Massimo
Bagnani
a,
Paride
Azzari
a,
Cristiano
De Michele
b,
Mario
Arcari
a and
Raffaele
Mezzenga
*ac
aETH Zurich, Department of Health Sciences and Technology, Schmelzbergstrasse 9, LFO E23 Zurich 8092, Switzerland
b“Sapienza” Universita' di Roma, Dipartimento di Fisica, P.le A. Moro 2, 00185 Roma, Italy
cETH Zurich, Department of Materials, Wolfgang-Pauli-Strasse 10, Zurich 8093, Switzerland. E-mail: raffaele.mezzenga@hest.ethz.ch
First published on 15th December 2020
Biological liquid crystals, originating from the self-assembly of biological filamentous colloids, such as cellulose and amyloid fibrils, show a complex lyotropic behaviour that is extremely difficult to predict and characterize. Here we analyse the liquid crystalline phases of amyloid fibrils, and sulfated and carboxylated cellulose nanocrystals and measure their Frank-Oseen elastic constants K1, K2 and K3 by four different approaches. The first two approaches are based on the benchmark of the predictions of: (i) a scaling form and (ii) a variational form of the Frank-Oseen energy functional with the experimental critical volumes at order–order liquid crystalline transitions of the tactoids. The third and the fourth methods imply: (iii) the direct scaling equations of elastic constants and (iv) a molecular theory predicting the elastic constants from the experimentally accessible contour length distributions of the filamentous colloids. These three biological systems exhibit diverse liquid crystalline behaviour, governed by the distinct elastic constants characterizing each colloid. Differences and similarities among the three systems are highlighted and interpreted based on the present analysis, providing a general framework to study dispersed liquid crystalline systems.
When these filamentous colloids are dispersed at concentrations within the isotropic–nematic coexistence regime,10,11 nematic droplets, named tactoids, nucleate and grow in the surrounding continuous isotropic phase.12–17 The director field configuration and the shape assumed by these tactoids change with their increasing volume.9 It is a fascinating problem that has been the object of intense study in recent decades and still today carries forward some unclear aspects.15,18,19
The shape and director field configurations are modeled using a free energy approach in which a tactoid of defined volume V is characterized by a bulk free energy defined by the Frank-Oseen free energy functional. The three main and independent bulk contributions are the splay, twist and bend, proportional to the elastic constants K1, K2 and K3, respectively. To this, an anisotropic surface free energy must be added, defined by the Rapini and Papoular interfacial tension, dependent on the isotropic-nematic surface tension γ and a dimensionless anchoring strength ω. The total free energy can then be written as:
![]() | (1) |
Multiple studies have focused on the theoretical estimation32–35 of the elastic constants or on their experimental measurements; however, these works are mostly focused on only one single constant or on the ratio between them36–39 and often rely on external stimuli such as electric or magnetic fields.40–42 Due to the very sensitive conditions, experimental measurements of the elastic constants remain nontrivial and time consuming. Moreover, water-based biological CLCs such as cellulose nanocrystals (CNCs) and amyloid fibrils are weakly sensitive to electric and magnetic fields, and the measurement of the elastic constant for these systems is extremely elusive.43–45
In this work, we shed light on the liquid crystalline behavior of three biological colloids: β-lactoglobulin amyloid fibrils (BLGs), sulfated cellulose nanocrystals (SCNCs) and carboxylated cellulose nanocrystals (CCNCs) and use four different ways to determine their relative Frank elastic constants K1, K2 and K3. A first approach consists in extracting these elastic constants from the transition volumes of the tactoids from one configuration to the other, during their growth. The predictions of the constants use scaling forms of the Frank-Oseen energy functional of eqn (1). A second approach consists in a more accurate prediction using a variational method to describe the effect of the volume in the Frank-Oseen energy functional. To support the experimental finding, we estimate with a third approach the elastic constants from the contour length distribution of the rod-like filaments via computational molecular theory and lastly, we compute the elastic constants relying on existing scaling theories but including electrostatic effects.
By combining these results together, we show that these filamentous rod-like colloidal particles have different elastic constants, reflected by the distinctive liquid crystalline behavior of each class of colloid.
The values of ω and q∞ come from the experimental values. The ratios are chosen as the best fits for the transition volumes observed experimentally between each phase. The value of
comes from the transition into uniaxial cholesteric; while
is estimated from the transition into radial configurations. The errors in the estimates come from the uncertainties in the transition volumes. Since the splay energy is not directly influencing a particular geometry, this quantity has the greatest uncertainty. The ratio
comes from tuning the homogenous to bipolar transition. This procedure and the comparison with the experimental data enable us to carefully tune each single constant of the free energy; the different classes of tactoids are then represented graphically based on the results of the minimization, using the extended Jones matrix to reproduce the PolScope results.46
In agreement with previous work,46 amyloid-based tactoids show, at increasing volumes, four different equilibrium director field configurations, namely homogenous, bipolar, uniaxial and radial cholesteric and therefore undergo three director field transitions (see Fig. 1) and the critical volumes at which these transitions occur provide insights on the elastic properties of the crystalline phases. In particular, scaling laws for these transitions have already been proposed previously9,46 and the homogeneous to bipolar phase transition,55 observed at volumes of 4.95 ± 0.35 × 102 μm3, occurs when the droplet volume reaches the critical value of while the bipolar to uniaxial cholesteric phase transition, observed at 3.7 ± 2.7 × 104 μm3, occurs when the droplet reaches a volume of
Unfortunately, scaling arguments fail to predict the exact values of the critical volumes for the uniaxial to radial cholesteric transition, due to the fact that the latter configuration always exists with defects at the core of the droplets due to radial configuration with spherical confinement.46 However, as we proposed previously,46 the dimension of the core in the radial cholesteric droplets, which has been found in this case to be equal to ∼12 μm, is independent of the volume and its radius can be expressed as
providing valuable information on the ratio
In particular, by using the measured pitch p∞ = 30 μm = 2π/q∞ we have a first estimate of
The variational theory46 applied for this system, results in structures shown in Fig. 1 with The aspect ratio of homogenous tactoids is smaller than the observed one. This comes from the choice of using prolate ellipsoids in the theory, which have a lower aspect ratio compared to a spindle-like shape of the same volume. The bipolar regime appears at larger volumes compared to experimental data, while the transitions into uniaxial and radial cholesteric are correctly predicted.
As reported in a previous study,8 the tactoids formed by SCNC liquid crystals show only two main director field configurations: the homogeneous and the uniaxial cholesteric (see Fig. 2). In this system, the homogeneous tactoids have aspect ratios of around 1.5, the bipolar configuration is missing and the radial configuration is not achieved even for very large droplet sizes (see Fig. 2(C)), suggesting a higher bending constant compared to amyloids.
In this system, the droplets undergo only one transition, which is from homogeneous to uniaxial cholesteric configuration, experimentally observed at volumes of V = 8 ± 6 × 103 μm3 (see Fig. 2). In this case, similarly to the bipolar to cholesteric transition, the transition happens when the energy of the homogeneous untwisted tactoid matches the energy of the anchoring term of the uniaxial cholesteric, γωαhomo−2V2/3 + K2(q∞)2V ≈ γωαuni−2V2/3, suggesting that this transition happens at volumes considering that during this transition the aspect ratio discontinuously changes from 1.5 to ∼1 (see Fig. 2).
In SCNC, the radial cholesteric configuration has not been experimentally observed, and this configuration is expected to appear only at very large droplet volumes, larger than the inner diameter of the couvette used (200 μm). As expected, in some cases, few droplets with diameters similar to the couvette height start to bend (see the ESI†) suggesting that the transition may indeed happen at larger volumes; the contribution of the bend term becomes therefore important for droplets of diameter ∼200 μm, corresponding to spherical droplet volumes ≥106 μm3 and so we expect the bend term to be in the order of
Similar to the amyloid case, we can verify this result by analyzing the core of the droplets, which is assumed to be undergoing radial cholesteric configuration (see the ESI†), equal in this case to ∼20 μm, which together with the measured pitch, p∞= 18 μm = 2π/q∞ allows us to show that the ratio is
In this case, the variational theory agrees with the experimental evidence and the best fit is found for ratios
(see Fig. 2). The two morphologies are correctly predicted. The aspect ratio of the homogeneous is lower for the use of ellipsoids, and higher in the uniaxial cholesteric, since α = 1.1 is the lowest, non-unitary aspect ratio used.
As already shown by Nyström et al.,64 the liquid crystalline behavior of CCNCs is characterized by the formation of classes of tactoids, which are different from the previous systems. In particular, detailed PolScope analysis allowed discriminating three main director field configurations at increasing volumes: homogeneous, radial nematic and uniaxial cholesteric, see Fig. 3.
The transition volume from homogeneous to radial nematic is experimentally identified at 4.5 ± 0.5 × 103 μm3, and is expected to take place when similar to the homogeneous to bipolar transition, since we can assume that the energy of a radial nematic droplet is the energy of a bipolar droplet with unitary aspect ratio. The second transition experienced by the tactoids in this system, the radial nematic to uniaxial cholesteric, is experimentally found for droplet volumes of 3.7 ± 2.7 × 104 μm3 and, considering the aspect ratio equal to one in both configurations is expected to take place when K2q∞2V + K3V1/3 = γωV2/3. However, also in the radial nematic configuration, a defect at the core of these droplets is always found due to spherical confinement with parallel anchoring, and thus the scaling might not be able to predict the exactly critical volume for this transition (see later for further discussion). For this reason, we can find a first estimation of K2 assuming a homogeneous to uniaxial cholesteric transition that, similar to SCNC, is expected at
and by setting this critical volume at 104 μm3 and
results in
The variational theory applied for this system is shown in Fig. 3 using
In this case, homogenous, radial nematic and uniaxial cholesteric aspect ratios and transition volumes are correctly predicted. However, the variational theory shows a bipolar regime in between the homogenous and radial morphologies, with a pseudo-continuous decrease of the aspect ratio, which has not been found experimentally. This is due to the fact that the variational theory is unable to completely predict this structure that must include a defect at the core caused by the spherical confinement combined with planar anchoring.
The length distribution of the rod-like molecules employed in this study is shown as log-normal distribution in Fig. 4. Additionally, in the case of amyloid fibrils and SCNCs, the distribution was extracted also from the isotropic and nematic phases separately, once the systems achieved macroscopic phase separation through sedimentation of the anisotropic particles (see Fig. 4 and the ESI,† for raw data distribution), allowing a more specific characterization of the composition of the anisotropic phase.
As expected theoretically65,66 and as reported previously,10,63 the fibril length distribution of the different phases clearly shows that the nematic phase is composed of more polydispersed fibril lengths than the isotropic counterpart (see Fig. 4(D) and (E)). Specifically, the nematic phase is composed of fibrils that are longer compared to the ones of the isotropic phase since, due to their higher aspect ratio, they undergo the isotropic–nematic transition earlier. In particular, the average length and height (corresponding to fibril diameter D) measured for amyloid fibrils are 〈L〉 =322 ± 134 nm and 〈D〉 =3.36 ± 0.78 nm in the isotropic phase (n = 417), while they are 〈L〉 =652 ± 400 nm and 〈D〉 =3.75 ± 0.83 nm in the nematic phase (n = 486), corresponding to aspect ratios (L/D) equal to 96 and 173 in the two phases, respectively. For SCNCs we found 〈L〉 =212 ± 115 nm in the isotropic (n = 429) and 〈L〉 =325 ± 168 nm in the nematic (n = 426) phase with a negligible difference in average height of 〈D〉 =4.60 ± 1.03 nm and 〈D〉 =4.58 ± 1.00 nm in the isotropic and nematic phase, corresponding to fibril aspect ratios equal to 46 and 71, respectively. In the case of CCNCs, the analysis of the compositions of rods in the two distinct phases was not possible since this system never reached macroscopic phase separation, probably due to the high viscosity of the solution or for smaller difference in concentration between the two phases and therefore an extremely slow sedimentation of tactoids. For CCNCs we therefore analyzed the mixture of the two phases (n = 400) finding 〈L〉 =464 ± 281 nm and 〈D〉 =3.76 ± 1.17 nm and corresponding to an aspect ratio of 123.
Moreover, for the amyloids, we measured a concentration of 1.9% for the isotropic and 2.8 wt% for the nematic phase, while for SCNCs we measured 2.3 wt% for the isotropic and 2.7 wt% for the nematic phase. The critical concentrations of the isotropic ϕI and nematic ϕN phases predicted using the classic Onsager theory, and considering the charges on the particles, can be estimated using and
corresponding to 2.2 wt% and 2.9 wt% for BLGs and to 2.4 wt% and 3.3 wt% for SCNCs, in good agreement with the concentrations measured experimentally, considering the polydispersity of the systems66,67 (see the length distribution data in the ESI,† and the discussion on the effective diameter below).
Under the experimental conditions reported, the dispersed rods are charged and the effect of electrostatic interactions on the diameter of the rods must be taken into account. The effective diameter proposed by Onsager is Deff = D + k−1(lnA + C + ln
2 − 1/2), where D is the diameter of the rod, k−1 is the Debye length, C is Euler's constant (C = 0.577) and A, for a weakly charged rod in the anisotropic phase, is given by eqn (5.1) in ref. 32. Under the present conditions (see the ESI†), we found that the effective diameters for the three systems become 5.54, 15.7 and 20.0 nm for amyloids, SCNCs and CCNCs, respectively. This finding highlights the strong effect of electrostatic interactions, especially for the two cellulose systems, where the diameter of the fibrils increases drastically.
In the theoretical calculations, the diameter of the hard cylinders is assumed to be equal to the effective diameter previously discussed, in order to account for electrostatic repulsion between fibrils. We account for electrostatic effects only through an effective diameter of fibrils, thus neglecting the so-called twisting effect (see ref. 32 for more details on this) as well as possible counter-ion condensation effects. In a forthcoming publication we will study a model of fibrils where charges are explicitly considered to accurately evaluate their role played in the estimate of elastic constants.
The elastic constants predicted by this model are plotted as functions of the order parameters S in Fig. 4(G)–(I) for BLGs, SCNCs and CCNCs, respectively.
BLGs | K 1 (pN) | K 2 (pN) | K 3 (pN) |
---|---|---|---|
Odijk (D2/Deff) | 5.1 ± 3.9 | 4 ± 0.8 | 15 ± 2 |
Tactoids | 1.6 ± 1.1 | 0.3 ± 0.2 | 5 ± 3 |
Var. th. | 2.0 ± 1.4 | 2.7 ± 0.7 | 8.5 ± 1.7 |
Mol. th. (S) | 0.8 ± 0.1 | 0.26 ± 0.04 | 1.6 ± 0.1 |
Mol. th. (Lp/L) | 0.4 ± 0.2 | 0.13 ± 0.07 | 1.1 ± 0.3 |
SCNCs | K 1 (pN) | K 2 (pN) | K 3 (pN) |
---|---|---|---|
Odijk (D2/Deff) | 1.4 ± 0.8 | 3.9 ± 0.3 | 15 ± 1 |
Tactoids | 0.09 ± 0.04 | 0.02 ± 0.01 | 0.9 ± 0.4 |
Var. th. | 2.6 ± 2.1 | 0.02 ± 0.01 | 2.3 ± 0.4 |
Mol. th. (S) | 0.2 ± 0.1 | 0.07 ± 0.04 | 1.7 ± 0.7 |
Mol. th. (Lp/L) | 0.2 ± 0.1 | 0.06 ± 0.04 | 1.4 ± 0.7 |
CCNCs | K 1 (pN) | K 2 (pN) | K 3 (pN) |
---|---|---|---|
Odijk (D2/Deff) | 0.9 ± 0.6 | 0.7 ± 0.06 | 2.7 ± 0.2 |
Tactoids | 0.7 ± 0.5 | 0.039 ± 0.004 | 2 ± 1 |
Var. th. | 0.6 ± 0.4 | 0.10 ± 0.01 | 1.0 ± 0.1 |
Mol. th. (S) | 0.10 ± 0.02 | 0.031 ± 0.008 | 0.20 ± 0.02 |
Mol. th. (Lp/L) | 0.05 ± 0.03 | 0.017 ± 0.012 | 0.15 ± 0.05 |
![]() | ||
Fig. 5 Elastic constants and anchoring strength for the three colloidal systems investigated. The elastic constants for BLGs, CCNCs and SCNCs are plotted in panels (A), (B) and (C), respectively. The elastic constants estimated from the theoretical scaling of Odijk are plotted as blue dots, from the tactoids director field transitions in grey dots, from the variational theory in yellow dots and the molecular theoretical prediction in orange dots. In panels (D), (E) and (F) the ratio K3/K1 obtained from the molecular theory are plotted as functions of the order parameter S values for BLGs, CCNCs, and SCNCs, respectively. The green arrows indicate the order parameters measured (see the ESI,† Fig. S7), and the corresponding K3/K1 values; the black arrows indicate the predicted order parameter S values, which are extracted by the imposed (experimentally accessible) Lp/L = K3/K1. Panel (G) contains examples of characteristic homogeneous tactoids, where the tip angle is shown, and the average anchoring value indicated, for BLGs in blue, CCNCs in yellow and SCNCs in red. The scale bar is equal to 10 μm. |
The previously discussed molecular theory predicts the elastic behavior of the anisotropic phases from the rod length distribution and effective fibril diameter as function of the order parameter S and so, this variable is needed to predict the elastic constant values. The order parameters estimated in the three systems through the analysis of birefringence (see the ESI†) are equal to 0.55 ± 0.03, 0.58 ± 0.06 and 0.89 ± 0.03 for BLGs, CCNCs and SCNCs, respectively (green arrows in Fig. 5(D)–(F)). The high value of S for the SCNCs might be explained by the fact that these fibrils have the lowest values of L/Lp ratio among the three systems studied in the present work and it is known that the order parameter S increases on decreasing L/Lp for semiflexible polymers as discussed in ref. 72. Since the molecular theory gives elastic constants and their ratios, such as as functions of the order parameters, it is also possible to back-extract an estimated order parameter S, with the assumption K3/K1 = Lp/L and compare it with the one measured by birefringence. As shown by the black arrows in Fig. 5 panels (D)–(F), by fixing K3/K1 to Lp/L, we obtain order parameters equal to 0.67 ± 0.13, 0.67 ± 0.13 and 0.92 ± 0.12, for BLGs, CCNCs and SCNCs, respectively, in good agreement with the S values estimated with the birefringence method. The values of order parameter and of the ratio K3/K1 estimated are in the range expected for rod-like lyotropic liquid crystals and in agreement with previous experimental18,55,73,74 and computational75 works. The three elastic constants predicted by this molecular theory and corresponding to the order parameters estimated by the birefringence measurements are therefore plotted for the three systems in Fig. 5 (orange dots in panels (A)–(C)).
The interfacial tension of hard rod colloids in isotropic–nematic suspensions has been studied by various approaches, all leading to the universal law76–79 The numerical prefactor b varies from 0.18 to 0.3280 and is fixed here as 0.3 for the estimation of γ. Using the average length of the fibrils and the Deff values estimated before we find interfacial tension values of 0.34 × 10−6 N m−1 for BLGs, of 0.24 × 10−6 N m−1 for SCNCs and 0.13 × 10−6 N m−1 for CCNCs.
Following the Wulff model,21 the adimensional anchoring strength ω value can be estimated by analyzing the morphology of homogeneous tactoids, where indeed the twist bend and splay contributions are zero and the free energy is composed of the surface energy only; in the case of chiral systems, this approach neglects the energy contribution associated with the unrelaxed twist of the rods, however when the length scale of the volume is much smaller than the pitch of the cholesteric, the twist energy contribution can be disregarded. As a result, exclusively for homogeneous tactoids, tip angle θ and the aspect ratio α (with where R and r are the major and minor axes of the tactoid, respectively,) are a function of the anchoring only, where (ω = 1 + (tan
θ))−2 and ω = (α/2)2 for ω > 1 and ω = α − 1 for ω ≤ 1. By measuring the tip angles in homogeneous tactoids, we obtained an average angle of 126 ± 6° for amyloids, 176 ± 6° for SCNCs and 156 ± 4° for CCNCs, corresponding to anchoring values of 1.26 ± 0.06 for amyloid, 1.00 ± 0.07 for SCNC and 1.04 ± 0.02 for CCNC. In agreement with a previous study, amyloids are characterized by the highest value of anchoring strength11 and the homogeneous tactoids formed by this system have a more elongated shape and sharper tips when compared, for example, with the ones composed by CNCs (see Fig. 5(G)). At the same time, the homogeneous tactoids formed by CCNC show smaller tip angles and more prolate shapes compared with SCNC, suggesting that SCNC leads to an even weaker anchoring strength. The expression relating the anchoring to the tip angle lacks precision for tip angle values close to 180° and therefore, for a more precise measurement of the anchoring, we analyzed the aspect ratios of homogeneous tactoids of all three systems. Measuring the average aspect ratio of homogeneous tactoids in the three systems (see also the plots in Fig. 1–3) allowed estimating anchoring strength values of 1.35 ± 0.28, 0.53 ± 0.04 and 0.83 ± 0.10 for BLG, SCNC and CCNC, respectively, confirming the trends based on the tip angle estimation. Interestingly, in the sample of SCNC, characterized by the weaker anchoring, homogeneous tactoids appear to be prolate spheroids and not spindle-like as typically observed experimentally in hard-rod systems. The anchoring values measured are in the range predicted theoretically from several works on hard rods and worm-like chains79 and smaller but in the same order of magnitude reported in the literature for carbon nanotubes.24,81 Interestingly, when comparing the length of the mesogens with the anchoring strength measured, we observe that the anchoring is a function of the rod length also when comparing different systems, extending the validity of our previous study based only on amyloid fibrils of different lengths.11
The values of surface tension γ and anchoring strength ω just calculated can now be used to compute the elastic constants from the scaling on the critical volumes at which the director field transitions are observed experimentally and from the variational theory discussed before (plotted in Fig. 5A–C as orange and grey dots, respectively).
For BLG in particular, using γ = 0.34 × 10−6 N m−1 and ω = 1.35 and with the assumption we can compute K3 and K1, from the critical volume of the homogeneous to bipolar transition using
and K2 from the bipolar to uniaxial cholesteric transition using
In particular, from these scaling arguments we obtain K1 = 1.6 ± 1.1 pN K2 = 0.3 ± 0.2 pN and K3 = 5 ± 3 pN and from the variational theory K1 = 2.0 ± 1.4 pN K2 = 2.7 ± 0.7 pN and K3= 8.5 ± 1.7 pN.
For SCNC, using γ = 0.24 × 10−6 N m−1 and ω = 0.53, the twist elastic constant K2 can also be readily computed from the critical volume observed experimentally for the homogeneous to uniaxial cholesteric transition using and it results in K2 = 0.02 ± 0.01 pN. The elastic constant K3, can be calculated in two different approaches, both based on the analysis of the droplets undergoing uniaxial to radial cholesteric transition; as discussed before, (I) the dimension of the core in radial cholesteric depends on the ratio
in this case ≈48, allowing one to estimate K3 = 0.96 ± 0.48 pN from K2, but additionally, (II) the critical volume at which the bend term starts to dominate, in this case suggesting that
allows one to compute that K3 ≈ 12.7 pN. The elastic constant K1 becomes available from the ratio K3/K1 = Lp/L ∼ 10 ± 5 for this system, resulting in 0.09 ± 0.04 pN ≤ K1 ≤ 1.2 ± 0.6 pN. The two approaches for the estimation of K3 result in values that differ by one order of magnitude, but we consider the first method to be more accurate and to have the best agreement with the variational theory that results in K1 = 2.6 ± 2.1 pN, K2 = 0.02 ± 0.01 pN and K3= 2.3 ± 0.4 pN.
In the case of CCNC, with the values γ = 0.13 × 10−6 N m−1 and ω = 0.83, using the assumption we can extract K3 and K1 from the critical volume for the homogeneous to bipolar transition using
and K2 from the radial nematic to uniaxial cholesteric, using K2q∞2V + K3V1/3 = γωV2/3, resulting in K1 = 0.7 ± 0.5 pN K2 = 0.03 ± 0.02 pN and K3 = 2 ± 1 pN. Alternatively, K2 can be computed by assuming the homogeneous to uniaxial cholesteric transition at intermediate volumes of 1.7 ± 1.3 × 104 μm3 (see Fig. 1), expected when
resulting in K2 = 0.039 ± 0.004 pN, in good agreement with the scaling expression on the radial to uniaxial transition, suggesting that the defect energy cost in the radial nematic configuration is, in this case, almost negligible. The variational theory, for CCNC results in K1 = 0.6 ± 0.4 pN K2 = 0.10 ± 0.01 pN and K3 = 1.0 ± 0.1 pN.
The large difference between SCNC (around 10) and CCNC (close to 1) in the ratio K3/K1 can be understood in terms of the difference between the nematic order parameter S values in these two systems. According to the molecular theory used in the present work, elastic constants depend on the effective stiffness of the semiflexible fibrils as quantified by the deflection length11,33 (see the discussion on L0 in the ESI†). As a consequence, even if two fibrils possess similar stiffness (Lp), their deflection length can vary a lot on changing α, thus inducing a large variation of K3/K1 according to the results of the molecular theory shown in panels D, E and F of Fig. 5.
Concerning the differences between the estimates of elastic constants obtained by the various approaches used in the present study, we note that the molecular theory overall underestimates the elastic constants when compared to the other methods, and this may depend on the simplified treatment of (i) bending fluctuations, which generally ease elastic deformations by inducing an effective reduced flexibility of the fibrils, and of (ii) electrostatic interactions, which play in these systems a significant role due to the largely unscreened charge of the fibrils. Indeed, (i) bending fluctuations enter the theoretical calculations through a deflection length, which is assumed to be inversely proportional to the degree of orientational order as quantified by the Onsager parameter αN (see the ESI†) and (ii) electrostatic interactions are crudely accounted for through the effective diameters of fibrils.
The variational theory, on the other hand, seems to display a higher bending constant, probably due to the assumptions used for parametrizing the nematic field, while K1 and K2 are quite close to the scaling argument on the critical volumes, since both approaches use the critical volume of transition between classes for tactoids for estimating the constants. The variational theory results are extremely useful in predicting the transition volumes, ultimately needed for the estimation of the elastic constants. However, since the shape of the tactoids used for the calculations is ellipsoidal and does not consider the tip angle of spindle-like tactoids, the variational theory necessarily presents deviations in predicting the correct droplet aspect ratios, especially when the anchoring strength is high, resulting in elongated droplets with narrow tip angles. As an example, the BLG system is characterized by the highest anchoring and we observed homogeneous tactoids with an average aspect ratio α of 2.3 and bipolar tactoids with average α of 1.7, while the variational theory results in average α values of 2.1 and 1.9 for homogeneous and bipolar tactoids, respectively, corresponding to a difference of ∼10%. In uniaxial and radial cholesteric tactoids, where the tip angles are absent, we measured average α values of 1.2 and 1.0, for uniaxial and radial cholesteric, respectively, in good agreement with the variational theory predicting α value of 1.1 and 1.0. In SCNC, the system characterized by the lowest anchoring, the average α values measured in homogeneous and in uniaxial cholesteric tactoids are equal to 1.4 and to 1.05, respectively, and the variational theory can correctly predict their aspect ratios equal to 1.4 for homogeneous and 1.1, respectively.
Lastly, the Odijk scaling approach displays for cellulose a higher twist constant compared to other estimates, possibly because of the polydispersity of the fibrils, which is not taken into account in the scaling argument. For the three systems, we also notice that the twist is generally weaker than the other elasticities, while the ratio between bend and splay depends on the flexibility of the fibrils, as generally accepted. In fact, when the fibrils behave as rigid rods the bend elasticity is found to be much greater than the splay elasticity, especially compared to systems in which the fibrils behave as semiflexible wormlike chains, where the two contributions may become comparable. Granted that different systems are characterized by similar elastic constants (or the ratio between them), their liquid crystalline behavior might strongly vary due to differences in anchoring strength, which is found to decrease with fibril length. Therefore, it is ultimately the ratio between the elastic constants and the anchoring strength (also referred to as extrapolation length, i.e.), which plays a pivotal role in the configuration of the nematic field: an illustrative sample is offered by the SCNC system, where the very high extrapolation length compared to the other systems penalizes bending of the director, and thus a radial orientation of the nematic field, imposing absolutely straight cholesteric bands within a perfectly uniaxial chiral nematic field.
Additionally, omitting of the saddle-splay contribution might result in an additional error in the estimation based on the experimental analysis and the variational theory.
The results show that the liquid crystalline structures originating from these chiral biological filaments with similar physical properties, present striking differences, reflected by distinct values of anchoring strength, elastic constants, and surface tension and provide a general framework to understand the morphologies in confined liquid crystalline systems.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0sm01886d |
This journal is © The Royal Society of Chemistry 2021 |