Pseudonematic Domains Research Articles

Nematic liquid crystals blended ferroelectric nanoparticles (BaTiO3): A perspective way for improving the response time and photoluminescence for electro-optical devices

We represent a detailed study of a dilute suspension of ferroelectric barium titanate nanoparticles (BaTiO3 NPs) in a nematic matrix by probing dielectric, electro-optical, and optical properties. The strong local surface electric field (order of ∼1010 Vm−1) of BaTiO3 produces pseudonematic domains within the matrix, which renders an enhancement in both the perpendicular and parallel components of dielectric permittivity that increases with increasing NP concentrations. Due to changes in molecular ordering, the dielectric anisotropy of pristine nematic increases by 10.7%, 24.6%, and 33.3% after the addition of 0.1, 0.3, and 0.5 wt. % BaTiO3 NPs with subsequent lowering in threshold voltage. Interestingly, the nematic-BaTiO3 blended cells showed 23.2%, 49.3% and 67.6% faster electro-optic response for 0.1, 0.3, and 0.5 wt. % of NPs, respectively. In the nematic matrix, BaTiO3 NPs act like chromophores resulting in a bathochromic shift in fluorescence spectra due to an enhanced degree of conjugation. This investigation reveals that the ferroelectric BaTiO3 NPs-nematic composites will be suitable for superior fluorescent electro-optical devices.

Nature of the glass transition in 2D colloidal suspensions of short rods

The glass transition, a many-body system undergoes an apparent dynamic arrest with no appearance of long-range order, is one of the most challenging problems in condensed matter physics. Recently attention has been focused on the puzzling two-step transition observed in colloidal suspensions of ellipsoids. It was observed that micro nematic domains appearing with increasing packing density suggesting that such micro domains may be the structure origin of the two-step glass transition. Here we report an experimental study of monolayers of colloidal rods which exhibits a two-step glass transition with no appearance of pseudo-nematic domains. Instead, it was found that parallel and perpendicularly packed rods form local free energy minima in configurational space, separated by an activation barrier. This barrier increases significantly when rotational glass transition is approached, thereby the rotational motion is frozen while the translational one remains diffusive. We propose that the activation barrier for rotation is the key to the two-step glass transition in suspensions of rods. Such an activation barrier between well-defined local configurations holds the key to understanding the two-step glass transition in general.

Dielectric and Electro-Optic Effects in a Nematic Liquid Crystal Doped with h-BN Flakes

A small quantity of hexagonal boron nitride (h-BN) flakes is doped into a nematic liquid crystal (LC). The epitaxial interaction between the LC molecules and the h-BN flakes rising from the π−π electron stacking between the LC’s benzene rings and the h-BN’s honeycomb structure stabilizes pseudo-nematic domains surrounding the h-BN flakes. Electric field-dependent dielectric studies reveal that the LC-jacketed h-BN flakes follow the nematic director reorientation upon increasing the applied electric field. These anisotropic pseudo-nematic domains exist in the isotropic phase of the LC+h-BN system as well, and interact with the external electric field, giving rise to a nonzero dielectric anisotropy in the isotropic phase. Further investigations reveal that the presence of the h-BN flakes at a low concentration in the nematic LC enhances the elastic constants, reduces the rotation viscosity, and lowers the pre-tilt angle of the LC. However, the Fréedericksz threshold voltage stays mostly unaffected in the presence of the h-BN flakes. Additional studies show that the presence of the h-BN flakes enhances the effective polar anchoring strength in the cell. The enhanced polar anchoring strength and the reduced rotational viscosity result in faster electro-optic switching in the h-BN-doped LC cell.

Measuring order in disordered systems and disorder in ordered systems: Random matrix theory for isotropic and nematic liquid crystals and its perspective on pseudo-nematic domains.

Surprisingly long-ranged intermolecular correlations begin to appear in isotropic (orientationally disordered) phases of liquid crystal forming molecules when the temperature or density starts to close in on the boundary with the nematic (ordered) phase. Indeed, the presence of slowly relaxing, strongly orientationally correlated, sets of molecules under putatively disordered conditions ("pseudo-nematic domains") has been apparent for some time from light-scattering and optical-Kerr experiments. Still, a fully microscopic characterization of these domains has been lacking. We illustrate in this paper how pseudo-nematic domains can be studied in even relatively small computer simulations by looking for order-parameter tensor fluctuations much larger than one would expect from random matrix theory. To develop this idea, we show that random matrix theory offers an exact description of how the probability distribution for liquid-crystal order parameter tensors converges to its macroscopic-system limit. We then illustrate how domain properties can be inferred from finite-size-induced deviations from these random matrix predictions. A straightforward generalization of time-independent random matrix theory also allows us to prove that the analogous random matrix predictions for the time dependence of the order-parameter tensor are similarly exact in the macroscopic limit, and that relaxation behavior of the domains can be seen in the breakdown of the finite-size scaling required by that random-matrix theory.

Insulator-to-conductor transition in liquid crystal-carbon nanotube nanocomposites

A small quantity of carbon nanotubes (CNTs) was dispersed in a liquid crystal (LC) and the LC + CNT hybrid in the isotropic phase was found to exhibit an insulator-to-conductor transition when an external electric field was applied. This effect was probed by measuring the resistance of the system as a function of applied voltage across the LC cell. In an LC + CNT hybrid, the LC molecules self-assemble themselves at the CNT surface due to π-π electron stacking, creating pseudonematic domains (PNDs) surrounding the CNTs. These CNT-embedded PNDs interact with the external electric field even in the isotropic phase of the LC. When the external field is applied, the PND-encapsulated CNTs start to rotate along the field and form wires due to their natural tendency of entanglement. The CNT-wires eventually short the two electrodes of the LC cell, manifesting an insulator-to-conductor transition in the LC + CNT hybrid. Additional studies revealed that the cell spacing and the CNT-concentration had a significant impact on the threshold voltage across the LC cell for the insulator-to-conductor transition process.

Graphene and liquid crystal mediated interactions

ABSTRACTThe two-dimensional graphene-honeycomb structure can interact with the liquid crystal’s (LC) benzene rings through π–π electron stacking. This LC–graphene interaction gives rise to a number of interesting physical and optical phenomena in the LC. In this paper, we present a combination of a review and original research of the exploration of novel themes of LC ordering at the nanoscale graphene surface and its macroscopic effects on the LC’s nematic and smectic phases. We show that monolayer graphene films impose planar alignment on the LC, creating pseudo-nematic domains (PNDs) at the surface of graphene. In a graphene-nematic suspension, these PNDs enhance the orientational order parameter, exhibiting a giant enhancement in the dielectric anisotropy of the LC. These anisotropic domains interact with the external electric field, resulting in a non-zero dielectric anisotropy in the isotropic phase as well. We also show that graphene flakes in an LC reduce the free ion concentration in the nematic media by an ion-trapping process. The reduction of mobile ions in the LC is found to have subsequent impacts on the LC’s rotational viscosity, allowing the nematic director to respond quicker on switching the electric field on and off. In a ferroelectric LC (smectic-C* phase), suspended graphene flakes enhance the spontaneous polarisation by improving the tilted smectic-C* ordering resulting from the π–π electron stacking. This effect accelerates the ferroelectric-switching phenomenon. Graphene can possess strain chirality due to a soft shear mode. This surface chirality of graphene can be transmitted into LC molecules exhibiting two types of chiral signatures in the LCs: an electroclinic effect (a polar tilt of the LC director perpendicular to, and linear in, an applied electric field) in the smectic-A phase, and a macroscopic helical twist of the LC director in the nematic phase. Finally, we show that a graphene-based LC cell can be fabricated without using any aligning layers and ITO electrodes. Graphene itself can be used as the electrodes as well as the aligning layers, obtaining an electro-optic effect of the LC inside the cell.

The inherent dynamics of isotropic- and nematic-phase liquid crystals.

The geodesic (shortest) pathways through the potential energy landscape of a liquid can be thought of as defining what its dynamics would be if thermal noise were removed, revealing what we have called the "inherent dynamics" of the liquid. We show how these inherent paths can be located for a model liquid crystal former, showing, in the process, how the molecular mechanisms of translation and reorientation compare in the isotropic and nematic phases of these systems. These mechanisms turn out to favor the preservation of local orientational order even under macroscopically isotropic conditions (a finding consistent with the experimental observation of pseudonematic domains in these cases), but disfavor the maintenance of macroscopic orientational order, even in the nematic phase. While the most efficient nematic pathways that maintain nematic order are indeed shorter than those that do not, it is apparently difficult for the system to locate these paths, suggesting that molecular motion in liquid-crystal formers is dynamically frustrated, and reinforcing the sense that there are strong analogies between liquid crystals and supercooled liquids.

Nano-electromechanical rotation of graphene and giant enhancement in dielectric anisotropy in a liquid crystal

A nematic liquid crystal (LC) is doped with dilute concentrations of pristine monolayer graphene (GP) flakes, and the LC + GP hybrids are found to exhibit a dramatic increase in the dielectric anisotropy. Electric field-dependent conductance studies reveal that the graphene flakes follow the nematic director that mechanically rotates on increasing an applied electric field. Further studies show that the π–π electron stacking, between the graphene's honeycomb structure and the LC's benzene rings, stabilizes pseudo-nematic domains that collectively amplify the dielectric anisotropy by improving the orientational order parameter in the nematic phase. These anisotropic domains interact with the external electric field, resulting in a nonzero dielectric anisotropy in the isotropic phase as well. The enhancement in dielectric anisotropy, due to the LC–graphene coupling, is found to have subsequent positive impacts on the LC's orientational threshold field and elasticity that allows the nematic director to respond quicker on switching the electric field off.

New divergent dynamics in the isotropic to nematic phase transition of liquid crystals measured with 2D IR vibrational echo spectroscopy.

The isotropic phase of nematogenic liquid crystals has nanometer length scale domains with pseudonematic ordering. As the isotropic to nematic phase transition temperature (TNI) is approached from above, the orientational correlation length, ξ, of the pseudonematic domains grows as (T - T(*))(-1/2), where T(*) is 0.5-1 K below TNI. The orientational relaxation, which is a collective property of the pseudonematic domains, was measured with optical heterodyne detected-optical Kerr effect (OHD-OKE). The orientational relaxation obeys Landau-de Gennes theory, as has been shown previously. To examine the environmental evolution experienced by molecules in the pseudonematic domains, two-dimensional infrared (2D IR) vibrational echo experiments on the CN stretching mode of the non-perturbative vibrational probes 4-pentyl-4(')-selenocyanobiphenyl (5SeCB) and 4-pentyl-4(')-thiocyanobiphenyl (5SCB) in the nematogen 4-cyano-4(')-pentylbiphenyl (5CB) were performed. The 2D IR experiments measure spectral diffusion, which is caused by structural fluctuations that couple to the CN vibrational frequency. Temperature dependent studies were performed just above TNI, where the correlation length of pseudonematic domains is large and changing rapidly with temperature. These studies were compared to 2D IR experiments on 4-pentylbiphenyl (5B), a non-mesogenic liquid that is very similar in structure to 5CB. The time constants of spectral diffusion in 5CB and 5B are practically identical at temperatures ≥5 K above TNI. As the temperature is lowered, spectral diffusion in 5B slows gradually. However, the time constants for spectral diffusion in 5CB slow dramatically and diverge as T(*) is approached. This divergence has temperature dependence proportional to (T - T(*))(-1/2), precisely the same as seen for the correlation length of pseudonematic domains, but different from the observed orientational relaxation times, which are given by the Landau-de Gennes theory. The data and previous results show that spectral diffusion in 5CB has no contributions from orientational relaxation, and the structural dynamics responsible for the spectral diffusion are likely a result of density fluctuations. The results suggest that the correlation length of the density fluctuations is diverging with the same temperature dependence as the pseudonematic domain correlation length, ξ. The isotropic-nematic phase transition in liquid crystals is described in the context of the slowing of orientational relaxation associated with divergent growth of the orientational correlation length. The results presented here show that there is another divergent dynamical process, likely associated with density fluctuations.

Length Scales and Structural Dynamics in Nematogen Pseudonematic Domains Measured with 2D IR Vibrational Echoes and Optical Kerr Effect Experiments

Nematogen liquids in the isotropic phase are macroscopically homogeneous but on multinanometer length scales have pseudonematic domains with correlation lengths that grow as the isotropic to nematic phase transition temperature (TNI) is approached from above. Orientational relaxation of nematogens in the isotropic phase manifests as two fast power laws and a slow exponential decay when measured by optical heterodyne detected optical Kerr effect (OHD-OKE) experiments. The long time exponential relaxation is associated with complete randomization of pseudonematic domains. We examine the effect of local orientational correlation on spectral diffusion (structural evolution) experienced by a vibrational probe molecule within the pseudonematic domains of 4-cyano-4'-pentylbiphenyl (5CB) using two-dimensional infrared (2D IR) vibrational echo spectroscopy. The addition of low concentration 4-pentyl-4'-thiocyanobiphenyl (5SCB) as a long-lived vibrational probe to 5CB is shown to lower TNI of the sample slightly, but the fast power law dynamics and exponential decays observed by OHD-OKE spectroscopy are unchanged. We compare the complete orientational relaxation and spectral diffusion for samples of 5SCB in 5CB to 5SCB in 4-pentylbiphenyl (5B) at four temperatures above TNI. 5B has a molecular structure similar to 5CB but is not a nematogen. At all but the lowest temperature, the spectral diffusion in 5CB and 5B is described well as a triexponential decay with very similar time constants. The results demonstrate that the presence of local orientational order at temperatures well above TNI does not affect the spectral diffusion (structural evolution) within pseudonematic domains when the correlation lengths are short. However, when the temperature of the sample is held very close to TNI, the spectral diffusion in 5CB slows dramatically while that in 5B does not. It is only as the correlation length becomes very long that its presence impacts the spectral diffusion (structural fluctuations) sensed by the vibrational probes located in pseudonematic domains. The orientational relaxation is modeled with schematic mode coupling theory (MCT). Fitting with MCT provides density and orientational correlation functions. The density correlation decays are similar for 5B and 5CB, but the orientational correlation decays are much slower for 5CB. Additionally, the time dependence of the spectral diffusion in 5CB is strikingly similar to that of the density correlation function decay, while the orientational correlation function decay is far too slow to contribute to the spectral diffusion. Therefore, density fluctuations are likely the source of spectral diffusion at temperatures at least 5 K above TNI.

Soft memory in a ferroelectric nanoparticle-doped liquid crystal.

A small quantity of BaTiO(3) ferroelectric nanoparticles (FNP) was doped in a liquid crystal (LC), and the LC + FNP hybrid was found to exhibit a nonvolatile electromechanical memory effect in the isotropic phase. The permanent dipole moment of the FNPs causes the LC molecule to form short-range pseudonematic domains surrounding the FNPs. The FNP-induced short-range orders become more prominent in the isotropic phase when the global nematic order is absent. These short-range domains, being anisotropic in nature, interact with an external electric field, exhibiting a Fréedericksz-type transition. When the field is turned off, these domains stay oriented, showing a hysteresis effect due to the absence of any long-range order and restoring forces in the isotropic phase. The hysteresis graph for this memory effect shows a significant pretransitional behavior on approaching the nematic phase from the isotropic phase.

Dynamics in the Isotropic Phase of Nematogens Using 2D IR Vibrational Echo Measurements on Natural-Abundance 13CN and Extended Lifetime Probes

The long time scale orientational relaxation of nematogens in the isotropic phase is associated with the randomization of pseudonematic domains, which have a correlation length that grows as the isotropic-to-nematic phase transition temperature is approached from above. Here we begin to address the fast dynamics of the nematogen molecules within the domains using two-dimensional infrared (2D IR) vibrational echo experiments. The problems of performing ultrafast IR experiments in pure liquids are discussed, and solutions are presented. In addition, the issue of short vibrational lifetimes, which limit the ability of 2D IR experiments to examine dynamics over a wide range of times, is addressed. The experiments were performed on the nematogen 4-cyano-4'-pentylbiphenyl (5CB), with the CN stretch initially used as the vibrational probe. Although the CN stretch has a small transition dipole, because the sample is a pure liquid it is necessary to use an exceedingly thin sample to perform the experiments. The small sample volume leads to massive heating effects that distort the results. In addition, the high concentration in the pure liquid can result in vibrational excitation transfer that interferes with the measurements of structural dynamics, and the CN vibrational lifetime is very short (3.6 ps). These problems were overcome by performing the experiments on the natural-abundance (13)CN stretch (5(13)CB), which greatly reduced the absorbance, eliminating the heating problems; also, this stretch has a longer lifetime (7.9 ps). Experiments were also performed on benzonitrile, which showed that the heating problems associated with pure liquids are not unique to 5CB. Again, the problems were eliminated by conducting measurements on the (13)CN stretch, which has an even longer lifetime (20.2 ps) compared with the (12)CN stretch (5.6 ps). Finally, to extend the range of the dynamical measurements, 4-pentyl-4'-thiocyanobiphenyl (5SCB) was synthesized and studied as a dilute solute in 5CB. The CN stretch of 5SCB has a vibrational lifetime of 103 ps, which permits dynamical measurements to 200 ps, revealing the full range of fast structural dynamics in the isotropic phase of 5CB. It is shown that the 5SCB probe reports essentially the same dynamics as 5(13)CB on the short time scale that is observable with the 5(13)CB vibrational probe.

Non-linear dielectric behaviour of 4,4′-n-hexylcyanobiphenyl near the nematic–isotropic phase transition

The non-linear dielectric spectra of 4,4′-n-hexylcyanobiphenyl (6CB) were recorded in the frequency range 1–100 MHz in the presence of a strong static electric field E 0 = 1 × 107 V/m. The measurements were performed in the isotropic phase and the isotropic and nematic coexistence temperature region. Two absorption bands, related to intermolecular interactions (chemical bands), were observed in the non-linear dielectric relaxation spectra. The dynamic phenomena of the band which appears in the frequency region below 10 MHz in the vicinity of the phase transition between the isotropic and nematic phases were interpreted within the framework of the Landau–de Gennes theory for weakly first-order phase transitions. The second effect, with relaxation times 2–20 ns (about one order of magnitude lower, as in a process where the molecules are involved in pseudonematic domains), is expected to characterise the interaction processes of individual molecules.

Glass Transitions in Quasi-Two-Dimensional Suspensions of Colloidal Ellipsoids

We observed a two-step glass transition in monolayers of colloidal ellipsoids by video microscopy. The glass transition in the rotational degree of freedom was at a lower density than that in the translational degree of freedom. Between the two transitions, ellipsoids formed an orientational glass. Approaching the respective glass transitions, the rotational and translational fastest-moving particles in the supercooled liquid moved cooperatively and formed clusters with power-law size distributions. The mean cluster sizes diverge in power law as they approach the glass transitions. The clusters of translational and rotational fastest-moving ellipsoids formed mainly within pseudonematic domains and around the domain boundaries, respectively.

Raman Spectroscopic Studies of Liquid Phase Ordering and Dynamics for Solutions of ME6N Liquid Crystal: The Approach to Simple Molecule Behavior at High Dilutions

We have measured the Raman isotropic profiles of the ν(C≡N) band at 2235 cm(-1) for five solutions of ME6N (4-cyanophenyl-4'-hexylbenzoate) liquid crystal dissolved in CCl(4) in the range from x = 0.12 to 0.007 (x, mole fraction of ME6N) and then obtained the corresponding vibrational correlation functions, C(v)(t), by time Fourier transformation. The increase with dilution of the dephasing times τ(v) complies with the behavior of the nonmonotonic concentration dependence predicted by the fluctuation concentration model for this concentration range (x < 0.5). The interpretation of C(v)(t) within the Kubo stochastic theory which, being based on the assumption that the environmental modulation arises from a single relaxation process, e(-t/τ(c)), applies to simple liquids, has proved to be inadequate except for the most diluted solutions where the environmental correlation time τ(c) amounted to 1.16 ps. On the other hand, we have found that the vibrational correlation functions for these ME6N/CCl(4) solutions comply, in the whole concentration range, with the approach proposed by Rothschild, which, being based on the assumption that the environmental modulation is described by a stretched exponential decay e(-(t/τ(0))(α)), is more appropriate for the interpretation of the vibrational correlation function arising from a distribution of relaxation processes caused, as in the present case, by the persistence of pseudonematic domains. With dilution the dispersion parameter α and the average correlation time τ(0) progressively increases and decreases, respectively, and tend to converge to the values appropriate for simple liquids, as given by the Kubo theory, i.e., α to unity and τ(0) to approximately 1.16 ps. The relaxation time probability distribution h(τ) progressively shrinks and shifts down with dilution. The evolution of α and τ(0) parameters and, contextually, of h(τ) with dilution offers a complete picture of the way a complex liquid attains the condition of a simple one.

Orientational coupling enhancement in a carbon nanotube dispersed liquid crystal

We present a detailed study of a dilute suspension of carbon nanotubes (CNTs) in a pentylcyanobiphenyl (5CB) liquid crystal (LC) by probing the dielectric properties as a function of applied ac voltage and frequency. In principle, to minimize the elastic distortion in the nematic matrix, the monodispersed CNTs follow the nematic director without disturbing the director field significantly. A strong anchoring energy due to π-π electron stacking between LC-CNT molecules results in an increase in the dielectric anisotropy for the hybrid system, indicating a significant enhancement in the orientational order parameter. The frequency-dependent dielectric anisotropy for the composite system reveals the intrinsic frequency response of the LC-CNT anchoring mechanism. As a matter of consequence, the extracted value of splay elastic constant suggests that LC-CNT anchoring has an impact on the structural modification of the hybrid LC+CNT system. This strong anchoring energy stabilizes local pseudonematic domains, giving rise to a nonzero dielectric anisotropy in the isotropic phase that also shows an intrinsic frequency response.

Dielectric hysteresis, relaxation dynamics, and nonvolatile memory effect in carbon nanotube dispersed liquid crystal

Self-organizing nematic liquid crystals (LCs) impart their orientational order onto dispersed carbon nanotubes (CNTs) and obtain CNT-self-assembly on a macroscopic dimension. The nanotube-long axis, being coupled to the nematic director, enables orientational manipulation via the LC nematic reorientation. Electric-field-induced director rotation of a nematic LC+CNT system is of potential interest due to its possible application as a nanoelectromechanical system. Electric field and temperature dependence of dielectric properties of a LC+CNT composite system have been investigated to understand the principles governing CNT assembly mediated by the LC. In the LC+CNT nematic phase, the dielectric relaxation on removing the applied field follows a single-exponential decay, exhibiting a faster decay response than the pure LC above a threshold field. The observed dielectric behaviors on field cycling in the nematic phase for the composite indicates an electromechanical hysteresis effect of the director field due to the LC-CNT anchoring mechanism. Observations in the isotropic phase coherently combine to confirm the presence of anisotropic pseudonematic domains stabilized by the LC-CNT anchoring energy. These polarized domains maintain local directors and respond to external fields, but do not relax back to the original state on switching the field off, showing nonvolatile memory effect.

Nematic anchoring on carbon nanotubes

A dilute suspension of carbon nanotubes (CNTs) in a nematic liquid crystal (LC) does not disturb the LC director. Due to a strong LC-CNT anchoring energy and structural symmetry matching, CNT long axis follows the director field, possessing enhanced dielectric anisotropy of the LC media. This strong anchoring energy stabilizes local pseudonematic domains, resulting in nonzero dielectric anisotropy in the isotropic phase. These anisotropic domains respond to external electric fields and show intrinsic frequency response. The presence of these domains makes the isotropic phase electric field-responsive, giving rise to a large dielectric hysteresis effect.

Carbon nanotube dispersed liquid crystal: A nano electromechanical system

Electric field induced director orientation of a nematic liquid crystal (LC)+carbon nanotube (CNT) system reveals insights on switching behavior for this anisotropic composite. Once the field goes off, the LC+CNT system relaxes back to the original orientation through a mechanical rotation, revealing the intrinsic dynamics. LC molecules and CNTs cooperatively form local pseudonematic domains in the isotropic phase due to strong LC-CNT interactions. These field-responsive anisotropic domains do not relax back to the original orientation on switching of the field off, which could find potential applications in memory devices.

Dynamics of the Self-Assembling of Mesogenic Molecules in the Prenematic Region of Isotropic Liquid

The linear and nonlinear dielectric relaxation spectra, recorded in the isotropic phase of mesogenic 4-(trans-4'-n-hexylcyclohexyl)isothiocyanatobenzene (6CHBT), reveal an equilibrium between the molecules that are involved and non-involved in the pseudo-nematic domains, which are spontaneously formed in the prenematic region. A perturbation of the equilibrium by a static electric field of high strength (E0 approximately 10(7) V/m) is followed by a nonlinear dielectric relaxation process that, measured with the probing electric field E(omega) of small amplitude and variable high frequency (up to 100 MHz), reflects the rate of the domains formation. The fraction of the mesogenic molecules involved in the prenematic domains at different temperatures is estimated.