Maier-Saupe mean field model

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. 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 2 1 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 0 T S m low T high T 1 -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= 1 -0.5 T 0.43 0 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. 43.0)(22019.02==NIBNITSkVATA reasonable value compared with experiment. 1-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

1. W. Maier and A. Saupe "EINE EINFACHE MOLEKULARE THEORIE DES NEMATISCHEN KRISTALLINFLUSSIGEN ZUSTANDES", Zeitschrift für Naturforschung A 13 pp. 564- (1958)
2. W. Maier and A. Saupe "EINE EINFACHE MOLEKULAR-STATISTISCHE THEORIE DER NEMATISCHEN KRISTALLINFLUSSIGEN PHASE .1", Zeitschrift für Naturforschung A 14 pp. 882- (1959)
3. W. Maier and A. Saupe "EINE EINFACHE MOLEKULAR-STATISTISCHE THEORIE DER NEMATISCHEN KRISTALLINFLUSSIGEN PHASE .2", Zeitschrift für Naturforschung A 15 pp. 287- (1960)
4. 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)
5. G. R. LUCKHURST and C. ZANNONI "Why is the Maier−Saupe theory of nematic liquid crystals so successful?", Nature 267 pp. 412 - 414 (1977)