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30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí. The Distortion free energy density is a quantity that describes the increase in the free energy density of a liquid crystal caused by distortions from its uniformly aligned configuration. It also commonly goes by the name Frank free energy density named after Frederick Charles Frank.

Nematic Liquid Crystal

The Distortion free energy density in a nematic liquid crystal is a measure of the increase in the Helmholtz free energy per unit volume due to deviations in the orientational ordering away from a uniformly aligned nematic director configuration. The total free energy density for a nematic is therefore given by:

${\mathcal {F}}_{T}={\mathcal {F}}_{0}+{\mathcal {F}}_{d}$

where ${\mathcal {F}}_{T}$ is the total free energy density of a liquid crystal, ${\mathcal {F}}_{0}$ is the free energy density associated with a uniformly aligned nematic, and ${\mathcal {F}}_{d}$ is the contribution to the free energy density due to distortions in this order. For a non-chiral nematic liquid crystals ${\mathcal {F}}_{d}$ is commonly taken to consist of three terms given by:

${\mathcal {F}}_{d}={\frac {1}{2}}K_{1}(\nabla \cdot \mathbf {\hat {n}} )^{2}+{\frac {1}{2}}K_{2}(\mathbf {\hat {n}} \cdot \nabla \times \mathbf {\hat {n}} )^{2}+{\frac {1}{2}}K_{3}(\mathbf {\hat {n}} \times \nabla \times \mathbf {\hat {n}} )^{2}$

The unit vector $\mathbf {\hat {n}}$ is the normalized director of the molecules $(|\mathbf {\hat {n}} |=1)$, which describes the nature of the distortion. The three constants $K_{i}$ are known as the Frank constants and are dependent on the particular liquid crystal being described. They are usually of the order of $10^{-6}$ dyn. Each of the three terms represent a type of distortion of a nematic. The first term represents pure splay, the second term pure twist, and the third term pure bend. A combination of these terms can be used to represent an arbitrary deformation in a liquid crystal. It is often the case that all three Frank constants are of the same order of magnitude and so its commonly approximated that $K_{1}=K_{2}=K_{3}=K$. This approximation is commonly referred to as the one-constant approximation and is used predominantly because the free energy simplifies when used to the much more computationally compact form:

${\mathcal {F}}_{d}={\frac {1}{2}}K((\nabla \cdot \mathbf {\hat {n}} )^{2}+(\nabla \times \mathbf {\hat {n}} )^{2})={\frac {1}{2}}K\partial _{\alpha }n_{\beta }\partial _{\alpha }n_{\beta }$

A fourth term is also commonly added to the Frank free energy density called the saddle-splay energy that describes the surface interaction. It is often ignored when calculating director field configurations since the energies in the bulk of the liquid crystal are often greater than those due to the surface. It is given by:

${\frac {1}{2}}K_{24}\nabla \cdot ((\mathbf {\hat {n}} \cdot \nabla )\mathbf {\hat {n}} -\mathbf {\mathbf {\hat {n}} } (\nabla \cdot \mathbf {\hat {n}} ))$

If inclusions are added to a liquid crystal an additional term contributes to the free energy density due to their presence, often characterized by a term known as the Rapini approximation:

${\mathcal {F}}_{s}=-\oint {\frac {1}{2}}W(\mathbf {\hat {n}} \cdot \mathbf {\hat {\nu }} )^{2}\mathrm {d} S$

The anchoring energy is given by $W$ and the unit vector $\mathbf {\hat {\nu }}$ is normal to the particles surface.

Chiral Liquid Crystal

For the case when the liquid crystal consists of chiral molecules an additional term to the distortion free energy density is added. The term changes sign when the axes are inverted and is given by:

${\mathcal {F}}_{Ch}=k_{2}(\mathbf {\hat {n}} \cdot \nabla \times \mathbf {\hat {n}} )$

The prefactor $k_{2}$ is dependent on the degree of molecular chirality. And so for the case of a chiral liquid crystal the total free energy density is given by:

${\mathcal {F}}_{T}={\mathcal {F}}_{0}+{\frac {1}{2}}K_{1}(\nabla \cdot \mathbf {\hat {n}} )^{2}+{\frac {1}{2}}K_{2}(\mathbf {\hat {n}} \cdot \nabla \times \mathbf {\hat {n}} +q_{0})^{2}+{\frac {1}{2}}K_{3}(\mathbf {\hat {n}} \times \nabla \times \mathbf {\hat {n}} )^{2}$

The quantity $q_{0}=2\pi /P_{0}$, describes the pitch $P_{0}$ of the cholesteric helix.

Electric and Magnetic Field Contributions

As a result of liquid crystal mesogens' anisotropic diamagnetic properties and electrical polarizability, electric and magnetic fields can induce alignments in liquid crystals. By applying a field one is effectively lowering the free energy of the liquid crystal.

To understand the effect a magnetic field produces on the distortion free energy density, a small region of local nematic order $\mathbf {\hat {n}}$ is often considered in which $\chi _{\perp }$ and $\chi _{\parallel }$ is the magnetic susceptibility perpendicular and parallel to $\mathbf {\hat {n}}$. The value $\Delta \chi \equiv \chi _{\parallel }-\chi _{\perp }=N<P_{2}(\cos {\theta })>$, where N is the number of mesogens per unit volume. The work per unit volume done by the field is then given by:

$W_{magnetic}=\int _{0}^{H}(-M_{\perp }\sin {\theta }-M_{\parallel }\cos {\theta })\,dH=-{\frac {H^{2}}{2}}(\chi _{\perp }+\Delta \chi \cos {\theta }^{2})$

where:

$M_{\parallel }=H\chi _{\parallel }\cos {\theta }$

$M_{\perp }=H\chi _{\perp }\sin {\theta }$

Since the $-{\frac {H^{2}\chi _{\perp }}{2}}$ term is spatially invariant it can be ignored and so the magnetic contribution to the distortion free energy density becomes:

$-{\frac {\Delta \chi }{2}}[\mathbf {H} \cdot \mathbf {\hat {n}} ]^{2}$

From similar arguments the electric field's contribution to the distortion free energy can be found and is given by:

$-{\frac {\Delta \epsilon }{8\pi }}[\mathbf {E} \cdot \mathbf {\hat {n}} ]$

The quantity $\Delta \epsilon \equiv \epsilon _{\parallel }-\epsilon _{\perp }$ is the difference between the local dielectric constants perpendicular and parallel to $\mathbf {\hat {n}}$.

Notes

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References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534