Edwards curve: Difference between revisions

Revision as of 11:18, 29 December 2013

The Fréedericksz transition is simply a transition from a uniform director configuration to a deformed director configuration upon application of a sufficiently strong magnetic or electric field, note that it is not a phase transition, because at any point in the liquid crystals the amount of order of the molecules relative to one another remains the same. ^[1] Below a certain field threshold the director remains undistorted. As the field value is gradually increased from this threshold, the director begins to twist until its aligned with the field. In this fashion the Fréedericksz transition can occur in three different configurations known as the twist, bend, and splay geometries. The phase transition was first observed by Fréedericksz and Repiewa in 1927.^[2] In this first experiment of theirs, one of the walls of the cell was concave so as to produce a variation in thickness along the cell.^[3] The phase transition is named in honor of the Russian physicist Vsevolod Frederiks.

Derivation

Twist Geometry

File:TwistGeometry.jpg

A diagram showing the twist geometry, where

E_{t}

is the threshold electric field.

If a nematic liquid crystal that is confined between two parallel plates that induce a planar anchoring is placed in a sufficiently high constant electric field then the director will be distorted. If under zero field the director aligns along the x-axis then upon application of an electric field along the y-axis the director will be given by:

\hat{n} = n_{x} \hat{x} + n_{y} \hat{y}

n_{x} = \cos θ (z)

n_{y} = \sin θ (z)

Under this arrangement the distortion free energy density becomes:

ℱ_{d} = \frac{1}{2} K_{2} {(\frac{d θ}{d z})}^{2}

The total energy per unit volume stored in the distortion and the electric field is given by:

U = \frac{1}{2} K_{2} {(\frac{d θ}{d z})}^{2} - \frac{1}{2} ϵ_{0} Δ χ_{e} E^{2} \sin^{2} θ

The free energy per unit area is then:

F_{A} = \int_{0}^{d} \frac{1}{2} K_{2} {(\frac{d θ}{d z})}^{2} - \frac{1}{2} ϵ_{0} Δ χ_{e} E^{2} \sin^{2} θ d z

Minimizing this using calculus of variations gives:

(\frac{\partial U}{\partial θ}) - \frac{d}{d z} (\frac{\partial U}{\partial (\frac{d θ}{d z})}) = 0

K_{2} (\frac{d^{2} θ}{d z^{2}}) + ϵ_{0} Δ χ_{e} E^{2} \sin θ \cos θ = 0

Rewriting this in terms of $ζ = \frac{z}{d}$ and $ξ_{d} = d^{- 1} \sqrt{\frac{K_{2}}{ϵ_{0} Δ χ_{e} E^{2}}}$ where $d$ is the separation distance between the two plates results in the equation simplifying to:

ξ_{d}^{2} (\frac{d^{2} θ}{d ζ^{2}}) + \sin θ \cos θ = 0

By multiplying both sides of the differential equation by $\frac{d θ}{d ζ}$ this equation can be simplified further as follows:

\frac{d θ}{d ζ} ξ_{d}^{2} (\frac{d^{2} θ}{d ζ^{2}}) + \frac{d θ}{d ζ} \sin θ \cos θ = \frac{1}{2} ξ_{d}^{2} \frac{d}{d ζ} ({(\frac{d θ}{d ζ})}^{2}) + \frac{1}{2} \frac{d}{d ζ} (\sin^{2} θ) = 0

\int \frac{1}{2} ξ_{d}^{2} \frac{d}{d ζ} ({(\frac{d θ}{d ζ})}^{2}) + \frac{1}{2} \frac{d}{d ζ} (\sin^{2} θ) d ζ = 0

\frac{d θ}{d ζ} = \frac{1}{ξ_{d}} \sqrt{\sin^{2} θ_{m} - \sin^{2} θ}

The value $θ_{m}$ is the value of $θ$ when $ζ = 1 / 2$ . Substituting $k = \sin θ_{m}$ and $t = \frac{\sin θ}{\sin θ_{m}}$ into the equation above and integrating with respect to $t$ from 0 to 1 gives:

\int_{0}^{1} \frac{1}{\sqrt{(1 - t^{2}) (1 - k^{2} t^{2})}} d t \equiv K (k) = \frac{1}{2 ξ_{d}}

The value K(k) is the complete elliptic integral of the first kind. By noting that $K (0) = \frac{π}{2}$ one finally obtains the threshold electric field $E_{t}$ .

E_{t} = \frac{π}{d} \sqrt{\frac{K_{2}}{ϵ_{0} Δ χ_{e}}}

As a result, by measuring the threshold electric field one can effectively measure the twist Frank constant so long as the anisotropy in the electric susceptibility and plate separation is known.

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
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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
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

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

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

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

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+The '''Fréedericksz transition''' is simply a transition from a uniform director configuration to a deformed director configuration upon application of a sufficiently strong magnetic or electric field, note that it is not a [[phase transition]], because at any point in the [[liquid crystals]] the amount of order of the molecules relative to one another remains the same. <ref>{{harvnb|Collings, P,J|Liquid Crystals Nature's Delicate Phase of Matter|2002|pp=42–43}}</ref>  Below a certain field threshold the [[liquid crystal#Director|director]] remains undistorted. As the field value is gradually increased from this threshold, the director begins to twist until its aligned with the field. In this fashion the Fréedericksz transition can occur in three different configurations known as the twist, bend, and splay geometries. The [[phase transition]] was first observed by Fréedericksz and Repiewa in 1927.<ref>{{harvnb|Fréedericksz|Repiewa|1927|pp=532–546}}</ref> In this first experiment of theirs, one of the walls of the cell was concave so as to produce a variation in thickness along the cell.<ref>{{harvnb|Priestley|Wojtowicz|Sheng|1975|p=115}}</ref> The phase transition is named in honor of the Russian physicist [[Vsevolod Frederiks]].
+==Derivation==
+===Twist Geometry===
+[[File:TwistGeometry.jpg|thumb|right|A diagram showing the twist geometry, where <math>E_t</math> is the threshold electric field.]]If a nematic liquid crystal that is confined between two parallel plates that induce a planar anchoring is placed in a sufficiently high constant electric field then the director will be distorted. If under zero field the director aligns along the x-axis then upon application of an electric field along the y-axis the director will be given by:
+:<math>\mathbf{\hat{n}}=n_x\mathbf{\hat{x}}+n_y\mathbf{\hat{y}}</math>
+:<math>n_x=\cos{\theta(z)}</math>
+:<math>n_y=\sin{\theta(z)}</math>
+Under this arrangement the [[distortion free energy density]] becomes:
+:<math>\mathcal{F}_{d}=\frac{1}{2}K_2\left(\frac{d\theta}{dz}\right)^2</math>
+The total energy per unit volume stored in the distortion and the electric field is given by:
+:<math>U=\frac{1}{2}K_2\left(\frac{d\theta}{dz}\right)^2-\frac{1}{2}\epsilon_0\Delta\chi_eE^2\sin^2{\theta}</math>
+The free energy per unit area is then:
+:<math>F_A=\int_0^d\frac{1}{2}K_2\left(\frac{d\theta}{dz}\right)^2-\frac{1}{2}\epsilon_0\Delta\chi_eE^2\sin^2{\theta}\,dz \,</math>
+Minimizing this using [[calculus of variations]] gives:
+:<math>\left(\frac{\partial U}{\partial \theta}\right)-\frac{d}{dz}\left(\frac{\partial U}{\partial\left(\frac{d\theta}{dz}\right)}\right)=0</math>
+:<math>K_2\left(\frac{d^2\theta}{dz^2}\right)+\epsilon_0\Delta\chi_eE^2\sin{\theta}\cos{\theta}=0</math>
+Rewriting this in terms of <math>\zeta=\frac{z}{d}</math> and <math>\xi_d=d^{-1}\sqrt{\frac{K_2}{\epsilon_0\Delta\chi_eE^2}}</math> where <math>d</math> is the separation distance between the two plates results in the equation simplifying to:
+:<math>\xi_d^2\left(\frac{d^2\theta}{d\zeta^2}\right)+\sin{\theta}\cos{\theta}=0</math>
+By multiplying both sides of the differential equation by <math>\frac{d\theta}{d\zeta}</math> this equation can be simplified further as follows:
+:<math>\frac{d\theta}{d\zeta}\xi_d^2\left(\frac{d^2\theta}{d\zeta^2}\right)+\frac{d\theta}{d\zeta}\sin{\theta}\cos{\theta}=\frac{1}{2}\xi_d^2\frac{d}{d\zeta}\left(\left(\frac{d\theta}{d\zeta}\right)^2\right)+\frac{1}{2}\frac{d}{d\zeta}\left ( \sin^2{\theta}\right)=0</math>
+:<math>\int\frac{1}{2}\xi_d^2\frac{d}{d\zeta}\left(\left(\frac{d\theta}{d\zeta}\right)^2\right)+\frac{1}{2}\frac{d}{d\zeta}\left ( \sin^2{\theta}\right)\,d\zeta \,=0</math>
+:<math>\frac{d\theta}{d\zeta}=\frac{1}{\xi_d}\sqrt{\sin^2{\theta_m}-\sin^2{\theta}}</math>
+The value <math>\theta_m</math> is the value of <math>\theta</math> when <math>\zeta=1/2</math>. Substituting <math>k=\sin{\theta_m}</math> and <math>t=\frac{\sin{\theta}}{\sin{\theta_m}}</math> into the equation above and integrating with respect to <math>t</math> from 0 to 1 gives:
+:<math>\int_0^1\frac{1}{\sqrt{(1-t^2)(1-k^2t^2)}}\,dt \,\equiv K(k)=\frac{1}{2\xi_d}</math>
+The value K(k) is the [[complete elliptic integral of the first kind]]. By noting that <math>K(0)=\frac{\pi}{2}</math> one finally obtains the threshold electric field <math>E_t</math>.
+:<math>E_t=\frac{\pi}{d}\sqrt{\frac{K_2}{\epsilon_0\Delta\chi_e}}</math>
+As a result, by measuring the threshold electric field one can effectively measure the twist [[Frank constant]] so long as the anisotropy in the electric susceptibility and plate separation is known.
+==Notes==
+{{reflist}}
+==References==
+*{{cite book |last1= Collings |first1= Peter J. | authorlink1=  | last2 = Hird | first2 = Michael | title= Introduction to Liquid Crystals: Chemistry and Physics | edition =  |publisher= Taylor & Francis Ltd. |year= 1997 | date =  |month=  |isbn= 0-7484-0643-3}}
+*{{cite book |last1= de Gennes |first1= Pierre-Gilles | authorlink1= Pierre-Gilles de Gennes | last2 = Prost | first2 = J. | title= The Physics of Liquid Crystals | edition = 2nd |publisher= Oxford University Press | date = 10 August 1995 |month=  |isbn= 0-19-851785-8}}
+*{{cite journal |last1= Fréedericksz | first1= V. | ref=harv | last2= Repiewa | first2= A. | title= Theoretisches und Experimentelles zur Frage nach der Natur der anisotropen Flüssigkeiten | journal= Zeitschrift für Physik Society | year= 1927 | volume=42 | issue=7 | pages = 532–546 | doi= 10.1007/BF01397711 |bibcode = 1927ZPhy...42..532F }}
+*{{cite journal |last1= Fréedericksz | first1= V. | ref= | last2=Zolina | first2= V. | title= Forces causing the orientation of an anisotropic liquid | journal= Trans. Faraday Soc. | year= 1933 | volume=29 | issue= | pages =919–930 | doi= 10.1039/TF9332900919 }}
+*{{cite book | ref=harv | last1= Priestley |first1= E. B. | authorlink1=  | last2 = Wojtowicz | first2 = Peter J. | last3=Sheng | first3=Ping | title= Introduction to Liquid Crystals | edition =  |publisher= Plenum Press |year= 1975 | date =  |month=  |isbn= 0-306-30858-4}}
+*{{cite journal |last1= Zöcher | first1= H. | title= The effect of a magnetic field on the nematic state | journal= Transactions of the Faraday Society | year= 1933 | volume=29 | pages = 945–957 | doi= 10.1039/TF9332900945 }}
+{{DEFAULTSORT:Freedericksz Transition}}
+[[Category:Liquid crystals]]
+[[Category:Phase transitions]]

Edwards curve: Difference between revisions

Revision as of 11:18, 29 December 2013

Contents

Derivation

Twist Geometry

Notes

References

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Edwards curve: Difference between revisions

Revision as of 11:18, 29 December 2013

Derivation

Twist Geometry

Notes

References

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