A simple and astonishing statement can be made about the formation of liquid crystal phases by bent-core molecules:chiralphases spontaneously grow fromachiralmolecules. This behavior is unique to bent-core mesogens and has fueled countless investigations into the behavior of bent-core phases, both theoretically and experimentally, required many to re-write the book on liquid crystal theories. The bent-core geometry reduces the symmetry otherwise present in calamitic (rod-like) molecules and induces a net polarization perpendicular to the molecular long axis, i.e. in the direction of the bend. This molecular polarity, paired with the bent-geometry, unfolds the range of bent-core phases; each with unique electrical, optical, and mechanical properties.
Of the eight unique phases, just two are known to possess just the right combination of layer-structure, polarity and phase chirality (a consequence of the polarization and bend) to allow stable fibers to form when the material is placed between two surfaces and strained along a single axis: the B2 and the B7 phases [3–5, 21–29]. Both the B2 and the B7 phases share a notable property: they both form a tilted smectic phase which seems to be vital to forming stable, free-standing filaments.
Bent-core phases which are not tilted, i.e. where the molecular long axis is parallel to the smectic layer normal, do not form stable fibers [3, 41]. The bent-core geometry, polar ordering, and tilt angle present in the B2 and B7 phases result in layer chirality and has been defined as
ˆ z×nˆ
|sinθ|=±bˆ (1.3)
where ˆnis the long axis of the molecule, ˆzthe layer normal,θis the angle between ˆnand ˆz, and ˆbis the bend direction which is parallel to the dipole moment for symmetric molecules [40].
It was the keen observation by J´akliet al.in 2000 that certain BLC mesogens (Figure 1.3) spontaneously formed helical structures within the bulk when confined between two glass slides, cooled slowly from the isotropic phase into the B2 or the B7 phase, and observed under cross polarizers [42] that led to the 2003 discovery that the same material – when heated to the same phases between two glass interfaces – would spontaneously form stable filaments when the two interfaces were moved apart to stretch the material [20]. Once it was established that these liquid crystalline fibers could be consistently pulled and studied, years of research quickly followed to understand everything from their layer structure and optical properties to their elastic and mechanical stability and potential activation response [3–5, 21–29].
Figure 1.3:The first bent-core compounds shown to form stable, free-standing fibers are shown along with their phase transitions. Only the B2 and B7 phases showed to form fibers. The nematic phase, N-1, did not.
Static Tension Investigation
In 2012, Moryset al.introduced a technique for measuring the static tension of a liquid crystal fiber formed by BLCs [4]. In their paper, they used a mesogen (Figure 1.4) which tended to form very thick fiber bundles – typically between 40 and 100µmin diameter compared to the 1 to 6µm thick single fibers studied before – composed of tightly packed single fibers. These fiber bundles were prepared in the same manner as other BLC fibers; material was placed between two glass rods inside a heating chamber with observation windows, the chamber was heated to a temperature exceeding the isotropic transition of the material and cooled into the smectic phase before the two glass rods were pulled apart to form a long, stable fiber. At this point, a small glass bead was lifted into the heating stage through a small hole in its bottom face. The glass bead was carefully brought into contact with the midpoint of the fiber where it adhered strongly due to capillary forces. The bead was then lowered until it detached from its delivery mechanism and hung freely from the fiber as schematically shown inFigure 1.5. The accompanying force balance was written as
Σ= F
sin(α1) +sin(α2) (1.4)
or, sinceα1=α2=α,
Σ= F
2sin(α) (1.5)
whereΣis the fiber’s tension – or restoring force – along its axis,αis the deflection angle of the fiber from the horizontal, andF=mgis the weight of the glass bead wheremis its mass4andgis the acceleration due to earth’s gravity.
Plotting this steady restoring force versus fiber perimeter revealed a quadratic increase as the fibers became thicker indicating a quadratic contribution to the restoring force instead of a linear one; this indicated that the restoring force was growing as the cross-sectional area and not as the circumference. Due to this result, it was understood that the bulk of the fibers – not only the surface tension – played a key role in their elastic response.
Figure 1.4:The mesogen used by Moryset al.in 2012 to measure the tension of bundles of BLC fibers
They noted that while pulling the initial fiber, its diameter was observed to remain constant indicating that material had to be flowing from the excess material at the boundaries to continuously form more fiber. This conclusion made sense for millimeter-long fibers being pulled from an initially fluid bridge with a length on the order of a few microns. The same assumption was then applied to the behavior of the fiber during bead hanging since no observable change in the fiber radius was observed during its deflection. The total tension, in terms of the surface and bulk contributions, was presented as
Σ=σP(a,b) +η πab (1.6)
whereσis the surface tension5,P(a,b)is the perimeter of the fiber bundle corrected for corrugation due to packing,a,bare the major and minor radii to account for fiber profiles deviating from the perfectly circular, andη represents the bulk contribution to the total fiber tension as a volume
4The mass of the glass bead was calculated using the known value of the density of glass, assuming a spherical shape and optically measuring its diameter.
5In this experiment, and many that followed, the temperature independent surface tension value of 0.02 N/m was used.
This value seems to have come from a 2010 paper by Baileyet al.[23]. Arguments against using this value are presented in theAnalysischapter
Figure 1.5:The force balance used by Morys et al.in their attempt to properly gain insight into the elastic nature of BLC fiber bundles as described inEquation 1.5. The red (checkerboard) sections indicate glass, while the yellow (plain) sections indicate the excess material left over on each glass rod from which each fiber was pulled.
energy density. They concluded that the bulk contribution to the fiber tension was coming from the energy needed to add new material to the bulk and that the thicker the fiber, the more total energy would be required to add material in proportion to its cross-sectional area. This, however, does not have to be true.
During bead hanging, the deflection angleα typically did not exceed 30◦6. Using a constant-volume approach with a Poisson ratio of 0.5, the radial strain (εr) of the fiber is related to its extensional strain (εext) by
εext= ∆L
L0 =−2∆R
R0 =−2εr (1.7)
Furthermore, after some algebraic and trigonometric manipulation, the deflection angleαcan be related toεext using the expression
εext=sec(α)−1. (1.8)
Forα=30◦,Equation 1.8yieldsεext≈0.15 andεr≈ −0.075. Therefore, in the most extreme case, a fiber bundle with an initial diameter of 40µmwould be compressed down to a diameter of 37µm upon hanging a glass bead. Since both interfaces holding the fiber were visible and fiber lengths were typically on the order of a millimeter, it can be assumed that the low magnification would not have revealed such radial compression (beyond reasonable error) giving the experimenter the
6Deduced from the images presented in the publication
illusion that the fiber diameter had remained constant.
This 2012 study by Moryset al.inspired the current work to focus on measuring the elastic properties of single fibers instead of fiber bundles to try to observe the limit where the surface effects and bulk energy effects converge. Similar bead hanging experiments were done by Kress in 2015 which showed that the assumed value of the surface tension given by Baileyet al.in 2010 which was used in Morys’ study was too large by about a factor of 2 [23, 43].
Dynamic Studies
Earlier, in 2005, the dynamic response of fiber bundles formed by the same mesogen later used by Moryset al.shown inFigure 1.4was examined by means of acoustic driving by Stannariuset al.at the O.v.G University in Magdeburg, Germany. [27]. Their study showed that these fibers responded similarly to rigid beams when driven by acoustic compression. Later, Petzoldet al.demonstrated another interested feature of bent-core fibers: they responded strongly to external electric fields.
Most recently, in 2016, the dynamic behavior of single bent-core fibers was probed by analyzing the kinematics of recoil after sudden rupture. Using the same mesogen as Kress in 2015 – a mesogen that consistently formed single B7 filaments – Saliliet al.showed that the fibers tended to recoil in a helical fashion when ruptured either thermally at one or or mechanical via impulse.
Albeit the many efforts to understand the material response of these complex fibers, the prior work was forced to implement indirect methods of studying their dynamic behavior. A more straight-forward experimental approach was necessary.