Maier-Saupe mean field model: Difference between revisions

Revision as of 13:48, 9 April 2009

The Maier-Saupe model is widely used as a model of the nematic phase. The Maier-Saupe model is based on long-range dispersion forces, and has the form

U(\cos \theta )={\overline {u}}_{2}{\overline {P}}_{2}P_{2}(\cos \theta )

where $\theta$ is the angle between the molecular symmetry axis and the director. ${\overline {P}}_{2}$ is the uniaxial order parameter.

References

W. Maier and A. Saupe "EINE EINFACHE MOLEKULARE THEORIE DES NEMATISCHEN KRISTALLINFLUSSIGEN ZUSTANDES", Zeitschrift für Naturforschung A 13 pp. 564- (1958)
W. Maier and A. Saupe "EINE EINFACHE MOLEKULAR-STATISTISCHE THEORIE DER NEMATISCHEN KRISTALLINFLUSSIGEN PHASE .1", Zeitschrift für Naturforschung A 14 pp. 882- (1959)
W. Maier and A. Saupe "EINE EINFACHE MOLEKULAR-STATISTISCHE THEORIE DER NEMATISCHEN KRISTALLINFLUSSIGEN PHASE .2", Zeitschrift für Naturforschung A 15 pp. 287- (1960)
P A Vuillermot and M V Romerio "Exact solution of the Maier-Saupe model for a nematic liquid crystal on a one-dimensional lattice", Journal of Physics C: Solid State Physics 6 pp. 2922-2930 (1973)
G. R. LUCKHURST and C. ZANNONI "Why is the Maier−Saupe theory of nematic liquid crystals so successful?", Nature 267 pp. 412 - 414 (1977)

@@ Line 5: / Line 5: @@
 where <math>\theta</math> is the angle between the molecular symmetry axis and the [[director]]. <math>\overline P_2</math>
 is the [[Order parameters | uniaxial order parameter]].
-Maier and Saupe Theory
-Aim is to calculate S as function of T.
-Maier and Saupe (1960) anisotropic attraction
-Onsager (1949) anisotropic repulsion
-We will look at MS theory and then consider its strengths and weaknesses.
-(i)
-Simplest attractive interaction between two polarizable rods. Instantaneous dipole interacts with induced dipole.
-⎟⎠⎞⎜⎝⎛−=21cos23)(),(12212121212ββrurU
-β12
-r12
-(ii)
-Too difficult to consider interaction of every molecule with every other molecule so we construct an average potential energy function that one molecule feels due to immersion in a sea of other similar molecules. Mean field approximation. )(cos)(21cos23)(1)()(21cos23)(222222iiiiiiiiPVASUVASUVUSUUββββββββ−=⎟⎠⎞⎜⎝⎛−−=∝∝⎟⎠⎞⎜⎝⎛−−∝
-i.e. potential proportional to cos squared of angle
-and order parameter
-and density squared
-n
-βi
-A defines strength of potential. Ignores fluctuations and SRO
-(iv)
-Now calculate orientational distribution function: angle. azimuthal theis and director theand axis longmolecular ebetween th anglepolar theis wheresin where1)(020)()(αβαβββππββddeZeZfiiTkUTkUiBiiBii∫∫−−==
-(v)
-The order parameter can now be calculated using the method outlined in lecture 2. The order parameter is just the average value of )(cos2β
-P. That is)cos(2βPS=.
-In more detail… αβββαββββαββββππππππddTkVASPddPTkVASPddPfSBBsin)(cosexpsin)(cos)(cosexpsin)(cos)(020220220220220∫∫∫∫∫∫⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛==
-This is just an equation with S on both sides. It is tricky to solve because S appears within an integral but solutions can be found using the following method. ()()ATkmVSTkVASmdxmxdxxmxSdxTkVxASPdxxPTkVxASPSdxTkVxASPdxxPTkVxASPSBBBBBB221022102102221022112221122or where21expexp23)(exp)()(exp)(exp)()(exp==−=∴⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=∫∫∫∫∫∫−−
-(vi)
-Two simultaneous equations in S and m. Integral may be done numerically and equations solved “graphically”. Slope of straight line is proportional to T.
-The results are:
-Tmax
-T
-S
-m
-low T
-high T
--0.5
-S
-)/(22284.02maxBkVAT×=
-At high T (ie T > Tmax) there is only 1 solution, S = 0
-At low T (ie T < Tmax) there are 3 self consistent solutions. S=0, S>0 and S<0
-(vii)
-Which one has lowest free energy? The one with lowest free energy! Use : Helmholtz free energy = energy – T × entropy Σ
-−=TU
-Recall from Statistical Mechanics:
-Probability of the system being in a state with energy Er : 1 and ==Σ−rrTkErPZePBr mean energy of system, UrrrEPΣ= entropy of system, rrrBPPklnΣ−=Σ
-(viii)
-Average energy of a molecule : )(cosS using and function,on distributi over the averagean represent where)(cos)(cos)(2222222ββββPVASPVASPVASUUii=−=−=−== Energy of phase of N molecules: 2221VASNU−= (Note the half)
-(ix)
-Entropy of a molecule is –kB times average of ln(distribution): ZkTUfkBiiBiln)(ln+=−=Σβ from (iv) Entropy of N average molecules: ⎟⎟⎠⎞⎜⎜⎝⎛−=−=Σ−=+−=+=Σ=ΣZTkVASNZTNkVASNTUFZNkTVASNZNkTUNNBBBBiiln21ln21lnln222222 Unfortunately, this too must be evaluated numerically: For each value of S find m then calculate Z
-ddmPZβββππsin))(cosexp( where0202∫∫−= Iit turns out that for BkVAT222019.0< the positive S solution has the lowest free energy. Hence BNIT22019.0=
--0.5
-T
-.43
-Tmax
-S
-S decreases steadily as T is increased until it suddenly drops to zero at TNI .
-(x)
-TNI is less than TMAX so have first order transition.
-.0)(22019.02==NIBNITSkVATA reasonable value compared with experiment.
--K Joules 5.3Joules 5.3=ΔΣ×=ΔNININITU Much weaker than crystal to liquid transition
-Strong angle dependant attraction (large A) increases TNI.
-Dilution (increasing V) decreases TNI.
-Why does it work? We have neglected the shape completely but it seems to give reasonable values.
 ==References==
 #W. Maier and A. Saupe "EINE EINFACHE MOLEKULARE THEORIE DES NEMATISCHEN KRISTALLINFLUSSIGEN ZUSTANDES", Zeitschrift für Naturforschung A '''13'''  pp. 564- (1958)

Maier-Saupe mean field model: Difference between revisions

Revision as of 13:48, 9 April 2009

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