin layer of LC with symmetric interfaces the spontaneous surface polarizations at the upper and lower sides exactly cancel, while for asymmetric interfaces a net polarization exists (dueoftoanthe dif2) even in the absence exterference of (W9) nal electric field. The near-surface gradients and the thickness of the surface layer in which p 5 exists do not directly enter in the proposed theory and therefore it is applicable even for very thin polarized layers [4]. The theory of electrodynamic properties of two-dimensional transient layers can be used for the calculation of the phase shift of the light transmitted across the thin bounded LC layer [10]. In conclusion we state that in the case in which a surface distortion of an average nematic molecular orientation is present, a surface electnc polarization may appear. The considered surface distortion is due to the surface second order elastic modulus of the K13 type, which is hence the source of the surface polarization. As shown this surface polarization exists only if the actual orientation imposed by the surface is tilted (neither homeotropic, nor planar). Of course other mechanisms could give rise to surface polarization, as discussed in ref. [11]. More precisely, surface polarization may be due to: (i) ferroelectric order at the surface, connected to the different chemical affinities of the two ends of the nematic molecules with the solid substrate, (ii) order electric polarization, coming from the spatial variation of the scalar order parameter imposed by the surface, (iii) selective ion adsorption. However, these types of surface polarizations are
44
19 October 1992
different from the one considered in our paper, because they can exist also in uniform samples. The surface polarization considered in the present Letter is similar to the one discussed by Tripathi et al. [12]. In a recent paper [12] an experimental observation of a spontaneous electric polarization very near a chiral nematic substrate interface with a non-zero surface tilt has been reported. From a phenomenological point of view this interesting observation is closely related to our prediction (see eq. (3)). However, far asofthe is oriented in thesoplane theobserved interfacepolarization and a net polarization exists in a cell with symmetric interfaces, this spontaneous two-dimensional electric polarization has a (symmetry breaking) nature different from that predicted in our paper.
References [1] PG.
de Gennes, The physics of liquid crystals (Clarendon, Oxford, 1974). 121 S. Chandrasekhar, Liquid crystals (Cambridge Univ. Press, Cambridge 1976). 131 G. Barbero, Z. Gabbasova and Yu.A. Kosevich, J. Phys. II (Paris) 1(1991)1505. [4] P. Guyot-Sionnest, H. Hsiung and YR. Shen, Phys. Rev. Lett. 57 (1986) 2963. [5] S.D. Lee and J.S. Pastel Phys. Rev. Lett. 65 (1990) 56. 161 G. Barbero, I. Dozov, J.F. Palierne and G. Durand, Phys. Rev. Lett. 56 (1986) 2056.
[71BE. Vugmeister and Yu.A.
Kosevich, Soy. Phys. Solid State 31(1989)1871. [8] Ping Sheng, Phys. Rev. A 26 (1982) 1610. [91A. Goossens, Mol. Cryst. Liq. Cryst. 124 (1985) 305; T.J. Sluckin and A. Poniewierski, in: Fluid interfacial phenomena, ed. C.A. Croxton (Wiley, New York, 1986); G. Barbero, Z. Gabbasova and M.A. Osipov, J. Phys. II (Paris) 1(1991)691. [10] Yu.A. Kosevich, Soy. Phys. JETP 69 (1989) 200. [11] G. Barbero, A.N. Chuvyrov, A.P. Krekhov andO.A. Scaldin, J. AppI. Phys. 69 (1991) 6343. [12] S. Tripathi, Mm Hua Lu, E.M. Terentjev, R.G. Petschek and C. Rosenblatt, Phys. Rev. Lett. 67 (1991) 3400.