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Particle-Stabilized Defect Gel in Cholesteric Liquid Crystals

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Science  08 Jan 1999:
Vol. 283, Issue 5399, pp. 209-212
DOI: 10.1126/science.283.5399.209

Abstract

Dispersions of colloidal particles in cholesteric liquid crystals form an unusual solid by stabilizing a network of linear defects under tension in the ideal layered structure of the cholesteric. The large length scales of the cholesteric liquid crystals allowed direct observation of the network structure, and its properties were correlated with rheological measurements of elasticity. This system serves as a model for a class of solids formed when particles are mixed with layered materials such as thermotropic and lyotropic smectic liquid crystals and block copolymers.

Lamellar phases (1), whose equilibrium state consists of a periodic stack of two-dimensional planes, occur in many materials, including thermotropic liquid crystals (LCs), surfactant-water and lipid-water mixtures, surfactant-water-oil mixtures, and block copolymers. In many cases, control of their rheology—their distortion, flow, and sedimentation under stress—is crucial to product performance in industries such as food, cosmetics, and coatings (2). The addition of colloidal particles can modify the rheological properties of a variety of complex fluids (3). Often, however, an incomplete understanding of structural modifications induced by inclusions has limited our ability to control and tune these properties. We show that the addition of a small volume fraction of colloidal particles to lamellar systems can efficiently control their rheology, and we present a detailed analysis of the relation between inclusion-induced microstructural changes and rheology modification.

We used a nearly ideal model lamellar system whose layer spacing can be varied and made sufficiently large to allow easy real-time optical visualization of both the lamellae orientation and the structure of any defects induced by the addition of colloidal particles. We studied cholesteric LCs (1), which have a twisted nematic structure in which anisotropic molecules rotate in a helical manner to form lamellae of equally spaced planes with a common molecular orientation (Fig. 1A) (4). The pure cholesteric structure behaves as a fluid when sheared along the lamellae. The addition of colloidal particles to a cholesteric LC, however, stabilized a network of linear defects under tension that is responsible for a solidlike response to shear stress. Because we can correlate direct visualization of the defect network with rheological measurements, we can probe quantitatively the mechanisms of elastic response. The resultant defect-mediated gel provides a model for predicting the consequences of the addition of colloidal particles to other lamellar phases. Indeed, the stabilization of a network of defects similar to that we describe here has been observed (5) in a lyotropic smectic system. In mesophases such as SmA liquid crystals, block co-polymers, and lyotropicL α phases, the formation of the defect network is expected to result in higher shear moduli than that in cholesteric LCs because of the smaller spacing between lamellae of these mesophases.

Figure 1

(A) The planar texture configuration of a cholesteric liquid crystal. Structural modifications occur when colloidal inclusions of sizeR inc exceeding the pitch length p are added into the system. (B) Arrangement of cholesteric layers in the cross section of a symmetric “oily streak” defect. The layers undergo a 180° rotation around each of the two constituent disclinations (shown as full circles) in the center of the streak. (C) An asymmetric oily streak, equivalent (at large distances from the center of the streak) to a dislocation with Burgers vector magnitude b = 4.

The defect-driven enhancement of the elastic modulus of lamellar systems, such as the one studied here, is intrinsically different from that obtained when colloidal particles are dispersed in systems, such as polymer solutions and melts (3, 6), which have no long-range order and do not support defect structures. It is also different from the solidlike rigidity observed in macroscopically disordered samples of layered phases (7, 8), whose origin is bulk regions with layers unfavorably aligned relative to the shear direction (1, 9). This rigidity cannot survive shear-induced alignment of these randomly oriented regions; in contrast, colloidally stabilized defect networks and their solid-like elasticity are not destroyed by such shear alignment. Lamellar hydrogels (10) produced by the addition of a polymer surfactant to the lamellar phase of a phospholipid in water have a defect structure, which was imaged by freeze-fracture electron microscopy, similar to that reported here for colloidal particle in cholesterics. The mechanism for stabilization of their defect networks, however, relies on specific interactions between polymer and phospholipid, unlike the more general, purely physical mechanism reported here. Colloidal particles dispersed in a nematic LC gives rise to defects (11), but they have no effect on the bulk rheological properties, at least at small volume fractions.

To determine the effects of defects in the ideal cholesteric structure shown in Fig. 1A, we quenched from the isotropic (66°C) to the cholesteric (55°C) phase over several seconds. A large concentration of “oily streak” defects (Fig. 1, B and C) emerged at the phase transition temperature (63°C) and formed a dense network (Fig. 2, A to C). This network coarsened quickly; after 2 min, almost no defects were left in the sample with thicknessD film = 40 μm. The dominant coarsening mechanism is the disconnection of defect lines from the nodes of the network and their subsequent shrinking with a constant velocity (a retracting oily streak is visible in the upper left corner of Fig. 2C).

Figure 2

(Upper row) Coarsening of the oily streak network in a pure cholesteric film with p = 7 μm andD film = 40 μm [at times (A)t = 15, (B) t= 30 s, and (C) t = 75 s after the quench] and in the same film with added colloidal inclusions [(D) t = 1 min, (E)t = 4 min, and (F)t = 7 min after the quench]. Imaging was performed with crossed polarizers; bar, 250 μm.

Dramatically different behavior was observed when the system was first doped with colloidal particles. We used silica particles with a diameter of ∼1 μm. The particles formed clusters of typical diameter R inc ≃ 5 to 20 μm, which then dispersed randomly. In the regime p <R inc < D film (wherep is the pitch length), cholesteric order (and consequently the organization of the layers) was strongly perturbed by the presence of the inclusions, but the boundary condition at the sample surface was not affected. The initial state, right after the quench, was indistinguishable from that of a pure cholesteric—a dense network of defects formed in both cases. Initially, the defects coarsened at the same rate as in the pure cholesteric. However, after about 1 min, the rate of coarsening dramatically decreased; subsequently, the characteristic mesh size d net remained ≤200 μm for several hours. The colloidal inclusions were located preferentially at the nodes of the defect network (Fig. 2, D to F) and stabilized it. This stabilization appears to be quite general: It was unaffected by the replacement of silica particles with water droplets or by the modification of boundary conditions on the director at inclusion surfaces from parallel to perpendicular alignment. It is this stabilized network of oily streaks that provides the solidlike elasticity in our samples.

The simplest, symmetric oily streak (Fig. 1B) (12) is composed of two disclinations of equal sign, separated by two-layer spacings. This structure does not introduce a change in the number of layers—it is not a dislocation, and it is not topologically stable. It can terminate in the bulk and retract at one end, as is indeed seen during the process of coarsening (Fig. 2, C and E). In contrast, an asymmetric oily streak (Fig. 1C) is topologically equivalent to a dislocation with nonzero Burgers vector when viewed on scales larger than the thickness of the streak, s, defined as the distance between the two constituent disclinations. Asymmetric streaks are topologically stable and cannot terminate in the bulk.

To explain the dynamics of the defect structures, we treat the cholesteric as an incompressible layered mesophase, with a free-energy density given byEmbedded Image(1)where R 1 andR 2 are the local values of the principal radii of curvature of the layers and K andK̅ are, respectively, the mean and Gaussian bending rigidities of the layers. Here K = (3/16)K 33, where K 33 is the standard elastic constant for bend deformations of the director (13). The Gaussian bending energy term, which integrates to boundaries, must be taken into account because our system contains internal surfaces and focal-conic type structures. The value ofK̅ can be estimated (14,15) from optical observations of focal conic structures in our samples as K̅ = (1.5 ± 0.3)K. In the following, we take K̅ =K ≃ 10−6 dyne.

In pure cholesterics and in cholesterics with inclusions at early times in the coarsening process, most of the defects are not stabilized by being connected to the nodes. The rate of coarsening is limited by the velocity of shrinking v of a typical disconnected streak, which was always observed to be constant in time. The value ofv can be estimated theoretically as 4K/(γ1 s), where γ1 is the twist viscosity of the LC (16). Consistent with these properties of individual oily streaks, the total normalized optical density A(t) of defects (defined as the ratio of dark to total area in digitized photographs such as Fig. 2) in pure cholesteric films decayed linearly with time (Fig. 3). As shown in the inset of Fig. 3, the time constant t dec in A(t) = 1 − t/t dec grew approximately exponentially with the film thickness, suggesting that the rate of coarsening becomes unobservably slow in bulk samples (D film ≥ 1 mm).

Figure 3

The optical densityA(t) of defects at timet after the quench from isotropic to cholesteric phase in a sample of thickness D film = 23 μm (crosses: pure cholesteric; squares: cholesteric with inclusions). (Inset) The time constant t decin the linear regime A(t) = 1 −t/t dec, plotted as a function of D film.

The behavior in thin samples with colloidal inclusions was dramatically different; a sharp crossover (Fig. 3) between the linear decay of A(t) and a nearly time-independent regime occurred at the time tc when the typical spacing between the oily streaks became comparable to the average separation of inclusions. After this crossover time (t c ≃ 50 to 100 s in our samples), most nodes of the oily streak network contained colloidal inclusions. The rate of coarsening was then limited by the rate at which the oily streaks disconnect from the nodes of the network, controlled in turn by the energy barrier for disconnection E disc that reflects the presence of the inclusions at the nodes. As a result, the defect network was stabilized by the inclusions.

The following simplified picture may be used to provide a qualitative understanding of the nature of the interaction of colloidal inclusions with the oily streaks. An isolated inclusion is expected to be surrounded by a focal conic domain (17) that can be viewed as arising from a disclination ring lying on the surface of the inclusion. When the inclusion is located at the intersection of several symmetric oily streaks, parts of the disclination ring are eliminated. It is thus energetically favorable for the colloidal inclusions to be located at the nodes of the oily streak network. Disconnection of an oily streak from the inclusion must overcome an energy barrier with magnitude on the order of (π/4)Ks, where s is the thickness of the disconnecting streak. We consequently obtain the estimate E discKp ≃ 10−9 erg for p ≃ 10 μm, and dimensional analysis leads us to expect the estimateE discKp ≃K̅ p to be generally valid.E disc exceeds the thermal energykBT by at least four orders of magnitude, so no thermally induced disconnections are expected to occur. Rather, it is the distribution of deformation stresses in the surrounding region of the network that will act as the driving force for overcomingE disc. The structure of the network is, therefore, strongly history-dependent.

The particle-stabilized defect network should substantially modify the rheological properties of the cholesteric fluid. On appropriate time scales (longer than the characteristic director relaxation time τdir ≃ (γ1/K)(p/4π)2at the cholesteric Brillouin zone edge and shorter than the typical lifetime of the network nodes), the defect structure can be viewed as a cross-linked network of elastic bonds that exert forces determined by the line tension T(the free energy per unit length) of the defects. Consequently, we expect the material to exhibit gel-like rheological behavior. This material exists solely because of the defect network; thus, it is a defect-mediated solid (18). The solid-like elasticity was evidenced in optical observations of a large air bubble moving through a thin sample: The defect network underwent strong transverse compression in the vicinity of the moving bubble; once the bubble moved away, however, the network rebounded to its original configuration (19).

The elastic shear modulus at low frequencies,G 0′, of this network can be estimated theoretically with arguments analogous to those used in the theory of rubber elasticity (20). These suggest thatG 0′ ≃ T/d net 2, whered net is the average mesh size of the defect network. Taking T ≃ 10K ≃ 10−5dyne (14, 16) andd net = 100 μm, we obtainG 0′ ≃ 0.1 dyne/cm2, which corresponds to an extremely weak gel.

We compared the experimentally measured rheological properties of a pure cholesteric material with those of a cholesteric containing a 0.3% volume fraction of silica particles (21). Through strong preshearing (applying a constant shear of rate 10 s−1 for 500 s), we created well-aligned bulk samples whose morphology is similar to that of samples studied in optical observations discussed above and whose elasticity is dominated by the linear defect network (22).

The pure cholesteric exhibited a liquid-like response in a wide range of frequencies, as shown in Fig. 4. For ω ≥ 10 rad/s, we observed typical Maxwell-fluid behavior:G" (ω) = ωη with effective viscosity η = 43cp and G′(ω) = ω2ητ with relaxation time τ = 5.6 ms. The value of τ agrees well with the characteristic director relaxation time τdir = (γ1/K)(p/4π)2≃ 6.1 ms.

Figure 4

Loss modulus (G", squares) and storage modulus (G′, circles) in the presheared cholesteric samples. The empty and filled symbols correspond to a sample without and with inclusions, respectively.

In a cholesteric with added colloidal particles, the Maxwell-fluid behavior persisted for ω ≥ 50 rad/s. At lower frequencies, however, the storage modulus G′(ω) exhibited a pronounced increase (by a factor of 3 to 4) compared with the pure cholesteric case (Fig. 4). This reflects the elasticity of the oily streak defects present in the sample with colloidal particles. Below ω = 1 rad/s, we observed a plateau in the storage modulus G′(ω) of magnitudeG 0′ = 0.2 dyne/cm2. Furthermore, the curves for G′(ω) and G"(ω) cross at ω ≃ ωg = 0.5 rad/s, and to within experimental precision, G′(ω) >G"(ω) for ω < ωg. This provides a clear signature (23) of gel-like behavior and is consistent with our model of a particle-stabilized defect network. The value ofG 0′ is consistent with the prediction given above. In addition, ωg agrees reasonably well with the estimated frequency T/(ud net3πγ1 R inc) ≃ 2 rad/s above which the maximum velocity ωud net of inclusions under oscillatory shear of amplitude u = 0.05 exceeds the velocity T/(3πγ1 R inc) with which an oily streak can pull the inclusion (24).

For clarity, we contrast again the origins of solid-like elasticity in fully or partially disordered homogeneous lamellar systems with that in prealigned lamellar systems containing colloidal inclusions. In the latter system, regions where the layers are oriented at a nonzero angle with respect to shear are restricted to the vicinity of the colloidal inclusions and the centers of the oily streak defects. Such misaligned regions are widely separated by the well-aligned regions between the streaks; these well-aligned regions do not transmit elastic stress. This results in a highly inhomogeneous medium whose macroscopic solid-like elasticity arises from a connected network of oily streak defects that transmits elastic stresses only along the defect lines.

  • * Present address: Centre de Recherche Paul Pascal, Avenue A. Schweitzer, F-33600, Pessac, France.

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