Effect Of Density Variation Research Articles

Spatial variations in the effective elastic thickness of the Indian Ocean lithosphere

The Indian Ocean lithosphere is a complex assemblage of large igneous provinces, seamounts, plateaus and ridges of different loading ages and tectono-thermal evolution. As a proxy for the strength of tectonic plates, effective elastic thickness (Te) illustrates the relationship between surface deformation and lithospheric rheology of the diverse provinces in response to long-term tectonic processes. Mapping the spatial variations in lithospheric rheology can aid in understanding the detailed tectono-thermal history of the Indian Ocean. In this paper, we perform an assessment of the spatial variation of Te for the Indian Ocean from the inversion of the real free-air admittance between free-air gravity anomalies and bathymetry corrected for the effect of density variations within sediments using a continuous wavelet spectral analysis. Incorporating the effect of sediments substantially reduces Te estimates and better corresponds with the tectonic units in the study region. The results show low overall Te over the Indian Ocean attributed to magmatism and temperature during a multistage opening process. We further demonstrate that temperature controls the strength of warm and young oceanic lithosphere, evidenced by the positive correlation between Te and geothermal proxies. Finally, moderately low Te values at the Southwest Indian Ridge suggest a relatively cold ultraslow lithosphere with sparse magmatism compared to typical mid-ocean ridges.

The effect of two main vortexes generator on heat transfer enhancement in mini rectangular ducts of airborne thermoelectric conversion system

To enhance the heat transfer process of supercritical pressure fluid in rectangular channels of the airborne thermoelectric conversion system, the combined effect of the Reynolds stress, properties variation and longitudinal fins on convective heat transfer process in ducts under supercritical pressure is investigated. Then, a method to induce the two main vortexes structure in rectangular ducts is proposed. Results show that, in smooth ducts, the two main vortexes structure is formed due to the effect of density variation. By comparing the flow and heat transfer process in the smooth duct and ridged duct with trapezoidal fins attached on the lower wall, it is found that, due to the effect of new vortexes induced by trapezoidal fins, the two main vortexes structure is inhibited, which makes local wall temperature increases about 50 K. Accordingly, the thermal performance enhancement factor of rectangular duct with trapezoidal fins is lower than 1. To enhance the two main vortexes structure, a method that using triangular fins to induce two main vortexes structure is proposed, and the duct with one triangular longitudinal fin attached on the upper wall is designed, which has better thermal performance than ducts with trapezoidal fins. Due to the effect of triangular fins, the dominant distance of two main vortexes structure increases by 416%. Accordingly, in ducts with triangular fins, local heat transfer coefficient is enhanced about 23.5%, and the maximum of thermal performance enhancement factor is about 1.29.

Effects of Density Variation on the Physical and Mechanical Properties of Empty Fruit Bunch Cement Board (EFBCB)

A cement board is a composite material mostly comprised natural fibre and cement. Cement board is mainly used in roofing, raised floors, dropped ceilings, prefabricated structures, office containers, and other building components. Fibres in cement composites from discarded palm oil fruit bunches have been used to increase the quality of construction materials. Therefore, the impact density of the natural fibre cement board is essential to enhance the physical and mechanical properties. However, research on untreated fibre at various densities has not been compressively discussed in previous studies. Therefore, this research used empty fruit bunch (EFB) fibre in manufacturing empty fruit bunch cement boards (EFBCB) with a cement-to-fibre ratio of 3:1 and thickness of 12 mm. Three target density variations, 1100 kg/m3, 1200 kg/m3 and 1300 kg/m3, were applied in this study to obtain their effect on physical and mechanical properties. The results revealed EFBCB sample with a target density of 1300 kg/m3 showed the most promising results. This sample’s average thickness is 12.38 mm after a 28-day curing period. Besides, at 1300 kg/m3 target density, EFBCB achieved the lowest thickness swelling (TS) value at 1.82%, highest internal bonding (IB) at 0.164 N/mm2, highest modulus of elasticity (MOE) and modulus of rupture (MOR) at 1398 N/mm2 and 3.51 N/mm2, respectively. Thus, increasing the cement board density and improving the physical and mechanical properties of EFBCB. This study demonstrates that EFB has the capacity to be one of the potential natural fibres for green building materials.

Effect of sintering time on phase, physical properties, microstructure and thermal conductivity of Y(Ba1-x La x )2Cu3O7-δ ceramics

This work investigates the effects of various sintering conditions on phase, physical properties microstructure and thermal conductivity of Y(Ba1-x La x )2Cu3O7-δ where x = 0, 0.05, 0.10, 0.15, 0.20 and 0.25 mol fraction. The sintering conditions involved the sintering temperature of 920 °C and sintering time of 6, 8, 10 and 12 h. It was found that pure phase could be obtained in the samples sintered for 6 h regardless of composition. However, small amount of BaCuO2 secondary phase was obtained at other sintering time. The relatively low density and significantly reduced grain size was obtained in the samples with high La content (e.g. x = 0.20 and 0.25) regardless of the sintering time. This showed that La addition had apparent effects on sintering ability and grain growth. Phase analysis indicated that the structure changed from orthorhombic to tetragonal when the La concentration reached 0.20 and 0.25 mol fraction. SEM images of all ceramics also indicated irregular shaped grains. The effects of variation in density, microstructure and composition on the thermal conductivity of all ceramics were also discussed in details.

Using Relative Projection Density for Classification of Terrestrial Laser Scanning Data with Unknown Angular Resolution

Point cloud classification is a key step for three-dimensional (3D) scene analysis in terrestrial laser scanning but is commonly affected by density variation. Many density-adaptive methods are used to weaken the impact of density variation and angular resolution, which denotes the angle between two horizontally or vertically adjacent laser beams and are commonly used as known parameters in those methods. However, it is difficult to avoid the case of unknown angular resolution, which limits the generality of such methods. Focusing on these problems, we propose a density-adaptive feature extraction method, considering the case when the angular resolution is unknown. Firstly, we present a method for angular resolution estimation called neighborhood analysis of randomly picked points (NARP). In NARP, n points are randomly picked from the original data and the k nearest points of each point are searched to form the neighborhood. The angles between the beams of each picked point and its corresponding neighboring points are used to construct a histogram, and the angular resolution is calculated by finding the adjacent beams of each picked point under this histogram. Then, a grid feature called relative projection density is proposed to weaken the effect of density variation based on the estimated angular resolution. Finally, a 12-dimensional feature vector is constructed by combining relative projection density and other commonly used geometric features, and the semantic label is generated utilizing a Random Forest classifier. Five datasets with a known angular resolution are used to validate the NARP method and an urban scene with a scanning distance of up to 1 km is used to compare the relative projection density with traditional projection density. The results demonstrate that our method achieves an estimation error of less than 0.001° in most cases and is stable with respect to different types of targets and parameter settings. Compared with traditional projection density, the proposed relative projection density can improve the performance of classification, particularly for small-size objects, such as cars, poles, and scanning artifacts.

Investigating the effect of density variation on pitch angle scattering events of runaway electrons as observed through electron cyclotron emission diagnostic at Aditya-upgrade tokamak.

Nascent observations about the influence of kinetic instabilities on electron cyclotron emission (ECE) from runaway electrons (REs) have been detected and explored at the Aditya-Upgrade (Aditya-U) tokamak. The developed broadband ECE radiometer system offers wideband measurements by integrating several radio frequency units with a fixed intermediate frequency receiver with multiple channels, which is a novel approach to meet the needs of the localized measurements at various toroidal fields and extend the system dynamic range. The low density (ne ≤ 1 × 10-19m-3) plasma discharges at Aditya-U are consistently accompanied by a 20%-40% increase in the ECE radiometer signal amplitude within 100 µs and sporadic step-like modulations. The Pitch Angle Scattering (PAS) of REs induced by kinetic instabilities is a potential candidate for their occurrence. This steep jump in the radiometer signals was detected due to its high temporal resolution of 10 µs. A "PREDICT" code that employs the relativistic test particle model validates these experimental findings of the radiometer diagnostic for the first time for Aditya-U tokamak. Preliminary observations of the ECE radiometer signals also show that additional gas puffs can be used to vary the trigger timings of such PAS events or even lead to their complete avoidance.

A Novel Approach to Quantifying the Effect of the Density of Sand Cores on Their Gas Permeability

The density of moulding mixtures used in the foundry industry plays a significant role since it influences the strength, porosity, and permeability of moulds and cores. The latter is routinely tested in foundries using different solutions to control the properties of the moulding materials that are used to make moulds and cores. In this paper, the gas permeability of sand samples was measured using a custom-made setup to obtain the gas permeability in standard units instead of the usual permeability numbers (PN) with calibrated units. The aim of the work was to explore the effect of density variations in moulding materials on their gas permeabilities. Permeability in this work is quantified in SI units, square metres [m2]. The setup works based on Darcy’s law and the numbers obtained from the measurements can be used to deduce the gas permeability, k, of a sample. Two furan resin bonded mixtures with the same grain size distribution were hand-rammed with varying compaction forces to obtain a variation in density. Cylindrical samples (50 × 50 mm) were prepared using a silica sand aggregate sourced from a Swedish lake. The results of the measurement provided the difference in gas permeability between the samples that have varying densities. The results of permeability were then extrapolated by modifying the viscosity value of the air passed through the sample. In order to find the effect of apparent density variation on the pore characteristics of the samples, mercury intrusion porosimetry (MIP) was also performed. The results were in line with the gas permeability measurements.

Density variation effects in α + Pb208 and O16 + Pb208 fusion reactions

We propose a hybrid approach between the sudden model and the adiabatic model to explain the fusion hindrance in heavy-ion fusion reactions at deep subbarrier energies. Due to the strong Pauli blocking effects at the density overlap region, the $\ensuremath{\alpha}$ particle approaching the target $^{208}\mathrm{Pb}$ becomes enlarged and its density distribution gets changed. By introducing energy dependence into this density variation effect, the potentials at different colliding energies are diverse. With the decrease of colliding energies, the significantly enhanced density variation effects make the nuclear potential more attractive and result in a deeper pocket in the inner part. A reasonable description of the experimental fusion cross sections is achieved for the $\ensuremath{\alpha} + ^{208}\mathrm{Pb}$ reaction. On account of highlighting the influence of density variation effects on the fusion process at deep subbarrier energies, we further investigate the density variation effects in the $^{16}\mathrm{O} + ^{208}\mathrm{Pb}$ fusion reaction based on the $\ensuremath{\alpha}$-cluster structures in $^{16}\mathrm{O}$ nuclei. The fusion hindrance at deep subbarrier energies is described well by considering the density variation effects. In addition, an astrophysical $S$ factor between the sudden model and adiabatic model is predicted at the energies below the experimental data.

On the use of conservative formulation of energy equation in hybrid compressible lattice Boltzmann method

Effect of density variations on mass conservation properties is widely recognized in the lattice Boltzmann method (LBM), thus non-conservative form of scalar transport equation was commonly adopted within the framework of hybrid LBM. Focusing on the compressible hybrid LBM, mass conservation and its effect on energy conservation equation are studied in this paper. Starting from the analysis on mass conservation law recovered by LBM, the consistency between conservative and non-conservative formulations of energy conservation equation based on various thermodynamic variables and lattice Boltzmann equation is addressed. Driven by the theoretical analysis, a set of modified consistent energy equations in entropy and internal energy form is derived to reduce the error terms and improve the consistency. The theoretical analysis and modified energy equations are intensively evaluated by several numerical test cases, e.g., the isentropic vortex convection, three-dimensional compressible Taylor-Green vortex and shock-vortex interaction.

The sensitivity of cosmogenic radionuclide analysis to soil bulk density: Implications for soil formation rates

AbstractImproving our knowledge of soil formation is critical so that we can better understand the first‐order controls on soil thickness and more effectively inform land‐management decisions. Cosmogenic radionuclide analysis has allowed soil scientists to more accurately constrain the rates at which soils form from bedrock. In such analysis, the concentration of an isotope, such as Beryllium‐10, is measured from a sample of bedrock. Because this concentration is partly governed by the lowering of the bedrock‐soil interface, a cosmogenic depth‐profile model can be fitted to infer the bedrock and surface lowering rates compatible with the measured concentrations. Given that the bedrock‐soil interface is shielded by soil, the cosmic rays responsible for the in‐situ production of the radionuclide are attenuated, with attenuation rates dependent on the density profile of this soil. Many studies have assumed that soil bulk density is either equal to that of the bedrock or constant with depth. The failure to acknowledge the variations in soil bulk density means that cosmogenically derived soil formation rates previously published may be under‐ or overestimates. Here, we deploy a new model called “CoSOILcal” to a global compilation of cosmogenic analyses of soil formation and, by making use of estimated bulk density profiles, recalculate rates of soil formation to assess the sensitivity to this important parameter. We found that where a soil mantle >0.25 m overlies the soil‐bedrock interface, accounting for the soil bulk density profile brings about a significantly slower rate of soil formation than that previously published. Moreover, the impact of using bulk density profiles on cosmogenically derived soil formation rates increases as soil thickens. These findings call into question the accuracy of our existing soil formation knowledge and we suggest that future cosmogenic radionuclide analysis must consider the bulk density profile of the overlying soil.Highlights The effect of heterogeneities in soil bulk density on cosmogenically derived soil formation rates is unknown. Soil formation rates are recalculated using a new model to analyse the effect of density variations. Accounting for density in soils >0.25 m thickness brings about significantly slower soil formation rates. Measuring soil bulk density is essential when cosmogenically deriving soil formation rates.

Influence of Board Density on the Physical and Mechanical Properties of Bamboo Oriented Strand Lumber

The process of bamboo-oriented strand lumber (BOSL) represents one of the best opportunities for automation, property control and consistency, and high utilization of material from abundant, fast-growing, and sustainable bamboo. In this study, BOSLs were prepared, with reference to the preparation process of bamboo scrimber, by compressing and densifying constituent units under the action of moisture-heat-force and resin polymerization, and then the effects of density variation on their physical and mechanical properties were investigated. The results revealed that the modulus of rupture, modulus of elasticity, compressive strength and shear strength of BOSL with density of 0.78–1.3 g/cm3 ranged from 124.42 to 163.2 MPa, 15,455 to 21,849 MPa, 65.02 to 111.63 MPa, and 9.88 to 18.35 MPa, respectively. The preparation of BOSL with bamboo as raw material could retain the good mechanical properties of natural bamboo, and produce bamboo-based structural products with different properties by controlling the density. The high strength of BOSL with high density was primarily due to the increased volume fraction of elementary fibers, the reduced porosity, and the enhanced gluing interface. The performance of BOSL can be comparable to, or surpass that of, wood or bamboo products. This study provided necessary basic research for the engineering design and application of BOSL.

A novel extrapolation method using OSL detectors for very small field output factor measurement for stereotactic radiosurgery.

Appropriate methods for the determination of very small X-ray beam output factors are essential to ensure correct clinical outcomes for stereotactic radiosurgery. To date, substantial work has been performed in identifying and quantifying suitable dosimeters for relative output factor (ROF) measurements including recent IAEA published recommendations. In this work, we provide a novel method using optically stimulated luminescent dosimeters (OSLDs) with different effective sizes of the readout area to determine ROFs. This involves applying an extrapolation technique to assess ROFs for 6MV SRS X-ray beams with field diameters ranging from 4 to 30mm as defined by the Brainlab SRS cones. By combining the use of multiple sized OSLDs and water droplets to remove air gaps located around the OSLD detectors, both volume averaging and density variation effects were minimised to estimate ROFs for an extrapolated zero volume detector. The measured results showed that for a 4mm diameter cone, the ROF was 0.660 ± 0.032 (2SD) as compared to 0.661 ± 0.01 and 0.651 ± 0.018 for the PTW 600019 microDiamond detector and Gafchromic EBT3 film respectively. Whilst the uncertainties were larger than conventional detectors, the technique shows promise and improvements in accuracy may be obtained by higher quality manufacturing techniques. Based on these results, using OSLDs with different effective sizes of readout area and an extrapolation technique shows promise for use as an independent verification tool for very small X-ray field ROFs in the clinical department.

Effect of Density Variation on Rarefied and Non-rarefied Gaseous Flows in Developing Region of Microtubes

The present study analyzes the rarefaction and non-rarefaction effects on heat transfer characteristics of hydrodynamically and thermally developing air flow through microtubes having a constant wall heat flux boundary condition. The problem is especially simulated with two microtubes having diameters 50 µm and 150 µm. The rarefaction effect is taken into consideration using Maxwell slip flow and temperature jump boundary conditions. The fluid property variation is considered as a non-rarefaction effect which plays a significant influence on convection through a microtube. It is observed that the rarefaction and non-rarefaction effects significantly affect the convective heat transfer. The non-rarefaction effect decreases the Nusselt number (Nu) as compared to constant property solution; however, the rarefaction effect increases the Nu compared to constant property solution.

Effect of variation in density on the stability of bilinear shear currents with a free surface

We perform the stability analysis for a free surface fluid current modeled as two finite layers of constant vorticity, under the action of gravity and absence of surface tension. In the same spirit as Taylor [“Effect of variation in density on the stability of superposed streams of fluid,” Proc. R. Soc. London, Ser. A 132, 499 (1931)], a geometrical approach to the problem is proposed, which allows us to present simple analytical criteria under which the flow is stable. A strong destabilizing effect of stratification in density is perceived when the results are compared with those obtained for the physical setting where the vorticity interface is also a density interface separating two immiscible fluids with constant densities. In contrast with the homogeneous case, the stratified bilinear shear current is mostly unstable and can only be stabilized when the background current in the upper layer is constant.

Density distribution effect on the analysis of the 6Li+ 58Ni reaction

The effect of density variation on the optical-model (OM) analysis of 6Li + 58Ni is investigated by using bare potentials extracted from different density distributions of the weakly-bound 6Li nucleus. For each of the bare potentials, the real and imaginaru parts of the direct-reaction and fusion polarization potentials are deduced from respective X2 fittings to the elastic scattering and fusion data. The results obtained show that variations in the density distribution of the weakly-bound projectile can change the strengths of the dynamic direct-reaction polarization potential, however, it cannot make a noticeable difference on the OM predictions of fusion cross sections.

A Study of Enhanced Heat and Mass Transfer From Variable Height Fin Array Undergoing Natural Convection

Abstract A numerical study is performed on simultaneous heat and mass transfer from a shrouded vertical nonisothermal variable height fin array, representing dehumidification process under natural convection. Fluid properties are treated as uniform, and the fluid is assigned to comply with Boussinesq approximation to include the effect of density variation with temperature and concentration. Semi-implicit method for the pressure linked equations revised (SIMPLER) algorithm is adopted to resolve pressure and velocity coupling. A detailed parametric investigation of fin spacing, variable fin height, and fin tip to shroud clearance for a range of thermal and mass Grashof number is undertaken. Results indicate that in case of smaller fin spacing, involving fin length of 0.3 m, coefficients of sensible and latent heat transfer increase with the decreasing variable height (H1*) of fin and become maximum at H1*=0.5, for all thermal and mass Grashof numbers considered presently. Further, total heat transfer analysis on a particular base length due to sensible heat shows a maximum of 24.4% enhancement, whereas same due to the latent heat shows a maximum of 25.8% enhancement, depending on the values of clearance. Induced velocities also increase with the decreasing variable height of fin (H1*), which influences the heat and mass transport. The output parameters of this analysis, like induced velocities and overall Nusselt numbers due to the sensible and latent heat, are correlated with the governing parameters. The correlation coefficients are found to be in a range from 0.97 to 0.99.

Simple 2D gravity–density inversion for the modeling of the basin basement: example from the Banat area, Serbia

We have developed a technique to calculate lateral density distribution of the sedimentary basin basement by combining linear gravity–density inversion and 2D forward modeling. The procedure requires gravity anomaly data, depth-to-basement data and density data for the sediments (density–depth distribution). Gravity effect of density variations in the basement was extracted from the total gravity anomaly by removing the joint effect of the sediments with vertical density variations and homogeneous basement of average density contrast (calculated by 2D modeling). Gravity effect of the sediments was calculated using depth-to-basement data and density–depth function, based on borehole logging data. Bouguer slab formula was used in order to simplify basement model and calculate the operator for the linear inversion of gravity effect of the basement into lateral density distribution. The results were tested by forward modeling, and differences between observed and modeled gravity data were used for the next iteration of the inversion and correction of density values in each point along the profile. The procedure was tested using synthetic models and field example, and both results were satisfying, especially taking into account the simplicity of the inversion technique. The main problem was the effect of abrupt changes in the basement topography on density distribution, but it was downsized by filtering. Basement density maps were compiled based upon the density distribution along the profiles.

Numerical solutions of compressible convection with an infinite Prandtl number: comparison of the anelastic and anelastic liquid models with the exact equations

We developed a numerical method for the set of equations governing fully compressible convection in the limit of infinite Prandtl numbers. Reduced models have also been analysed, such as the anelastic approximation and the anelastic liquid approximation. The tests of our numerical schemes against self-consistent criteria have shown that our numerical simulations are consistent from the point of view of energy dissipation, heat transfer and entropy budget. The equation of state of an ideal gas has been considered in this work. Specific effects arising because of the compressibility of the fluid are studied, like the scaling of viscous dissipation and the scaling of the heat flux contribution due to the mechanical power exerted by viscous forces. We analysed the solutions obtained with each model (fully compressible model, anelastic and anelastic liquid approximations) in a wide range of dimensionless parameters and determined the errors induced by each approximation with respect to the fully compressible solutions. Based on a rationale on the development of the thermal boundary layers, we can explain reasonably well the differences between the fully compressible and anelastic models, in terms of both the heat transfer and viscous dissipation dependence on compressibility. This could be mostly an effect of density variations on thermal diffusivity. Based on the different forms of entropy balance between exact and anelastic models, we find that a necessary condition for convergence of the anelastic results to the exact solutions is that the product $\unicode[STIX]{x1D716}q$ must be small compared to unity, where $\unicode[STIX]{x1D716}$ is the ratio of the superadiabatic temperature difference to the adiabatic difference, and $q$ is the ratio of the superadiabatic heat flux to the heat flux conducted along the adiabat. The same condition seems also to be associated with a convergence of the computed heat fluxes. Concerning the anelastic liquid approximation, we confirm previous estimates by Anufriev et al. (Phys. Earth Planet. Inter., vol. 152, 2005, pp. 163–190) and find that its results become generally close to those of the fully compressible model when $\unicode[STIX]{x1D6FC}T{\mathcal{D}}$ is small compared to unity, where $\unicode[STIX]{x1D6FC}$ is the isobaric thermal expansion coefficient, $T$ is the temperature (here $\unicode[STIX]{x1D6FC}T=1$ for an ideal gas) and ${\mathcal{D}}$ is the dissipation number.

Inertial Alfvén waves in auroral acceleration region with H+, He+ and O+ ions plasma

The propagation of inertial Alfven wave is investigated in cold, low- $\beta $ , homogeneous and bi-Maxwellian plasma consisting of multi-ions (H+, He+ and O+). Kinetic approach is adopted to derive the dispersion relation, damping rate, group velocity and growth/ damping length of the wave. Figures are exhibited with respect to ${ck_{\bot }} / {\omega _{pe}}$ . Effects of density variation with multi-ions are analysed on frequency, damping rate, parallel and perpendicular components of group velocity and growth/ damping length of the inertial Alfven wave. It is found that varying densities of multi-ions significantly influence the frequency, damping rate and group velocity of inertial Alfven wave. The wave frequency is observed between 0.5 to $18~\mbox{s}^{- 1}$ pertaining to observational data. The increasing density of heavy ions reduces the frequency of waves. The presence of He+ and O+ enhances the damping of wave showing more transfer of energy from wave to particles leading to increase in electron acceleration. The order of parallel and perpendicular group velocity is found to be $10^{9}~\mbox{cm}/\mbox{s}$ and $10^{5}~\mbox{cm}/\mbox{s}$ respectively. Maximum perpendicular growth length is observed corresponding to the minimum damping rate of wave at ${ck_{\bot }} / {\omega _{pe}} <1$ , signifying the dynamics in transverse direction to the magnetic field which are more significant in the present analysis. The parameters relevant to auroral acceleration region are used for graphical analysis. The applications of present study may be towards the electron acceleration in the dynamics of auroral acceleration region consisting of heavy ions in background plasma.

Effect of density variation on the Sm A–I phase transition properties

ABSTRACTBased on a thermodynamic model, we investigate how the density variation influences the smectic (Sm A)–isotropic (I) phase transition. We find that the density variation shifts the smectic A–isotropic transition temperature, however, first-order nature of the transition remains the same. Here, we also examine the effect of high pressure on the thermodynamic quantities. The present analysis shows that the pressure strongly influences the enthalpy and density step at the clearing transition. The pressure dependence of the thermodynamic properties is incorporated through the pressure dependent second Landau coefficient and the coefficients which couple the nematic and smectic order parameters to the density. We find a close agreement between theoretical and experimental results. Furthermore, we discuss the implications of the results for achiral liquid crystals.