DOI: 10.1002/cphc.201301030

From the Molecular Structure to Spectroscopic andMaterial Properties: Computational Investigation ofa Bent-Core Nematic Liquid CrystalCristina Greco,[a] Alberto Marini,[b] Elisa Frezza,[a] and Alberta Ferrarini*[a]

This paper is dedicated to Alberto Marini, whose enthusiasm was crucial to undertake this study

1. Introduction

Bent-core mesogens, also denoted as banana or V-shaped, arecharacterized by a fairly rigid aromatic core and two flexiblewings.[1] Initially, they attracted attention mainly because oftheir polar and chiral smectic phases,[2–4] but more recently itbecame clear that also their nematic phase has distinctive fea-tures, different from those of conventional rod-like systems(see ref. [5] for a review). Most of these features remain theobject of debate; they include phase biaxiality,[6–13] giant flexo-electricity,[14] unusually large viscosity,[15, 16] and small twist- andbend-elastic constants,[15, 17–23] as well as nonstandard dielectricrelaxation,[24] and electroconvection regimes.[25]

The liquid crystal behavior of bent-core mesogens hasa subtle dependence on the molecular structure; in particular,the existence of the nematic phase can be controlled throughsubstituents in the core and in the lateral chains.[26, 27] A keyrole is generally ascribed to the so-called bend angle betweenthe two arms, which can assume values between 908 and1608,[13, 28] also in relation to the formation of the biaxial nemat-ic phase.[6, 29, 30] The existence of two kinds of conformers withbend angles of about 1408 (in strict sense banana-shaped) andabout 1158 (hockey-stick-shaped), the relative stability of whichwould change with temperature, was proposed as the driving

force for the uniaxial-biaxial transition in some bent-core sys-tems.[31] Moreover, the bend angle has been shown to be cru-cial for the bend elasticity.[19–21, 23, 32]

Here we present a study of the uniaxial nematic phase ofthe bent-core mesogen known as A131 (Figure 1), using com-

putational tools that allow us to address molecular level andmacroscopic properties on the same footing:[33] A131 is a goodrepresentative system, as it has been the object of several in-vestigations by different experimental techniques, probing thephase properties both at the molecular and the macroscopiclevel.[17, 34–39] In particular, we focus on molecular orientationalorder parameters, which are directly related to 13C NMR chemi-cal shifts, and on the Frank elastic constants.

Realistic predictions of the nematic phase properties fora given compound require the use of accurate molecular geo-metries, with a suitable account of the conformational degreesof freedom. This has been shown to be important for the elas-tic properties, and crucial for the bend-elastic con-stant.[20, 21, 32, 40–43] Of course, a detailed molecular-level descrip-tion is a necessary requirement also for the prediction of NMR

We present a computational investigation of the nematicphase of the bent-core liquid crystal A131. We use an integrat-ed approach that bridges density functional theory calculationsof molecular geometry and torsional potentials to elastic prop-erties through the molecular conformational and orientationaldistribution function. This unique capability to simultaneouslyaccess different length scales enables us to consistently de-

scribe molecular and material properties. We can reassign13C NMR chemical shifts and analyze the dependence of phaseproperties on molecular shape. Focusing on the elastic con-stants we can draw some general conclusions on the uncon-ventional behavior of bent-core nematics and highlight thecrucial role of a properly-bent shape.

Figure 1. Chemical structure of A131 with carbon numbering. I, Nu, Nb andSmC denote the isotropic, the uniaxial nematic, the (presumed) biaxialnematic and the smectic C phase, respectively.

[a] C. Greco, E. Frezza, Prof. A. FerrariniDepartment of Chemical SciencesUniversity of Padovavia Marzolo 1, 35131 Padova (Italy)E-mail : alberta.ferrarini@unipd.it

[b] Dr. A. MariniDepartment of Chemistry and Industrial ChemistryUniversity of Pisavia Risorgimento 35, 56126 Pis (Italy)

Supporting Information for this article is available on the WWW underhttp://dx.doi.org/10.1002/cphc.201301030.

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observables, which reflect the different orientational averagingof magnetic tensors located at different sites in the mole-cule.[44, 45] In this study, geometric parameters and torsional po-tentials of A131 are obtained by density functional theory(DFT) calculations in vacuum, along the lines proposed inref. [46]. Monte Carlo sampling of the conformational space isthen performed and equilibrium properties are obtained asaverages over the torsional and orientational degrees of free-dom.[47, 48] The full account of the molecular flexibility (includinglateral chains) is the main difference between ours and previ-ous computational investigations of A131, which were restrict-ed to a few conformers of the core;[31] and indeed, the need ofa deeper analysis of the conformational distribution has beenpointed out in that context.

In the next section the computational methods are summar-ized. Then the results are reported and discussed in the lightof the available experimental data: we examine the conforma-tional distribution of A131, the 13C NMR chemical shifts, the ori-entational order parameters, and finally the elastic constants.In the last section we present the conclusions of our study.

2. Theoretical and Computational Methods

Molecular Geometry and Torsional Potentials

Figure 2 shows the molecular structure of A131 with the rotat-ing bonds located in the five-ring core and in the alkyl chains.The overall shape of the aromatic core depends on the valueof the dihedral angles flanking the central aromatic ring, whichare denoted as c24 (C9�N�C10�C11) and c25 (C13�C14�C=O). Aspecial role of the geometry around the central ring, which isdetermined by the constraints imposed by the local chemicalstructure,[49] is recognized as a typical feature of bent-coremesogens.[26, 27] According to previous computations,[31, 46] thetorsional potential relative to the dihedral angle c24 is charac-terized by two degenerate minima (c24 ~ �458) ; for each ofthem four states were identified for c25 (~ �158 and ~ �1658).Thus, there is a total of eight symmetry-related configurationsof the core, henceforth denoted as G1–G8. They are representedin Figure 2 (right) and their relevant structural and energeticparameters[31] are summarized in Table 1.

The description of core configurations arising from rotationaround c24 and c25 in terms of the rotational isomeric state ap-proximation (RIS)[51] is justified by the fact that the potentialenergy minima are relatively narrow and separated by highbarriers. This description is also beneficial for the presentstudy, as the identification of a small number of core geome-tries simplifies the analysis of the relationship between the mo-lecular structure and the physical properties of interest. There-fore, it is kept throughout this work, and molecular conforma-tions are grouped into discrete sets, each corresponding toa given core configuration. Thus, when speaking of the Gi set,we refer to the group of molecular conformations having thecore in the Gi configuration and any possible value of the dihe-drals different from c24 and c25.

Atomic coordinates for each of the Gi core configurationswere obtained by geometry optimization using DFT at theB3LYP/6-31 + G* level.[31] Torsional energy profiles for thebonds, indicated by arrows in Figure 2, were derived by re-laxed scans on selected fragments. DFT calculations were per-formed using different levels of the theory as explained in thefollowing.[52] For the alkyl chain dihedrals (c0-c4, c5, c6 and theiranalogues in the other side arm), calculations were done atthe M06-2X/6-31 + G** level of theory. According to a recentcomputational analysis of the conformational preferences ofbutylbenzene, the M06-2X functional gives results that are in

Figure 2. Left : molecular structure of A131; ci are the rotating dihedrals and {x,y,z} is the molecular frame, with x parallel to the para axis of the central ringand y perpendicular to the plane of this ring. Right: representation of the different configurations of the core of A131. The hexagon represents the centralring (C) ; open and filled symbols are used for the iminic hydrogen, linked to the carbon C9 (triangle) and the carbonylic oxygen (circle), above and below theplane of ring C, respectively. Core configurations are defined as in ref. [31] but, according to the IUPAC recommendation,[50] the opposite convention is usedfor the sign of c24 and c25 (therefore the sign of the two dihedrals is the opposite of that of f4 and f5 in Figure 3 of ref. [31]).

Table 1. Structural and energetic parameters for the eight core configu-rations of A131, as obtained by DFT calculations at the B3LYP/6-31 + G*level in vacuum (from ref. [31]).

Configuration c24 [8][a] c25 [8][a] DV [kJ mol�1][b]

G1 �44 + 165 0.0G2 �45 + 18 4.5G3 �45 �17 5.9G4 �45 �166 1.7G5 + 44 + 166 1.7G6 + 45 + 17 5.9G7 + 45 �18 4.5G8 + 44 �165 0.0

[a] Dihedral angles are defined as positive for clockwise rotation[50] (oppo-site to the convention used in ref. [31]). [b] Relative energies are definedwith respect to the most stable configuration.

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better agreement with experiments than the B3LYP function-al.[53] The torsional potential for c8 and c10 was calculated atthe M06-2X/6-31 + G** level, using 4-methylphenyl 4-methyl-benzoate as a model compound. The potential energy surfaceof phenyl benzoate has been the subject of intense investiga-tion, and some discrepancies exist between the different calcu-lations as to the exact position of the minima and the relativeheight of barriers.[54–56] However, in all cases four equivalentenergy minima are found (~ �608, ~ �1208) and the energybarriers at 08 and 1808 are higher than those at �908. The tor-sional profile calculated in this work substantially agrees withthose obtained at the B3LYP/6-31 + G*[54, 55] or the MP2/6–31 +

G** and MP2/6–31 + G* levels.[54–56] The c7 and c9 dihedrals arecharacterized by relatively simple torsional potentials, forwhich the standard B3LYP functional was used with the 6–31 +

G* basis set. For c11 and c12 the same torsional potential wasassumed, taken from ref. [46] . All the torsional potential pro-files are shown in the Supporting Information.

Molecular Field Theory for Order Parameters and ElasticConstants in the Uniaxial Nematic Phase

Our molecular field approach is based on the so-called surfaceinteraction model, wherein the orientational distribution ofa molecule in the nematic phase is parameterized according tothe amount of its surface that is aligned parallel to the direc-tor.[57] The potential of mean torque, U, experienced by a mole-cule in the uniaxial nematic phase is expressed as [Eq. (1)]:

UðWÞ ¼ kBTe

ZS

P2ðn � sÞdS ð1Þ

where the integral is over the molecular surface; kB is theBoltzmann constant, T is the temperature, P2 is the second Leg-endre polynomial, s is the normal to the surface element dS,and n is the mesophase director at the position of the surfaceelement. Finally, e is a parameter, which represents the orient-ing strength of the nematic environment and, within the mo-lecular field theory, is related to the reduced temperature.[57]

The potential of mean torque [Eq. (1)] is a function of the ori-entation of the molecule with respect to the director, definedby the Euler angles W = (a,b,g), and accounts for director de-formations through the position dependence of the director n.

The singlet orientational distribution function, p(W), is relat-ed to the potential of mean torque by Equation (2):

pðWÞ ¼ exp �UðWÞ=kBT½ �=Z

exp �UðWÞ=kBT½ �dW ð2Þ

and orientational averages are defined as [Eq. (3)]:

:::h i ¼Z:::pðWÞdW ð3Þ

Thus, the second-rank order parameter for an arbitrary axis(i) in the molecule is defined as [Eq. (4)]:

Sii ¼ 3 cos2 bih i � 1ð Þ=2 ð4Þ

with bi being the angle between the i axis and the mesophasedirector.

Within this framework we can also calculate the elastic con-stants.[40] The Frank expression for the elastic energy density ofnon-chiral uniaxial nematics reads [Eq. (5)]:[58]

fel ¼12

K1 r � nð Þ2þ 12

K2 n � r � nð Þ½ �2þ 12

K3 n� r� nð Þj j2 ð5Þ

where K1, K2, and K3 are the elastic constants for splay, twist,and bend deformations, respectively. Under the approxima-tions of the molecular field theory,[59] the Helmholtz free-energy difference between the nematic and the isotropicphase is evaluated as the sum of an internal energy contribu-tion, obtained from the average potential of mean torque, anda term accounting for the orientational entropy, expressed inthe Gibbs form. The elastic constants are then obtained as thesecond derivatives of the Helmholtz free-energy density withrespect to the three director deformations.[40]

The model just outlined can be easily extended to take intoaccount the molecular flexibility, which is a typical feature ofmesogenic systems. For this purpose, the orientational distri-bution must be replaced by the conformational-orientationaldistribution function, p(c,W), where c are the torsional degreesof freedom [Eq. (6a)]:

pðc;WÞ ¼ exp � Uðc;WÞ þ VðcÞ½ �=kBTf g=Z ð6aÞ

with the conformational-orientational partition function[Eq. (6b)]:

Z ¼Z

exp � Uðc;WÞ þ VðcÞ½ �=kBTf gdcdW ð6bÞ

where V(c) is the torsional potential in the isotropic phase,whereas U(c,W) is the potential of mean torque experiencedby a molecule in the W orientation and conformation specifiedby the set of c dihedrals.

Equilibrium properties are then calculated as conformation-al-orientational averages [Eq. (7)]:

:::h i ¼Z:::pðc;WÞdcdW ð7Þ

this equation can be rewritten in the form [Eq. (8)]:

:::h i ¼Z

:::ðcÞh iWpðcÞdc ð8Þ

where h. .(c)iW denotes the orientational average for a givenconformation and p(c) is the conformational distribution func-tion in the nematic phase, defined as [Eq. (9)]:

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pðcÞ ¼ ð1=ZÞ exp �VðcÞ=kBTf gZ

exp �Uðc;WÞ=kBTf gdW ð9Þ

This distribution function clearly depends on the degree oforder in the nematic phase and is different from that in theisotropic phase. In general, the potential of mean torque stabil-izes elongated conformations, which can be better accommo-dated in the nematic environment.

13C NMR Chemical Shifts

In 13C NMR experiments, the relative shieldings of 13C nuclei,known as chemical shifts, are measured. Henceforth these aredenoted by hdi, that is, the average value of the symmetricpart of the chemical shift tensor, d. This average is convenient-ly expressed in terms of the order parameters for the principalaxes of the tensor.[60] Hence, for the ith nucleus we can writeEquation (10):

di� �¼ di

iso þ23

di33 �

di22 þ di

11

2

� �Si

33 þ12

di22 � di

11

� �Si

22 � Si11

� ��

ð10Þ

where dijj (j = 1–3) are the principal values of the chemical shift

tensor, Sijjare the orientational order parameters for the corre-

sponding principal axes, and diiso ¼ Trdi

� �=3.

Computational Details

Given the high number of variables, the integrals in [Eq. (7)]cannot be calculated with conventional quadrature algorithms.We followed the method proposed in refs. [47, 48] , which com-bines standard integration for the orientational variables, withMonte Carlo sampling of the conformational space. For thelatter, the Metropolis criterion was adopted, with the purelytorsional probability distribution, p0ðcÞ / exp �VðcÞ=kBTf g.[61]

Each Monte Carlo move consists in the random rotation ofa certain number of dihedral angles (c). The c24 and c25 dihe-drals were not sampled since, for convenience of analysis andcomputation, separate calculations were performed for each ofthe core configurations schematized in Figure 2. Actually, inthe light of the molecular symmetry, only the G1–G4 sets wereconsidered. The final averages were calculated including allcore configurations. Only in some cases partial averages, re-stricted to conformations having a specific core configuration,were considered; this is explicitly said in Section 3.

Over 50 000 conformations were sampled for each core con-figuration, to guarantee convergence of all the investigatedproperties. Actually, even a lower number would have beenenough for most properties, with the exception of the bend-elastic constant of the G1 and G4 sets, which at high orderingexhibits relatively large fluctuations between negative and pos-itive values. The need of accurate calculations was alreadydemonstrated for other bent-core systems[20, 32] and for meso-genic dimers with very low bend-elastic constants.[43, 62]

For each molecular conformation, calculation of the meanfield potential, [Eq. (1)] , requires the definition of the molecularsurface.[63] This was generated using the fast molecular surfacecalculation library MSMS,[64] assuming the following atomicradii : rC = 0.185 nm, rO = 0.15 nm, rN = 0.15 nm, rH = 0.1 nm.[65]

Moreover, a rolling sphere radius equal to 0.3 nm and a densityof vertices equal to 5 ��2 were taken. Average properties inthe nematic phase were calculated as a function of the orient-ing strength e, in Equation (1). To make the comparison withexperimental data easier, some of the results are reported asa function of temperature. To relate e to temperature, a lineardependence of the chemical shifts calculated for C5 on tem-perature was assumed, as observed experimentally for thiscarbon in the range between 445 and 406 K.

3. Results and Discussion

Molecular Shape and Conformational Distribution

Figure 3, showing the molecular geometry of conformers withthe core in G1–G4 configurations and all-trans alkyl chains, illus-

trates the importance of the c24 and c25 dihedrals. The overallshape of the G2 and G3 structures is clearly more bent thanthat of the G1 and G4 structures. For this reason they were de-noted as hockey-stick-shaped and banana-shaped, respective-ly.[31] The bend angle, which we defined as the angle betweenthe para axes of the AE rings (see Figure 1), is about 1358 forthe G1 and G4 core configurations and 1158 for G2 and G3.Analogous values were found for the bend angle between thepara axes of the BD rings in the G1 and G4 and in the G2 and G3

core configurations, respectively.The torsional freedom within the aromatic core does not sig-

nificantly change the core shape within a given Gi set. Due tothe form of the torsional potentials for bonds within the core,the B ring, on one side, and the D, E rings, on the other side,tend to be twisted with respect to the central C ring, withtwist angles of about �458 and �508, respectively. A wider an-gular distribution is predicted for ring A, as relatively lowenergy barriers oppose its rotation with respect to ring B. Ac-cording to the torsional potentials for the c6 and c13 dihedrals,alkyl chains tend to lie perpendicular to the benzene ring to

Figure 3. Superposition of four conformers of A131, with the core in G1–G4

configurations and all-trans alkyl chains.

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which they are attached, so protruding out of the approximatemolecular plane, defined by the aromatic core.

Rotations around chain bonds can significantly modify theoverall molecular shape. To quantify these changes we calcu-lated the size and shape distribution of molecular conforma-tions. To this purpose the smallest rectangular box containingeach conformer, with the edges parallel to its principal inertiaaxes, was considered. Thus, the length (L), breadth (B), andwidth (W) were taken equal to the largest differences betweenthe atomic coordinates along each side. Figure 4 shows the L,

B, W distributions calculated for the G1 and the G2 sets, in theisotropic and in the nematic phase. The outer edges of the Ldistributions correspond to the end-to-end distance (measuredbetween the terminal methyl carbons) in all-trans conformers:depending on the values of the c6 and c13 dihedrals, this dis-tance can reach up to 5.2 nm for the G1 and 4.8 nm for the G2

core configuration. In Figure 4 we can see that, even thoughthe distributions are broadened by the chain mobility, the mo-lecular shape remains strongly anisometric, with some differ-ence between the G1 and G2 sets. On the average, conforma-tions having the core in the G2 configuration are less elongat-ed and more biaxial (larger difference between breadth andwidth) than those with the core in the G1 configuration. In thenematic phase, the maxima of the L distributions are shiftedtowards higher values, as a consequence of the stabilization ofthe more elongated molecular conformations within a given Gi

set. The average values obtained for G1 are L = 4.02 nm, B =

1.32 nm, W = 0.76 nm in the isotropic phase at T = 450 K, andL = 4.28 nm, B = 1.22 nm, W = 0.70 nm in the nematic phase atT = 395 K. For G2 we have obtained L = 3.71 nm, B = 1.62 nm,W = 0.80 nm in the isotropic phase, and L = 3.93 nm, B =

1.53 nm, W = 0.74 nm in the nematic phase.The effect of the nematic environment does not dramatically

modify the relative population of Gi core configurations.Table 2 reports the molar fractions calculated according to[Eq. (9)] , which takes into account the effect of both the tor-sional potential and the potential of mean torque. We can seethat the population of the G2 and G3 configurations, alreadylow in the isotropic phase at 450 K, becomes even lower withdecreasing temperature, in favor of the G1 and G4 configura-tions. These do not only have a lower energy in vacuum (seeTable 1), but also, being more elongated, are stabilized by theinteraction with the nematic environment. Thus, according to

our calculations, hockey-stick-shaped conformers would havea low relative weight in the uniaxial nematic phase of A131,which becomes even smaller with decreasing temperature.

13C NMR Chemical Shifts and Order Parameters

13C NMR chemical shifts were calculated according to [Eq. (10)] ,using experimental chemical shielding tensors for all carbonswith the exception of C1, C15, and C24, for which only the iso-tropic value, di

iso, is available.[66] For these sites, theoreticaltensor components (GIAO-DFT) were used.[31] The NMR spectralassignments reported in ref. [66] were revised in the light ofthe results of our calculations. It should be noted that the13C NMR spectrum of a complex molecule such as A131 is char-acterized by crowding of several peaks in a reduced spectralregion,[66] and the assignment necessarily has some degree ofarbitrariness. The plots in Figure 5 show the calculated and themeasured chemical shifts as a function of the shifted tempera-ture T–TNI, where TNI is the isotropic-nematic transition temper-ature. It may be worth stressing that our approach is methodo-logically different from other methods used for previous analy-ses of the 13C NMR chemical shifts of A131, in which the experi-mental data were fitted to a model containing a number of fit-ting parameters (typically order parameters for different partsof the molecule), which were then used a posteriori to inter-pret the orientational behavior.[31, 66] Here, on the contrary, westarted modeling the orientational and conformational distri-bution of the mesogen and the effects of the molecular geom-etry are introduced a priori. Chemical shifts, as well as anyother property of the nematic phase, are obtained as a functionof the orienting strength e in [Eq. (1)] , without free parameters.Here we used the experimental chemical shifts of C5 to definethe relation between e and temperature, to report our resultsas a function of temperature, which makes the comparisonwith experimental data easier. However, this choice does notimply the agreement between theoretical and calculatedchemical shifts, as the pattern of hdii values is determined bythe molecular geometry and the specific orientational prefer-ences of the conformers. Therefore the results shown inFigure 5 can be taken as an assessment of the adequacy of ourdescription of the molecular structure and our account of theconformational and orientational distribution for A131. Somediscrepancy between theoretical and experimental results canbe ascribed to uncertainty in the molecular geometry and inthe principal values and axes of the chemical shift tensorsused in the calculations. In general, relatively higher discrepan-

Figure 4. Distribution of length L, breadth B and width W calculated sepa-rately for the G1 (left) and G2 (right) sets of conformations, in the isotropicphase at T = 450 K (a) and in the nematic phase at T = 395 K (c).

Table 2. Molar fraction of the Gi core configurations of A131, calculatedat different temperatures in the isotropic (I) and in the nematic (N) phase,according to [Eq. (9)] .

Configuration 450 K (I) 425 K (N) 395 K (N)

G1 =G8 0.23 0.26 0.29G2 =G7 0.07 0.05 0.03G3 =G6 0.05 0.03 0.02G4 =G5 0.15 0.16 0.16

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cies are found for carbons for which theoretical, rather than ex-perimental shielding tensors were used. An example is repre-sented by C15, the chemical shifts of which were overestimat-ed, probably because of the too large d tensor values used inthe calculations (theoretical diso values equal to 142.4 for G1

and 137.6 for G4 compared to the experimental value of134.5). We can discern in Figure 5 that the discrepancies be-tween theoretical and experimental chemical shifts tend to in-crease on approaching the nematic-smectic C transition, whichcould be a sign of deviations from the standard nematic or-ganization assumed in our description. Indeed, the transitionto a biaxial nematic phase has been debated;[31, 34–39] more re-cently the onset of smectic C-like short-range fluctuations hasbeen suggested.[17]

We can also calculate quantities that are not directly accessi-ble to the experiment. For instance, it may be interesting tocompare the order parameters separately calculated for the G1

and G2 sets of conformations. Figure 6 shows significant differ-ences between the two group of conformers, amenable totheir different shape (see Figures 3–4). For G1 conformationshigher ordering is predicted, with a strong tendency to aligntheir long z axis to the director (high Szz) and very small biaxial-ity of molecular order (low Sxx–Syy). This is the typical behaviorof rod-like mesogens of relatively big size. The ordering behav-ior of G2 conformations is featured by a lower tendency toalign their z axis to the director and higher biaxiality of molec-ular order. The positive sign of the difference Sxx–Syy, for bothG1 and G2 conformations, indicates their preference to keepthe director on the xz plane, which can be identified as themolecular plane.

Elastic Constants

Figure 7 shows the elastic constants calculated for A131, asa function of temperature. Compared to the typical behaviorof conventional thermotropic liquid crystals, remarkable differ-ences appear. The elastic anisotropy, K3�K1, is negative, ratherthan positive. K3 is unusually low and its temperature depend-ence cannot be described by the relationship K3/ Szz

2, custom-arily assumed for rod-like mesogens.[42] Using our computation-al methodology, which allows us to take into account the realmolecular shape, we already showed that there may be signifi-cant deviations from such a relationship.[20, 41] The special sensi-tivity of K3 to the molecular curvature has already been evi-denced for bent-core systems[20, 21, 32] and for bent mesogenic

Figure 5. 13C NMR chemical shifts for the aromatic rings of A131 as a function of the shifted temperature, T–TNI : calculated (lines) and experimental values(symbols).[66] Some of the assignments reported in ref. [66] have been changed; the chemical shifts of C15 and C11 are now identical. The vertical lines indi-cate the presumed uniaxial–biaxal nematic transition and the nematic–smectic C transition. Site labelling is shown in Figure 1.

Figure 6. Order parameters, separately calculated for conformations of A131having the core in the G1 (c) and in the G2 (a) configuration, as a func-tion of the shifted temperature, T–TNI. For both configurations the {x,y,z} axesof the molecular frame, shown in Figure 2, are close to the principal axis sys-tems of the molecular Saupe matrix.

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dimers even negative K3 values have been predicted.[43, 62] Wecould also highlight strong effects of the conformational de-grees of freedom, as rotations around bonds can lead to signif-icant changes of the molecular shape. The non-monotonictemperature dependence of K3 for A131, shown in Figure 7,can be interpreted in the following way: the sample containsa mixture of conformers with a wide range of shapes, fromrod-like to bent-shaped, which can give highly different contri-butions to the bend elastic constants. The relatively low K3 athigher temperatures results from the average between highand positive values for elongated conformers, and small oreven negative values for bent-shaped conformers. The steepdecrease of the latter values at high ordering is the reason forthe decrease of K3 at low temperature.

To illustrate the effect of the molecular structure, we showin Figure 7 also the elastic constants calculated for the singleset of molecular conformations having the core in the G2 con-figuration; similar results were obtained for the single G3 set.The elastic constants predicted for single G1 and G4 sets arenot shown in the figure, because they are close to those ob-tained after the averaging over all core configurations (due tothe high statistical weight of G1 and G4 conformations). Forhockey-stick-shaped conformers (G2 and G3) we predict signifi-cantly lower splay- and twist-elastic constants than for banana-shaped conformers (G1 and G4). K2, already small for the latter,becomes even smaller for hockey-stick-shaped conformers, butagain it is K3 that exhibits the most striking differences. Strong-ly negative values are predicted for G2 and G3 conformations,which would be incompatible with the existence of a stablenematic phase.[67] Our calculations may somehow overestimatethe magnitude of these negative K3 values, but they givea strong indication of the different contributions of banana-shaped and hockey-stick-shaped conformers to the bendingstiffness. Due to the low relative weight of the G2 and G3 con-formations, the bend elastic constant obtained after averagingover all core configurations remains positive; however, itwould become rapidly negative upon increasing the amountof hockey-stick-shaped conformers.

The dramatic effect of a narrow bend angle on the bend-elastic constant is probably an example of a general behaviorof bent-core mesogens. The role of the bend angle has been

widely investigated and it is well known that substituents insuitable core positions, which stabilize conformations withwider bend angles, such as the methyl in the central ring ofA131, must be introduced to promote the formation of thenematic phase.[26, 27] The values of K3 calculated for the G2 con-formations, shown in Figure 7, suggest that the bend anglecan also have a critical role, which was up to now not suffi-ciently highlighted, for the stability of the nematic phase withrespect to bend distortions.

The elastic constants calculated for A131 can be comparedwith the measured K1 and K3 values, reported in ref. [17] . Forthe splay constant there is good agreement between theoryand experiment: K1 is high and shows an approximately lineardependence on temperature. For the bend-elastic constant,theoretical predictions are in good agreement with the experi-mental trend in the temperature region far from the smectic Ctransition: K3<K1 is found, with K3 scarcely dependent on tem-perature. Indeed, there are an increasing number of experi-mental observations pointing to a low bending stiffness ofbent-core nematics. Bent-core mesogens were predicted topromote spontaneous bend of the nematic director,[68] andlowering of the bend elastic constant has been evidencedupon doping conventional nematics with bent-core meso-gens.[69–71] Very recently, the temperature dependence of theelastic constants was measured for the nematic phase ofa number of bent-shaped mesogens,[18–23, 72–74] with results simi-lar to those for A131. In ref. [15] , unusually low values of K2

and K3 were reported for a bent-core mesogen at 2 K belowthe isotropic–nematic transition. It was argued that the molec-ular shape would not be sufficient to explain such findings andthe presence of cybotactic clusters was invoked. The resultsobtained here for A131 and in previous works for other bent-core systems[20, 21] show that the bent molecular shape canindeed explain their unusual elastic properties.

At low temperatures experimental data show a rapid in-crease of K3, which has been ascribed to pre-transitional smec-tic C fluctuations.[17, 75–77] The results reported in Figure 7 showthat this divergence cannot be predicted for a simple uniaxialnematic phase. Phase biaxiality, which has been proposed forthe A131,[31, 34–39] could neither be the reason for the experi-mental behavior, as relatively small contributions to the elasticconstants are expected to arise from biaxial ordering.[78, 79]

4. Conclusions

In this work, we have presented a computational investigationof the nematic phase formed by the bent-core mesogen A131,based on an atomistic representation of the molecule com-bined with a molecular field model. We have shown that this isan affordable task and the capability to simultaneously analyzeproperties at different length scales can provide considerableinsight, allowing us to check the consistency of hypothesesotherwise difficult to ascertain. The main results can be sum-marized as follows.

We have calculated the 13C NMR chemical shifts in the uniax-ial nematic phase of A131 over a wide temperature range thatextends from a few degrees below the nematic-isotropic tran-

Figure 7. Elastic constants calculated for the nematic phase of A131 (c).Dashed lines represent the results obtained for the restricted set of molecu-lar conformations having the core in the G2 configuration (from top tobottom: K1, K2, K3).

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sition downwards close to the transition to the smectic Cphase. The comparison with the spectral signals reported inref. [66] has allowed us to revise some assignment and to pro-pose a consistent description of the molecular order in thissystem. We have found good agreement between experimen-tal and theoretical chemical shifts over almost all the investi-gated temperature range. Some increase of the discrepanciesupon lowering the temperature could be a sign of a change inthe phase organization, which is also suggested by the low-temperature behavior of the bend-elastic constant.

The elastic constants predicted for A131, in agreement withthe experimental data far from the smectic C transition, are dif-ferent from those of typical low molar mass rod-like nematics.The most relevant feature is that K3 is smaller than K1 and re-mains almost constant over a wide temperature range. Analo-gous behavior has recently been found experimentally forother bent-core systems.[15, 18, 19, 22, 23, 74] and calculations haveshown that it can be ascribed to the bent molecularshape.[20, 21] The experimental data for A131 show also a diver-gence of K3 on moving towards the smectic C transition, whichcannot be explained within the same framework. This kind ofbehavior, however, does not seem to be a general feature forbent-core mesogens[18, 20, 22] .

A valuable feature of our approach is the capability to relatephase properties to the conformational preferences of themesogen. We have found that the two kinds of conformers ofA131, evidenced by previous theoretical studies[31] and denot-ed as banana-shaped and hockey-stick-shaped, give quite dif-ferent contributions not only to the molecular order parame-ters, but also to the elastic constants. However, the hockey-stick-shaped conformers do not significantly affect the proper-ties of the system, given their low relative weight, which isaround 10 % and is even predicted to decrease with increasingorder in the uniaxial nematic phase. A significant increase ofthese conformers, which has been proposed in ref. [31] in con-nection with the presumed transition to a biaxial nematicphase, appears difficult to justify at the mean field level, unlessshort-range correlations are invoked. Thus, the features of thenematic phase are essentially determined by the banana-shaped conformers, which are elongated enough to allow forthe existence of a nematic phase over a wide temperaturerange, and at the same time are bent enough to give thisphase unconventional properties.

Acknowledgements

We are very grateful to prof. Ronald Y. Dong for sharing with ushis NMR results and to Prof. Benedetta Mennucci for enlighteningdiscussions. A.F. acknowledges financial support from the Univer-sity of Padova (ex 60 % grants). C.G. thanks Merck Chemicals Ltdfor a scholarship. Computational resources and assistance wereprovided by the Department of Chemical Sciences, University ofPadova (LICC).

Keywords: bent-core · chemical shifts · elastic constants ·liquid crystals · molecular modelling

[1] T. Niori, F. Sekine, J. Watanabe, T. Furukawa, H. Takezoe, J. Mater. Chem.1996, 6, 1231 – 1233.

[2] A. J�kli, C. Bailey, J. Harden in Thermotropic Liquid Crystals : Recent Ad-vances (Ed. : A. Ramamoorthy), Springer-Verlag, Dordrecht, 2007.

[3] R. A. Reddy, C. Tschierske, J. Mater. Chem. 2006, 16, 907 – 961.[4] H. Takezoe, Y. Takanishi, Jap. J. Appl. Phys. 2006, 45, 597 – 625.[5] A. J�kli, Liq. Cryst. Rev. 2013, 1, 65 – 82.[6] G. R. Luckhurst, Thin Solid Films 2001, 393, 40 – 52.[7] C. Tschierske, D. J. Photinos, J. Mater. Chem. 2010, 20, 4263 – 4294.[8] R. Y. Dong, Int. J. Mod. Phys. B 2010, 24, 4641 – 4682.[9] R. Berardi, L. Muccioli, S. Orlandi, M. Ricci, C. Zannoni, J. Phys. Condens.

Matter 2008, 20, 463101.[10] L. A. Madsen, T. J. Dingemans, M. Nakata, E. T. Samulski, Phys. Rev. Lett.

2004, 92, 145505.[11] B. R. Acharya, A. Primak, S. Kumar, Phys. Rev. Lett. 2004, 92, 145506.[12] E. T. Samulski, Liq. Cryst. 2010, 37, 669 – 678.[13] M. Lehmann, Liq. Cryst. 2011, 38, 1389 – 1405.[14] J. Harden, B. Mbanga, N. �ber, K. Fodor-Csorba, S. Sprunt, J. T. Gleeson,

A. J�kli, Phys. Rev. Lett. 2006, 97, 157802.[15] M. Majumdar, P. Salamon, A. J�kli, J. Gleeson, S. Sprunt, Phys. Rev. E

2011, 83, 031701.[16] V. Domenici, Soft Matter 2011, 7, 1589 – 1598.[17] P. Sathyanarayana, M. Mathews, Q. Li, V. S. S. Sastry, B. Kundu, K. V. Le, H.

Takezoe, S. Dhara, Phys. Rev. E 2010, 81, 010702.[18] P. Tadapatri, U. S. Hiremath, C. V. Yelamaggad, K. S. Krishnamurthy, J.

Phys. Chem. B 2010, 114, 1745 – 1750.[19] P. S. Salter, C. Tschierske, S. J. Elston, E. P. Raynes, Phys. Rev. E 2011, 84,

031708.[20] S. Kaur, J. Addis, C. Greco, A. Ferrarini, V. Gçrtz, J. W. Goodby, H. F. Glee-

son, Phys. Rev. E 2012, 86, 041703.[21] S. Kaur, H. Liu, J. Addis, C. Greco, A. Ferrarini, V. Gçrtz, J. W. Goodby,

H. F. Gleeson, J. Mater. Chem. C 2013, 1, 6667 – 6676.[22] N. Avci, V. Borshch, D. D. Sarkar, R. Deb, G. Venkatesh, T. Turiv, S. V.

Shiyanovskii, N. V. S. Rao, O. D. Lavrentovich, Soft Matter 2013, 9, 1066 –1075.

[23] R. Balachandran, V. P. Panov, J. K. Vij, A. Lehmann, C. Tschierske, Phys.Rev. E 2013, 88, 032503.

[24] Y. Jang, V. P. Panov, A. Kocot, A. Lehmann, C. Tschierske, J. K. Vij, Phys.Rev. E 2011, 84, 060701.

[25] D. Wiant, J. T. Gleeson, N. �ber, K. Fodor-Csorba, A. J�kli, T. T�th-Katona,Phys. Rev. E 2005, 72, 041712.

[26] W. Weissflog, H. Nadasi, U. Dunemann, G. Pelzl, S. Diele, A. Eremin, H.Kresse, J. Mater. Chem. 2001, 11, 2748 – 2758.

[27] C. Keith, A. Lehmann, U. Baumeister, M. Prehm, C. Tschierske, SoftMatter 2010, 6, 1704 – 1721.

[28] M. Lehmann, J. Seltmann, Beilstein J. Org. Chem. 2009, 5, 73.[29] P. Grzybowski, L. Longa, Phys. Rev. Lett. 2011, 107, 027802.[30] M. A. Bates, G. R. Luckhurst, Phys. Rev. E 2005, 72, 051702.[31] R. Y. Dong, A. Marini, J. Phys. Chem. B 2009, 113, 14062 – 14072.[32] S. Kaur, L. Tian, H. Liu, C. Greco, A. Ferrarini, J. Seltmann, M. Lehmann,

H. F. Gleeson, J. Mater. Chem. C 2013, 1, 2416 – 2425.[33] M. Cestari, PhD thesis, Universit� di Padova (IT), 2009.[34] V. Prasad, S.-W. Kang, K. A. Suresh, L. Joshi, Q. Wang, S. Kumar, J. Am.

Chem. Soc. 2005, 127, 17224 – 17227.[35] H. G. Yoon, S.-W. Kang, R. Y. Dong, A. Marini, K. A. Suresh, M. Srinivasar-

ao, S. Kumar, Phys. Rev. E 2010, 81, 051706.[36] a) M. S. Park, B.-J. Yoon, J. O. Park, V. Prasad, S. Kumar, M. Srinivasarao,

Phys. Rev. Lett. 2010, 105, 027801; b) H. F. Gleeson, P. D. Brimicombe,Phys. Rev. Lett. 2011, 107, 109801; c) M. S. Park, B.-J. Yoon, J. O. Park, V.Prasad, S. Kumar, M. Srinivasarao, Phys. Rev. Lett. 2011, 107, 109802.

[37] K. V. Le, M. Mathews, M. Chambers, J. Harden, Q. Li, H. Takezoe, A. J�kli,Phys. Rev. E 2009, 79, 030701.

[38] B. Senyuk, H. Wonderly, M. Mathews, Q. Li, S. V. Shiyanovskii, O. D. Lav-rentovich, Phys. Rev. E 2010, 82, 041711.

[39] T. Ostapenko, C. Zhang, S. N. Sprunt, A. J�kli, J. T. Gleeson, Phys. Rev. E2011, 84, 021705.

[40] M. Cestari, A. Bosco, A. Ferrarini, J. Chem. Phys. 2009, 131, 054104.[41] M. Cestari, A. Ferrarini, Soft Matter 2009, 5, 3879 – 3887.[42] A. Ferrarini, Liq. Cryst. 2010, 37, 811 – 823.[43] M. Cestari, E. Frezza, A. Ferrarini, G. R. Luckhurst, J. Mater. Chem. 2011,

21, 12303 – 12308.

� 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ChemPhysChem 2014, 15, 1336 – 1344 1343

CHEMPHYSCHEMARTICLES www.chemphyschem.org

[44] A. Ferrarini, G. J. Moro, P. L. Nordio, Liq. Cryst. 1990, 8, 593 – 621.[45] R. Y. Dong, Nuclear Magnetic Resonance in Liquid Crystals, Springer-

Verlag, New York, 1994.[46] A. Marini, V. Prasad, R. Y. Dong in Nuclear Magnetic Resonance Spectros-

copy of Liquid Crystals (Ed. : R. Y. Dong), World Scientific, Singapore,2009.

[47] A. Ferrarini, G. R. Luckhurst, P. L. Nordio, Mol. Phys. 1995, 85, 131 – 143.[48] M. Stocchero, A. Ferrarini, G. J. Moro, D. A. Dunmur, J. Chem. Phys. 2004,

121, 8079 – 8097.[49] I. Cacelli, G. Prampolini, Chem. Phys. 2005, 314, 283 – 290.[50] G. P. Moss, Pure Appl. Chem. 1996, 68, 2193 – 2222.[51] P. W. Flory, Statistical Mechanics of Chain Molecules, Interscience, New

York, 1969[52] Gaussian 09 (Revision B.01), M. J. Frisch, G. W. Trucks, H. B. Schlegel, G.

E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Men-nucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A.F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K.Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O.Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr. , J. E. Peralta, F. Ogliaro,M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, T.Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C.Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene,J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochter-ski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J.J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V.Ortiz, J. Cioslowski, D. J. Fox, Gaussian, Inc. , Wallingford CT, 2010.

[53] S. Halbert, C. Clavagu�ra, G. Bouchoux, J. Comput. Chem. 2011, 32,1550 – 1560.

[54] R. Wrzalik, K. Merkel, A. Kocot, J. Mol. Model. 2003, 9, 248 – 258.[55] G. Cinacchi, G. Prampolini, J. Phys. Chem. A 2005, 109, 6290 – 6293.[56] S. Kawauchi, S.-W. Choi, K. Fukuda, K. Kishikawa, J. Watanabe, H. Take-

zoe, Chem. Lett. 2007, 36, 750 – 751.[57] A. Ferrarini, G. J. Moro, P. L. Nordio, G. R. Luckhurst, Mol. Phys. 1992, 77,

1 – 15.[58] P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, Clarendon Press,

Oxford, 1993.[59] G. R. Luckhurst in The Molecular Physics of Liquid Crystals (Eds. : G. R.

Luckhurst, G. W. Gray), Academic, New York, 1979.[60] P. L. Nordio, U. Segre in The Molecular Physics of Liquid Crystals (Eds. :

G. R. Luckhurst, G. W. Gray), Academic, New York, 1979.

[61] N. Metropolis, A. Rosenbluth, M. Rosenbluth, N. Marshall, A. Teller, E.Teller, J. Chem. Phys. 1953, 21, 1087 – 1092.

[62] M. Cestari, S. Diez-Berart, D. A. Dunmur, A. Ferrarini, M. R. de La Fuente,D. A. Jackson, D. O. Lopez, G. R. Luckhurst, M. A. Perez-Jubindo, R. M. Ri-chardson, J. Salud, B. A. Timimi, H. Zimmermann, Phys. Rev. E 2011, 84,031704.

[63] A. Ferrarini, F. Janssen, G. J. Moro, P. L. Nordio, Liq. Cryst. 1999, 26, 201 –210.

[64] M. F. Sanner, J. C. Spehner, A. J. Olson, Biopolymers 1996, 38, 305 – 320.[65] Handbook of Chemistry and Physics (Ed. : D. R. Lide), CRC Press, Boca

Raton, 1996.[66] R. Y. Dong, S. Kumar, V. Prasad, J. Zhang, Chem. Phys. Lett. 2007, 448,

54 – 60.[67] J. L. Ericksen, Phys. Fluids 1966, 9, 1205 – 1207.[68] I. Dozov, Europhys. Lett. 2001, 56, 247 – 253.[69] M. R. Dodge, R. G. Petschek, C. Rosenblatt, M. E. Neubert, M. E. Walsh,

Phys. Rev. E 2003, 68, 031703.[70] B. Kundu, R. Pratibha, N. V. Madhusudana, Phys. Rev. Lett. 2007, 99,

247802.[71] P. Sathyanarayana, B. K. Sadashiva, S. Dhara, Soft Matter 2011, 7, 8556 –

8560.[72] P. Sathyanarayana, S. Radhika, B. K. Sadashiva, S. Dhara, Soft Matter

2012, 8, 2322 – 2327.[73] A. Chakraborty, M. K. Das, B. Das, U. Baumeister, W. Weissflog, J. Mater.

Chem. C 2013, 1, 7418 – 7429.[74] A. Chakraborty, M. K. Das, B. Das, A. Lehmann, C. Tschierske, Soft Matter

2013, 9, 4273 – 4283.[75] N. Vaupotic, J. Szydlowska, M. Salamonczyk, A. Kovarova, J. Svoboda, M.

Osipov, D. Pociecha, E. Gorecka, Phys. Rev. E 2009, 80, 030701.[76] O. Francescangeli, F. Vita, C. Ferrero, T. Dingemans, E. T. Samulski, Soft

Matter 2011, 7, 895 – 901.[77] S. H. Hong, J. C. Williams, R. J. Twieg, A. J�kli, J. Gleeson, S. Sprunt, B.

Ellman, Phys. Rev. E 2010, 82, 041710.[78] Y. Singh, K. Rajesh, V. J. Menon, Phys. Rev. E 1994, 49, 501 – 512.[79] A. Kapanowski, Phys. Rev. E 1997, 55, 7090 – 7104.

Received: November 6, 2013

Published online on January 27, 2014

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