Hendrik
Bartsch
*ab,
Markus
Bier
abc and
Siegfried
Dietrich
ab
aMax Planck Institute for Intelligent Systems, Heisenbergstr. 3, 70569, Stuttgart, Germany. E-mail: hbartsch@is.mpg.de; bier@is.mpg.de
bInstitute of Theoretical Physics IV, University of Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany
cUniversity of Applied Sciences Würzburg-Schweinfurt, Ignaz-Schön-Str. 11, 97421, Schweinfurt, Germany
First published on 30th April 2019
Ionic liquid crystals (ILCs) are anisotropic mesogenic molecules which additionally carry charges. This combination gives rise to a complex interplay of the underlying (anisotropic) contributions to the pair interactions. It promises interesting and distinctive structural and orientational properties to arise in systems of ILCs, combining properties of liquid crystals and ionic liquids. While previous theoretical studies have focused on the phase behavior of ILCs and the structure of the respective bulk phases, in the present study we provide new results, obtained within density functional theory, concerning (planar) free interfaces between an isotropic liquid L and two types of smectic-A phases (SA or SAW). We discuss the structural and orientational properties of these interfaces in terms of the packing fraction profile η(r) and the orientational order parameter profile S2(r) concerning the tilt angle α between the (bulk) smectic layer normal and the interface normal. The asymptotic decay of η(r) and of S2(r) towards their values in the isotropic bulk is discussed, too.
A specific example of an ILC system, which has been studied, e.g., in ref. 7 and 8, is composed of cations with long alkyl-chains attached (1-dodecyl-3-methylimidazolium) and significantly smaller anions (iodide). For such an ILC system, one observes a liquid crystalline structure, in particular the smectic-A phase SA. (The SA phase is characterized by layers of particles which are well aligned with the layer normal and the layer spacing is of the size of the particle length.) The layer structure of the large cations leads to a locally increased concentration of anions in between the layers of cations.7 Thereby, the nanostructure of the cations gives rise to “pathways” for the anions, which increase the conductivity measurable in the direction parallel to the layers. Therefore this particular type of an ILC system is a promising candidate for technological applications, e.g., as electrolyte in dye-sensitized solar cells (DSSCs).7,9
While the complexity of the underlying interactions gives rise to these interesting properties of ILCs, it is at the same time very challenging to study these systems within theory or simulations. Previous theoretical studies10,11 of ILC systems have been able to reduce this complexity by considering a simplified description of ILC systems, which incorporates, however, the generic properties of ILCs. They rely on an effective one-species description in which one of the ion species (referred to as counterions) is not accounted for explicitly, but is incorporated as a continuous background, giving rise to the screening of the coions. On the contrary, the coions are modeled as ellipsoidal particles. Thus, the anisotropic molecular shape, which gives rise to the formation of mesophases, and the (screened) electrostatic interaction are both incorporated by this approach. Of course, this is a simplified representation of any realistic ionic liquid crystalline system. However, it allows one to study the interplay of the two key features, i.e., an anisotropic molecular shape and the presence of charges, which are omnipresent in ILC systems. Yet it should be noted, that ILC systems exhibiting a significant difference in size of the cations and of the anions (e.g., the aforementioned example of 1-dodecyl-3-methylimidazolium) might be candidates which come closest to the present theoretical representation of ILCs, as the size difference rationalizes in parts the idea of structureless point-like counterions.
As a first step, such a model allows one to study the phase behavior of ILC systems and thereby to gain insight about how molecular properties, e.g., the aspect-ratio or the charge distribution of the molecules, affect the phase behavior of such types of ILCs. A comprehensive understanding of the relation between the underlying molecular properties and the resulting phase behavior is inter alia, necessary for a systematic synthesis of ILCs, which should meet specific material properties. Furthermore, theoretical guidance is beneficial for finding and exploring novel materials properties which might occur in ILC systems. For instance, in ref. 11 a new smectic-A structure (SAW) has been observed, which exhibits an alternating layer structure. In between layers of elongated particles, which prefer to be oriented parallel to the layer normal, like in the ordinary SA phase, one observes secondary layers in which the particles prefer to be oriented perpendicular to it. Due to this alternating structure the layer spacing of this new SAW phase is significantly wider compared to the ordinary SA phase. The SAW structure is stabilized by charges which are located at the tips of the molecules. This shows in an exemplary way how the combination of liquid-crystalline behavior and electrostatics can lead to an interesting and novel phenomenology.
The aim of the present investigation is to extend the analysis by studying spatially inhomogeneous systems of ILCs. This is done by investigating how the structural and orientational properties of ILC systems are affected by the presence of a free interface between coexisting bulk states. Both smectic-A phases, SA and SAW, observed in ref. 11 can be in coexistence with the isotropic liquid phase L. This is of intrinsic interest, because it allows one to investigate interfaces which interpolate between a structured and orientationally ordered (i.e., smectic) phase and an isotropic, homogeneous, and thus structure-less, fluid phase. In particular, the transition in the structural and in the orientational order allows one to study the interplay of both properties while they build up at the interface. Although there are theoretical analyses12–18 concerning related types of free interfaces, in these studies the constituent particles are plain liquid crystals without any charges. On the other hand, there is a vast number of theoretical studies on ionic fluids. The thermodynamic behavior19–22 as well as the structure23–27 of these types of fluids, in which long-ranged Coulomb interactions are present, have been intensively studied. However, ionic systems are often analyzed assuming a simple geometry of the particles, such as a spherical shape of the particles like in the restricted primitive model.28–30 In this regard, the present study attempts to analyze the aforementioned type of interface between an isotropic and a smectic phase by accounting for an anisotropic particle shape combined with the presence of charges.
Moreover, different orientations between the interface normal and the smectic layer normal are possible. In this context, an interesting question addresses the equilibrium tilt angle between the interface and the smectic layer normal. This angle may provide insight into nucleation and growth phenomena which are affected by the dependence of the interfacial tension on the orientation of the considered structure.31,32
The present study is structured as follows: in Section 2 the model and the employed density functional theory approach are presented. Our results for the interfaces between the isotropic liquid L and the considered smectic-A phases SA or SAW are discussed in Section 3. Finally, in Section 4 we summarize the results and draw our conclusions.
This model is studied by (classical) density functional theory (DFT), which will be applied to spatially inhomogeneous systems, in particular free interfaces formed between coexisting bulk phases. The methodological and technical details of the present DFT approach are described in Section 2.2.
The two-body interaction potential consists of a hard core repulsive and an additional contribution UGB + Ues beyond the contact distance Rσ, the sum of which can be attractive or repulsive:
![]() | (1) |
![]() | (2) |
![]() | (3) |
![]() | (4) |
The second contribution Ues(r12,ω1,ω2) in eqn (1) is the electrostatic repulsion of ILC molecules. Within the scope of the present study, the counterions are not modeled explicitly. They will be considered to be much smaller in size than the ILC molecules such that they can be treated as a continuous background. On the level of linear response, this background gives rise to the screening of the pure Coulomb potential between two charged sites on a length scale given by the Debye screening length λD such, that the effective electrostatic interaction of the ILC molecules is given by
![]() | (5) |
![]() | (6) |
In Fig. 2 we illustrate the full pair potential (eqn (1)) beyond the contact distance for certain choices of the parameters. The two top panels, (a) and (b), show the pure Gay–Berne potential (uncharged liquid crystals), which is predominantly attractive in the space outside the overlap volume (salmon-colored area). The shape of this overlap volume changes by varying the particle orientations as well as by changing the length-to-breadth ratio L/R. However, these dependences are not apparent from Fig. 2, because L/R = 4 and the particle orientations ωi are kept fixed for all panels. In panel (b) the anisotropy parameter εR/εL = 4 is chosen to be two times larger than for panel (a) (εR/εL = 2). Thus, the ratio of the well depth at the tails and at the sides is increased. The two bottom panels, (c) and (d), show the same choices for the Gay–Berne parameters as for panel (a), but the electrostatic repulsion of the charged groups on the molecules, illustrated by blue dots, is included (γ/(Rε0) = 0.25). In panel (c) the loci of the two charges of the particles coincide at their centers (i.e., D/R = 0) while in panel (d) they are located near the tips (D/R = 1.8). For both cases with charge, the effective interaction range is significantly increased compared with the uncharged case and is governed by the Debye screening length, chosen as λD/R = 5.
It is worth mentioning, that the present model cannot be considered as a quantitatively valid description of any realistic ionic liquid crystal system. A screened electrostatic pair interaction of the Yukawa form (eqn (5)) is the extreme case of the effective pair potential between ions in a (dilute) electrolyte at high temperatures. Nonetheless, for the purpose of the present theoretical study, which is concerned with the basic microscopic mechanisms and the generic molecular properties present in ILC systems, the employed model is appropriate as it incorporates the following key properties of ILCs: first, a sufficiently anisotropic shape (prolate) of the particles, i.e., they can be considered as (calamitic) mesogenes. In this context, an assessment of the bulk phase behavior, depending on the length-to-breadth ratio of the particles, is provided in ref. 11. In particular, the relevance of a sufficiently anisotropic shape (i.e., L/R > 2) for observing genuine smectic phases is discussed. Second, the ionic properties of ILCs are incorporated such that they reflect the main feature of ionic fluids, i.e., the effective interaction of the ionic compounds via a screened electrostatic pair interaction. Although the chosen functional form given by eqn (5) cannot be considered as a quantitatively reliable representation, it still accounts for the fact that the actual ion–ion pair interaction in an ionic fluid is indeed short-ranged, rather than long-ranged, as it is the case for the bare Coulomb interaction.
In conclusion, eqn (5) is characterized by an effective interaction strength γ/R (which will be numerically expressed as the relative interaction strength γ/(Rε0) compared to the interaction strength ε0 of the Gay–Berne potential), an effective interaction range λD, and an effective location D of the charge sites inside the coions. In order to represent specific ILC molecules by a particular set of parameters of the present model, one would tune the independent model parameters, i.e., L/R, εR/εL, γ/(Rε0), D/R, and λD/R such that the resulting total pair potential U(r12,ω1,ω2)/ε0 (compare eqn (1) and Fig. 2) resembles (qualitatively) the actual pair potential of the considered ILC molecules. In this regard, it is worth mentioning that in principle comparisons of our effective theory with particle simulations can be made, related to the study by Saielli et al.,35 who performed molecular dynamics (MD) simulations for a mixture of (ellipsoidal) Gay–Berne and (spherical) Lennard-Jones particles. Additionally, both species carry charges and therefore resemble cations and anions, respectively. Our ad hoc pair potential (eqn (1)) of the coions can be compared with the effective interaction, which can be determined as the logarithm of the particle–particle distribution function of the elongated cations in the MD simulations.
We note, that the choices L/R = 4 and εR/εL = 2, which are used throughout our analysis, are comparable to those used in previous studies (see, e.g., ref. 10, 11, 35 and 36) for similar kinds of particles. While these values of the Gay–Berne parameters give rise to the formation of smectic phases, the occurrence of nematic phases is typically observed for much larger values of L/R and εR/εL.37
![]() | (7) |
The last term is the excess free energy β[ρ] in units of kBT, which incorporates the effects of the inter-particle interactions. Minimizing eqn (7) leads to the Euler–Lagrange equation, which implicitly determines the equilibrium density profile ρ(r,ω):
![]() | (8) |
![]() | (9) |
The excess free energy functional is the characterizing quantity of the underlying many-body problem. However, in general it is not known exactly so that one has to adopt appropriate approximations of it. Following the approach of our previous study11 concerning the bulk phase behavior of ILCs, in the spirit of ref. 38 a weighted density expression for β[ρ] is considered:
![]() | (10) |
![]() | (11) |
The effective one-particle potential βψ(r,ω) consists of two contributions. The first one incorporates the hard-core interactions via the well-studied Parsons–Lee functional,39,40
![]() | (12) |
![]() | (13) |
The second contribution to the effective one-particle potential βψ[] takes into account the interactions beyond the contact distance (see the case |r12| ≥ Rσ in eqn (1)) within the modified mean-field approximation,44 which is a variant of the extended random phase approximation (ERPA):45
![]() | (14) |
The present study is devoted to the analysis of free interfaces which are formed between coexisting bulk phases. In particular, the planar interfaces between the isotropic liquid L and the two different types of smectic-A phases (SA or SAW, see Section 3) will be considered, for which the interface normal is expected to be parallel to the z-direction (see Fig. 3). Due to the isotropy of the liquid phase L, the direction of the smectic layer normal
![]() ![]() | (15) |
![]() | ||
Fig. 3 Sketch of the interface structure under consideration. Consider a planar interface, illustrated by the horizontal red line, between the isotropic bulk liquid L, imposed as the boundary condition at z → −∞, and the smectic-A phase SA (or SAW), imposed as the boundary condition at z → +∞. Thus, the interface normal (red vertical arrow) points into the z-direction. At the top, the tails of four layers of particles of the (ordinary) SA phase are visible, which are well aligned with the smectic layer normal ![]() ![]() ![]() ![]() |
As mentioned above, the coefficients Qi(r) in eqn (11) arise from expanding ρ(r,ω) in a second-order Legendre- and Fourier-series:11
![]() | (16) |
![]() | (17) |
![]() | (18) |
For 0 < α ≤ π/2 one could also consider an integration domain which has a non-vanishing extent in z-direction. However, such a choice has at least two disadvantages: first, unlike dx, which corresponds to the periodicity of the system in x-direction, for 0 < α ≤ π/2 there is no obvious choice for the extent of d parallel to the interface normal. Additionally, there is no unique choice for the geometrical shape of the integration domains; besides using a rectangular form, one could also use any other (two-dimensional) geometrical object as integration domain
d. In this sense the slice of length dx perpendicular to the interface normal is a simple but also consistent choice. Second, this choice renders the evaluation numerically less demanding, because it requires only a one-dimensional integration (exploiting the translational invariance in y-direction), instead of evaluating a two-dimensional integral. We note, that an infinite extent of the integration domain parallel to the interface normal leads to coefficients Qi which are independent of the position r and therefore cannot be used to obtain interface profiles.
If α = 0, i.e., the smectic layer normal = ẑ is parallel to the interface normal, dx diverges and the system is translationally invariant in x- and y-direction. In this case, the integration domain
d has an extent of length d in z-direction, i.e.,
(r − r′) = Θ(d/2 − |z − z′|) with arbitrary extent in the lateral dimensions x and y. As before, the coefficients Qi(z) depend only on the z-coordinate. It is worth mentioning, that for all tilt angles 0 ≤ α ≤ π/2 the correct (constant) bulk values of the coefficients Qi are recovered, although for 0 < α < π/2 the orientation of the integration domain
d (recall that
d is a slice of width dx in x-direction for all α ∈ (0,π/2]) changes with respect to the direction of the smectic layer normal
(α). However, because the integration domain covers a full period dx in x-direction, it gives the same values for the coefficients Qi in the bulk phases, as for evaluating the coefficients Qi with an integration domain parallel to the smectic layer normal
, which is the case for α = 0 and π/2.
Finally, the one-particle direct correlation function c(1)(r,ω,[ρ]) can be derived by considering eqn (9) which leads to the following (modified) expression for c(1)(r,ω,[ρ]) (compare eqn (21) in ref. 11):
![]() | (19) |
Eqn (8) has been solved numerically (utilizing a Picard scheme with retardation) by using eqn (19) as well as the (constant) bulk values of the coefficients Qi,L = Qi(z → −∞) in the isotropic liquid phase L and Qi,S = Qi(z → ∞) in the smectic-A phase (SA or SAW)) at coexistence (T,μ) = (Tcoex,μcoex). The structural properties and the orientational order at the free interface are analyzed in terms of the interface profiles of the packing fraction
![]() | (20) |
![]() | (21) |
![]() | (22) |
zη = −hη(0)/(η0,SA − η0,L), | (23) |
zS2 = −hS2(0)/(S20,SA − S20,L), | (24) |
Note, that instead of using the mean packing fraction η0 or the mean orientational order parameter S20 in eqn (22), for determining the interface positions, in principle, one could also use the profiles η(r) and S2(r) directly. However, the disadvantage of this latter approach is that in the smectic-A bulk phase SA (or SAW) the profiles η(r) and S2(r) are still functions of the position r (via the projection r· onto the layer normal
). Typically, this prevents the use of the latter generalized eqn (23) and (24) for determining zη and zS2. Instead, one has to solve eqn (22) numerically, which requires many iterations depending on the desired accuracy.
Nevertheless, in the particular case α = π/2 the interface normal and the smectic layer normal are perpendicular. Due to the translational invariance of the smectic phases perpendicular to their layer normal, here the density profile η(z → ∞) and the orientational order parameter profile S2(z → ∞) do not depend on z for z → ∞ in the smectic bulk, but they depend only on the x-coordinate. Thus, for α = π/2 one can define interface contours η(x) and
S2(x), analogously to zη and zS2:
![]() | (25) |
![]() | (26) |
![]() | ||
Fig. 4 Bulk phase diagrams for (a) ionic liquid crystals with L/R = 4, εR/εL = 2, γ/(Rε0) = 0.045, λD/R = 5, and D = 0 and (b) with L/R = 4, εR/εL = 2, γ/(Rε0) = 0.045, λD/R = 5, and D/R = 1.8. For D = 0, i.e., the charges being concentrated in the center of the molecules, solely a first-order phase transition from the isotropic liquid phase L to the ordinary smectic-A phase SA occurs at sufficiently high mean packing fractions η0. The ordinary smectic-A phase SA is characterized by a layer structure with smectic layer spacing d/R ≈ 4.3 ≳ L/R = 4 comparable with the particle length L. The particles in the layers are well aligned with the layer normal ![]() ![]() |
The L–SA-interface is shown in Fig. 5 for T* = 1.3. In the phase diagram in Fig. 4(a) the corresponding two coexisting bulk states are marked by black dots (•). Panels (a) and (b) show the packing fraction profile η(z) along the interface normal and the orientational order parameter profile S2(x), respectively. The black dashed vertical line in panel (a) marks the position zη of the Gibbs dividing surface, which is defined by eqn (23). Correspondingly, the black dashed vertical line in panel (b) marks the position zS2 (eqn (24)). Apparently, the two interface positions zη and zS2, which are related to the interfacial transition in the structure and in the orientational order, respectively, differ from each other. In Fig. 6, these differences zη − zS2 are plotted as function of the reduced temperature T* for three different kinds of liquid-crystalline systems. The violet curve corresponds to ILCs with all charges concentrated in the molecular centers, i.e., D = 0, while the green curve shows data points for D/R = 1.8. The blue curve corresponds to a system of ordinary (uncharged) liquid crystals (OLCs) described by L/R = 4, εR/εL = 2, and γ/(Rε0) = 0. The phase diagram for OLCs is not shown here; it is presented in Fig. 4(a) of ref. 11. Within the considered temperature ranges, in all three cases the differences are at most as large as the length of the particle diameter R, which in turn is much smaller than the smectic layer spacing d/R ≈ 4.3 which is comparable to the particle length L, because the particles within the smectic layers are well aligned with the z-direction, indicated by S2(z) > 0.8 in the centers of the smectic layers. Thus, the small size of the differences shows that in these cases the transition in the orientational order and in the fluid structure go along with each other. As soon as the smectic layer structure dies out, the orientational order vanishes as well.
![]() | ||
Fig. 5 The L–SA-interface profile of the packing fraction η(z), panel (a), and the orientational order parameter S2(z), panel (b), are shown for an ionic liquid crystal with L/R = 4, εR/εL = 2, γ/(Rε0) = 0.045, λD/R = 5, and D = 0, i.e., the charges are concentrated in the center of the molecules. The free interface between the isotropic liquid L (imposed as boundary condition for z → −∞) and the ordinary smectic-A phase SA (i.e., z → ∞) is considered for the reduced temperature T* = 1.3. The corresponding coexisting bulk states are marked by the black dots (•) in the phase diagram in Fig. 4(a). The tilt angle is α = 0, i.e., the smectic layer normal ![]() |
![]() | ||
Fig. 6 The difference (zη − zS2)/R between the Gibbs dividing surface position zη (eqn (23)), and the surface position zS2 (eqn (24)), which corresponds to the transition of the orientational order at the interface, are shown for three cases. First, an ordinary (uncharged) liquid crystal (OLC; blue curve); second, ILCs with charges in their center, i.e., D = 0 (violet curve); and, third, ILCs with charges at the tips, i.e., D/R = 1.8 (green curve). Here, the smectic layer normal ![]() |
While for ILCs with charges in their center, within the considered temperature range, only L–SA-coexistence is observable (see Fig. 4(a)). For ILCs with the charges at the tips, such as in the case L/R = 4, εR/εL = 2, γ/(Rε0) = 0.045, λD/R = 5, and D/R = 1.8, the bulk phase behavior changes significantly at low temperatures, i.e., for T* < 1.23. The bulk phase diagram in Fig. 4(b) shows that in this case the distinct smectic-A phase SAW occurs for intermediate mean packing fraction η0. The SAW phase is characterized by an alternating layer structure of smectic layers with a majority of particles being oriented parallel to the smectic layer normal and a minority of particles localized in secondary layers which prefer orientations perpendicular to the smectic layer normal. Due to this alternating layer structure the smectic layer spacing d/R ≈ 7.5 is increased for the SAW phase. A detailed discussion of the structural and orientational properties of this new and peculiar smectic-A phase, in particular concerning the bulk density and the orientational order parameters profiles, is given in ref. 11.
In Fig. 7 the L–SAW-interface profiles η(z) and S2(z) are shown for α = 0 and T* = 0.9. In the phase diagram in Fig. 4(b) the corresponding coexisting bulk states are marked by red dots (). On the right hand side of Fig. 7 the alternating layer structure of the bulk SAW phase is evident. In the main layers the majority of the particles (η(z) > 2) has orientations parallel to the z-axis (S2(z) > 0.8) and in the secondary layers, formed by less of them (η(z) ≈ 0.6), the particles prefer orientations perpendicular to the z-axis (S2(z) < 0). For the L–SAW-interface the difference (zη − zS2)/R ≈ 2.6 of the two interface positions is increased compared to the L–SA-interface (see Fig. 6), because the smectic layer spacing d/R ≥ 7.5 in the SAW phase is enlarged, too. As before, the orientational order directly vanishes with the disappearance of the layer structure. Furthermore, the inset in Fig. 6 shows that (zη − zS2)/R decreases upon lowering the temperature. Thus the difference zη − zS2 becomes smaller relative to the layer spacing d, such that the direct vanishing of the orientational order associated with the disappearance of the layer structure is observable for the whole temperature range considered here.
![]() | ||
Fig. 7 For α = 0, the L–SAW-interface profiles η(z) and S2(z) are shown for ILCs with charges at the tips (L/R = 4, εR/εL = 2, γ/(Rε0) = 0.045, λD/R = 5, and D/R = 1.8) at the reduced temperature T* = 0.9 (see the red dots (![]() |
![]() | ||
Fig. 8 The L–SA-interface profiles η(x,z), panel (a), and S2(x,z), panel (b), are shown for T* = 1.3 (see the black dots (•) in Fig. 4(a)) and α = π/2. Accordingly, the smectic layer normal ![]() ![]() ![]() ![]() |
The black dashed lines in Fig. 8 show the interface positions zη and zS2 calculated from eqn (23) and (24), while the white dotted lines show the interface contours η(x) and
S2(x) obtained from eqn (25). The contour lines
η(x) and
S2(x) at the centers of the tails of the smectic layers, e.g., at x/R ≈ 2.14, are very close to zη and zS2, respectively. This suggests that the two distinct definitions of the interface positions, i.e., using either eqn (23) and (24) or eqn (25), are consistent with each other, because the majority of the particles in the smectic phase are located close to the centers of the smectic layers. In Fig. 8(a) the packing fraction interface contour
η(x) exhibits discontinuities for lateral positions
; at which the smectic bulk packing fraction ηSA(
) := η(
, z → ∞) takes the same value ηL = η(
, z → −∞) as in the isotropic liquid L, i.e., ηSA(
) = ηL. Thus, the numerical calculation of the Gibbs dividing surface viaeqn (25) leads to a divergence due to the vanishing denominator. This can be considered as an artifact, which, however, occurs only at the particular lateral positions
. Nevertheless, the benefit of considering
η(x) and
S2(x) as interface positions is their dependence on the lateral coordinate x. In particular, for the case of the L–SAW-interface it is necessary to consider
η(x) and
S2(x) in order to study the interface at the main layers and at the secondary layers separately (see below).
Interestingly, if the layer normal and the interface normal are perpendicular, one observes a significant difference (zη − zS2)/R ≈ 0.72 − (−1.76) = 2.48 between the interface position zη, corresponding to the structural transition, and zS2 corresponding to the transition in the orientational order between the coexisting phases. Hence, the alignment of the particles with the x-axis persists a few particle diameters deeper into the liquid phase L than the layer structure of the SA phase is maintained – unlike in the case α = 0, i.e., in which the smectic layer normal is parallel to the interface normal, for which the orientational order directly vanishes when the smectic layers disappear (see Section 3.1). We note, that the vanishing of the orientational order significantly after (upon approaching the interface from the orientational ordered phase) the structural transition associated with the density profile, has already been observed previously18 in the case of the interface between an isotropic liquid and a plastic-triangular crystal (PTC).
For the type of ILCs with the charges at the tips, at low temperatures the new wide smectic-A phase SAW can be observed (see Fig. 4(b)). It is characterized by an alternating structure of layers in which the particles are predominantly parallel to the layer normal =
(like in the SA phase) and layers of particles which are preferentially perpendicular to the layer normal. The free interface formed between the isotropic liquid L and the SAW phase for T* = 0.9 and α = π/2 is shown in Fig. 9. The red regions in Fig. 9(a) show the layers of particles (at x = 0 and x/R ≈ ±d/R = ±7.5) being parallel to the layer normal, while in between (at x/R ≈ ±d/(2R) = ±3.75) in light blue color the secondary layers are visible. The dark blue color at x/R ≈ ±d/(2R) = ±3.75 in panel (b) shows that the orientational order parameter S2(x,z) is negative at the location of the secondary layers, because there the particles are preferentially perpendicular to the layer normal. The interface at the parallel layers behaves very much like the L–SA interface, as can be inferred from the (white) interface contours
η(x/R = 0, ±7.5)/R ≈ 0.81 and
S2(x/R = 0, ±7.5)/R ≈ −1.83 which show that the orientational ordering of the SAW phase persists into the liquid phase L for a few particle diameters. This is also apparent from the interface positions zη/R ≈ 1.0 and zS2/R ≈ −2.3, depicted by the black dashed lines in Fig. 9. Conversely, at lateral positions x/R ≈ d/(2R) = ±3.75 associated with the centers of the intermediate layers, it turns out that the orientational order undergoes the transition before the layer structure vanishes if one approaches the interface from the SAW side (
S2(x/R = ±3.75)/R ≈ 3.39 and
η(x/R = ±3.75)/R ≈ −0.34; in order to guide the eye the magenta dots (
) in Fig. 9 mark these positions). This behavior is opposite to the above one and is presumably related to the fact, that the secondary layers consist of particles being preferentially perpendicular to the layer normal; unlike the particles in the main layers of the SAW phase or the particles in the SA layers, these particles do not align with the layer normal
=
. Instead they are avoiding an orientation parallel to it. While the transition across the L–SA interface – from alignment with the layer normal towards an isotropic orientational distribution – results in an increase of the effective particle diameter in the y- and z-direction, for the secondary SAW layers the effective diameter is decreased from the SAW phase towards the isotropic liquid L. In Fig. 9 there are discontinuities in the (white) interface contour lines
η(x) and
S2(x), as in Fig. 8. These discontinuities occur at lateral positions
at which the packing fraction η(
, z → ±∞) or the orientational order parameter S2(
, z → ±∞) take the same value in the isotropic bulk, i.e., for z → −∞, as in the SAW bulk, i.e., for z → ∞.
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Fig. 9 The interface profiles η(x,z) and S2(x,z) for T* = 0.9 and α = π/2. Here the L–SAW interface (see the red dots (![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Interestingly, while for D = 0 the periodic structure of the profiles η(x,z) and S2(x,z) in x-direction is clearly apparent also in the decays ln|η(x,z) − ηL| and ln|S2(x,z) − S2,L| far away from the L–SA-interface (z/R < −20 in Fig. 10(a) and (b)), for D/R = 1.8 (panels (c) and (d)) the decays vary only little as function of x. This distinct behavior can be a signature of the respective molecular charge distributions, because if the charges are localized at the centers of the molecules, due to the layer structure in the SA phase the charges are also localized at the centers of the smectic layers, while for D/R = 1.8 the charges are less localized along the lateral direction x. Close to the interface (z/R > −20) the structure is very similar in both cases and, as will be discussed later, it is the hard-core repulsion which is the dominant contribution here.
Turning the view parallel to the x-axis, one obtains projected representations of the logarithmic plots in Fig. 10, which are shown in Fig. 11 keeping the order of panels as in Fig. 10. Hence, Fig. 11(a) and (b) correspond to the case D = 0 presenting ln|η(x,z) − ηL| and ln|S2(x,z) − S2,L|, respectively. Similarly, Fig. 11(c) and (d) show the case D/R = 1.8. In both cases, at large distances, i.e., z/R < −20, the decay of the density profiles is dominated by the electrostatic contribution Ues to the total interaction potential U (see Fig. 11(a) and (c)). Accordingly, the decay of the envelope is determined by the Debye screening length λD/R = 5, highlighted by the orange lines in Fig. 11. It is worth mentioning that a DFT study46 of the asymptotic behavior of the liquid–vapor interface has yielded, unlike the present findings, a decay length lb larger than the Debye screening length λD for a hard sphere system with additional Yukawa interaction. While in the present study the Yukawa potential is purely repulsive, in ref. 46 using an attractive Yukawa potential is indispensable, because a sufficiently strong attraction is needed for liquid–vapor coexistence to occur.
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Fig. 11 The same quantities as shown in Fig. 10. Panels (a) and (b) correspond to the case D = 0 presenting ln|η(x,z) − ηL| and ln|S2(x,z) − S2,L|, respectively, whereas panels (c) and (d) correspond to the case D/R = 1.8. However, here the direction of view is parallel to the x-axis so that the manifold from Fig. 10 is projected onto the plane spanned by the vertical axis and the z axis. Away from the interface, i.e., for z/R < −20, the decay length for ln|η(x,z) −ηL| can be identified as the Debye screening length λD/R = 5 for both cases (a) D = 0 and (c) D/R = 1.8. From the inset in panel (a), which shows ln|η(x,z) − ηL| for the corresponding (uncharged) ordinary liquid crystal with L/R = 4 and εR/εL = 2, it is apparent that the contributions due to the Gay–Berne potential (the asymptotics of which is indicated by the blue line) and due to the hard-core interaction (the asymptotics of which is depicted by the black line) are much weaker than the (screened) electrostatic contribution and do not play a role within the range of ln|η(x,z) − ηL| considered here. (In order to guide the eye, the blue and black lines are also shown in the main plots. Apparently, in (a) and (c) the blue and black lines are far below the respective profiles.) However, for ln|S2(x,z) − S2,L|, i.e., for panels (b) and (d), one observes crossovers – indicated by the intersection of the orange and blue lines at z/R ≈ −67 in (b) and z/R ≈ −45 in (d) (compare the red arrows in the respective plots) – from the electrostatic regime towards the decay governed by the Gay–Berne contribution with decay length ξGB/R ≈10. Such crossovers occur within the considered range z/R ∈ [−80,0], because for the orientational order parameter the amplitude of the decay, due to the Gay–Berne interaction, is larger than for the packing fraction (compare the intersections of the blue lines with the ordinates in panels (a) and (b)). Due to the hard-core interaction, for z/R > −20 the decay length ξPL/R ≈ 1.9 (Parsons–Lee, black lines) is visible for the ordinary liquid crystal in the insets of (a) and (b) as well as for ln|S2(x,z) − S2,L| of the two considered ILCs. (Due to the small amplitudes of the hard-core contributions to ln|η(x,z) − ηL|, for the ILC considered here, this decay has not been observed.) In order to confirm, that the decay length ξPL/R ≈ 1.9 is indeed due to the hard-core interaction, the insets of the panels (c) and (d) show ln|η(x,z) − ηL| and ln|S2(x,z) − S2,L| of the pure hard-core system (βψ := βψPL). Interestingly, ln|η(x,z) − ηL| and ln|S2(x,z) − S2,L| behave very similarly close to the interface, i.e., z/R > −10, for all three kinds of systems studied here. This suggests that the structure and the orientational properties close to the interface are governed by the hard-core interaction which enters into the present DFT approach (see Sections 2.1 and 2.2). |
Interestingly, the asymptotic behavior of the orientational order parameter at far distances, i.e., for z/R < −60, differs from the electrostatic decay and another regime (highlighted by blue lines in Fig. 11) with a larger decay length ξGB/R ≈ 10 sets in. This longer-ranged decay is due to the Gay–Berne interaction UGB which is verified by calculating the interface profile for an ordinary liquid crystal (OLC) without charges (compare the insets of panels (a) and (b) of Fig. 11). For the OLC, at far distances, i.e., z/R < −30, the same large decay length ξGB/R ≈ 10 is observed. However, the amplitudes of the decay of the packing fraction and of the orientational order parameter differ significantly. (The blue line in panel (a) intersects the ordinate at ln|η − ηL| ≈ −25, whereas the blue line in (b) intersects the ordinate at ln|S2 − S2,L| ≈ −20.) For D = 0, it turns out that for the orientational order parameter the crossover from the electrostatic decay towards the Gay–Berne decay occurs at z/R ≈ −67 (this position is marked by the red arrow in Fig. 11(b)), whereas for the case D/R = 1.8 the crossover occurs at z/R ≈ −45 (see the red arrow in Fig. 11(d)). Ultimately, the larger Gay–Berne decay length ξGB/R ≈ 10 will also become apparent in the decay profile of the packing fraction. However, due to the smaller amplitude of the Gay–Berne decay of the density compared with the decay of the orientational order parameter (compare the insets in Fig. 11(a) and (b)), in the present case the crossover occurs further away from the interface (in Fig. 11(a) the intersection of the orange line and the blue line is located at z/R ≈ −121 (not visible) and in Fig. 11(c) at z/R ≈ −97 (also not visible)). However, at very far distances z/R < −80, the magnitudes ln|η − ηL| ≲ −25 are very small and cannot be resolved numerically. For this reason, in Fig. 11(a) and (c) crossovers from the electrostatic regime to the Gay–Berne regime are not shown.
We note that, although the Gay–Berne potential UGB decays algebraically ∝(r12/R)−6 (see eqn (2)), here the Gay–Berne decay is exponential, because solving the Euler–Lagrange equation in eqn (8) requires the evaluation of the ERPA contribution βψERPA of the effective one particle potential βψ (see eqn (14) and (19)). The numerical calculation of this integral (which extends over the whole volume of the system) requires a truncation in terms of a cut-off distance of the integral which leads to an exponential decay of this contribution, instead of a power law decay ∝(z/R)−3,46–48 as it is expected for the full Gay–Berne potential UGB. (The exponent 3 arises because the asymptotic behavior of an interfacial density profile, generated by long-ranged forces, varies proportional to the corresponding (total) potential, which acts on a test particle at a distance z from the interface and which is due to the pair interaction between the particles in one of the two coexisting phases (which are separated by the considered interface) and the test particle. Thus, via an integration of the Gay–Berne pair interaction, which decays ∝(r12/R)−6, over a half-space, one obtains the corresponding total potential decaying ∝(z/R)−3.47–49)
For z/R → −∞ the algebraic decay of the Gay–Berne interaction potential always dominates the exponential decay due to the screened electrostatic interaction, independent of the relative strength of the electrostatic and the Gay–Berne interaction potential. A variation of their relative strength γ/(Rε0) would only lead to a shift of the location of the corresponding crossovers in the density and the order parameter profiles (see the red arrows in Fig. 11) caused by altering the amplitudes of the respective decays of the two interactions.
Close to the interface, i.e., for −20 < z/R < −5, in the insets of Fig. 11 one can observe an exponential decay with a decay length ξPL/R ≈ 1.9 (depicted by the black lines) which arises from the pure hard-core Parsons–Lee contribution βψPL. Thus ξPL can be identified as the isotropic-liquid bulk correlation length of the pure hard-core system. Interestingly, while the hard-core correlation length ξPL is observable in OLCs – within both the η and the S2 profiles (at distances z/R ∈ [−20,−5] the respective decays closely follow the black lines which depict the hard-core decay in the insets of Fig. 11(a) and (b)), for ILCs this decay is visible only within the S2 profile. Only for the S2 profile the amplitude of the hard-core decay is large enough, such that the hard-core correlation length ξPL is observable before the electrostatic decay becomes dominant. The insets in Fig. 11(c) and (d) show the interface profiles calculated for the pure hard-core system (βψ := βψPL) in order to verify that the decay close to the interface, i.e., for −20 < z/R < −5, is governed by the hard-core interaction.
Finally, it is worth mentioning that for all cases shown in Fig. 11, the structural and orientational properties close to the interface, i.e., for z/R > −10, agree very well. Thus, it is the hard-core interaction which determines the structural and orientational properties close to the interface, while the electrostatic and the Gay–Berne contributions dominate further away from the interface. At intermediate distances electrostatics dominates the decay of the interface profiles whereas far away from the interface ultimately the attractive Gay–Berne interaction dominates. Furthermore, the positions of the crossovers between these regimes are distinct for the packing fraction profile and the orientational order parameter profile.
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Fig. 12 The L–SA interface profiles η(x,z) (eqn (20)) and S2(x,z) (eqn (21)) for α = π/4 and T* = 1.3 are shown. Here, an ILC with charges localized at its center is considered (L/R = 4, εR/εL = 2, γ/(Rε0) = 0.045, λD/R = 5, and D = 0). For z → −∞ the isotropic liquid bulk L is approached and for z → ∞ the bulk of the SA phase is attained, i.e., the interface normal is parallel to the z-axis. The red stripes at the top of the contour plots show the tails of the smectic layers. The black dashed lines mark the interface positions zη/R ≈ 5.56 and zS2/R ≈ 2.79 calculated viaeqn (23) and (24). Similar to the case α = π/2 (see Fig. 8), to a certain extent the orientational order persists into the liquid phase L. |
In Fig. 13 the interfacial tension Γ*(α) given by eqn (26) and the distance zη − zS2 between the interface positions associated with the mean packing fraction η0(x) and the mean orientational order parameter S20(x) are shown as function of the tilt angle α. In Fig. 13(a) the case of the L–SA-interface for ILCs with the charges at their center is considered for T* = 1. Both the interfacial tension Γ*(α) (black dots, •) and the distance zη − zS2 (orange dots, ) exhibit a global minimum at α = 0 and a second, local minimum at α = π/2. Thus, the equilibrium tilt angle αeq = 0 corresponds to the configuration in which the interface normal and the smectic layer normal
= ẑ are parallel, whereas the corresponding perpendicular orientation α = π/2 is metastable. This increase in the interfacial tension Γ* below α = π/2 suggests that the configuration, in which the interface normal and the layer normal are orthogonal, should be observable without resorting to any external stabilizing field which could be provided, e.g., by a suitably structured substrate. This metastability of the tilt angle α = π/2 can be checked also via computer simulations. Interestingly, the increase of the interfacial tension below α = π/2 is accompanied by an increase in the distance zη − zS2, suggesting that maintaining the local orientational order in the isotropic liquid beyond the smectic layers costs free energy. Consistently, in the case αeq = 0, for which the orientational order vanishes directly with the disappearance of the smectic layers, the cost in free energy is lowest. Apparently, for α = 0 the interfacial tension Γ*(α = 0) ≈ 0.006 is significantly smaller than for all other angles α shown in Fig. 13(a). For technical reasons we did not study small tilt angles α > 0 and hence cannot comment on the functional form of Γ*(α) for 0 < α < π/6 in the case D/R = 0 or for 0 < α < π/4 in the case D/R = 1.8. This is indicated by connecting the data points at α = 0 and π/6 by dashed lines. (For the same reason, in (b) the data points at α = 0 and π/4 are connected by dashed lines.) It has been pointed out in Section 2.2, that due to the crossover at the tilt angle α = 0 from a periodic system towards one which is translationally invariant in lateral direction x, the integration domain
d for evaluating the coefficients Qi(r) (see eqn (16)) is not continuously evolving at α = 0. For α > 0 it is a slice of length dx = d/sin(α) in x-direction, while for α = 0 it is the subsystem of length d in z-direction at position r. (For α = 0 the extent in x- and y-direction is arbitrary due to the translational invariance in lateral direction.) In order to describe a continuous variation of the interfacial tension Γ*(α) for all tilt angles α ∈ [0,π/2], one thus needs to consider a different approach, which does not rely on a projected density and thereby on the direction of the bulk smectic layer normal
throughout the whole interface structure. Nonetheless, our above approach still allows one to compare the interfacial tension Γ*(α) for the extreme cases α = 0 and π/2, thus predicting which one of the two is preferred. Furthermore, our approach provides an understanding of the local increase in Γ*(α) below α = π/2, as one observes an increasing distance zη − zS2 between the transition in the structural and the orientational order at the interface.
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Fig. 13 The (reduced) interfacial tension Γ*(α) (eqn (26), black line) and the distance zη − zS2 between the transition in the structural and the orientational order (orange line) as function of the tilt angle α. In panel (a) the L–SA interface at T* = 1 is considered for ILCs with their charges localized at the center (L/R = 4, εR/εL = 2, γ/(Rε0) = 0.045, λD/R = 5, and D = 0 (see Fig. 4(a)). There are two minima: the global minimum at the equilibrium tilt angle αeq = 0 (i.e., interface normal and smectic layer normal are parallel) and a local minimum at α = π/2 which shows that the orthogonal orientation of the smectic layer normal and the interface normal is a metastable configuration. The increase of the interfacial tension below α = π/2 is accompanied by an increase of the distance zη − zS2. This suggests that maintaining to a certain extent the local orientational order in the isotropic liquid beyond the smectic layers costs free energy. For technical reasons we did not study small tilt angles α > 0. Hence we cannot comment on the functional form of Γ*(α) for 0 < α < π/6 in the case D/R = 0 or for 0 < α < π/4 in the case D/R = 1.8. This is indicated by connecting the data points at α = 0 and π/6 by dashed lines in (a) (see the discussion in the main text of Section 3.4). In panel (b) the L–SAW interface, which is accessible for ILCs with their charges at the tips (L/R = 4, εR/εL = 2, γ/(Rε0) = 0.045, λD/R = 5, and D/R = 1.8), is considered for T* = 0.9 (see Fig. 4(b)). Also in this case the equilibrium tilt angle αeq = 0 corresponds to the parallel orientation of the interface normal and the layer normal. Below α = π/2, as function of α the interfacial tension is rather flat, taking the value Γ* ≈ 0.07. Thus, for the L–SAW interface the perpendicular orientation of the interface normal and of the smectic layer normal corresponds to a labile configuration. (Analogously to panel (a), the data points at α = 0 and π/4 in (b) are connected by a dashed line.) We note that Γ*(α) is symmetric around α = π/2, due to the mirror-symmetry of the particles. |
Fig. 13(b) shows data for the L–SAW-interface at T* = 0.9 for ILCs with charges located at the tips. Around α = π/2 the interfacial tension (black squares, ■) is a rather flat function of α taking values around Γ* ≈ 0.07. The slight variations in Γ* for α ∈ [π/4,π/2] might be caused by the numerical evaluation of eqn (8) which has to be done separately for each tilt angle α. Consistently, the distance zη − zS2 (orange squares, ) does not vary much as function of the tilt angle α. As above, the equilibrium tilt angle αeq = 0 corresponds to the configuration in which the interface normal and the smectic layer normal
= ẑ are parallel.
Finally, in Fig. 14, we show the contour plot of the L–SAW-interface for α = π/3 and T* = 0.9 for an ILC system with D/R = 1.8, illustrating the structure of this type of interface.
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Fig. 14 Same as Fig. 12. Here, the L–SAW interface profiles η(x,z) (eqn (20)) and S2(x,z) (eqn (21)) are shown for α = π/3 and T* = 0.9. To this end, an ionic liquid crystal with charges at the tips is considered (L/R = 4, εR/εL = 2, γ/(Rε0) = 0.045, λD/R = 5, and D/R = 1.8). For z → −∞ the isotropic liquid bulk L is approached and for z → ∞ the bulk of the SAW phase, i.e., the interface normal is parallel to the z-axis. The transition in the structure occurs at zη/R ≈ 2.28 and the transition in the orientational order does so at zS2/R ≈ −0.68. |
For D = 0 coexistence between the isotropic liquid L and the ordinary smectic-A phase SA can be observed at a sufficiently large mean packing fraction η0 (see Fig. 4(a)). The SA phase is characterized by a layered structure in the direction of the smectic layer normal with a smectic layer spacing d ≈ L comparable to the particle length L. Within the smectic layers the particles are well aligned with the smectic layer normal. The phase behavior of ILCs is altered by varying the molecular charge distribution, as can be inferred from comparing the case D = 0 (i.e., charges at the center) and D/R = 1.8 (i.e., charges at the tips, see Fig. 4(b)). At sufficiently low temperatures a new smectic-A phase has been observed, which is referred to as the SAW phase.11 The SAW phase shows an alternating structure of layers with the majority of the particles being oriented parallel to the smectic layer normal
and the minority of the particles localized in secondary layers which prefer orientations perpendicular to
. Due to the alternating layer structure, the smectic layer spacing d/R ≈ 7.5 in the SAW phase is increased compared with the spacing in the SA phase.
For a parallel orientation of the smectic layer normal = ẑ and the L–SA-interface normal, i.e., for α = 0 (see Fig. 3), it turns out that the interface locations zη and zS2, associated with the transition in the structural and in the orientational order, respectively, are very close to each other (see Fig. 5). In fact, Fig. 6 shows that for the whole temperature range considered here, the difference zη − zS2 < d in the two interface positions is smaller than the smectic layer spacing d. Hence, for α = 0 the orientational order vanishes within the last smectic layer at the L–SA-interface. Concerning the interface positions, Fig. 6 demonstrates that ILCs with D/R = 1.8 and ordinary (uncharged) liquid crystals with L/R = 4 and εR/εL = 2 exhibit qualitatively the same results. Considering the L–SAW-interface (see Fig. 7) one observes an increase in zη − zS2, but it remains significantly smaller than the smectic layer spacing d/R ≈ 7.5. Thus, for α = 0 it turns out that the loss of orientational order coincides with the disappearance of the layer structure of the respective smectic-A phase at the interface towards the isotropic liquid. This holds for all parameter values studied here.
Interestingly, for α = π/2, i.e., changing the relative orientation of the smectic layer normal =
and the interface normal such that they are perpendicular to each other, leads to qualitative changes in the interfacial properties: a periodic structure of the interface in lateral direction x can be observed, which is a direct consequence of the periodicity in the bulk smectic-A phase with the smectic layer spacing d (see Fig. 3, 8, and 9). For the L–SA-interface (see Fig. 8) one observes considerable differences (zη − zS2)/R ≳ 2 between the interface positions. Thus, the (nearly) parallel orientations of particles in the SA layers persists a few particle diameters R into the liquid phase L, unlike the case α = 0, for which the orientational order vanishes directly with the breakdown of the SA layer structure at the interface, i.e., within the last smectic layer. Due to the periodicity in (lateral) x-direction, in the case α = π/2 one indeed observes a qualitative change in the structure of the L–SAW-interface compared to the L–SA-interface. While at the tails of the SAW main layers the interface also features an orientational order which continues further into the liquid phase L than the layer structure ((
η(x) −
S2(x))/R ≈ 2.6). For the secondary layers it is the layer structure that persists deeper into the L phase than the orientational order ((
η(x) −
S2(x))/R ≈ −3.73). The opposite behavior at the main, respectively secondary, layers is presumably driven by the orientational properties of the respective kinds of layers: in the main layers the particles are well aligned with the smectic layer normal
=
and therefore show an effective diameter in the y−z-plane which is comparable to the particle diameter R. However, in the secondary layers (here with S2(x,z) < 0) the particles avoid orientations parallel to the x-axis, giving rise to an considerably larger effective radius. Upon approaching the liquid phase L, this effective radius increases for the main layers of the SAW phase, whereas it decreases for the secondary layers.
In Section 3.3 the asymptotic behavior of the interface profiles has been studied. In particular, in Fig. 10 and 11 the L–SA-interface for α = π/2 has been considered for the two ILC systems with D/R = 0 and 1.8. For D = 0, i.e., with the charges being localized at the center, the periodic structure of the interface is apparent from the quantities ln|η(x,z) − ηL| and ln|S2(x,z) − S2,L|, showing the logarithmic deviations of the profiles η(x,z) and S2(x,z) from their respective liquid bulk values ηL and S2,L (Fig. 11(a) and (b)), which can be resolved even at far distances z/R < −20 from the L–SA-interface. Conversely, for D/R = 1.8, i.e., the charges being fixed at the tips, far from the interface ln|η(x,z) − ηL| and ln|S2(x,z) − S2,L| vary only marginally as function of the lateral coordinate x. While for D = 0 the charges are strongly localized at the centers of the smectic layers, thus promoting the periodic structure, for D/R = 1.8 the charges are less localized and more distributed along the x-direction.
The asymptotic decays of the interface profiles towards the isotropic liquid L show an interesting and rich behavior. We have found three distinct spatial regimes, which are associated with the three contributions to the underlying pair potential (see eqn (1)). Although the presence of charges is the distinctive feature of ILCs, the (screened) electrostatic contribution to the interaction (eqn (5)) governs the asymptotic decay only at intermediate distances from the interface (see Fig. 11). In this regime, the decay length is given by the Debye screening length, here λD/R = 5. Ultimately, it is the attractive Gay–Berne contribution to the interaction (eqn (2)) which dominates the outermost asymptotic behavior; for the system studied here a considerably large decay length ξGB/R ≈ 10 is observed, which is due to the truncated power law decay of the GB potential. Close to the interface, the hard-core interaction, which leads to the Parsons–Lee contribution to the DFT expression (eqn (12)), dominates the profiles η(x,z) and S2(x,z). The corresponding decay length ξPL/R ≈ 1.9 is comparable to the particle diameter R. This is plausible, because for the case considered here the tilt angle is α = π/2, i.e., the smectic layer normal is perpendicular to the interface normal, and thus the particles in the SA layers are oriented preferentially perpendicular to the interface normal as well. Interestingly, the crossovers between these three different regimes occur at distances characteristic for the packing fraction η(x,z) and the orientational order parameter S2(x,z). While for both types of ILCs considered in Fig. 11 all three decay lengths ξPL, ξGB, and λD are apparent from ln|S2(x,z) − S2,L|, from ln|η(x,z) − ηL| only the decay length λD can be inferred within the considered range z/R > −80. This situation is caused by the relative magnitudes of the respective decay amplitudes: for the packing fraction profile the decay amplitudes due to the Gay–Berne and the hard-core interaction are too small, compared to the corresponding amplitude due to the electrostatic interaction, to be observable.
Since the structural and orientational properties directly at the interface position are determined by the hard-core interaction, i.e., the Parsons–Lee contribution βψPL (eqn (12)), to the effective one-particle potential βψ, close to the interface the profiles for ordinary liquid crystals (OLCs) and ILCs with the same length-to-breadth ratio L/R are very similar. In particular, this includes the interface positions zη and zS2 (see Fig. 6) associated with the transition in the structural and orientational order, respectively. Nevertheless the asymptotic behavior, as discussed above, is distinct for the different kinds of particles (hard ellipsoids, OLCs, and ILCs) and shows a rich phenomenology, specifically for ILCs, due to the cross-overs between the distinct spatial regimes corresponding to the various contributions to the pair potential. Additionally, the bulk phase behavior is crucially affected by the type of particles, because only for the ILCs with charges at the tips, the phase SAW is observed.
Finally, the dependence of the structural and orientational properties of liquid–smectic interfaces on the tilt angle α between the interface normal and the smectic layer normal has been discussed. For the L–SA-interface (see Fig. 13(a)), it turns out, that the parallel orientation of the interface normal and of the smectic layer normal is the one in thermal equilibrium, i.e., αeq = 0. The perpendicular orientation α = π/2 is metastable. Interestingly, the increase in the interfacial tension below α = π/2 is accompanied by an increase in the distance zη − zS2, suggesting that maintaining the local orientational order beyond the smectic layers towards the isotropic liquid costs free energy. Consistently, in the case αeq = 0, for which the orientational order vanishes directly with the disappearance of the smectic layers, the cost of free energy for forming the interface is lowest. For the L–SAW-interface (see Fig. 13(b)) again the equilibrium tilt angle αeq = 0 corresponds to the parallel orientation of the interface and smectic layer normal. However, in this case, around α = π/2, the interfacial tension Γ*(α) varies only weakly so that here the perpendicular orientation is labile. Additional contributions to the surface tensions might arise from elastic deformations of the director field, i.e., spatial variations of the director :=
(r), or deviations from a rotational-symmetric distribution of particle orientations around the director, i.e., f(r,ω) ≠ f(r,
·ω). These contributions are neglected by our approach. Elastic effects can be considered through an explicit dependence of the free energy functional on the director field n(r), i.e., via an elastic energy contribution.50 Alternatively, giving up the assumption of a rotational symmetric distribution of orientations around a particular axis (and thereby enforcing a prescribed homogeneous director field) would also allow one to study the deformations of the director field. However, incorporating these effects would lead to a drastic increase of the computational effort.
Lastly, we emphasize that although here we have focused solely on free interfaces between coexisting bulk phases of ILCs, the DFT framework in Section 2.2 can be extended to inhomogeneous systems of ILCs exposed, e.g., to external fields or ILC–electrolytes in contact with an electrode.
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Fig. 15 Same as Fig. 14. Here, the L–SAW interface profiles η(x,z) (eqn (20)) and S2(x,z) (eqn (21)) are calculated for α = π/3 and T* = 0.9 by using the projected density containing odd Fourier-modes up to second order. The profiles are qualitatively equivalent to those obtained without using the odd modes in the projected density ![]() |
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