Special issue mini-review Received: 14 April 2014
Revised: 14 July 2014
Accepted: 15 July 2014
Published online in Wiley Online Library: 20 August 2014
(wileyonlinelibrary.com) DOI 10.1002/mrc.4124
Conformational problem of alkanes in liquid crystals by NMR spectroscopy: a mini-review† Adrian C. J. Webera* and Daniel H. J. Chenb Recent discoveries of the role of alkane flexibility in determining liquid-crystal behaviour are surveyed. With the impetus for understanding the alkane conformational problem established, recent model dependent 1 H NMR work on the topic will be reviewed where progress is made but the need to circumvent models eventually becomes evident. A closer look at the rigid basic units of alkanes will provide the way forward where it is shown that the orientational ordering and anisotropic potentials of these molecules dissolved in liquid crystals scale with each other. Once this relationship is established, a series of works using anisotropic and isotropic 1 H NMR spectroscopy to study alkane conformational statistics will be covered, wherein the influence of the gas, isotropic condensed and anisotropic condensed phases will be described. Copyright © 2014 John Wiley & Sons, Ltd. Keywords: NMR; H; liquid crystals; alkanes; conformational statistics; n-butane; anisotropic model; dipolar coupling; nematic; Etg
Introduction
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While it is true that liquid crystals are a rich source of fascinating physical and chemical behaviour, they also touch many aspects of our daily lives. Therefore, the importance of liquid-crystal studies is twofold; they are a testing ground for discerning fundamental chemical and physical phenomena and represent a wealth of potential in a wide variety of applications such as the rapidly developing field of liquid-crystal display technologies (used in smart phone, tablet, computer and television screens) and are even being used to create better welding masks, temperature sensors and windows while of course being essential to life vis-a-vis biological membranes.[1] The identified supramolecular architectures formed in mesogenic phases are numerous and characterized by long-range orientational and positional order of component molecules. The fluid nature of these phases is typically the result of flexible segments on the liquid-crystal molecules, which are usually hydrocarbon chains. The flexibility of these segments is manifested because of the possibility that each saturated C–C bond can be in a trans or gauche conformation. Therefore, the behaviour of the alkane chain conformational statistics becomes an important question in terms of what these segments bestow onto the properties of the multitude of liquid-crystal phases, which can often yield technological applicability. Dendrite audiophiles, water-soluble liquid crystals, have demonstrated anti-HIV, anti-STD, antifungal and antimycobacterial properties and have recently been studied via molecular dynamics simulation to gain insight into how they interact with phospholipid bilayer membranes.[2] These simulations show that alkyl chains facilitate entry into target cell membranes and change the molecular dynamics significantly. At the same time, data on the deuterium order parameter, bilayer thickness and tail tilt angle show increased disorder and tail kinking by the addition of dendritic amphiphiles. Further understanding of these phenomena involving the alkane conformational problem in a liquid crystal is said to have the potential to result in a new line of pharmaceuticals.
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Another important type of biologically active flexible chain molecules are the polypeptides, which fold up into proteins and exist in isotropic condensed phases as well as anisotropic liquid-crystal phases (membrane proteins). The magnitude of the internal rotational barriers about the backbone single covalent bonds of these molecules is crucial in determining conformations, folding, dynamics and functioning of proteins. For decades now, conventional wisdom has it that the internal barriers to rotation in polypeptides are small (around 0.7 kcal mol1 ) and of an order comparable to kB T at physiological temperatures.[3] However, some recent semi-empirical and ab initio calculations on model dipeptides have shown that the conventional estimates for non-glycine single bond backbone rotations could be 10 to 20 times higher.[4] It would be of value to verify these calculations experimentally because such an observation would have great implications for elucidating the stability, functioning, folding and structural organization of proteins. The development of new dendritic polymers has gained much attention for their use and potential use in drug delivery among other biomedical applications.[5] The usual way to obtain the desired liquid crystallinity is to introduce ‘promesogenic units’ to the periphery, but more recently, this has been circumvented by exploiting the natural tendency of these macromolecules to display phase separation when functionalized with aliphatic chains,[6] which leads to improved liquid-crystalline properties
* Correspondence to: Adrian C. J. Weber, Chemistry Department, Brandon University, 270-18th St, Brandon, MB R7A 6A9, Canada. E-mail: [email protected]; homepage, http://people.brandonu.ca/webera/ †
This article is published in Magnetic Resonance in Chemistry as a special issue on the NMR of Liquid Crystals by Ronald Y. Dong (Department of Physics and Astronomy, UBC, Vancouver, Canada).
a Chemistry Department, Brandon University, 270-18th St, Brandon, MB, R7A 6A9, Canada b Chemistry Department, University of British Columbia, 2036 Main Mall, Vancouver, BC, V6T 1Z1, Canada
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over which mesophases could be observed. Although alkyl chains spacers did very well in this regard, it was remarkable to observe that adding an ether linkage could result in a mesophase temperature range of 225 K. The extensive mesophase stability was rationalized by considering that the insertion of a flexible ether spacer leads to an increase in the degree of rotational and vibrational freedoms, and therefore entropy, while at the same time allowing the ether oxygen to take part in a three-dimensional hydrogen bonding network adding yet further stability. This kind of phase stability has great potential for studying the kinetics of conformational change via dynamic NMR[14] (which were previously complicated by the need for multiple liquid-crystal systems to span the required temperature range) as well as for the type of liquid-crystal NMR conformational statistics studies that follow.
Conformational Probability Extraction by Modelling In order to better understand the condensed phase conformational problem by NMR in general and how it pertains to liquid crystals in particular, it seems pragmatic to consider the simplest flexible alkane n-butane, which possesses a single degree of rotational freedom. In the earlier NMR studies of n-butane dissolved in anisotropic phases, it was more difficult to obtain the necessary dipolar or quadrupolar couplings because selective/random deuteration or multiple quantum methods needed to be used. Even once the single quantum 1 H NMR spectrum of n-butane was solved,[15] another problem arises because the conformational probability distribution function, pn , appears multiplied by the Saupe order matrix for the nth conformer in the expression n
Dij D
2 X nX n n p S˛ˇ Dij,˛ˇ 3 n
(1)
˛ˇ
for the dipolar couplings between nuclear spins i and j, and there is no way to unambiguously separate the two at a single temperature. The Saupe order matrix, Sn˛,ˇ , for the nth conformer is defined by 3 1 n n cos ˛,Z Sn˛ˇ D cos ˇ,Z ı˛ˇ (2) 2 2 n where ˛,Z is the angle between the molecular-˛ axis and the N director, which is aligned with the static magnetic field along the laboratory fixed Z-axis. Dnij,˛ˇ is a tensor defined by
Dnij,˛ˇ D
0 2 „ 8 2 rij3
3 1 ij,n cos ˛ij,n cos ˇ ı˛ˇ 2 2
(3)
ij,n
where ˛ is the angle between the internuclear vector rij and the molecular ˛ axis of the nth conformer. In order to proceed at the time, some approximation needed to be made. One of these is the rotational isomeric state (RIS) model,[16] which allows only three n-butane conformations to be accessible, namely, the trans (located at the global minimum of the intramolecular potential) and the twofold degenerate ˙gauche (located at the local minima). Additionally, it was found that allowing for fluctuations about the dihedral angles in an extended RIS model yielded slightly better fits to dipolar couplings. But to separate the conformer probabilities, it was necessary to use a variety of mean-field models to describe the elements of the Saupe order tensor via the expression
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and a larger mesophase temperature range. To better understand these observations, thermogravimetric analysis, differential scanning calorimetry and X-ray diffraction studies were carried out, and it was found that not only does the introduction of the alkyl chains lead to non-polar interaction of these components but also resulted in hydrogen bond formation in the core of the polymer.[7] These two effects lead to microphase separation, which is regarded as the main factor inducing mesomorphic character. It was then determined that the interplay between conformational change of the alkyl chains and the hydrogen bonding networks of the core was responsible for the hysteresis in phase transitions over heating–cooling cycles. Isotactic polypropylene is an alkyl chain widely used in industry for its intermediate crystallinity and flexibility and has been shown to display smectic phases under certain conditions.[8] Depending on different crystallization conditions, this polymer can take on different crystalline morphological forms. To uncover the nature of this process, Monte Carlo simulations were carried out wherein it was found that conformational order is the precondition for a smectic intermediate phase formation, which then leads to crystallization.[9] More specifically upon lowering the temperature (but before the liquid to crystal transition temperature), an increasing amount of trans-gauche pairs emerges resulting in the formation of helices and perhaps some local nematic (N) ordering in the melt. Once enough of this conformational ordering has occurred upon approaching the crystallization temperature, a critical amount of helices of a critical length appear to give rise to an intermediate smectic phase, which then leads to crystallization. So it seems as though the crystallization is ‘nucleated’ by a change in the alkyl chain conformational statistics. This change then brings about the orientational and translational ordering required for crystallization of this ubiquitous polymer. Over the years, a variety of ferroelectic liquid crystals (FLC) forming chiral smectic phases (Sm-C*) have been produced for potential applications in fast electro-optical devices. However, there is still no straightforward explanation of the relationship between the structure of FLC molecules and their behaviour in the Sm-C* phase. The molecular-statistical theory of ferroelectric ordering in the Sm-C* phase[10] and indigenous polarization theory[11] both describe the chiral centre and polarizable core as key factors for the emergence of the Sm-C* phase. However, principal-component analysis and two-dimensional correlation infrared spectroscopy of a particular FLC[12] showed that 92% of the spectral variance at the smectic A (SmA) to Sm-C* phase transition is due to the first principal component, which is dominated by the stretching modes of the methyl and methylenes of the FLC alkyl chains. So it seems that some change in the alkyl chains of at least some FLCs are of particular importance for the emergence of the technologically valuable Sm-C* phase. Ionic liquids (ILs) have matured as a class of materials over the last decade with the emergence of several commercial applications. Bis-cationic ILs in particular have been shown capable of mesophase formation when endowed with a sufficiently long alkyl chain. The mesomorphic properties show strong dependence on side chain length and the type of spacer linking the cations, which was recently investigated for a series of bis-cationic imidazolium based ILs via thermogravimetric analysis, differential scanning calorimetry and X-ray diffraction in the small angle range.[13] The authors then found that the ability of the ILs studied to form mesophases was not only related to the van der Waals forces between alkyl chains but also that the flexibility of the spacers seemed to be responsible for the temperature range
A. C. J. Weber and D. H. J. Chen R 3 Sn˛ˇ
D
2
n n 12 ı˛ˇ exp.Unaniso ./=kT/d cos ˇ,Z cos ˛,Z R exp Unaniso ./=kT d (4)
where Unaniso is the anisotropic N ordering potential of the nth conformer to be modelled. It was then found that the chord model[17] and size-and-shape model[18] gave the best fits to dipolar couplings.[15] The chord model is designed for the orientational order of molecules consisting of repeating identical units in a uniaxial phase and has the form UnCd,aniso ./ D
X
Q 0 P2 .si , si / C w Q 1 P2 .si , siC1 / w
(5) iso iso int ext Etg D Ugauche Utrans D Etg C Etg
iD1
Q i D 32 Swi and wi scales are the strength of the chord’s where w interaction with the mean field and S is the liquid-crystal order parameter. The si is a unit vector describing the orientation of the ith C–C bond of the hydrocarbon chain, and the sum is over all bonds in the chain. The factors P2 .si , siCm / are given by P2 .si , siCm / D
1 3 cos Zi cos ZiCm si siCm 2 2
(6)
where Zi is the angle between the ith bond and the N director. The model postulates that orientational ordering arises from an orientational torque about each C–C bond and the lines joining their midpoints. The second term incorporates correlations between adjacent-bond orientations, which distinguishes between conformations that may have equal numbers of trans and gauche bonds but different shapes. The size-and-shape model potential is the sum UnCI ./ D
1 1 k.Cn .//2 ks 2 2
Z
Zmax,n
Cn .Z, /dZ
(7)
Zmin,n
the first term of which involves a Hooke’s law restoring force where Cn ./ is the minimum circumference traced out by the projection of the solute onto a plane perpendicular to the N director. The second term represents the anisotropic surface potential where the area of the infinitesimally thin ribbon Cn .Z, /dZ is summed over its projection onto the plane parallel to the N director. But because UnCI ./ contains an isotropic component in practice, UnCI,aniso ./ D UnCI ./ hUnCI i
(8) hUnCI i
where Gn, is a rotational kinetic energy factor, which is dependent on the principal values of the moment of inertia tensor for each conformer and dihedral angle. The isotropic potential, Uiso , is the sum of the intramolecular gas phase component (calculated with GAUSSIAN 03) and the intermolecular component due iso to the condensed phase environment, Uext,n , which is represented ext , which raises or lowers the potential by a step function of size Etg energy of the n-butane geometries in the gauche potential well relative to those in the trans as can be seen in Fig. 1. The trans-gauche energy (Etg) difference in a condensed phase is then also a sum
is the is used to model the anisotropic potential where isotropic average over all angles. Later on when the spectral analysis of highly congested and complex anisotropic NMR spectra (like n-butane) became routine[19] (vide infra) with the use of an evolutionary strategy (ES), it became possible to revisit the problem while fitting to multiple liquid-crystal n-butane NMR spectra simultaneously.[20] In addition to calculations using the RIS and extended RIS model, the iso , was calculated using gas phase intramolecular potential, Uint,n GAUSSIAN 03 (Gaussian, Inc., Wallingofrd, Connecticut, USA).[21] This allowed for the calculation of a continuous dihedral angle probability distribution function via the expression
gas
int Etg D 643 cal mol1 is taken from the Gauswhere Etg ext sian calculations and Etg from fits to n-butane dipolar couplings in four different liquid crystals simultaneously. Despite fitting to four times the amount of equations, better fits are obtained with respect to the earlier study, likely due to the use of high quality GAUSSIAN 03 geometries in the latter compared to a more crude geometry in the former. When the continuous dihedral angle population distributions were plotted for all the calculation variants, it was found that only two distinct distributions emerged (as can be seen in Fig. 2), and that results depended only on the model chosen to describe the orientational potential. Specifically, calculations with the modified chord model, where Q 0 and w Q 1 are no longer held equal to each other and allowed to w vary independently, were found to favour the gauche conformation more so than those with the two parameter size-and-shape model where k and ks were also allowed to vary independently as opposed to being kept to their optimized ratio.[18] To better understand this observation, the calculated average mean square orientational ordering for the trans and gauche conformers was determined by the expression
q p n hS2 i D S2xx C S2yy C S2zz
(11)
where it was found that the modified chord model ascribed greater ordering to the trans conformer than the two paramp does g eter size-and-shape model. Furthermore, hS2 i was larger when calculated by the modified chord model. This in turn explains why calculations with the modified chord model have enhanced gauche populations relative to calculations with the two parameter size-and-shape model because the order matrix multiplies the conformer probability function in Eqn (1) and both methods are fitting the same four sets of dipolar couplings. In order to sort out which (if either) model was more or less physically correct, a subsequent study on n-butane sought to exploit the conformational probability temperature dependence. Such a study was made possible by the robust nature of an ES to obtain dipolar couplings with relative ease over (what was at the time) a very large 80ı temperature range.[22] In these fits to dipolar couplings, it was found that the continuous intramolecular potential resulted in root mean square (RMS) values 1–3 Hz lower than the RIS model. Both anisotropic models yield Etg ’s lower than the experimental gas phase range of 788–884 cal mol1 [23–27]
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iso iso R =kT exp Uint,n ./=kT exp Unaniso ./=kT d Gn, exp Uext,n iso iso R pn, D P P n, exp Uext,n =kT exp Uint,n ./=kT exp Unaniso ./=kT d n G
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iso iso ext Figure 1. (a) Uext is a step function, which raises or lowers the potential energy of n-butane geometries in the gauche wells by the amount Etg . (b) Uint is the n-butane internal rotational potential calculated by GAUSSIAN 03. The gap along the energy axis between the local (gauche) minima and the global gas int Etg . (c) The total isotropic potential, Uiso , is the sum of the latter two functions. While there is a slight discontinuity (trans) minimum is the quantity Etg in this function, the populations corresponding to the dihedral angles in the vicinity are very small as can be seen in Fig. 2. The gap between the local int ext and global minima is now the sum Etg D Etg C Etg . Thus, the model assumes that when n-butane is in one particular conformer, the internal motions are iso only plays a role in raising or lowering the potential for gauche conformers. governed completely by the intramolecular (gas phase) potential. Hence, Uext [22] Fig. 1 from Ref.
Figure 2. The probability P./ of finding n-butane at dihedral angle for the modified chord model (solid line) and the two parameter size-and-shape model (dashed line) in a N liquid-crystal phase. Fig. 3 from Ref.[20]
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the experimental condensed phase range of 502–681 cal mol1 whereas the modified chord model yielded values for the most part below this range. Perhaps most remarkable was that both models seemed to indicate, although to varying degrees, that Etg
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consistent with the idea that condensed phases favour the gauche conformer relative to the gas phase as found in previous experiments in condensed phases. It was found that the two parameter size-and-shape model gave Etg values towards the top end of
A. C. J. Weber and D. H. J. Chen
Figure 3. Calculated spectra of ethane (A) and propane (B) from using an evolutionary strategy on the experimental spectrum (C) of these solutes in 5CB at 298 K.
accounting for the effect of the condensed phase. Although this result appeared intriguing, the values and variation of Etg , and therefore the conformational statistics, were still model dependent. Because the orientational models used only predict order parameters at approximately the ˙10 % level,[28] the physics associated with the conformational change is masked by the uncertainty. The precise nature of the ordering potential is a very old and recurring problem, and it becomes evident that a model-free method is necessary in order to unambiguously sort out the conformational problem of alkanes in liquid crystals by NMR spectroscopy.
terms of how to correctly separate the Saupe order matrix from the conformational probability distribution function. To this end, the solutes ethane, propane and 1,3,5-trichlorobenzene (tcb) (added to provide an orientational and chemical shift reference) were co-dissolved in the liquid-crystal mixture Merck (White House Station, New Jersey, USA) ZLI-1132 (1132) and 4-n-pentyl-40 -cyanobiphenyl (5CB). Because these alkanes are gases at room temperature and ambient pressure, they were allowed to flow into a vacuum and then condensed into an NMR tube, which was pre-filled with liquid crystal and tcb and submerged into liquid nitrogen. The concentrations of tcb and ethane in 1132 are 0.5 and 5 mol % while that of propane is 2.5 mol %. In 5CB, the solute concentrations of tcb and ethane were 0.5 and 5.0 mol % while that of propane was 2.5 mol %. After these samples were sealed and thoroughly mixed in the isotropic phase, they were placed in turn into a Bruker Avance 400 MHz NMR spectrometer magnet (Bruker Biosciences Corporation, Billerica, Massachusetts, USA). With the temperature controlled by the Bruker air-flow system, 1 H NMR spectra were acquired every 5ı from 253.0 to 333.0 K for 1132 and from 273.0 to 303.0 K for 5CB; an example of which can be found in Fig. 3, which includes the calculated spectra for the alkanes.
Experimental
Results
Rather than continue on with mean-field models, it becomes apparent that a more basic look at the actual ordering of the fundamental units that flexible alkanes are composed of, namely ethane and propane, can hopefully be instructive in
In order to obtain the spectral parameters defining the anisotropic spectra with an ES, one has to first choose reasonable upper and lower limits for each parameter, which defines the search space. A complete set of spectral parameters is called a chromosome, and
is temperature dependent. Of course, it is anticipated from the Boltzmann statistics that the trans conformer population should increase as the thermal energy needed to populate the gauche conformation decreases with decreasing temperature. However, an Etg rising with decreasing temperature implies an additional source enhancing the trans probability. If the Etg difference is a function of temperature, it would have to result from the second term in the sum gas
ext Etg .T/ D Etg C Etg .T/
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Table 1. Experimental dipolar couplings of 1,3,5-trichlorobenzene, ethane and propane in 1132 (top) and 5CB (bottom) as a function of temperature. The numbers in brackets are the errors in Hertz in the last one or two digits prop
prop
prop
prop
T/K
Dtcb
Deth 12
Deth 14
253.5 258.5 263.5 268.5 273.5 278.5 283.5 288.5 293.5 298.5 303.5 308.5 313.5 318.5 323.5
274.63 270.82 265.33 260.02 254.84 249.21 243.86 237.61 230.85 223.17 215.50 207.22 195.12 187.82 175.81
866.40(16) 848.25(13) 830.93(13) 811.17(13) 790.69(13) 769.43(13) 749.18(13) 725.46(13) 700.48(13) 671.31(13) 643.37(13) 613.51(13) 581.14(13) 545.31(13) 504.21(13)
343.28(15) 336.20(11) 329.41(11) 321.71(11) 313.77(11) 305.37(11) 297.44(11) 288.15(11) 278.31(11) 266.82(11) 255.82(11) 244.06(11) 231.25(11) 217.08(11) 200.81(11)
922.25(9) 904.66(7) 887.55(6) 867.57(6) 846.66(7) 824.49(6) 803.17(7) 778.20(6) 751.61(6) 720.59(7) 690.79(7) 658.90(6) 624.19(6) 585.92(6) 541.70(7)
208.26(7) 204.37(6) 200.68(5) 196.35(5) 191.83(6) 187.00(5) 182.42(5) 177.03(5) 171.29(5) 164.48(6) 157.96(6) 150.95(5) 143.27(5) 134.74(5) 124.91(5)
439.40(8) 431.07(6) 422.73(5) 413.18(5) 403.13(6) 392.43(5) 382.09(6) 369.97(5) 357.12(5) 342.12(6) 327.70(7) 312.31(5) 295.61(5) 277.15(5) 256.04(6)
2154.35(22) 2113.49(18) 2073.26(16) 2026.60(17) 1978.00(17) 1927.24(17) 1878.49(17) 1821.18(16) 1760.48(16) 1689.49(18) 1621.24(18) 1548.23(17) 1467.82(17) 1379.63(17) 1277.69(17)
268.5 273.5 278.5 283.5 288.5 293.5 298.5 303.5
233.91 214.68 205.35 194.94 183.00 168.83 149.39 112.45
616.57(9) 588.64(13) 559.86(13) 528.44(13) 492.85(13) 451.22(13) 395.03(13) 292.31(13)
244.96(8) 233.90(11) 222.54(11) 210.13(11) 196.06(11) 179.59(11) 157.29(11) 116.46(11)
668.15(5) 638.58(6) 607.86(6) 574.05(6) 535.57(6) 490.42(6) 429.41(6) 317.92(6)
153.0(4) 146.51(5) 139.70(5) 132.16(5) 123.57(5) 113.40(5) 99.56(5) 73.96(5)
319.99(4) 302.73(5) 287.94(5) 271.77(5) 253.30(5) 231.72(6) 202.65(5) 149.84(6)
1580.54(12) 1511.46(16) 1439.90(17) 1361.09(17) 1271.24(17) 1165.72(17) 1022.45(17) 758.20(17)
D12
a population of these is initially spread out randomly across the search space. To evaluate the goodness of each member of the population, a fitness function needs to be defined Ffg D
.f g/ k f kk g k
(13)
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D16
D45
energy parameters. It is then interesting to compute the propane to ethane ratios of these parameters, which can be seen in Fig. 5. The invariance of these ratios over a wide temperature range provides evidence that ethane and propane are ‘magic solutes’ whose orientational order is governed by a single mechanism because if there were more, we should expect the orientation of these molecules to display different temperature dependencies, which would result in non-constant ratios. The order parameters of ‘magic solutes’ only depend on solute size-and-shape that are essentially temperature independent. Therefore, these ratios are expected to be temperature independent so long as such a solute does not undergo conformational change. This key result provides the way forward towards the separation of the orientational order matrix from the conformational probabilities because any ratio temperature dependence (whether order parameter, dipolar coupling or Maier–Saupe potential energy parameter) in a flexible alkane can now be completely attributed to the temperature dependence of the conformer probabilities. Therefore, if one knows the order parameters as a function of temperature of say ethane in some partially ordered environment, then one can make a given flexible alkane conformers ordering proportional to this known so as to then extract the desired conformational statistics.
Conformational Probability Extraction by Scaling A study of this nature[36] was carried out using the ethane dipolar couplings in Table 1. First, the n-butane to ethane dipolar coupling ratios were calculated and found to be temperature dependent as a result of changes in conformer populations as a function of temperature. It was then found that as the details of
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where f and g are the vector representations of the experimental and calculated spectra. Ffg is a measure of the extent to which a calculated spectrum overlaps with the experimental, which is maximum at the global minimum of the error surface where the solution is obtained.[29–32] However, before all this can be carried out, the liquid-crystal background needs to be removed because the ES iterates on both the frequency and intensity of transitions. The usual way to remove the background in the experimental spectrum is by a manual cubic base spline, which was performed here. Once the dipolar couplings are obtained as shown in Table 1, one can go on to calculate the chord and size-and-shape model parameters from least squares fitting, which are shown in Fig. 4. While both model potentials fit equally well for ethane in both liquid crystals, the case of propane appears more discriminating. For propane across all temperatures, the size-and-shape model RMS is 1.5 Hz in 5CB and 2.8 Hz in 1132 while those of the chord model are 13.2 and 17.2 Hz. The experimental order parameters are obtained via Eqn (1) for which pn will be one for rigid solutes like ethane and propane and from Eqn (4), the classic Maier–Saupe potential[33,34] energy parameters are calculated. The energy parameters are composed of independent second-rank tensorial liquid crystal (G) and solute (ˇ ) terms, which interact with each other giving rise to orientational order.[35] The symmetry of the solute governs the number of unique order parameters and hence
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Figure 4. Chord (top) and size-and-shape (bottom) model parameters for ethane (filled circles) and propane (open circles) as a function of temperature in 1132 (left) and 5CB (right).
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calculations fitting to n-butane dipolar couplings were improved that not only were better fits obtained but also more realistic physical parameters are gotten. The first challenge is to separate the orientational order from conformer probability. To this end, it was assumed that the six unknown order parameters of n-butane (three for the trans and three for the gauche) for each temperature scale to the ethane dipolar coupling or anisotropic potential energy parameter. Therefore, all spectra collected every 5ı in a given liquid-crystal solvent are fit to six proportionality constants so as to determine the orientational order of the n-butane conformers via the presence of ethane. In order to use the continuous intramolecular potential n-butane geometries residing in gauche, potential wells were given the same orientational order proportionality constants (and similarly for geometries in the trans well) so as to separate the internal motion from reorieniso tation. It was then found that using the Uint gave improved fits compared to calculations using the RIS approximation. Further improvement was gained when the anisotropic energy contribution to the conformer probabilities was accounted for. Even better fits to dipolar couplings are obtained when allowing Etg to have a temperature dependence. In fact when scaling to ethane dipolar couplings (labelled ST in Fig. 6), an RMS of 6.2 Hz over a 75ı 1132 temperature range and 2.7 Hz over the 40ı 5CB temperature range is amazing and unprecedented in earlier studies of orientationally ordered hydrocarbons. Just as important is the result that only when an Etg temperature dependence is allowed does one obtain orientational ordering consistent with simple size-and-shape arguments involved in the single orientational mechanism. Better fits are obtained when scaling to the ethane
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anisotropic potential parameters (labelled UT in Fig. 6), but in this case, the 1132 Etg range becomes shifted down from that of 5CB. These Etg values can be seen in Fig. 6 and are clearly consistent ext with a temperature variation of Etg due to the condensed phase, which turns out to be very similar to the previous result obtained with the modified chord model.[22] So while the size-and-shape model seems to reproduce the ordering of hydrocarbon chain units somewhat better than the chord model, it appears that the modified chord model better allocates order among the different conformers in a flexible hydrocarbon than does the two parameter size-and-shape model. All the values for Etg are lower than the GAUSSIAN 03 gas phase value of 643 cal mol1 indicating that the condensed phase favours the gauche conformer, which is consistent with what had been found theoretically[37,38] and experimentally (see Refs [15,20] and references therein). More recently, a study in this journal[39] employed indirect spin couplings of n-butane through n-heptane as a function of temperature and in different isotropic environments in order to shed light on the conformational problem of alkanes by 1 H NMR spectroscopy. The Etg was found to be close to 681 cal mol1 , which is somewhat higher than the GAUSSIAN 03 value but lower than the experimental gas phase range. Furthermore, the trans–trans to trans-gauche conformer energy difference of n-pentane was found to be 24–48 cal mol1 higher than the Etg for n-butane. It was also found that more polar and spherical solvent molecules clearly favoured the gauche conformation, which is more spherical than the trans; however, the more sophisticated analysis of the temperature dependencies could not provide unambiguous estimates for the energetics.
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Conformational problem of alkanes in liquid crystals
Figure 5. Ratio of propane to ethane order parameters (top) and potential energy parameters (bottom) as a function of temperature in the liquid-crysta ls 1132 (right) and 5CB (left). An expression for Uaniso ./ in terms of Gˇ can be found in Ref.[35]
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Figure 6. Etg as a function of temperature where ‘CCd chord’ is the modified chord model fit in 1132 from Ref.[22] The ST lines result from ethane dipolar coupling scaling and the UT result is from scaling to the anisotropic potential of ethane. Fig. 5 from Ref.[36]
A. C. J. Weber and D. H. J. Chen
Figure 7. Temperature dependence of n-butane trans and gauche total (including the anisotropic contribution in 1132), isotropic and gas phase conformer probabilities. Fig. 6 from Ref.[36]
Figure 8. Probability of trans-n-butane conformer as a function of reduced temperature in a liquid-crystal mixture possessing an N, reentrant nematic (RN) and smectic A (SmA) phase (upper line and points) and in the N and SmA phase of cyano-octyloxybiphenyl (lower line and points). Filled symbols are N and RN, while the open symbols are SmA. The line uses only the isotropic potential, whereas the points include anisotropic orientational and positional considerations. The dotted line is a smooth curve connecting the N and RN phases. Bottom half of Fig. 6 in Ref. [40]
568
The temperature dependencies of the energetics do seem to yield to the dipolar couplings of n-butane as can be seen in Fig. 6 as well as in Fig. 7[36] where the total (including the anisotropic), isotropic and gas phase trans probabilities are shown for 1132. Clearly, the anisotropic contribution leads to an increase in the trans population, and this effect increases with decreasing tem-
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perature and increasing orientational order, which makes intuitive sense because a more elongated solute is likely more easily accommodated in an N-type environment. This observation seems analogous to those in the isotropic phase[39] where the more globular gauche n-butane conformation was better accommodated by spherical solvent molecules. Observing these various
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Conformational problem of alkanes in liquid crystals effects of condensed phases on the conformational statistics of flexible molecules gives rise to the question of what role, if any, additional ordering like the layering in biological membranes could have? The effect of layering on n-butane conformational statistics has recently been studied in two SmA liquid-crystal phases; one of which was a mixture of mesogens that displays a lower temperature RN phase.[40] The same scaling approach described earlier for n-butane in 1132 and 5CB was used for the N phases of the liquid-crystal mixture, and once again, an excellent fit with an RMS of 1.9 Hz across the entire temperature range is obtained. These parameters were then transferred to the N phase of 4-n-octyloxy-4’-cyanobiphenyl using no adjustable parameters, and an RMS of 2.7 Hz is obtained, which is of the order of the spectral line width. The fact that this transfer can be carried out so readily is strong confirmation that the same orientational mechanism is in operation in the N phases of both liquid crystals, that the hydrocarbons are indeed ‘magic solutes’ and that the single mechanism is to do with size-and-shape entropic effects. This amazing result demonstrates the validity of the general approach used to account for chain flexibility in an N phase. Once again, the effect of the ordered phases on the conformer probabilities can be seen in Fig. 8 where the gap between the isotropic and full potential trans probabilities increases with decreasing temperature because the anisotropic potential increases. The N potential in the SmA phase is then interpolated between the N and reentrant nematic phases, which allows for the determination that the ethane, propane and n-butane partitioning across the layer shows a small preference for the intralayer region. The effect of the SmA phase can be seen to enhance the gauche probabilities as indicated in Fig. 8 by the depressed slope of the trans probabilities upon entry into the SmA phases.
Conclusion
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Acknowledgements The authors would like to acknowledge E. Elliott Burnell for fruitful discussions as well as W. Leo Meerts for the use of the Nijmegen cluster and his help with various algorithms over the years.
References [1] P. J. Collins, Liquid Crystals, Princeton University Press, Princeton, 2010. [2] R. J. Muckom, F. Stanzione, R. D. Gandour, A. K. Sum, J. Phys. Chem. B 2013, 117, 1810–1818. [3] G. N. Ramachandran, V. Sasisekharan, Adv. Prot. Chem. 1968, 23, 283–437. [4] M. A. Basharov, Eur. Biophys. J. 2012, 41, 53–61. [5] E. R. Gillies, J. M. J. Fréchet, Drug Discov. Today 2005, 10, 35–43. [6] V. Percec, M. Imam, M. Peterca, W. D. Cho, P. Heiney, Isr. J. Chem. 2009, 49, 55–70. [7] D. Tsiourvas, M. Arkas, Polymer 2013, 54, 1114–1122. [8] D. B. Li, W. H. de Jeu, Phys. Rev. Lett. 2004, 92, 075506. [9] X. Chen, S. K. Kumar, R. Ozisik, J. Polym. Sci. Part B 2006, 44, 3453–3460. [10] L. A. Beresnev, L. M. Blinov, M. A. Osipov, S. A. Pikin, Mol. Cryst. Liq. Cryst. 1998, 158A, 3–150. [11] D. J. Photinos, E. T. Samilski, Science 1995, 270, 783–786. [12] J. Zhao, K. Tatani, Y. Ozaki, Appl. Spectrosc. 2005, 59, 620–629. [13] M. Yang, K. Stappert, A. V. Mudring, J. Mater. Chem. 2014, 2, 458–473. [14] Z. Luz, in NMR of Ordered Liquids (Eds: E. E. Burnell, C. A. de Lange), Kluwer Academic, Dordrecht, The Netherlands, 2003, pp. 419–448. [15] J. M. Polson, E. E. Burnell, J. Chem. Phys. 1995, 103, 6891–6902. [16] P. J. Flory, Statistical Mechanics of Chain Molecules, Wiley-Interscience, New York, 1969. [17] D. J. Photinos, E. T. Samulski, H. Toriumi, J. Phys. Chem. 1990, 94, 4688–4694. [18] D. S. Zimmerman, E. E. Burnell, Mol. Phys. 1993, 78, 687–702. [19] W. L. Meerts, C. A. de Lange, A. C. J. Weber, E. E. Burnell, in Encyclopedia of Magnetic Resonance (Eds: R. K. Harris, R. E. Wasylishen), John Wiley & Sons Ltd, New York, 2013, pp. 437–449. [20] A. C. J. Weber, C. A. de Lange, W. L. Meerts, E. E. Burnell, Chem. Phys. Lett. 2010, 496, 257–262. [21] H. Nakatsuji, et al., Gaussian, Inc. 03, revision E.01. [22] A. C. J. Weber, E. E. Burnell, Chem. Phys. Lett. 2011, 506, 196–200. [23] A. L. Verna, W. F. Murphy, H. J. Bernstein, J. Chem. Phys. 1974, 60, 1540–1544. [24] J. R. Durig, D. A. C. Compton, J. Phys. Chem. 1979, 83, 265–268. [25] D. A. C. Compton, S. Montero, W. F. Murphy, J. Phys. Chem. 1980, 84, 3587–3591. [26] R. K. Heenan, L. S. Bartell, J. Chem. Phys. 1983, 78, 1270–1274. [27] K. B. Wiberg, M. A. Murcko, J. Am. Chem. Soc. 1988, 110, 8029–8038. [28] E. E. Burnell, C. A. de Lange, Chem. Rev. (Washington, DC) 1998, 98, 2359–2387. [29] W. L. Meerts, M. Schmitt, Int. Rev. Phys. Chem. 2006, 25, 353–406. [30] W. L. Meerts, C. A. de Lange, A. C. J. Weber, E. E. Burnell, J. Chem. Phys. 2009, 130, 044504–044504-8. [31] A. C. J. Weber, X. Yang, R. Y. Dong, W. L. Meerts, E. E. Burnell, Chem. Phys. Lett. 2009, 476, 116–119. [32] E. E. Burnell, C. A. de Lange, W. L. Meerts, in Nuclear Magnetic Resonance Spectroscopy of Liquid Crystals (Ed: R. Y. Dong), World Scientific, Singapore, 2010, pp. 1–35. [33] W. Maier, A. Saupe, Z. Naturforsch. A. 1959, 14, 882–900. [34] W. Maier, A. Saupe, Z. Naturforsch. A. 1960, 15, 287–292. [35] E. E. Burnell, L. C. ter Beek, Z. Sun, J. Chem. Phys. 2008, 128, 164901–164901-10. [36] E. E. Burnell, A. C. J. Weber, C. A. de Lange, W. L. Meerts, R. Y. Dong, J. Chem. Phys. 2011, 135, 234506–234506-10. [37] L. R. Pratt, C. S. Hsu, D. Chandler, J. Chem. Phys. 1978, 68, 4202–4212. [38] C. S. Hsu, L. R. Pratt, D. Chandler, J. Chem. Phys. 1978, 68, 4213–4217. [39] T. Tynkkynen, T. Hassinen, M. Tiainen, P. Soininen, R. Laatikainen, Magn. Reson. Chem. 2012, 50, 598–607. [40] A. C. J. Weber, R. Y. Dong, W. L. Meerts, X. Yang, E. E. Burnell, J. Phys. Chem. A 2013, 117, 9224–9234.
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It is clear that the alkane conformational problem is integral to understanding a wide breadth of liquid-crystal behaviour from biologically active pharmaceuticals to polymers commonly used in industry. Of course, there is a flip side of the coin because just as alkane flexibility bestows properties onto liquid crystals so to do liquid crystals (and isotropic phases) affect conformational probabilities. In order to deduce these effects via NMR spectroscopy, one can make progress with orientational models, but greater confidence is achieved by scaling flexible alkane conformer ordering to their basic rigid units so as to extract conformer probabilities. When this is carried out, a number of sensible results fall out. Firstly, the isotropic condensed phase clearly favours the gauche conformer relative to the gas phase, and this is even more so the case for spherical solvent molecules. An anisotropic phase favours the trans conformer, and this is even more so the case when the rod like molecules are more ordered at lower temperatures. These observations can be seen as a manifestation of a very basic truth in chemistry that ‘like dissolves like’ and perhaps could have been anticipated. Finally, the presence of layering seems to enhance the gauche conformers perhaps as a result of being more easily accommodated in the more flexible and less ordered interlayer region. Future studies might consider applying the ethane scaling to the dipolar couplings of n-pentane to explore the behaviour of the simplest example of chain conformational statistics involving the ‘n-pentane effect’ (steric repulsion
of methyls in the gauche plus gauche minus conformer) and helical conformer.
Revised: 14 July 2014
Accepted: 15 July 2014
Published online in Wiley Online Library: 20 August 2014
(wileyonlinelibrary.com) DOI 10.1002/mrc.4124
Conformational problem of alkanes in liquid crystals by NMR spectroscopy: a mini-review† Adrian C. J. Webera* and Daniel H. J. Chenb Recent discoveries of the role of alkane flexibility in determining liquid-crystal behaviour are surveyed. With the impetus for understanding the alkane conformational problem established, recent model dependent 1 H NMR work on the topic will be reviewed where progress is made but the need to circumvent models eventually becomes evident. A closer look at the rigid basic units of alkanes will provide the way forward where it is shown that the orientational ordering and anisotropic potentials of these molecules dissolved in liquid crystals scale with each other. Once this relationship is established, a series of works using anisotropic and isotropic 1 H NMR spectroscopy to study alkane conformational statistics will be covered, wherein the influence of the gas, isotropic condensed and anisotropic condensed phases will be described. Copyright © 2014 John Wiley & Sons, Ltd. Keywords: NMR; H; liquid crystals; alkanes; conformational statistics; n-butane; anisotropic model; dipolar coupling; nematic; Etg
Introduction
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While it is true that liquid crystals are a rich source of fascinating physical and chemical behaviour, they also touch many aspects of our daily lives. Therefore, the importance of liquid-crystal studies is twofold; they are a testing ground for discerning fundamental chemical and physical phenomena and represent a wealth of potential in a wide variety of applications such as the rapidly developing field of liquid-crystal display technologies (used in smart phone, tablet, computer and television screens) and are even being used to create better welding masks, temperature sensors and windows while of course being essential to life vis-a-vis biological membranes.[1] The identified supramolecular architectures formed in mesogenic phases are numerous and characterized by long-range orientational and positional order of component molecules. The fluid nature of these phases is typically the result of flexible segments on the liquid-crystal molecules, which are usually hydrocarbon chains. The flexibility of these segments is manifested because of the possibility that each saturated C–C bond can be in a trans or gauche conformation. Therefore, the behaviour of the alkane chain conformational statistics becomes an important question in terms of what these segments bestow onto the properties of the multitude of liquid-crystal phases, which can often yield technological applicability. Dendrite audiophiles, water-soluble liquid crystals, have demonstrated anti-HIV, anti-STD, antifungal and antimycobacterial properties and have recently been studied via molecular dynamics simulation to gain insight into how they interact with phospholipid bilayer membranes.[2] These simulations show that alkyl chains facilitate entry into target cell membranes and change the molecular dynamics significantly. At the same time, data on the deuterium order parameter, bilayer thickness and tail tilt angle show increased disorder and tail kinking by the addition of dendritic amphiphiles. Further understanding of these phenomena involving the alkane conformational problem in a liquid crystal is said to have the potential to result in a new line of pharmaceuticals.
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Another important type of biologically active flexible chain molecules are the polypeptides, which fold up into proteins and exist in isotropic condensed phases as well as anisotropic liquid-crystal phases (membrane proteins). The magnitude of the internal rotational barriers about the backbone single covalent bonds of these molecules is crucial in determining conformations, folding, dynamics and functioning of proteins. For decades now, conventional wisdom has it that the internal barriers to rotation in polypeptides are small (around 0.7 kcal mol1 ) and of an order comparable to kB T at physiological temperatures.[3] However, some recent semi-empirical and ab initio calculations on model dipeptides have shown that the conventional estimates for non-glycine single bond backbone rotations could be 10 to 20 times higher.[4] It would be of value to verify these calculations experimentally because such an observation would have great implications for elucidating the stability, functioning, folding and structural organization of proteins. The development of new dendritic polymers has gained much attention for their use and potential use in drug delivery among other biomedical applications.[5] The usual way to obtain the desired liquid crystallinity is to introduce ‘promesogenic units’ to the periphery, but more recently, this has been circumvented by exploiting the natural tendency of these macromolecules to display phase separation when functionalized with aliphatic chains,[6] which leads to improved liquid-crystalline properties
* Correspondence to: Adrian C. J. Weber, Chemistry Department, Brandon University, 270-18th St, Brandon, MB R7A 6A9, Canada. E-mail: [email protected]; homepage, http://people.brandonu.ca/webera/ †
This article is published in Magnetic Resonance in Chemistry as a special issue on the NMR of Liquid Crystals by Ronald Y. Dong (Department of Physics and Astronomy, UBC, Vancouver, Canada).
a Chemistry Department, Brandon University, 270-18th St, Brandon, MB, R7A 6A9, Canada b Chemistry Department, University of British Columbia, 2036 Main Mall, Vancouver, BC, V6T 1Z1, Canada
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over which mesophases could be observed. Although alkyl chains spacers did very well in this regard, it was remarkable to observe that adding an ether linkage could result in a mesophase temperature range of 225 K. The extensive mesophase stability was rationalized by considering that the insertion of a flexible ether spacer leads to an increase in the degree of rotational and vibrational freedoms, and therefore entropy, while at the same time allowing the ether oxygen to take part in a three-dimensional hydrogen bonding network adding yet further stability. This kind of phase stability has great potential for studying the kinetics of conformational change via dynamic NMR[14] (which were previously complicated by the need for multiple liquid-crystal systems to span the required temperature range) as well as for the type of liquid-crystal NMR conformational statistics studies that follow.
Conformational Probability Extraction by Modelling In order to better understand the condensed phase conformational problem by NMR in general and how it pertains to liquid crystals in particular, it seems pragmatic to consider the simplest flexible alkane n-butane, which possesses a single degree of rotational freedom. In the earlier NMR studies of n-butane dissolved in anisotropic phases, it was more difficult to obtain the necessary dipolar or quadrupolar couplings because selective/random deuteration or multiple quantum methods needed to be used. Even once the single quantum 1 H NMR spectrum of n-butane was solved,[15] another problem arises because the conformational probability distribution function, pn , appears multiplied by the Saupe order matrix for the nth conformer in the expression n
Dij D
2 X nX n n p S˛ˇ Dij,˛ˇ 3 n
(1)
˛ˇ
for the dipolar couplings between nuclear spins i and j, and there is no way to unambiguously separate the two at a single temperature. The Saupe order matrix, Sn˛,ˇ , for the nth conformer is defined by 3 1 n n cos ˛,Z Sn˛ˇ D cos ˇ,Z ı˛ˇ (2) 2 2 n where ˛,Z is the angle between the molecular-˛ axis and the N director, which is aligned with the static magnetic field along the laboratory fixed Z-axis. Dnij,˛ˇ is a tensor defined by
Dnij,˛ˇ D
0 2 „ 8 2 rij3
3 1 ij,n cos ˛ij,n cos ˇ ı˛ˇ 2 2
(3)
ij,n
where ˛ is the angle between the internuclear vector rij and the molecular ˛ axis of the nth conformer. In order to proceed at the time, some approximation needed to be made. One of these is the rotational isomeric state (RIS) model,[16] which allows only three n-butane conformations to be accessible, namely, the trans (located at the global minimum of the intramolecular potential) and the twofold degenerate ˙gauche (located at the local minima). Additionally, it was found that allowing for fluctuations about the dihedral angles in an extended RIS model yielded slightly better fits to dipolar couplings. But to separate the conformer probabilities, it was necessary to use a variety of mean-field models to describe the elements of the Saupe order tensor via the expression
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561
and a larger mesophase temperature range. To better understand these observations, thermogravimetric analysis, differential scanning calorimetry and X-ray diffraction studies were carried out, and it was found that not only does the introduction of the alkyl chains lead to non-polar interaction of these components but also resulted in hydrogen bond formation in the core of the polymer.[7] These two effects lead to microphase separation, which is regarded as the main factor inducing mesomorphic character. It was then determined that the interplay between conformational change of the alkyl chains and the hydrogen bonding networks of the core was responsible for the hysteresis in phase transitions over heating–cooling cycles. Isotactic polypropylene is an alkyl chain widely used in industry for its intermediate crystallinity and flexibility and has been shown to display smectic phases under certain conditions.[8] Depending on different crystallization conditions, this polymer can take on different crystalline morphological forms. To uncover the nature of this process, Monte Carlo simulations were carried out wherein it was found that conformational order is the precondition for a smectic intermediate phase formation, which then leads to crystallization.[9] More specifically upon lowering the temperature (but before the liquid to crystal transition temperature), an increasing amount of trans-gauche pairs emerges resulting in the formation of helices and perhaps some local nematic (N) ordering in the melt. Once enough of this conformational ordering has occurred upon approaching the crystallization temperature, a critical amount of helices of a critical length appear to give rise to an intermediate smectic phase, which then leads to crystallization. So it seems as though the crystallization is ‘nucleated’ by a change in the alkyl chain conformational statistics. This change then brings about the orientational and translational ordering required for crystallization of this ubiquitous polymer. Over the years, a variety of ferroelectic liquid crystals (FLC) forming chiral smectic phases (Sm-C*) have been produced for potential applications in fast electro-optical devices. However, there is still no straightforward explanation of the relationship between the structure of FLC molecules and their behaviour in the Sm-C* phase. The molecular-statistical theory of ferroelectric ordering in the Sm-C* phase[10] and indigenous polarization theory[11] both describe the chiral centre and polarizable core as key factors for the emergence of the Sm-C* phase. However, principal-component analysis and two-dimensional correlation infrared spectroscopy of a particular FLC[12] showed that 92% of the spectral variance at the smectic A (SmA) to Sm-C* phase transition is due to the first principal component, which is dominated by the stretching modes of the methyl and methylenes of the FLC alkyl chains. So it seems that some change in the alkyl chains of at least some FLCs are of particular importance for the emergence of the technologically valuable Sm-C* phase. Ionic liquids (ILs) have matured as a class of materials over the last decade with the emergence of several commercial applications. Bis-cationic ILs in particular have been shown capable of mesophase formation when endowed with a sufficiently long alkyl chain. The mesomorphic properties show strong dependence on side chain length and the type of spacer linking the cations, which was recently investigated for a series of bis-cationic imidazolium based ILs via thermogravimetric analysis, differential scanning calorimetry and X-ray diffraction in the small angle range.[13] The authors then found that the ability of the ILs studied to form mesophases was not only related to the van der Waals forces between alkyl chains but also that the flexibility of the spacers seemed to be responsible for the temperature range
A. C. J. Weber and D. H. J. Chen R 3 Sn˛ˇ
D
2
n n 12 ı˛ˇ exp.Unaniso ./=kT/d cos ˇ,Z cos ˛,Z R exp Unaniso ./=kT d (4)
where Unaniso is the anisotropic N ordering potential of the nth conformer to be modelled. It was then found that the chord model[17] and size-and-shape model[18] gave the best fits to dipolar couplings.[15] The chord model is designed for the orientational order of molecules consisting of repeating identical units in a uniaxial phase and has the form UnCd,aniso ./ D
X
Q 0 P2 .si , si / C w Q 1 P2 .si , siC1 / w
(5) iso iso int ext Etg D Ugauche Utrans D Etg C Etg
iD1
Q i D 32 Swi and wi scales are the strength of the chord’s where w interaction with the mean field and S is the liquid-crystal order parameter. The si is a unit vector describing the orientation of the ith C–C bond of the hydrocarbon chain, and the sum is over all bonds in the chain. The factors P2 .si , siCm / are given by P2 .si , siCm / D
1 3 cos Zi cos ZiCm si siCm 2 2
(6)
where Zi is the angle between the ith bond and the N director. The model postulates that orientational ordering arises from an orientational torque about each C–C bond and the lines joining their midpoints. The second term incorporates correlations between adjacent-bond orientations, which distinguishes between conformations that may have equal numbers of trans and gauche bonds but different shapes. The size-and-shape model potential is the sum UnCI ./ D
1 1 k.Cn .//2 ks 2 2
Z
Zmax,n
Cn .Z, /dZ
(7)
Zmin,n
the first term of which involves a Hooke’s law restoring force where Cn ./ is the minimum circumference traced out by the projection of the solute onto a plane perpendicular to the N director. The second term represents the anisotropic surface potential where the area of the infinitesimally thin ribbon Cn .Z, /dZ is summed over its projection onto the plane parallel to the N director. But because UnCI ./ contains an isotropic component in practice, UnCI,aniso ./ D UnCI ./ hUnCI i
(8) hUnCI i
where Gn, is a rotational kinetic energy factor, which is dependent on the principal values of the moment of inertia tensor for each conformer and dihedral angle. The isotropic potential, Uiso , is the sum of the intramolecular gas phase component (calculated with GAUSSIAN 03) and the intermolecular component due iso to the condensed phase environment, Uext,n , which is represented ext , which raises or lowers the potential by a step function of size Etg energy of the n-butane geometries in the gauche potential well relative to those in the trans as can be seen in Fig. 1. The trans-gauche energy (Etg) difference in a condensed phase is then also a sum
is the is used to model the anisotropic potential where isotropic average over all angles. Later on when the spectral analysis of highly congested and complex anisotropic NMR spectra (like n-butane) became routine[19] (vide infra) with the use of an evolutionary strategy (ES), it became possible to revisit the problem while fitting to multiple liquid-crystal n-butane NMR spectra simultaneously.[20] In addition to calculations using the RIS and extended RIS model, the iso , was calculated using gas phase intramolecular potential, Uint,n GAUSSIAN 03 (Gaussian, Inc., Wallingofrd, Connecticut, USA).[21] This allowed for the calculation of a continuous dihedral angle probability distribution function via the expression
gas
int Etg D 643 cal mol1 is taken from the Gauswhere Etg ext sian calculations and Etg from fits to n-butane dipolar couplings in four different liquid crystals simultaneously. Despite fitting to four times the amount of equations, better fits are obtained with respect to the earlier study, likely due to the use of high quality GAUSSIAN 03 geometries in the latter compared to a more crude geometry in the former. When the continuous dihedral angle population distributions were plotted for all the calculation variants, it was found that only two distinct distributions emerged (as can be seen in Fig. 2), and that results depended only on the model chosen to describe the orientational potential. Specifically, calculations with the modified chord model, where Q 0 and w Q 1 are no longer held equal to each other and allowed to w vary independently, were found to favour the gauche conformation more so than those with the two parameter size-and-shape model where k and ks were also allowed to vary independently as opposed to being kept to their optimized ratio.[18] To better understand this observation, the calculated average mean square orientational ordering for the trans and gauche conformers was determined by the expression
q p n hS2 i D S2xx C S2yy C S2zz
(11)
where it was found that the modified chord model ascribed greater ordering to the trans conformer than the two paramp does g eter size-and-shape model. Furthermore, hS2 i was larger when calculated by the modified chord model. This in turn explains why calculations with the modified chord model have enhanced gauche populations relative to calculations with the two parameter size-and-shape model because the order matrix multiplies the conformer probability function in Eqn (1) and both methods are fitting the same four sets of dipolar couplings. In order to sort out which (if either) model was more or less physically correct, a subsequent study on n-butane sought to exploit the conformational probability temperature dependence. Such a study was made possible by the robust nature of an ES to obtain dipolar couplings with relative ease over (what was at the time) a very large 80ı temperature range.[22] In these fits to dipolar couplings, it was found that the continuous intramolecular potential resulted in root mean square (RMS) values 1–3 Hz lower than the RIS model. Both anisotropic models yield Etg ’s lower than the experimental gas phase range of 788–884 cal mol1 [23–27]
562
iso iso R =kT exp Uint,n ./=kT exp Unaniso ./=kT d Gn, exp Uext,n iso iso R pn, D P P n, exp Uext,n =kT exp Uint,n ./=kT exp Unaniso ./=kT d n G
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(10)
Copyright © 2014 John Wiley & Sons, Ltd.
(9)
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iso iso ext Figure 1. (a) Uext is a step function, which raises or lowers the potential energy of n-butane geometries in the gauche wells by the amount Etg . (b) Uint is the n-butane internal rotational potential calculated by GAUSSIAN 03. The gap along the energy axis between the local (gauche) minima and the global gas int Etg . (c) The total isotropic potential, Uiso , is the sum of the latter two functions. While there is a slight discontinuity (trans) minimum is the quantity Etg in this function, the populations corresponding to the dihedral angles in the vicinity are very small as can be seen in Fig. 2. The gap between the local int ext and global minima is now the sum Etg D Etg C Etg . Thus, the model assumes that when n-butane is in one particular conformer, the internal motions are iso only plays a role in raising or lowering the potential for gauche conformers. governed completely by the intramolecular (gas phase) potential. Hence, Uext [22] Fig. 1 from Ref.
Figure 2. The probability P./ of finding n-butane at dihedral angle for the modified chord model (solid line) and the two parameter size-and-shape model (dashed line) in a N liquid-crystal phase. Fig. 3 from Ref.[20]
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the experimental condensed phase range of 502–681 cal mol1 whereas the modified chord model yielded values for the most part below this range. Perhaps most remarkable was that both models seemed to indicate, although to varying degrees, that Etg
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consistent with the idea that condensed phases favour the gauche conformer relative to the gas phase as found in previous experiments in condensed phases. It was found that the two parameter size-and-shape model gave Etg values towards the top end of
A. C. J. Weber and D. H. J. Chen
Figure 3. Calculated spectra of ethane (A) and propane (B) from using an evolutionary strategy on the experimental spectrum (C) of these solutes in 5CB at 298 K.
accounting for the effect of the condensed phase. Although this result appeared intriguing, the values and variation of Etg , and therefore the conformational statistics, were still model dependent. Because the orientational models used only predict order parameters at approximately the ˙10 % level,[28] the physics associated with the conformational change is masked by the uncertainty. The precise nature of the ordering potential is a very old and recurring problem, and it becomes evident that a model-free method is necessary in order to unambiguously sort out the conformational problem of alkanes in liquid crystals by NMR spectroscopy.
terms of how to correctly separate the Saupe order matrix from the conformational probability distribution function. To this end, the solutes ethane, propane and 1,3,5-trichlorobenzene (tcb) (added to provide an orientational and chemical shift reference) were co-dissolved in the liquid-crystal mixture Merck (White House Station, New Jersey, USA) ZLI-1132 (1132) and 4-n-pentyl-40 -cyanobiphenyl (5CB). Because these alkanes are gases at room temperature and ambient pressure, they were allowed to flow into a vacuum and then condensed into an NMR tube, which was pre-filled with liquid crystal and tcb and submerged into liquid nitrogen. The concentrations of tcb and ethane in 1132 are 0.5 and 5 mol % while that of propane is 2.5 mol %. In 5CB, the solute concentrations of tcb and ethane were 0.5 and 5.0 mol % while that of propane was 2.5 mol %. After these samples were sealed and thoroughly mixed in the isotropic phase, they were placed in turn into a Bruker Avance 400 MHz NMR spectrometer magnet (Bruker Biosciences Corporation, Billerica, Massachusetts, USA). With the temperature controlled by the Bruker air-flow system, 1 H NMR spectra were acquired every 5ı from 253.0 to 333.0 K for 1132 and from 273.0 to 303.0 K for 5CB; an example of which can be found in Fig. 3, which includes the calculated spectra for the alkanes.
Experimental
Results
Rather than continue on with mean-field models, it becomes apparent that a more basic look at the actual ordering of the fundamental units that flexible alkanes are composed of, namely ethane and propane, can hopefully be instructive in
In order to obtain the spectral parameters defining the anisotropic spectra with an ES, one has to first choose reasonable upper and lower limits for each parameter, which defines the search space. A complete set of spectral parameters is called a chromosome, and
is temperature dependent. Of course, it is anticipated from the Boltzmann statistics that the trans conformer population should increase as the thermal energy needed to populate the gauche conformation decreases with decreasing temperature. However, an Etg rising with decreasing temperature implies an additional source enhancing the trans probability. If the Etg difference is a function of temperature, it would have to result from the second term in the sum gas
ext Etg .T/ D Etg C Etg .T/
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Table 1. Experimental dipolar couplings of 1,3,5-trichlorobenzene, ethane and propane in 1132 (top) and 5CB (bottom) as a function of temperature. The numbers in brackets are the errors in Hertz in the last one or two digits prop
prop
prop
prop
T/K
Dtcb
Deth 12
Deth 14
253.5 258.5 263.5 268.5 273.5 278.5 283.5 288.5 293.5 298.5 303.5 308.5 313.5 318.5 323.5
274.63 270.82 265.33 260.02 254.84 249.21 243.86 237.61 230.85 223.17 215.50 207.22 195.12 187.82 175.81
866.40(16) 848.25(13) 830.93(13) 811.17(13) 790.69(13) 769.43(13) 749.18(13) 725.46(13) 700.48(13) 671.31(13) 643.37(13) 613.51(13) 581.14(13) 545.31(13) 504.21(13)
343.28(15) 336.20(11) 329.41(11) 321.71(11) 313.77(11) 305.37(11) 297.44(11) 288.15(11) 278.31(11) 266.82(11) 255.82(11) 244.06(11) 231.25(11) 217.08(11) 200.81(11)
922.25(9) 904.66(7) 887.55(6) 867.57(6) 846.66(7) 824.49(6) 803.17(7) 778.20(6) 751.61(6) 720.59(7) 690.79(7) 658.90(6) 624.19(6) 585.92(6) 541.70(7)
208.26(7) 204.37(6) 200.68(5) 196.35(5) 191.83(6) 187.00(5) 182.42(5) 177.03(5) 171.29(5) 164.48(6) 157.96(6) 150.95(5) 143.27(5) 134.74(5) 124.91(5)
439.40(8) 431.07(6) 422.73(5) 413.18(5) 403.13(6) 392.43(5) 382.09(6) 369.97(5) 357.12(5) 342.12(6) 327.70(7) 312.31(5) 295.61(5) 277.15(5) 256.04(6)
2154.35(22) 2113.49(18) 2073.26(16) 2026.60(17) 1978.00(17) 1927.24(17) 1878.49(17) 1821.18(16) 1760.48(16) 1689.49(18) 1621.24(18) 1548.23(17) 1467.82(17) 1379.63(17) 1277.69(17)
268.5 273.5 278.5 283.5 288.5 293.5 298.5 303.5
233.91 214.68 205.35 194.94 183.00 168.83 149.39 112.45
616.57(9) 588.64(13) 559.86(13) 528.44(13) 492.85(13) 451.22(13) 395.03(13) 292.31(13)
244.96(8) 233.90(11) 222.54(11) 210.13(11) 196.06(11) 179.59(11) 157.29(11) 116.46(11)
668.15(5) 638.58(6) 607.86(6) 574.05(6) 535.57(6) 490.42(6) 429.41(6) 317.92(6)
153.0(4) 146.51(5) 139.70(5) 132.16(5) 123.57(5) 113.40(5) 99.56(5) 73.96(5)
319.99(4) 302.73(5) 287.94(5) 271.77(5) 253.30(5) 231.72(6) 202.65(5) 149.84(6)
1580.54(12) 1511.46(16) 1439.90(17) 1361.09(17) 1271.24(17) 1165.72(17) 1022.45(17) 758.20(17)
D12
a population of these is initially spread out randomly across the search space. To evaluate the goodness of each member of the population, a fitness function needs to be defined Ffg D
.f g/ k f kk g k
(13)
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D16
D45
energy parameters. It is then interesting to compute the propane to ethane ratios of these parameters, which can be seen in Fig. 5. The invariance of these ratios over a wide temperature range provides evidence that ethane and propane are ‘magic solutes’ whose orientational order is governed by a single mechanism because if there were more, we should expect the orientation of these molecules to display different temperature dependencies, which would result in non-constant ratios. The order parameters of ‘magic solutes’ only depend on solute size-and-shape that are essentially temperature independent. Therefore, these ratios are expected to be temperature independent so long as such a solute does not undergo conformational change. This key result provides the way forward towards the separation of the orientational order matrix from the conformational probabilities because any ratio temperature dependence (whether order parameter, dipolar coupling or Maier–Saupe potential energy parameter) in a flexible alkane can now be completely attributed to the temperature dependence of the conformer probabilities. Therefore, if one knows the order parameters as a function of temperature of say ethane in some partially ordered environment, then one can make a given flexible alkane conformers ordering proportional to this known so as to then extract the desired conformational statistics.
Conformational Probability Extraction by Scaling A study of this nature[36] was carried out using the ethane dipolar couplings in Table 1. First, the n-butane to ethane dipolar coupling ratios were calculated and found to be temperature dependent as a result of changes in conformer populations as a function of temperature. It was then found that as the details of
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where f and g are the vector representations of the experimental and calculated spectra. Ffg is a measure of the extent to which a calculated spectrum overlaps with the experimental, which is maximum at the global minimum of the error surface where the solution is obtained.[29–32] However, before all this can be carried out, the liquid-crystal background needs to be removed because the ES iterates on both the frequency and intensity of transitions. The usual way to remove the background in the experimental spectrum is by a manual cubic base spline, which was performed here. Once the dipolar couplings are obtained as shown in Table 1, one can go on to calculate the chord and size-and-shape model parameters from least squares fitting, which are shown in Fig. 4. While both model potentials fit equally well for ethane in both liquid crystals, the case of propane appears more discriminating. For propane across all temperatures, the size-and-shape model RMS is 1.5 Hz in 5CB and 2.8 Hz in 1132 while those of the chord model are 13.2 and 17.2 Hz. The experimental order parameters are obtained via Eqn (1) for which pn will be one for rigid solutes like ethane and propane and from Eqn (4), the classic Maier–Saupe potential[33,34] energy parameters are calculated. The energy parameters are composed of independent second-rank tensorial liquid crystal (G) and solute (ˇ ) terms, which interact with each other giving rise to orientational order.[35] The symmetry of the solute governs the number of unique order parameters and hence
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Figure 4. Chord (top) and size-and-shape (bottom) model parameters for ethane (filled circles) and propane (open circles) as a function of temperature in 1132 (left) and 5CB (right).
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calculations fitting to n-butane dipolar couplings were improved that not only were better fits obtained but also more realistic physical parameters are gotten. The first challenge is to separate the orientational order from conformer probability. To this end, it was assumed that the six unknown order parameters of n-butane (three for the trans and three for the gauche) for each temperature scale to the ethane dipolar coupling or anisotropic potential energy parameter. Therefore, all spectra collected every 5ı in a given liquid-crystal solvent are fit to six proportionality constants so as to determine the orientational order of the n-butane conformers via the presence of ethane. In order to use the continuous intramolecular potential n-butane geometries residing in gauche, potential wells were given the same orientational order proportionality constants (and similarly for geometries in the trans well) so as to separate the internal motion from reorieniso tation. It was then found that using the Uint gave improved fits compared to calculations using the RIS approximation. Further improvement was gained when the anisotropic energy contribution to the conformer probabilities was accounted for. Even better fits to dipolar couplings are obtained when allowing Etg to have a temperature dependence. In fact when scaling to ethane dipolar couplings (labelled ST in Fig. 6), an RMS of 6.2 Hz over a 75ı 1132 temperature range and 2.7 Hz over the 40ı 5CB temperature range is amazing and unprecedented in earlier studies of orientationally ordered hydrocarbons. Just as important is the result that only when an Etg temperature dependence is allowed does one obtain orientational ordering consistent with simple size-and-shape arguments involved in the single orientational mechanism. Better fits are obtained when scaling to the ethane
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anisotropic potential parameters (labelled UT in Fig. 6), but in this case, the 1132 Etg range becomes shifted down from that of 5CB. These Etg values can be seen in Fig. 6 and are clearly consistent ext with a temperature variation of Etg due to the condensed phase, which turns out to be very similar to the previous result obtained with the modified chord model.[22] So while the size-and-shape model seems to reproduce the ordering of hydrocarbon chain units somewhat better than the chord model, it appears that the modified chord model better allocates order among the different conformers in a flexible hydrocarbon than does the two parameter size-and-shape model. All the values for Etg are lower than the GAUSSIAN 03 gas phase value of 643 cal mol1 indicating that the condensed phase favours the gauche conformer, which is consistent with what had been found theoretically[37,38] and experimentally (see Refs [15,20] and references therein). More recently, a study in this journal[39] employed indirect spin couplings of n-butane through n-heptane as a function of temperature and in different isotropic environments in order to shed light on the conformational problem of alkanes by 1 H NMR spectroscopy. The Etg was found to be close to 681 cal mol1 , which is somewhat higher than the GAUSSIAN 03 value but lower than the experimental gas phase range. Furthermore, the trans–trans to trans-gauche conformer energy difference of n-pentane was found to be 24–48 cal mol1 higher than the Etg for n-butane. It was also found that more polar and spherical solvent molecules clearly favoured the gauche conformation, which is more spherical than the trans; however, the more sophisticated analysis of the temperature dependencies could not provide unambiguous estimates for the energetics.
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Figure 5. Ratio of propane to ethane order parameters (top) and potential energy parameters (bottom) as a function of temperature in the liquid-crysta ls 1132 (right) and 5CB (left). An expression for Uaniso ./ in terms of Gˇ can be found in Ref.[35]
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Figure 6. Etg as a function of temperature where ‘CCd chord’ is the modified chord model fit in 1132 from Ref.[22] The ST lines result from ethane dipolar coupling scaling and the UT result is from scaling to the anisotropic potential of ethane. Fig. 5 from Ref.[36]
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Figure 7. Temperature dependence of n-butane trans and gauche total (including the anisotropic contribution in 1132), isotropic and gas phase conformer probabilities. Fig. 6 from Ref.[36]
Figure 8. Probability of trans-n-butane conformer as a function of reduced temperature in a liquid-crystal mixture possessing an N, reentrant nematic (RN) and smectic A (SmA) phase (upper line and points) and in the N and SmA phase of cyano-octyloxybiphenyl (lower line and points). Filled symbols are N and RN, while the open symbols are SmA. The line uses only the isotropic potential, whereas the points include anisotropic orientational and positional considerations. The dotted line is a smooth curve connecting the N and RN phases. Bottom half of Fig. 6 in Ref. [40]
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The temperature dependencies of the energetics do seem to yield to the dipolar couplings of n-butane as can be seen in Fig. 6 as well as in Fig. 7[36] where the total (including the anisotropic), isotropic and gas phase trans probabilities are shown for 1132. Clearly, the anisotropic contribution leads to an increase in the trans population, and this effect increases with decreasing tem-
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perature and increasing orientational order, which makes intuitive sense because a more elongated solute is likely more easily accommodated in an N-type environment. This observation seems analogous to those in the isotropic phase[39] where the more globular gauche n-butane conformation was better accommodated by spherical solvent molecules. Observing these various
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Conformational problem of alkanes in liquid crystals effects of condensed phases on the conformational statistics of flexible molecules gives rise to the question of what role, if any, additional ordering like the layering in biological membranes could have? The effect of layering on n-butane conformational statistics has recently been studied in two SmA liquid-crystal phases; one of which was a mixture of mesogens that displays a lower temperature RN phase.[40] The same scaling approach described earlier for n-butane in 1132 and 5CB was used for the N phases of the liquid-crystal mixture, and once again, an excellent fit with an RMS of 1.9 Hz across the entire temperature range is obtained. These parameters were then transferred to the N phase of 4-n-octyloxy-4’-cyanobiphenyl using no adjustable parameters, and an RMS of 2.7 Hz is obtained, which is of the order of the spectral line width. The fact that this transfer can be carried out so readily is strong confirmation that the same orientational mechanism is in operation in the N phases of both liquid crystals, that the hydrocarbons are indeed ‘magic solutes’ and that the single mechanism is to do with size-and-shape entropic effects. This amazing result demonstrates the validity of the general approach used to account for chain flexibility in an N phase. Once again, the effect of the ordered phases on the conformer probabilities can be seen in Fig. 8 where the gap between the isotropic and full potential trans probabilities increases with decreasing temperature because the anisotropic potential increases. The N potential in the SmA phase is then interpolated between the N and reentrant nematic phases, which allows for the determination that the ethane, propane and n-butane partitioning across the layer shows a small preference for the intralayer region. The effect of the SmA phase can be seen to enhance the gauche probabilities as indicated in Fig. 8 by the depressed slope of the trans probabilities upon entry into the SmA phases.
Conclusion
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Acknowledgements The authors would like to acknowledge E. Elliott Burnell for fruitful discussions as well as W. Leo Meerts for the use of the Nijmegen cluster and his help with various algorithms over the years.
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It is clear that the alkane conformational problem is integral to understanding a wide breadth of liquid-crystal behaviour from biologically active pharmaceuticals to polymers commonly used in industry. Of course, there is a flip side of the coin because just as alkane flexibility bestows properties onto liquid crystals so to do liquid crystals (and isotropic phases) affect conformational probabilities. In order to deduce these effects via NMR spectroscopy, one can make progress with orientational models, but greater confidence is achieved by scaling flexible alkane conformer ordering to their basic rigid units so as to extract conformer probabilities. When this is carried out, a number of sensible results fall out. Firstly, the isotropic condensed phase clearly favours the gauche conformer relative to the gas phase, and this is even more so the case for spherical solvent molecules. An anisotropic phase favours the trans conformer, and this is even more so the case when the rod like molecules are more ordered at lower temperatures. These observations can be seen as a manifestation of a very basic truth in chemistry that ‘like dissolves like’ and perhaps could have been anticipated. Finally, the presence of layering seems to enhance the gauche conformers perhaps as a result of being more easily accommodated in the more flexible and less ordered interlayer region. Future studies might consider applying the ethane scaling to the dipolar couplings of n-pentane to explore the behaviour of the simplest example of chain conformational statistics involving the ‘n-pentane effect’ (steric repulsion
of methyls in the gauche plus gauche minus conformer) and helical conformer.
