On the influence of an orienting field on the static twist-splay periodic distortions of nematic liquid crystals

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On the influence of an orienting field on the static twist-splay periodic distortions of nematic liquid crystals

H.M. Zenginoglou

To cite this version:

H.M. Zenginoglou. On the influence of an orienting field on the static twist-splay peri- odic distortions of nematic liquid crystals. Journal de Physique, 1987, 48 (10), pp.1599-1603.

�10.1051/jphys:0198700480100159900�. �jpa-00210595�

On the influence of an orienting ^{field on the static} twist-splay periodic distortions of nematic

liquid crystals

H.M. Zenginoglou

University ^of Patras, Physics Department, Patras, ^Greece

(Reçu ^{Ie 26 mai} 1987, accepti ^{le 21} juillet 1987)

Résumé.- On considère un film nématique d’anisotropie diélectrique positive, ^{soumis à un} champ électrique ^{normal à}

sa direction initiale d’orientation et à un champ magnétique parallèle ^{à cette} direction. On suppose que le rapport ^des

constantes élastiques pour les déformations en torsion et en éventail est assez faible _pour permettre, ^au dessus d’un

seuil, l’apparition d’une déformation périodique torsion-éventail. On décrit la variation avec le champ magnétique

du seuil de champ électrique ^{et de la} longueur d’onde de la distortion. On prédit également l’existence d’une valeur limite.

Abstract.- A homogeneously aligned ^nematic liquid crystal layer ^with positive dielectric anisotropy ^is considered to

be acted _upon by ^{an electric} field normal to and a magnetic ^field parallel ^{to its initial} orientation direction. The ratio of the twist to splay ^{elastic constants of the} nematic is supposed sufficiently ^{small so} that static twist-splay periodic

distortion is exhibited by ^{it as a threshold effect. The} dependence ^{of the} threshold values of the electric field and the spatial wavenumber of the distortion on the orienting magnetic ^{field is} described. Also the existence of a limiting

value of the orienting field, beyond ^{which no} periodic distortion has to be expected, ^is predicted. ^The dependence

of this limiting ^{field on the twist to} splay ^{elastic constants ratio is} given ^{in the two cases of the} boundary ^conditions

considered.

Classification Physics ^Abstracts

61.30

1. Introduction.

A homogeneously aligned non-conducting ^nema-

tic liquid crystal layer ^with positive dielectric aniso- tropy (PNLC), ^{may exhibit a static} twist-splay peri-

odic distortion (TSP) ^{as a} threshold effect when ex-

cited with ^an electric field normal to the layer, pro- vided the ratio of its twist elastic constant to the splay

one (TSR) ^{is smaller than some} critical value. This

effect, ^{which was} discovered experimentally ^and jus-

tified theoretically by Lonberg ^and Meyer [1] ^{for the}

case where the excitation is by ^a magnetic field, ^has

been analysed ^further by ^{several authors for a vari-} ety ^{of field} geometries ^and boundary conditions [2-6].

The analyses ^have shown, particularly, that for the effect to be observed, ^the TSR should be less than

0.30325056..., ^{for a nematic} phase rigidly ^{oriented at}

the boundaries [2,3], ^{and less than 0.5} when the condi- tions ^are such that the director in immediate contact to the boundaries is, simply, parallel ^{to them} [3].

In this article a PNLC layer ^is considered to be under the action of a reorienting ^electric field, ^normal

to the layer, ^{and an} orienting magnetic field, parallel

to the original orientation direction of the layer ^and

the influence of the orienting ^{field on the threshold}

values of the reorienting field and the wavenumber

characterizing the TSP is examined in the two cases

of boundary conditions mentioned above.

2. Theory.

The PNLC sample ^is considered to be confined between two planes, parallel ^to zy-plane, ^{at z=0 and} z=L, the thickness L of the sample being ^{much small-}

er than its other dimensions (Fig.1). ^The sample,

with its original alignement parallel ^to x-axis, ^{is sub-} jected ^{to an} orienting magnetic ^{field H} along ^{the z-}

axis and a reorienting ^{electric field E} parallel ^{to +axis.}

The dielectric anisotropy ea ⁼ ep - en of the sample

is supposed sufficiently ^{small so that} for the small dis- tortions to be considered here, ^{the electric field inside}

the layer ^is nearly ^uniform.

As the aim of the foregoing ^{is the} calculation of threshold quantities, ^{it is} sufficient to use the well- known formulae of nematostatics [7] ⁱⁿ their linear- ized form. The results for the y and ^z components ^of the director field, ^{which are} considered as functions of _y and _z, are as follows :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480100159900

1600

with

where the following definitions are used

ilL : ^{the TSP} wavenumber. In what follows ç ^will

be referred, loosely, ^{as the TSP} wavenumber.

Kl’ K2 : ^{1 the} splay and twist elastic constants, ^re- spectively.

a ⁼ K21K, : ^{Twist to} Splay ^Ratio (TSR).

X a : ^: the anisotropy ^{of the} magnetic susceptibility which, ^as usual, ^is supposed positive.

- -

which are real, provided ^a1 [8], ^{and are taken al-}

ways in the order _ri > r2.

Fig.l.- ^The geometry ^{of the} problem.

Furthermore, using ^{the above} definitions, ^one may show that

and that in any case, rl is an increasing ^{and r2 is a} decreasing ^{function of} QE-

The solution described by equations (2) ^and (3)

is obtained by imposing ^two conditions to nx : ^: it is zero at the two boundaries and it is a symmetric

function of t7 ^with respect ^{to the} q=1/2 plane. ^This

last condition ^comes from the physical demand that

when ç ^{tends to} zero, equation (3) ^{has to describe}

the director field in the one dimensional case of the Freedericsz effect.

Two extremes cases of boundary condition for ny will be considered : the rigid boundary ^condition

(RBC) ^where ny vanishes ^{at the two} boundaries, ^and

the degenerate boundary ^condition (DBC) ^{where all}

orientations of the director parallel ^{to the} boundary planes ^are equally permissible.

1) ^{In the RBC case the demand for zero} ny ^at the two boundaries and equation (2) ^give

The already ^stated properties ^{of rl and r2} suggest that the equation (4) ^{has a solution} only ^for imagi-

nary A2, ^{that is} A2 = jl2, (j2 = -1) , ^{and, so, QE >}

u. Thus equation (4) ^becomes

which is the condition for the distortion ^to persist.

Equation (5) establishes an implicit ^{relation between} QE and ^with QH ^{and a as} parameters. ^{The min-} imum of QE ^{and the} corresponding ^value of ç give

the threshold voltage ^{and TSP} wavenumber, respec-

tively, for the used values of QH ^{and a.} Expanding QE ⁱⁿ powers of ç ^and retaining ^the leading ^{term one}

obtains

where

and

A sufficient condition for the threshold TSP wave-

number to be non-zero is my0. ^When QH ^{tends to}

zero this condition leads to the result [2,3]

On the other hand for values of a smaller than _aR the condition 10 gives ^the limiting ^values QHR ^of QH beyond ^{which no TSP is observed.} Assuming ^{a 1}

and using equation (6), ^{one finds that QHR is given} approximately by

As ^a calculation would show, the difference between

the values of QHR ^obtained using equation (7) ^and

those obtained numerically using equation (6) ^is

smaller than three percent ^{of the exact value} for val-

ues of a up to 0.285. On the other hand for values of a

close _{to aR,} the condition y0 ^{can be} approximated

with the formula

which, ^for a>0.285, gives ^an agreement ^{with the ex-}

act values better than five percent.

2) ^{In the DBC case, the condition} for the dis- tortion to persist ^{is that the} difference between the free _energy of the system ⁱⁿ the distorted state and that in the undistorted _one, with the external fields

unchanged, ^{is lesser than zero.} Using ^{the formulae of}

the nematic state, this difference is found to be, ^aside

from a positive proportionality constant, equal ^to

and, using ^once again ^the properties ^of rl and _r2, one

easily deduces that for G to be negative, A 2 ^{must be} imaginary. So, equation (9) ^becomes

Now, ^as equation (10) ^{suggests, the smallest value of}

d2 for which G is non-positive ^is l2 ^{= 7r.} d2, ^{on the}

other hand, ^{is an} increasing ^{function of} Q E. Thus,

the threshold value of QE ^is given by ^{the solution of} equation ^{12 = 7r. The result is}

From (11), ^{QE takes on} its minimum value

when

For the TSP to be the threshold effect we must have

Çmin > 0, ^and consequently, ^from equation (13)

where QHD ^{is the} limiting ^{value of} QH, beyond ^of

which no TSP is observed. From (14), ^{it is seen that}

the _necessary condition for TSP in the DBC case is

a1/2 [3].

3. Results.

The results of the calculations based on the _pre- sented theory ^{are shown in the} plots ^of figures ^2-4.

The limiting ^{value of} QH, ^is plotted, ⁱⁿ figure 2, ^as

a function of a for the RBC (R) ^{and the DBC} (D)

cases, using ^formula (6) for the condition ’10 ^and formula (14), respectively. ^{As the} figure shows, ^for

values of a much smaller than unity ^{the two curves}

tend to coincide, ^a fact which is expected ^{from a com-} parison ^of equations (7) ^and (14). ^The limiting QH

becomes _zero, of course, at ^{a =} aR and a=1/2 ^for

the (R) ^and (D) ^curves, respectively.

Fig.2.- ^The dependence ^{of the} limiting ^{value of} QH ^{on the TSR}

in the RBC(R) ^and DBC(D) ^cases.

In figures ^{3 and 4 the} dependences of threshold values for the reorienting field and the corresponding

wavenumber are given ^as functions of the orienting

field for the RBC _case, using equation (5), ^{and for}

the DBC case using equations (12) ^and (13).

In figure ^{3 the} dependence of the difference be- tween the one dimensional Freedericsz threshold value of QE, ^{which is} equal ^to QH ⁺ 7r , ^{and TSP threshold}

value is plotted ^{as a} function of Off ^{for two different}

values of _{a, ai} ⁼ 0.1 and _a2 ⁼ 0.2, and for the RBC

(C) ^{and DBC} (D) ^cases. The difference tends rapidly

1602

Fig.3.- ^The dependence ^of the difference between the one di- mensional Freedericsz threshold and TSP threshold values of

Qj ^on QH for the TSR values _ai = 0.1 and _a2 ⁼ 0.2 in the RBC (Ri ^and R2) and the DBC (Di ^and D2) ^cases.

Fig.4.- ^The dependence ^of the threshold TSP wavenumber on

QH ^{for the} same cases as in figure ^3.

to zero and, except ^for very small values of the orient-

ing field, ^{it is a} small fraction of the threshold value

of QE.

Finally, ⁱⁿ figure 4, the threshold value of the TSP wavenumber ^is plotted ^as function of QH ^for

the same values of a and cases as figure ^{3. As it is}

evident from the plots, the distortion period 2?rL/

is of the order of the thickness L of the PNLC layer, except for values of QH very close to its limiting ^value.

The most-striking ^effect predicted by ^the plots ^of figure ⁴ is that the threshold wavenumber, ^considered

as a function of QH, ^{exhibits maxima for} sufficiently

small values of a. This fact _may be understood for the DBC ^case immediately using ^the equation (13) : ²

has a maximum value equal ^to p2 7r2/2 ^when QH ^{= 1r2}

(1-2a-3a 2) / ^{4a, and in} order for this value of QH ^to

be positive ^{we must have} a1/3. ^{A similar argument}

can be used for the RBC case : for the correct value of 12 ^{which is obtained from the} solution of equation (5) ^and corresponds ^to the threshold condition, ^the

threshold value of QE ^is given by ^{the same formula as}

(11) by simply replacing 1r2 by lfl. Thus, ^the depen-

dence of the threshold value of 2 ^on QH ^{is similar to}

that of the DBC ^case. Nevertheless, ^{there are} quan- titative differences because the threshold value of l2

is a function of both QH and a and, so, the value of

a for the maximum off ^to be exhibited turns out to be smaller than 1/3.

4. Conclusion.

A homogeneously aligned ^{PLNC cell with a TSR} sufficiently ^small for the TSP to be the threshold ef- fect when excited with an external electric field, ^is supposed ^{to be acted} upon by ^an additional orient-

ing magnetic field. The calculations show that there is an upper limit for the orienting ^{field for the TSP}

to be observed and this _upper limit is a function of

both, ^{the TSR and the} boundary conditions. Fur-

thermore, the threshold value of the reorienting ^field

increases with the orienting field, and the relation be-

tween the two fields tends rapidly ^to that of the one-

dimensional Freedericsz effect. The distortion period,

on the other hand, ^{is a} sensitive function of the ori-

enting ^{field and is of the} order of the thickness of the

sample.

The experimental ^test of the results reported ^he-

re appears ^to present ^{no serious} difficulties, provided

one is sure that the boundary conditions fit the cases

examined. The relation between E2 and H2 tends

rapidly ^{to a linear} one, permitting the determination of the quantities e a / K 1 ^and xa /Ki . ^The experimen-

tal data on the dependence ^{of the threshold wavenum-}

ber on the orienting ^field may be fitted with relative

ease to a curve such as the ones plotted ⁱⁿ figure 4,

because the calculations leading ^{to these curves are} quite rapid.

As the results show, ^for relatively large ^{values of}

a, the _response of the PNLC layer ^{to the} reorienting

and orienting ^{fields is} largely dependent ^{on the condi-}

tions persisting ^{on the} boundaries of the layer. ^This

means that a more general ^{treatment of the} prob-

lem where the anchoring ^{effects at the} boundaries are

taken into account is worth.

References.

[1] ^{LONBERG, F. and MEYER,} R.B., Phys. ^Rev.

Lett. 55 (1985) ^718.

[2] OLDANO, C., Phys. Rev. Lett. 56 (1986) ^1098.

[3] ZIMMERMANN, ^{W. and} KRAMER, L., Phys.

Rev. Lett. 56 (1986) ^1655.

[4] KINI, U.D., ^J. Physique ⁴⁷ (1986) ^693.

[5] KINI, U.D., ^J. Physique ⁴⁷ (1986) ^1829.

[6] MIRALDI, E., OLDANO, ^{C. and} STRIGAZZI, A.,

Phys. ^{Rev. A 34} (1986) ^4348.

[7] ^DE GENNES, P.G., ^The Physics of Liquid Crys- tals, (Clarendon ^Press, Oxford) ^1974.

[8] ^{This is not a} real restriction because the values of 03B1 of interest here are smaller than 1/2. ^Be-

sides, ^{the occurence of a twist} distortion which

demands that K2 K1/2, ^is hardly ^{to be ex-}

pected ^when K2 > K1.