Sep 15th Il Palazzone SNS - Cortona (AR)

Transcription

1 Repertoire o Sep 5th 25 Il Palazzone SNS - Cortona (AR)

2 Repertoire of nematic biaxial phases. Fulvio Bisi SMMM - Soft Matter Mathematical Modeling Dip. di Matematica - Univ. di Pavia (Italy) fulvio.bisi@unipv.it Sep 5th 25 Il Palazzone SNS - Cortona (AR)

3 Summary Introduction Biaxial Interactions Mean-Field Theory Free Energy and Stability Bogolubov Principle Tricritical and Triple Points Repertoire of Phases Sep 5th 25 Il Palazzone SNS - Cortona (AR) 2

4 2 Introduction For more than 3 years has the hunt for nematic biaxial phases been going on. Strong evidence for existence of this phase recently emerged: Acharya et al. (24) Madsen et al. (24) Merkel et al. (24) This should open a new era for research, in LC field, eventually ending in new applications. We need a complete model for uniaxial and biaxial nematic phases, and a thorough understanding of how they are related as a function of temperature. Sep 5th 25 Il Palazzone SNS - Cortona (AR) 3

5 3 Biaxial Interactions Biaxial Molecules: described by platelets in which we distinguish three axes: m, major axis, and e, e (eigenvectors of any molecular polarizability tensor). Two components for the anisotropic part of every molecular biaxial tensor: q := m m 3 I, b := e e e e () m e e Sep 5th 25 Il Palazzone SNS - Cortona (AR) 4

6 Interaction Energy between two molecules (Straley s potential): Remark : λ = γ = Remark 2: λ = γ 2 V = U [q q + γ(q b + b q ) + λb b ], (2) Maier-Saupe interaction energy. London s dispersion forces approximation. There exist V invariant transformations (Longa et al., 25; De Matteis et al., 25) Sep 5th 25 Il Palazzone SNS - Cortona (AR) 5

7 Conjugation Chart λ γ Sep 5th 25 Il Palazzone SNS - Cortona (AR) 6

8 Ensemble Averages: Q := q =: S (e z e z 3 I ) + T (e x e x e y e y ) (3a) B := b =: S ( e z e z 3 I ) + T (e x e x e y e y ) (3b) Remark : unlike q and b, Q and B are NOT orthogonal. Terminology: Uniaxial Phases have T = T = (both Q and B uniaxial. Phase Biaxiality occurs when S = T = (cylindrical molecules). Intrinsic Biaxiality occurs whenever T. Sep 5th 25 Il Palazzone SNS - Cortona (AR) 7

9 Order Parameter Manifold: Euler Angles (ϑ, ϕ, ψ) representing the rotation that take {e x, e y, e z } into {m, e, e }. Then: S = 3 2 < cos2 ϑ /3 > /2 S T = 2 < sin2 ϑ cos 2ϕ > (/3)( S) T (/3)( S) S = 3 2 < sin2 ϑ cos 2ψ > ( S) S ( S) T = 2 < ( + cos2 ϑ) cos 2ϕ cos 2ψ 2 cos ϑ sin 2ϕ sin 2ψ > T (4a) (4b) (4c) (4d) Sep 5th 25 Il Palazzone SNS - Cortona (AR) 8

10 Different set of order parameters may describe the same state: we recast (S, T, S, T ) in a selected region of the order parameter manifold by using the mappings: (S, T, S, T ) (S, T, S, T ) (S, T, S, T ) ( ±3T S 2, T ± S 2, ±3T S 2, T ± S 2 (5a) ) (5b) T S Sep 5th 25 Il Palazzone SNS - Cortona (AR) 9

11 4 Mean-Field Model Pseudopotential: Partition Function: Z := U = U [q Q + γ(q B + b Q) + λb B], (6) T exp[β(q Q + γ(q B + b Q) + λb B], (7) where T = S 2 S is the toroidal manifold, β := U /(k B T ), with k B being the Boltzmann constant and T the absolute temperature. Sep 5th 25 Il Palazzone SNS - Cortona (AR)

12 Free Energy: F := F/U = 2 (Q Q + 2γQ B + λb B) β log Z 8π 2 3 S2 + T 2 + 2γ( 3 SS + T T ) + λ( 3 (S ) 2 + (T ) 2 ) β log Z 8π 2 (8) The states corresponding to equilibrium point of F in (8) satisfy the compatibility conditions Q = T fq =< q > B = T fb =< b > (9) Sep 5th 25 Il Palazzone SNS - Cortona (AR)

13 5 Bogolubov Principle We can write the the interaction potential V as a function of new tensor variables: V = U (a + q + q + + a q q ), () with q ± := q ± γ ± b and q + q =. Hence: F (Q +, Q ) = (a + Q + Q + + a Q Q ) β log Z 8π 2 ; () Above the parabola, both a + and a are positive: stable states are local minima. Below the parabola, one coefficient is negative: a <. It can be shown that for all stationary states F has a maximum in Q at given Q +. Sep 5th 25 Il Palazzone SNS - Cortona (AR) 2

14 In this latter case, according to Bogolubov principle, the best approximation to the stable equilibrium state are given by: min Q + max Q F(Q+, Q ) ; (2) As the dimension of the eigenspace associated to Q is 2, assessing the stability of an equilibrium solution below the parabola boils down to the third eigenvalue criterion. Sep 5th 25 Il Palazzone SNS - Cortona (AR) 3

15 Essential Triangle λ γ We explore some points inside the essential triangle to assess the effect of several flavours in the interaction potential. Sep 5th 25 Il Palazzone SNS - Cortona (AR) 4

16 Experimental Evidence Recently experimental results obtained on tetrapodes seem to be in agreement with the prediction of the model with γ = (Merkel, Vij et al., 24). What if γ? Sep 5th 25 Il Palazzone SNS - Cortona (AR) 5

17 6 Bifurcation plots γ =.4667, λ =.25, τ = UP LP S UP BP BP LP LP UP.5 UP LP.5 T.5 β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 6

18 S γ =.4667, λ =.25, τ = β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 7

19 T γ =.4667, λ =.25, τ = LP.5 LP UP β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 8

20 γ =.62934, λ =.2746, τ = S.5 UP LP BP UP BP UP LP LP UP UP.5.5 T.5 β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 9

21 γ =.9667, λ =.5, τ =.483 S.5 UP BP UP BP UP LP UP.5 T 2 β 4 6 Sep 5th 25 Il Palazzone SNS - Cortona (AR) 2

22 γ =.9667, λ =.5, τ = UP BP.6 S.4 UP LP.2 UP β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 2

23 γ =.9667, λ =.5, τ = T UP LP UP UP BP UP BP β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 22

24 γ =.3486, λ =.6365, τ = BP UP UP BP.5 S.5 UP LP UP.5 T.5 2 β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 23

25 γ =.3486, λ =.6365, τ = UP BP.5 UP LP S UP β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 24

26 γ =.3486, λ =.6365, τ = T UP LP BP UP UP BP β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 25

27 γ =.3486, λ =.6365 F β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 26

28 γ =.547, λ =.23932, τ = BP BP UP.6 S UP LP UP UP.5 T β Sep 5th 25 Il Palazzone SNS - Cortona (AR) 27

29 γ =.698, λ =.93654, τ = BP U2 S LP U2.5.5 T U2.5 2 β 4 6 Sep 5th 25 Il Palazzone SNS - Cortona (AR) 28

30 7 Tricritical and Triple Lines λ.3 I B.25 I U B.2 I U=B γ Sep 5th 25 Il Palazzone SNS - Cortona (AR) 29

31 8 Final Remarks : above the parabola, the sequence of phases somehow mimicks the one predicted for γ = 2: upon crossing the high-degeneracy line of the parabola, the bifurcation diagramme unfolds. 3: more detail on some aspects of the model can be discussed during this afternoon s Round Table on Biaxial Nematics (see E.G. Virga s and G. De Matteis presentations). Sep 5th 25 Il Palazzone SNS - Cortona (AR) 3

32 9 Acknowledgements Giovanni De Matteis Georges Durand Eugene C. Gartland Epifanio G. Virga Co-authors Institutions Royal Society of London (Southampton-Pavia Collaborative Project) Italian MIUR (PRIN Grant No M. Osipov, R. Rosso Sep 5th 25 Il Palazzone SNS - Cortona (AR) 3

33 References G.R. Luckhurst, C. Zannoni, P.L. Nordio, and U. Segre, Mol. Phys. 3, 345 (975). A.M. Sonnet, E.G. Virga, and G.E. Durand, Phys. Rev. E 67, 67 (23). B.R. Acharya, A. Primak, and S. Kumar, Phys. Rev. Lett. 92, 673 (24). L. A. Madsen, T. J. Dingemans, M. Nakata, and E. T. Samulski, Phys. Rev. Lett. 92, 4555 (24). K. Merkel, A. Kocot, J. K. Vij, R. Korlacki, G. H. Mehl, and T. Meyer, Phys. Rev. Lett. 93, 2378 (24). G. De Matteis, and E.G. Virga, Phys. Rev. E 7, 673 (25). Sep 5th 25 Il Palazzone SNS - Cortona (AR) 32