A Symmetry considerations

Polar liquid crystals in two spatial dimensions: the bridge from microscopic to macroscopic modeling

Abstract

Two-dimensional polar liquid crystals have been discovered recently in monolayers of anisotropic molecules. Here, we provide a systematic theoretical description of liquid-crystalline phases for polar particles in two spatial dimensions. Starting from microscopic density functional theory, we derive a phase-field-crystal expression for the free-energy density which involves three local order-parameter fields, namely the translational density, the polarization, and the nematic order parameter. Various coupling terms between the order-parameter fields are obtained which are in line with macroscopic considerations. Since the coupling constants are brought into connection with the molecular correlations, we establish a bridge from microscopic to macroscopic modeling. Our theory provides a starting point for further numerical calculations of the stability of polar liquid-crystalline phases and is also relevant for modeling of microswimmers which are intrinsically polar.

pacs:
64.70.mf, 64.70.M-, 61.30.Dk

I Introduction

The study of liquid-crystalline phases formed by banana-shaped molecules opens the door to generate polar directions in a partially or completely fluid system due to a collective alignment of the polar axis of the banana-shaped (or bent-core) molecules Brand et al. (1992). So far, most of the liquid-crystalline phases formed by banana-shaped molecules are smectic Niori et al. (1996); Link et al. (1997); Brand et al. (1998, 2003); Pelzl et al. (1999), but there have also been a few reports of nematic phases in this area Pelzl et al. (1999); Shen et al. (1999); Weissflog et al. (2001); Niori et al. (2004). In parallel, there has been a considerable amount of work in Watanabe’s group to generate polar nematic and cholesteric phases in liquid-crystalline polymers Choi et al. (1998); Watanabe et al. (1998); Yen et al. (2006); Koike et al. (2007); Taguchi et al. (2009). Among the polar nematic phases, a nematic phase with a symmetry as low as (or ) was found Koike et al. (2007) confirming earlier predictions about polar nematic phases with low symmetry Brand et al. (2000).

About 25 years ago, there has already been an early effort to synthesize polar nematics in systems composed of fairly large plate-like molecules Hsiung et al. (1987) (to avoid the flipping and thus to generate a lack of symmetry, with being the average preferred direction usually called the director de Gennes and Prost (1995)). About the same time, compounds composed of pyramidic molecules were synthesized with the same goal Zimmermann et al. (1985), but clear-cut evidence for a polar nematic could not be provided in either case. This early work, however, triggered early modeling in the framework of a Ginzburg-Landau description Pleiner and Brand (1989) and it was pointed out that phases with defects, in particular with spontaneous splay, should play an important role in such systems. It was predicted that a phase with defects would occur first in the vicinity of the phase transition to the polar nematic phase.

In 2003 the group of Y. Tabe Tabe et al. (2003) found a two-dimensional polar nematic phase in Langmuir monolayers using the measurements of ferroelectric response and optical investigations in a low molecular weight compound composed of rod-like molecules. Very recently, there were two additional reports on a ferroelectric response of a nematic phase in three-dimensional samples in compounds composed of bent-core molecules Francescangeli et al. (2009); Francescangeli and Samulski (2010), but it is open whether the ferroelectric response was due to a field-induced reorganization of cybotactic clusters – as suggested by the authors – or due to a bulk polar nematic behavior of a phase containing defects of the type outlined above.

Triggered by the reports of nematic phases in banana-shaped molecules, a macroscopic description of polar nematic phases in three spatial dimensions was derived Brand et al. (2006, 2009). It turned out that the absence of parity symmetry leads in such a fluid system to a number of cross-coupling terms between the macroscopic polarization and the other hydrodynamic variables, both statically and in the dissipative dynamic regime. In addition, it was found, both for reversible as well as for irreversible dynamics, that there are new cross-coupling terms not present in typical liquid-crystalline systems not breaking parity symmetry, such as, for example, reversible dynamic cross-coupling terms between flow and temperature or concentration gradients.

Therefore, it is of high interest to have a more microscopic description evaluating the new cross-coupling terms quantitatively in order to aid synthesis of new materials for which corresponding effects can be substantial. In this paper we start such a program using a phase-field-crystal (PFC) model Elder et al. (2007); Löwen (2010); Wittkowski et al. (2010) to analyze the static behavior of polar phases in two spatial dimensions. This approach can be used as a bridge from microscopic to macroscopic modeling. We will systematically compare the results obtained from the PFC model to those obtained using symmetry based approaches such as the Ginzburg-Landau approach, a mean-field description of phase transitions neglecting fluctuations, and the approach of generalized hydrodynamics or macroscopic dynamics Pleiner and Brand (1996).

While in the former only variables are taken into account that lead to an infinite lifetime for excitations in the long wavelength limit, the approach of macroscopic dynamics also incorporates variables, which relax on a sufficiently long, but finite time scale in the limit of vanishing wave number. On realizing our program we strongly build on the foundations given for the static PFC model for nematics and other phases with orientational order in two Löwen (2010) and three Wittkowski et al. (2010) spatial dimensions. In carrying out this program it turns out that it is of crucial importance for polar orientational order to go beyond the Ramakrishnan-Yussouff approximation Ramakrishnan and Yussouff (1979), which is usually used in the area of PFC models. As a matter of fact many of the cross-coupling terms would not be obtained if the Ramakrishnan-Yussouff approximation were implemented. The proposed model can be used as a starting point to explore phase transitions and interfaces for various polar liquid-crystalline sheets, in particular including plastic and full crystalline phases where the translational density shows a strong ordering.

The paper is organized as follows: in Sec. II, we derive a PFC model for polar liquid crystals. Then, in Sec. III, we discuss the relation of the two symmetry-based approaches with the PFC model studied in Sec. II and we show that many of the coefficients arising in the symmetry-based approaches can be linked to microscopic expressions via the PFC model. We finally discuss possible extensions of the model to more complicated situations and give final conclusions in Sec. IV.

Ii Phase-field-crystal model for polar liquid crystals

In general, a theory for polar liquid-crystalline phases can be constructed on three different levels. First of all, a full microscopic theory where the particle interactions and the thermodynamic conditions are the only input is provided by classical density functional theory (DFT) Evans (1979); Singh (1991); Löwen (1994); Roth (2010). DFT is typically used for isotropic particles Ramakrishnan and Yussouff (1979); Rosenfeld et al. (1997); Roth et al. (2002); Hansen-Goos and Mecke (2009) but analogously holds for anisotropic particle interactions Poniewierski and Holyst (1988); Graf and Löwen (1999, 1998); Hansen-Goos and Mecke (2009); Fus (). The second level which may be called mesoscopic is the phase-field approach where lowest-order gradients of an order-parameter field are considered Emmerich (2008). This can be performed up to fourth-order gradients in order to describe a stable crystalline state with order-parameter oscillations leading to the seminal PFC model of Elder and co-workers Elder et al. (2002); Elder and Grant (2004); Emmerich (2009). The prefactors can be brought into relation with the microscopic DFT approach Elder et al. (2007); van Teeffelen et al. (2009). The PFC model has been extensively used to study numerically freezing and melting phenomena on microscopic length but much larger (diffusive) time scales Tegze et al. (2009); Mellenthin et al. (2008); McKenna et al. (2009); Huang and Elder (2008); Wu and Voorhees (2009); Yu et al. (2009); Jaatinen et al. (2009); Jaatinen and Ala-Nissila (2010). Finally, the third level are continuum approaches de Gennes (1971, 1973); de Gennes and Prost (1995); Chaikin and Lubensky (1995); Pleiner and Brand (1996) which respect the basic symmetries. Here, the prefactors are phenomenological elastic constants. PFC modeling can be used to assign a microscopic meaning to the prefactors thus linking the microscopic DFT approach to the symmetry-based approach.

In this section, a PFC model for polar liquid crystals in two spatial dimensions is derived from DFT by a systematic gradient expansion of various coarse-grained order-parameter fields. As a result, we get a free-energy functional which involves the order-parameter fields and their spatial derivatives. The prefactors of various contributions are expressed as generalized moments of direct correlation functions in the isotropic state which provides a bridge between microscopic density functional theory and macroscopic approaches.

ii.1 Static free-energy functional

We consider a two-dimensional system of anisotropic particles with the center-of-mass positions and orientations that are characterized by the unit vectors with . To provide uniaxiality, we assume the existence of a symmetry axis for the anisotropic particles. Furthermore, we assume a broken head-tail-symmetry, i. e., we assume polar particles. This polar system is restricted to the domain with the total area

(1)

and kept at a finite temperature . The polar particles are supposed to interact in accordance with a prescribed pair-interaction potential . Typical examples include particles with an embedded dipole moment Lomba et al. (2000); Froltsov et al. (2003); Alvarez et al. (2008) modeled by a dipolar hard disk potential, colloidal pear-like particles Kegel et al. (2006); Hosein et al. (2009) with corresponding excluded volume interactions, Janus particles Hong et al. (2006); Ho et al. (2008) which possess two different sides, and asymmetric brush polymers modeled by Gaussian segment potentials Rex et al. (2007).

We define the one-particle density field as

(2)

with the mean particle number density

(3)

where

(4)

is the classical canonical average of the observable . Here, we introduced the notation for an arbitrary vector and . denotes the classical canonical partition function and guarantees correct normalization such that . Furthermore, is the inverse temperature with the Boltzmann constant and is the unit circle. The one-particle density describes the probability density to find a particle with orientation at position . Due to the restriction on two spatial dimensions, the orientation is entirely defined by the polar angle . A collective ordering of a set of particles may lead to a macroscopic polarization whose local direction can be expressed by the space-dependent dimensionless unit vector , that is parametrized by a scalar order-parameter field .

Under the assumption of small anisotropies in the orientation, it is now possible to expand the one-particle density with respect to the angle between the particular orientation and the macroscopic polarization into a Fourier series. Throughout this paper we will assume explicitly that the preferred direction associated with dipolar order, , and the direction associated with quadrupolar order, , are parallel. We will therefore use in the following. In general, these two types of order can be associated with two different preferred directions (compare, e. g., reference Brand et al. (2000)). The expansion with respect to orientation results in the approximation

(5)

where the Fourier series is truncated at second order. Here, we introduced three additional dimensionless order-parameter fields , , and . These order-parameter fields are the reduced orientationally averaged translational density

(6)

the strength of the polarization

(7)

and the nematic order parameter

(8)

that measures the local degree of orientational order. The strength of the polarization and the director are modulus and orientation of the polarization . Note that for apolar particles Löwen (2010) such that apolar particles result as a special limit from the present theory.

Now we refer to microscopic density functional theory which is typically formulated for spherical systems Evans (1979); Singh (1991); Löwen (1994) but can also be constructed for anisotropic particle interactions (which dates back to Onsager) Poniewierski and Holyst (1988); Graf and Löwen (1999, 1998); Hansen-Goos and Mecke (2009); Fus (). Density functional theory establishes the existence of a free-energy functional of the one-particle density which becomes minimal for the equilibrium density. The total functional can be split into an ideal rotator gas functional and an excess functional:

(9)

The ideal gas functional is local and nonlinear, it is exactly given by

(10)

where denotes the thermal de-Broglie-wavelength. The excess functional , on the other hand, is in general (i. e., for a non-vanishing ) unknown and approximations are needed. However, there is a formally exact expression gained from a functional Taylor expansion in the density variations around a homogeneous reference density Evans (1979):

(11)

with the -th order contributions

(12)

Here, denotes the -particle direct correlation function, and the notation for an arbitrary vector is used. The first term on the right-hand side of Eq. (11) corresponds to and is an irrelevant constant that can be neglected. We remark that also the first-order term ( in Eq. (12)) vanishes since in a homogeneous reference state must be constant due to translational and orientational symmetry.

For isotropic particles, various approximations based on expression (11) have been proposed. The theory of Ramakrishnan and Yussouff Ramakrishnan and Yussouff (1979) keeps only second-order terms in the expansion. This provides a microscopic theory for freezing both in three Ramakrishnan and Yussouff (1979) and two spatial dimensions van Teeffelen et al. (2006). More refined approaches include the third-order term Barrat (1987) with an approximate triplet direct correlation function Barrat et al. (1987, 1988), but a perturbative fourth-order theory has never been considered. Complementary, non-perturbative approaches like the recently proposed fundamental-measure theory for arbitrarily shaped hard particles Hansen-Goos and Mecke (2009) include direct correlation functions of arbitrary order.

We now insert the parametrization (5) of the one-particle density into Eqs. (10) and (11) in order to obtain a free-energy functional of the order-parameter fields , , , and . First, after inserting the density parameterization (5) into the ideal gas functional (10), we expand the logarithm and truncate the expansion of the integrand at fourth order. This order guarantees stabilization of the solutions (similar to the traditional Ginzburg-Landau theory of phase transitions). Performing the angular integration results in the approximation

(13)

with the local ideal rotator gas free-energy density

(14)

and the abbreviation

(15)

for a constant and therefore irrelevant term.

Secondly, we insert the density parametrization (5) into Eq. (11). We will truncate this expansion at fourth order. Since the -th order direct correlation function in Eq. (11) is not known in general, we expand it into a Fourier series with respect to its orientational degrees of freedom. By considering the translational and rotational invariance of the direct correlation function, we can use the parametrization with , , and for the direct correlation function to reduce its orientational degrees of freedoms from to . Here, the new variables are related to the previous ones by , , , and . With this parametrization, the Fourier expansion of the direct correlation function reads

(16)

with the expansion coefficients

(17)

Next, we set and perform a gradient expansion Löwen et al. (1989, 1990); Ohnesorge et al. (1991); Lutsko (2006); Elder et al. (2007) in the order-parameter fields. For the term (12) corresponding to , this gradient expansion is performed up to fourth order in to allow stable crystalline phases and up to second order in all other order-parameter products, where we assume that the highest-order gradient terms ensure stability. However, for and we truncate the gradient expansion at first and zeroth order, respectively. This results in the components

(18)

of the static excess free-energy functional. In this equation, the excess free-energy densities are local and given by

(19)
(20)
(21)

with the coefficients

(22)

in the gradient expansion in , that also appear – in a different form – in the traditional PFC model of Elder and co-workers Elder et al. (2007). The coefficients

(23)
(24)
(25)

belong to the terms that contain gradients and the modulus of the polarization in first order or that describe the coupling between gradients in the translational density and gradients in the nematic order parameter , respectively. The following three coefficients

(26)
(27)
(28)

appear in the gradient expansion regarding and

(29)

are the coefficients of the gradient expansion in . So far, all these coefficients can also be obtained by using the second-order Ramakrishnan-Yussouff functional for the excess free energy. The remaining coefficients, however, result from higher-order contributions in our functional Taylor expansion. In third order, we find for the homogeneous terms the coefficients

(30)
(31)
(32)
(33)

and for the terms containing a gradient we find the coefficients

(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)

In fourth order, we only kept homogeneous terms. The corresponding coefficients are

(42)
(43)
(44)
(45)
(46)
(47)
(48)

All the coefficients from above are linear combinations of moments of the direct correlation functions. These moments are defined through

(49)

To shorten the notation, we introduced the abbreviations and and used some symmetry considerations that are outlined in appendix A. The moments over expansion coefficients of the direct correlation functions depend on the particular thermodynamic conditions and therefore on the parameters and .

For stability reasons, we assume that the coefficients of the highest-order terms in the gradients and order-parameter fields are positive in the full free-energy functional. If this appears not to be the case for a certain system, it is necessary to take into account further terms of the respective order-parameter field up to the first stabilizing order.

Eqs. (19)-(21) constitute the main result of the paper: it is a systematic gradient expansion of order-parameter fields in the free-energy functional. The prefactors are moments of various direct correlation functions and therefore provide the link towards microscopic correlations. This is similar in spirit to PFC models Elder et al. (2007); Elder and Grant (2004); Jaatinen and Ala-Nissila (2010); Tegze et al. (2009); Emmerich (2009); van Teeffelen et al. (2009) for spherical particles.

ii.2 Special cases of the phase-field-crystal model

We now discuss special cases of our model. First of all, Eqs. (19)-(21) are an extension of the excess free-energy density for apolar particles, that was recently proposed in reference Löwen (2010). This extension comprises a possible polarization of liquid-crystalline particles as well as an enlarged functional Taylor expansion that is beyond the scope of the second-order (Ramakrishnan-Yussouff) approximation. Because of that, our free-energy functional contains a few simpler models as special cases and is therefore the main result of this paper. Two special models that are known from literature and can be obtained from our model by setting some of the order-parameter fields to zero are the traditional PFC model of Elder and co-workers Elder et al. (2007) for isotropic particles without orientational degrees of freedom and the PFC model of Löwen Löwen (2010) for apolar anisotropic liquid crystals in two spatial dimensions. In comparison with our free-energy functional, the two mentioned models base on the Ramakrishnan-Yussouff approximation. The traditional PFC model has only one order-parameter field. This is the translational density which corresponds to in our model. If we set all order-parameter fields that are related to orientational degrees of freedom in our PFC model to zero, i. e., and , and neglect all higher-order contributions for in the functional Taylor expansion (11), then we obtain the traditional PFC model of Elder and co-workers. The PFC model of Löwen considers anisotropic particles with one orientational degree of freedom but no polarization. Therefore, this PFC model results from our model for a vanishing polarization . Also here, we have to neglect all contributions (12) for .

Iii Macroscopic approaches

In this section, we investigate the bridge between the PFC model presented in detail in the last section for polar liquid crystals in two spatial dimensions and the symmetry-based macroscopic approaches: Ginzburg-Landau and generalized continuum description. In addition, we can also compare these results obtained for polar liquid crystals in two spatial dimensions with those obtained previously for non-polar liquid crystals in two Löwen (2010) as well as in three Wittkowski et al. (2010) spatial dimensions.

The general PFC results of this paper have been summarized in Eqs. (19)-(21). We first analyze the terms given in Eq. (19), which are quadratic in the variables and their gradients.

We start with terms containing only the translational density and its gradients in Eq. (19). In the vicinity of the smectic-A-isotropic transition one has for the smectic order parameter de Gennes (1973)

(50)

and for the density

(51)

with the average homogeneous density (compare also section 6.3 of reference Chaikin and Lubensky (1995) for a detailed discussion). Since the total free energy must be a good scalar, the smectic order parameter can enter the free energy only quadratically. For the lowest-order terms in the energy density , which we define as the integrand of the free-energy functional

(52)

we have Mukherjee et al. (2001)

(53)

Comparing Eq. (53) and the first three terms in Eq. (19), we can make the identifications , , and with , , and , respectively. This situation is similar for non-polar nematics in three spatial dimensions Wittkowski et al. (2010), where , , and are defined with different signs, and for non-polar nematics in two spatial dimensions Löwen (2010), where one must identify , , and with , , and , respectively.

For the terms containing only the non-polar orientational order in Eq. (19), we have two contributions to compare to other approaches. One is spatially homogeneous and the other one is quadratic in the gradients of the orientational order . The first contribution can be directly compared with the term in de Gennes’ pioneering paper de Gennes (1971). Using the structure for the conventional nematic order parameter in two spatial dimensions, we find using the original notation of reference de Gennes (1971). For the gradient terms in the Ginzburg-Landau approximation one has at first sight two contributions to the energy density just using the three-dimensional expression de Gennes (1971)

(54)

for two spatial dimensions. A straightforward calculation shows that the two contributions are in two dimensions identical, however, with and thus one independent coefficient just as for the PFC model where one has the contribution .

For the terms associated exclusively with orientational order we have, when specialized to two spatial dimensions, in the continuum description in the energy density

(55)

where is the transverse Kronecker symbol projecting onto the direction perpendicular to the preferred direction . In Eq. (55), the first line is connected to gradients of the director field . It contains in two spatial dimensions only splay and bend and no twist and goes back to Frank’s pioneering paper Frank (1958); de Gennes and Prost (1995). Lines 2 and 3 in Eq. (55) are associated with gradients of the nematic modulus, , and with a coupling term between gradients of the director and gradients of the modulus Brand and Kawasaki (1986); Brand and Pleiner (1987). We finally note that the gradient terms in Eq. (19) are identical to the ones given in reference Löwen (2010), where we must identify in the present paper with in reference Löwen (2010). This must indeed be the case, since polar nematics contain the case of non-polar nematics as a special case in the PFC approach.

Next, we come to the terms containing only contributions of the macroscopic polarization , or equivalently, its magnitude (modulus) and its direction . The term in Eq. (19) is the standard quadratic term for a Landau expansion near, for example, the paraelectric-ferroelectric transition Kittel (1995). It also emerges when the phase transition isotropic to polar nematic is studied in Ginzburg-Landau approximation Pleiner and Brand (1989). The terms that are quadratic in gradients of , i. e., the contributions and in Eq. (19), can be compared to the result of a Ginzburg-Landau approach

(56)

and contain two independent contributions even in the isotropic phase Pleiner and Brand (1989) in two spatial dimensions as is easily checked explicitly.

The gradient terms for the macroscopic polarization, or equivalently, for its magnitude and its direction , can also be compared to the macroscopic description of polar nematics Brand et al. (2006, 2009). For the corresponding terms we have

(57)

where denotes deviations from the equilibrium value, in particular and where the tensors are of the form

(58)
(59)
(60)

Eq. (57) represents the analogue of the Frank orientational elastic energy () with splay and bend, the energy associated with gradients of the modulus (), and a cross-coupling term between gradients of the preferred direction to gradients of the order-parameter modulus () – the analogue of the corresponding term in non-polar nematics Brand and Kawasaki (1986); Kawasaki and Brand (1985).

The contributions and in Eq. (19) are the PFC analogues of the contributions and in Eq. (57). Instead of four independent coefficients in the macroscopic description in two spatial dimensions, the PFC model gives rise to two. The contribution has no direct analogue in the PFC model.

Next, we start to compare cross-coupling terms between gradients of the variables. The discussion for the coupling terms between gradients of the density and gradients of the orientational order closely parallels that for the three-dimensional non-polar nematic case. In Eq. (19), the terms of interest are proportional to . In reference Wittkowski et al. (2010), these are the terms . A comparison of these two expressions reveals that they are identical in structure and that one has just to take into account the change in dimensionality. For spatial gradients in the director field coupling to spatial variations in the density we find in the energy density Pleiner and Brand (1980); Brand and Pleiner (1987)

(61)

By comparison with Eq. (19) we find . Finally, we have for the terms coupling gradients of the order-parameter modulus to gradients of the density Brand and Pleiner (1987)

(62)

where the second rank tensor is of the standard uniaxial form . A comparison with Eq. (19) yields and . The coupling terms listed in Eqs. (61) and (62) exist in both two and three spatial dimensions. Thus, in comparison to the hydrodynamic description of the bulk behavior, which is characterized by three independent coefficients, we find one independent coefficient in the PFC model. In the framework of a Ginzburg-Landau approach using the orientational order parameter we find in the isotropic phase

(63)

and thus one independent coefficient – as has also been the case for the non-polar PFC model in three dimensions Wittkowski et al. (2010) as well as in two dimensions Löwen (2010).

The contributions and are containing gradients of the macroscopic polarization and couple to density and quadrupolar order. They are unique to systems with polar order, or more generally, to systems with broken parity symmetry, since they contain one gradient and one factor . Such coupling terms are not possible, for example, in non-polar nematics or smectic A phases. The term can easily be compared with the macroscopic description of polar nematics given in reference Brand et al. (2006). The relevant terms from Eq. (1) of reference Brand et al. (2006) read

(64)

where . We thus read off immediately that when comparing to the PFC model we have and , that is one independent coefficient in the PFC model and two in the macroscopic description. For the term the situation is similar. One has to replace in Eq. (64) by , where is the modulus of the quadrupolar nematic order parameter with coefficients denoted by and . Then one makes the identifications and . For the contribution we can also make easily contact with the Ginzburg-Landau picture. For the coupling of and its gradients to quadrupolar orientational order we obtain to lowest order in the Ginzburg-Landau energy density

(65)

with . This term has been given before for the isotropic-smectic-C phase transition in liquid crystals Mukherjee et al. (2005) for which the polarization is a secondary-order parameter. We note that the contribution in Eq. (19) can be brought into a form identical to that of Eq. (65), when it is rewritten in terms of and . This shows once more the close structural connection between PFC modeling and the Ginzburg-Landau approach.

The spatially homogeneous contributions in Eq. (20) can all be interpreted in the symmetry-based framework as well. The term arises near the smectic-C-isotropic phase transition Mukherjee et al. (2005): . The terms and can be interpreted as the density dependence of the terms and in the Landau description of the polar nematic-isotropic Pleiner and Brand (1989) and the non-polar nematic-isotropic de Gennes (1971) phase transitions. Finally, the contribution would arise in a macroscopic description as a term cubic in the density variations: . Typically, such terms are considered to be of higher order in a macroscopic approach. The physical interpretation of this term is a density dependence of the compressibility.

Most of the terms in Eq. (20) containing one gradient, namely all terms containing , except for , can be interpreted in the framework of macroscopic dynamics as higher-order corrections to the terms , , , and discussed above. They correspond in this picture to the dependence of the coefficients and on the density changes and variations in the modulus of the quadrupolar order parameter . There is one exception to this picture and this is the term in Eq. (20). It is also this term, which has an analogue in the field of the Ginzburg-Landau description of ferroelectric materials:

(66)

This nonlinear gradient term has been introduced in reference Aslanyan and Levanyuk (1978) and it was demonstrated by Felix et al. Felix et al. (1986) that this term leads to qualitative changes in the phase diagram near the paraelectric-ferroelectric transition giving rise also to incommensurate structures.

In Eq. (21), spatially homogeneous terms that are of fourth order in the order parameters are presented. Most of them are familiar from Landau energies near phase transitions. The first contribution, the term , arises for all isotropic-smectic phase transitions Mukherjee et al. (2001, 2002, 2005) as well as for the nematic-smectic-A and the nematic-smectic-C transitions de Gennes (1973); de Gennes and Prost (1995): . The contribution arises near the paraelectric-ferroelectric phase transition Kittel (1995); Felix et al. (1986) and has also been used near the isotropic-polar-nematic transition Pleiner and Brand (1989): . The term is familiar from the non-polar nematic to isotropic de Gennes (1971) and the smectic A to isotropic Mukherjee et al. (2001) transitions: . The cross-coupling term corresponds to an analogous term for isotropic-smectic transitions Mukherjee et al. (2001, 2002, 2005): . For the Ginzburg-Landau description of the smectic-C-isotropic transition, the term arises Mukherjee et al. (2005): . The term has also an analogue at the smectic-C-isotropic transition, where it has not been discussed before. However, for the non-polar nematic to isotropic phase transition in an electric field one has shown in reference Brand (1986) that there are two contributions:

(67)

The same contributions are relevant here when the external electric field is replaced by the polarization . Finally, the term can be viewed as the density dependence of the term as it emerges near the isotropic-smectic-C phase transition Mukherjee et al. (2005).

Iv Conclusions and possible extensions

In conclusion, we systematically derived a phase-field-crystal model for polar liquid crystals in two spatial dimensions from microscopic density functional theory. Two basic approximations are involved: first, the density functional is approximated by a truncated functional Taylor expansion which we considered here up to fourth order. Then a generalized gradient expansion in the order parameters is performed which leads to a local free-energy functional. The density is parameterized by four order-parameter fields, namely the translational density which corresponds to the scalar phase-field variable in the traditional phase-field-crystal model, the strength of polarization , an orientational direction given by a two-dimensional unit vector , and the nematic order parameter . In the three latter quantities, the gradient expansion is performed up to second order, while it is done to fourth order in for stability reasons. The traditional phase-field-crystal model Elder et al. (2002); Elder and Grant (2004) and the recently proposed phase-field-crystal model for apolar liquid crystals Löwen (2010) are recovered as special cases. The additional terms are all in accordance with macroscopic approaches based on symmetry considerations Pleiner and Brand (1996); Brand and Pleiner (1987). The prefactors are generalized moments of various direct correlation functions and therefore provide a bridge between microscopic and macroscopic approaches.

As a general feature, we find that typically the number of independent coefficients for the phase-field-crystal and the Ginzburg-Landau approaches is the same, while in many cases the macroscopic hydrodynamics description valid inside the two-dimensional polar phase leads to a larger number of independent coefficients. This appears to be a general trend, which was also found to hold before for the comparison of phases with three-dimensional non-polar orientational order Wittkowski et al. (2010). In fact, it also applies to the two-dimensional phase-field-crystal model for systems with orientational order studied in reference Löwen (2010).

The proposed functional, as embodied in Eqs. (19)-(21), can be used to study phenomenologically phase transformations, for example, in polar nematic sheets, interfaces between coexisting phases McDonald et al. (2001); van der Beek et al. (2006); Bier et al. (2005), and certain biological systems that exhibit polar order Verkhovsky et al. (1999); Cisneros et al. (2006). Since our model has more parameters, we expect even more complicated phase diagrams than recently numerically discovered in the apolar phase-field-crystal model Achim et al. (2011).

One could also do in principle microscopic calculations of the bulk phase diagram for a given interparticle potential