Ferroelectrics Physical Effects Part 6 pot

Compounds with polar group 1, 3-5 have virtual SmC*A phase.. The placement of molecules of compounds with virtual SmC*A phase, which have a tendency to form the antiferroelectric phase b

composed only of compounds with fluoroalkyl group, giving the induction of SmC*A phase Compound 1 was dopped with compound 13, having terphenyl rigid core lateraly substituted by four fluorine atoms, Fig 18 This examples is an exeption from the rule that induction of SmC*A phase can appear in mixtures of compounds differing in the polarity The steric factors play also crucial role in this behaviour, because the induction of SmC*Aphase was not found in mixtures of both nonchiral compounds For the last system, it cannot be also excluded that the appearance of SmC*A phase could be a result of interactions between nonfluorinated and fluorinated rigid core

4 Conclusion

There are many systems in which the induced antiferroelectric phase can be observed Compounds with polar group (1, 3-5) have virtual SmC*A phase They have similar structure to compounds forming SmC*A phase by themselvs; e.g the structure of compound

1 and 4 is similar to the structure of the first antiferroelectric compound MHPOBC; they have the same rigid core and chiral terminal chain The placement of molecules of compounds with virtual SmC*A phase, which have a tendency to form the antiferroelectric phase but cannot fulfil all conditions, in a suitable matrix of less polar compounds causes the appearance of an antiferroelectric phase

The ability of more polar compounds with virtual SmC*A phase for induction of this phase decreases in the following order: compounds 1, 3, 4 and 5 Compounds with cyano group in terminal chain have smaller ability than compounds with fluoroalkyl group Compounds with biphenylate core have bigger ability than compounds with benzoate core The same is observed in group of less polar compounds with alkyl group in the terminal chain, i.e the biphenylate core is more preferable, as well as the increase of the terminal chain causes the inrease of tendency to induce SmC*A phase

In systems of similar polarity nonadditive behaviuor can be observed when fluorinated part

of the non-branched chain is short

The appearance of liquid crystalline phases is possible due to intermolecular interactions Interactions between permanent dipoles play a crucial role but interactions between induced dipoles cannot be neglectible The steric interactions are also important

The possibility of obtaining antiferroelectric phase from compounds forming SmC* phase or only SmA phase broadens the range of compounds useful for preparation of antiferroelectric mixtures for display application

5 References

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Chiral Smectic Phases Responsible for the Trislable Switching in MHPOBC,

Jpn.J.Appl.Phys., Vol.28, No.7, (July 1989), pp L1265-L1268, ISSN 0021-4922

Czupryński, K., Skrzypek, K., Tykarska, M & Piecek, W (2007) Properties of induced

antiferroelectric phase, Phase Transitions, Vol.80, No.6-7, (June 2007), pp 735-744,

ISSN 0141-1594

Dąbrowski, R (2000) Liquid crystals with fluorinated terminal chains and antiferroelectric

properties, Ferroelectrics, Vol.243, No.1, (May 2000), pp 1-18, ISSN 0015-0193

Dąbrowski, R., Czupryński, K., Gąsowska, J., Otón, J., Quintana, X., Castillo, P & Bennis, N

(2004) Broad temperature range antiferroelectric regular mixtures, Proceedings of SPIE, Vol.5565, (2004), pp 66-71, ISSN 0277-786X

Drzewiński, W., Czupryński, K., Dąbrowski, R & Neubert, M (1999) New antiferroelectric

compounds containing partially fluorinated terminal chains Synthesis and

mesomorphic properties, Mol.Cryst.Liq.Cryst., Vol.328, (August 1999), pp 401-410,

ISSN 1058-725X

Drzewiński, W., Dąbrowski, R & Czupryński, K (2002) Synthesis and mesomorphic

properties of optically active (S)-(+)-4-(1-methylheptyloxycarbonyl)phenyl (fluoroalkanoyloxyalkoxy)biphenyl-4-carboxylates and 4’-(alkanoyloxyalkoxy)

4’-biphenyl-4-carboxylates, Polish J.Chem., Vol.76, No.2-3, (February 2002), 273-284,

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Dziaduszek, J., Dąbrowski, R., Czupryński, K & Bennis, N (2006) , Ferroelectrics, Vol.343,

No.1, (November 2006), pp 3-9, ISSN 0015-0193

Fukuda, A., Takanishi, Y., Isozaki, T., Ishikawa, K & Takezoe, H (1994) Antiferroelectric

chiral smectic liquid crystals, J.Mater.Chem., Vol.4, No.7, (January 1994), pp

997-1016, ISSN 0959-9428

Gauza, S., Czupryński, K., Dąbrowski, R., Kenig, K., Kuczyński, W & Goc, F (2000)

Bicomponent systems with induced or enhanced antiferroelectric SmCA* phase,

Mol.Cryst.Liq.Cryst., Vol.351, (November 2000), 287-296, ISSN 1058-725X

Gauza, S., Dąbrowski, R., Czypryński, K., Drzewiński, W & Kenig, K (2002) Induced

antiferroelectric smectic CA* phase Structural correlations, Ferroelectrics, Vol.276,

(January 2002), pp 207-217, ISSN 0015-0193

Gąsowska, J (2004) PhD thesis, WAT, Warsaw

Gąsowska, J., Dąbrowski, R., Drzewiński, W., Filipowicz, M., Przedmojski, J & Kenig, K

(2004) Comparison of mesomorphic properties in chiral and achiral homologous

series of high tilted ferroelectrics and antiferroelectrics, Ferroelectrics, Vol.309, No.1,

(2004), pp 83-93, ISSN 0015-0193

Kobayashi, I., Hashimoto, S., Suzuki, Y., Yajima, T., Kawauchi, S., Imase, T., Terada, M &

Mikami, K (1999) Effects of Conformation of Diastereomer Liquid Crystals on the

Preference of Antiferroelectricity, Mol.Cryst.Liq.Cryst.A, Vol.328, (August 1999), pp

131-137, ISSN 1058-725X

Kula, P (2008) PhD thesis, WAT, Warsaw

Mandal, P.K., Jaishi, B.R., Haase, W., Dąbrowski, R., Tykarska, M & Kula, P (2006) Optical

and dielectric studies on ferroelectric liquid crystal MHPO(13F)BC: Evidence of SmC* phase presence, Phase Transitions, Vol.79, No.3, (March 2006), pp 223-235,

ISSN 0141-1594

Matsumoto, T., Fukuda, A., Johno, M., Motoyama, Y., Yuki, T., Seomun, S.S & Yamashita

M (1999) A novel property caused by frustration between ferroelectricity and antiferroelectricity and its application to liquid crystal displays-frustoelectricity and

V-shaped swiching, J.Mater.Chem., Vol.9, No.9, (1999), pp 2051-2080 ISSN

0959-9428

Skrzypek, K & Tykarska, M (2006) The induction of antiferroelectric phase in the

bicomponent system comprising cyano compound, Ferroelectrics, Vol.343, No.1,

(November 2006), pp 177-180, ISSN 0015-0193

Skrzypek, K., Czupryński, K., Perkowski, P & Tykarska, M (2009) Dielectric and DSC

Studies of the Bicomponent Systems with Induced Antiferroelectric Phase

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154-163, ISSN 1542-1406

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Guillon, D (2008) Physical properties of two systems with induced antiferroelectric

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Piezoelectrics

et al., 2004)) and in understanding of mechanisms of the piezoelectric coupling in ferroelectricpiezoelectrics (Fu & Cohen, 2000; Guo et al., 2000) This progress was triggered in particular bythe wide use of piezoelectric effect in a variety of devices (resonators, tactile sensors, bandpassfilters, ceramic discriminators, SAW filters, piezoresponse force microscopes and others).What concerns theoretical study of the piezoelectric effect, significant efforts were made

a relatively simple structure, in particular for simple and complex perovskites (Bellaiche

et al., 2000; Garcia & Vanderbilt, 1998) For compounds with a complex structure oftenonly the research within the Landau theory is possible The structure complexity justifiesthe application of semimicroscopic models considering only that characteristic feature ofthe microscopic structure which is crucial in explaining the ferroelectric transition or thepiezoelectric effect Such models are adequate for the crystal under study if they are able

to explain the wide range of physical properties

RS) is studied on the base of the semimicroscopic Mitsui model

The microscopic mechanism of ferroelectric phase transitions in RS was the subject ofnumerous investigations Studies based on x-ray diffraction data (Shiozaki et al., 1998) arguedthat these were the order-disorder motions of O9 and O10 groups, coupled with the displacivevibrations of O8 groups, which were responsible for the phase transitions in Rochelle salt aswell as for the spontaneous polarization Later it was confirmed by the inelastic neutronscattering data (Hlinka et al., 2001) Respective static displacements initiate the emergence

of dipole moments in local structure units in ferroelectric phase Such displacements can

be interpreted also as changes in the population ratio of two equilibrium positions of sites

in the paraelectric structure (revealed in the structure studies (Noda et al., 2000; Shiozaki

et al., 2001)) The order-disorder pattern of phase transitions in RS forms the basis of thesemimicroscopic Mitsui model (Mitsui, 1958) In this model the asymmetry of occupancy

of double local atomic positions and compensation of electric dipole moments occurring inparaelectric phases were taken into account

Piezoelectric Effect in Rochelle Salt

9

Recently (Levitskii et al., 2003) Mitsui model as applied to RS was extended by accounting

was extended to the four sublattice type (Levitskii et al., 2009; Stasyuk & Velychko, 2005)that gives more thorough consideration of real RS structure We performed our research ofpiezoelectric effect in Rochelle salt on the basis of the Mitsui-type model containing additionalterm of transverse field type responsible for dynamic flipping of structural elements (Levitskii,Andrusyk & Zachek, 2010; Levitskii, Zachek & Andrusyk, 2010) Originally, this term wasadded with the aim to describe resonant dielectric response which takes place in RS insubmillimeter region First, we provide characteristics of ferroelectric phase transitions in

RS and experimental data for constants of physical properties of RS Then, we present ourstudy results (thermodynamic and dynamic characteristics) obtained within Mitsui model for

RS Specifically, we calculate permittivity of free and clamped crystals, calculate piezoelectric

phenomenon of piezoelectric resonance

2 Physical properties of Rochelle salt

known ferroelectrics RS has been the subject of numerous studies over the past 60 years

The crystalline structure of RS proved to be complex It is orthorhombic (space group

ferroelectric phase (Solans et al., 1997) Spontaneous polarization is directed along the a crystal

112 atoms) in the unit cell of Rochelle salt in both ferroelectric and paraelectric phase Inrecent study (Görbitz & Sagstuen, 2008) the complete Rochelle salt structure in paraelectricphase was described

Due to the symmetry of RS crystal structure some elements of material tensors are zeroes In

RS case material tensors in Voigt index notations are of the form presented below (Shuvalov,1988)

Elastic stiffnesses or elastic constants (c E ij= (∂σ i/∂ε j)E):

respectively We will also call them tensor of free crystal dielectric permittivity (zero stress isassumed) and of clamped crystal dielectric permittivity Hereinafter coefficients equal to zero

in paraelectric phases are presented in bold

Experimental data for physical constants are presented in Figs 1, 2, and 3

-0,2 -0,1 0,0 0,1

c

44 E

(1010 Nm-2)

T (K)

0,04 0,08 0,12 0,16 0,20

estimated theoretically (Levitskii et al., 2005) and the result of estimation is presented in Fig 1

was derived

Fig 2 presents temperature dependencies of piezoelectric stress coefficients As one can see

Free and clamped crystal longitudinal susceptibilities are presented in Fig 3 Free crystalsusceptibility has singularities in transition point, whereas clamped susceptibility remainsfinite

3 Thermodynamic characteristics of Rochelle salt

3.1 Theoretical study of the Mitsui model

We give consideration to a two-sublattice order-disorder type system with an asymmetricdouble-well potential Hamiltonian of such system is referred to as the Mitsui Hamiltonian

We assume this system has essential piezoelectric coupling of the order parameter with

directed along x-axes and arises due to the structural units ordering in the one of two possible

equilibrium positions Precisely this case occurs in RS and such modified Mitsui model wasconsidered earlier (Levitskii et al., 2003) We complement this model with transverse field totake into account the possibility of dynamic ordering units flipping between two equilibrium

positions The resulting Hamiltonian is of the following form:

U0=N

1

represent the elastic, piezoelectric, and electric energies attributed to a host lattice, in which

between pseudospins belonging to the same and to different sublattices, respectively The firstterm in the second sum is the transverse field; the second term describes a) energy, associated

interaction energy of pseudospin with the field, arising due to the piezoelectric deformation

We conduct the study within the mean field approximation (MFA) Performing identicaltransformation

 S q f  =Sp(S q f ρMFA), (9)

1 Actual unit cell of the Rochelle salt crystal contains two pairs of pseudospins of two lattice sites and different sublattices; therefore, we should set the value of the model unit cell volume to be half of the crystal unit cell volume.

In homogeneous external field the system of 6N equations (11) has a lot of solutions, with a

In this case, system of equations reduces into system

Having introduced new variables

σ z = 12

(17)

notations are used:

properties of the Rochelle salt

Independent variable is stress rather than deformation, so we need express local fields in

defines which of the solutions is actually realized at each particular T The solution of first

type describes paraelectric phase and the solution of second type describes ferroelectric phase

Coefficient of the piezoelectric stress:

calculations (Levitskii et al., 2003), where transverse field was not taken into account

3.2 Results of calculations for Rochelle salt

The proposed model was used for analysis of physical properties of Rochelle salt crystal that is

of all to derive theory model parameters for calculations Deriving procedure was described

in (Levitskii, Zachek & Andrusyk, 2010) and here we will restrict ourselves to parameterspresenting:

Besides, we derived that dielectric permittivity of the free crystal has singularity in thetransition points while dielectric permittivity of the clamped crystal doesn’t Elastic constant

doesn’t have singularity in the transition point All these results agree with the prediction ofthe Landau theory for the behaviour of physical characteristics in the vicinity of the transitionpoints However, presented semimicroscopic approach has an advantage over the Landautheory: it allowed to explain physical properties of Rochelle salt in wide temperature ragecontaining both transition points in natural way Besides that Mitsui model gives some insightinto microscopical mechanism of the phase transition of Rochelle salt

240 255 270 285 300 315 1

2 3 4 5 6

T (K)

Fig 4 Theoretical and experimental physical characteristics of Rochelle salt Solid line

11(T):(Sandy &

11(T):(Taylor et al., 1984), e14(T):(Beige & Kühnel,1984)

4 Dynamic properties of Rochelle salt

4.1 Order parameter dynamics Dielectric susceptibility of a clamped crystal

We consider dynamic properties of the system with Hamiltonian (4) within the Blochequations method

¯h d  S q f  t

dt =  S q f  t × H q f(t ) − ¯h

T1  S q f  t −  S q f  t (32)Right part of this equation consists of two terms

The first term is Heisenberg part of the motion equation, calculated within random phase

−i[ , H ] = S  × H (t).

to the instantaneous value of the local field) towards its quasiequilibrium value with a

2k B T H q f(t)

situation, a pseudospin system is not an isolated system, whereas it is a part of a larger system.That part of extended system which is not a pseudospin subsystem appears as thermostat thatbehaves without criticality Respectively, pseudospin excitations relax due to the interaction

far as a phase transition is a collective effect and the relaxation term in Eq (32) describes

no singularity at the Curie point Relaxation time can be derived ab initio but we consider it to

be a model parameter and take it to be independent from temperature

In the same way it can be explained why relaxation in Eq (32) occurs towards

describes individual relaxation of pseudospin, which ‘is not aware’ of the state ofthermodynamic equilibrium but ‘is aware’ of the state of its environment at a particularmoment At every moment this environment creates instantaneous molecular fields which

operators are defined from Eq (9), (10) but with molecular fields Eq (33), (34) Making

Eventually, of course, quasiequilibrium values follow to equilibrium ones and relaxation leadsexcited system to thermodynamic equilibrium state

As we are interested in linear response of the system to a small external variable electric field

δE 1q(t) E 1q(t) =E1+δE 1q(t),

 S q f  t =  S q f 0+δ  S q f  t (36)

4 Sometimes one writes third term− ¯h

T2 S q f  t ⊥, describing decay process of the transverse component of pseudospin S q f  t ⊥, though it can be shown (Levitskii, Andrusyk & Zachek, 2010) that its impact on Rochelle salt dynamics is negligible.

H q f(t ) = H(0)f +δ H q f(t),  S q f  t =  S q f 0+δ  S q f  t (37)Now, we can linearize motion equation (32) by retaining terms, which are linear in deviations

δ  S q f  t,δ H q f(t),δ  S q f  t:

¯h dδ  S q f  t

dt =δ  S q f  t × H(0)f +  S q f 0× δ H q f(t ) − ¯h

T1 δ  S q f  t − δ  S q f  t , (38)where

for values dependent on q (like δ S z

q1  tand others) and

for interaction constants dependent on q1and q2(like J qq  and others) Here the dependency of M q1q2

on difference q − q2was used.

The following notations are used in this equation: I is identity matrix, i is the imaginary unit,

± ε˜2

˜λ2 2

± Ω˜ε˜ 2

˜λ2 2

, G1,2=K1ε˜2

˜λ2 1

± K2ε˜2

˜λ2 2,

±Ω˜2

˜λ2 2



˜λ2 1

± K2Ω˜ε˜ 2

˜λ2 2

δP 1k(ω) =e014δε 4k(ω) +χ11ε0 · δE 1k(ω) +μ˜

˜v · δξ z

dynamic coefficient of the piezoelectric stress:

The best agreement between theory and experiment for RS is reached at (Levitskii, Andrusyk

& Zachek, 2010)

polynomial function of degree not higher than 5 and denominator is polynomial function of

in total The first sum in Eq (52) is a contribution of Debye (relaxation) modes into orderparameter dynamics, and the second sum is a contribution of resonance modes In RS case we

below Fig 5 presents frequency dependencies of dielectric permittivity of clamped crystal in

figure shows, Mitsui model is able to describe dielectric permittivity in dispersion region.The correspondence between theory and experimental data for dynamic dielectricpermittivity deserves special attention Methods for experimental measurements of dynamicdielectric permittivity does not allow to assert that namely clamped crystal permittivity was