Anomalous heat capacity for nematic MBBA near clearing point

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1 Journal of Physics: Conference Series Anoalous heat caacity for neatic MA near clearing oint To cite this article: D A Lukashenko and M Khasanov J. Phys.: Conf. Ser. 394 View the article online for udates and enhanceents. Related content - The uzzle of the neatic-isotroic hase transition Prabir K Mukherjee - Distortion-sensitive insight into the retransitional behavior of 4-n-octyloxy- 4rie-cyanobihenyl (8OC) Sylwester J Rzoska, Aleksandra Drozd- Rzoska, Prabir K Mukherjee et al. - Liquid crystals S Chandrasekhar This content was downloaded fro IP address on /5/8 at :4

2 SPMCS Journal of Physics: Conference Series 394 () doi:.88/ /394// Anoalous heat caacity for neatic MA near clearing oint D A Lukashenko and.m. Khasanov Theoretical Physics Deartent, Institute of Physics, Kazan university, 48, Krelevskaya, 8; Kazan, Russia E-ail: Shadow-LD@yandex.ru, E-ail: bulat.khasanov@ksu.ru Abstract. The retransition roerties of the heat caacity on the both sides of the neatic isotroic hase transition for the liquid crystal MA (n-ethoxybenzylidene-n butylaniline) are discussed on the basis of the Landau-de Gennes theory. Agreeent between theory and exerient allows to obtain the critical exonents gaa and beta.. Introduction Exeriental and theoretical investigation of the statistical roerties of heat caacity in neatic liquid crystals, esecially near the hase transition to isotroic hase, is iortant both for the hysics of hase transition and for neatic liquid crystal theselves. The analysis of the teerature deendence of the secific heat showed that the fluctuations of the OP are not sall near hase transition oint. Exerients on heat caacity are usually interreted fro the oint of view of fluctuations of the odulus order araeter (OP), which are the rincial cause of anoalously teerature deendence near hase transition oint. ut in addition to these odulus OP fluctuations there are the fluctuations of the orientation and biaxial fluctuations that disturb the uniaxiality of the OP. Critical henoena in hase transitions in liquid crystals, accoanied by the vanishing of longrang orientational order of the long axes of the olecules, have been sufficiently studied. The existing exeriental data [] indicate that, as the neatic-isotroic transition oint is aroached the secific heat increases. This characteristic anoalies in the teerature are connected with the satiallyinhoogeneous fluctuations of the colex order araeter of neatic. In this work we have atteted to carry out quantitative analysis of the ossibility of the alication of the Landau-de Gennes theory for the descrition of critical anoalies both above and below the neatic-isotroic (NI) hase transition.. Landau-de Gennes theory and fluctuations In the henoenological Landau-de Gennes theory [] the free energy associated with the long- Q = Q n n /3 δ fluctuations for a uniaxial neatic in the wavelength art of the tensor OP αβ ( α β αβ ) neighborhood of the hase transition ( n r is a unit vector, director) in the one constant aroxiation, is given by Published under licence by Ltd

3 SPMCS Journal of Physics: Conference Series 394 () doi:.88/ /394// where 3 4 F = V AQ Q + CQ + d r[ ( A Q + 3CQ ) P P + L P P 3 4 r, () ( ) ( ) + A Q + CQ P P + L P P + A + Q + CQ P P + L P P ] A a T T = ( ) and, C are teerature indeendent constants, Q is a odule of OP, and T - the lowest ossible teerature for isotroic hase, L is an elastic constant. The first art of F is the Landau exansion in ters of the scalar OP. neatic hase: T T /4ac N Q = corresonds to the isotroic hase, while in the / ( τ ) /, τ ( )( ) Q = + C = T T T T. () = + is the uer liit of stability for the neatic hase. Fro the first art of Here exression () we can easily find the teerature and latent heat of the transition: = + a 9 ac = C (a) T, H T c as well as the suscetibility and heat caacity above and below the transition oint: χ at ( T ), T> T, c =, C = at ( T )( τ + τ), T< ( ) T > T / a ( + τ ) c <, T The suscetibility in the isotroic hase shows the linear divergence. At the sae tie the suscetibility in the neatic hase has a linear lus a square-root divergence, i.e. a deviation fro the Curie-Weiss law in the ean field aroxiation and γ exonent is less than one [3]. Let us consider neatic as a continuous hoogeneous ediu. Characteristic lengths of r considerable change of P ust be greater than the distance between olecules, i. e. k < k ax. Now we use the equiartition theore to find the theral averages of the fluctuations of the longitudinal, transverse and biaxial odes. In the isotroic hase one can find While in the neatic hase we obtain Here χ ( χ ) P P = P P = P P = k T + Lk (4) kt kt kt PP =, PP =, PP = χ + Lk Lk 3Q + Lk is deterined by the exression (3), and 3Q 6 a( T T )( τ ) = + (6) In the isotroic hase the fluctuations of all the five odes are degenerate. They increase in the longwave liit ( k ) as T T according to the Curie-Weiss law. In the neatic hase the fluctuations sectru slits. PP is the OP longitudinal fluctuations. PP is the transverse fluctuations (director), which are not critical and reresent Goldstone odes c (3) (5)

4 SPMCS Journal of Physics: Conference Series 394 () doi:.88/ /394// in the syetry-broken hase. PP deterine biaxial fluctuations, where the biaxial T = T ( τ = ). suscetibility (3 Q), contrary to the longitudinal χ, tends to a finite value at In the isotroic hase the excess secific heat caacity er unit volue due to OP fluctuations is given by 3 / χ x ( χ ) ( ) 5kT Δ C = L arctg x, (7) 8π T + x / where x= kξ, ξ ξ T / L = and ξ = is the bare OP correlating length. T T at Here k is the uer liit of integration with resect to the wave vector of fluctuation odes. The continuous odel we are used assues averaging over a sall but acroscoic volue. Therefore all our exressions are correct u to wave vector k. To coare the exression (7) with the exeriental data, it is convenient to rewrite it in the for T A AT C =Δ C + arctg T T T T T T + A / 3/ kl 5 ka M A= = k / ξ T, = 3/ a 8π L ρ ecause direct exeriental observation of T is not ossible, it have to be treated as adjustable araeter (together with A and ) in the fits. Furtherore, no data oints in the range T T < was covered in the fits. We consider the Gaussian aroxiation only that is correct not to close to the transition oint. The best agreeent with exerient was obtained for the araeter values A =., K Δ C = 499. J ol K 4, =.4. The fitting of T =, ( ) exeriental data [4] result is resented at figure. The fitting araeters A and are connected with each other through the equation (8), which iediately allows to get ( k ) = 5.5A, ξ = 6.8 A. Our value of ξ is close to the value obtained in [5].For density and the ρ 3 =.49 g c, = g ol. olecular weight of MA we used The treatent of the fluctuation art of the secific heat at the neatic hase is ore colex. The entroy density associated with fluctuations is obtained by differentiating () with resect to the teerature. The subsequent enseble average gives the entroy change due to fluctuations. The final exression for the fluctuation induced heat caacity has the for M ( 3Q) T Δ C = P + T P P χ T k T k ( 3Q) P T P P k T k T χ T, (8) (9) 3

5 SPMCS Journal of Physics: Conference Series 394 () doi:.88/ /394// Figure. Secific heat above the transition teerature. Solid line show theoretical data and circles show the exeriental data of Anisiov [4] Note that in (9) there are resent coonents which are roortional to the second derivative with resect to reverse suscetibilities. It is caused by the fact that the longitudinal suscetibility in neatics does not obey Curie-Weiss law, and the reverse biaxial suscetibility is directly roortional to OP. The forula (9) can be used to describe the exeriental data [4]. The results of the exeriental research of the heat caacity, OP and suscetibility deonstrate that the use of exressions (), (3) and (6) have the difficulties of describing the data in the neatic hase. Changing to soe other teerature scale T > T we can aroxiate the exression () in the corresonding teerature range to get the following forula: Q N = CT T T t β, () where t = ( T T) T. β and T are fitting araeters. The equation () has the for of widely used the Haller relation [6].The reverse of the longitudinal and biaxial suscetibilities in this case have γ the following fors ( ) χ = χ t, χ = at, 3Q = 3χ t. β Using these definitions, we find the exression for heat caacity in the neatic hase: 4

6 SPMCS Journal of Physics: Conference Series 394 () doi:.88/ /394// T C C C C C H t β =Δ + landau +Δ= Δ + κ β + T 3 3γ 4 kt x γ + t ( 3 arctg x) γγ ( ) x arctg( x), () π ξt + x kt 6 x + π 3 3γ 4 t 3 arctg( x) β β( β ) x arctg( x ξt + x kξ χ k β ξ T LN where x = t, x = t, κ =, ξ = 6 T at Here H is the transition latent heat and for the neatic MA its exeriental value is H =.78 J g []. L - is the elastic constant in neatic hase which is not equal to one in N isotroic hase. The second ter in equation () is the heat caacity obtained in the Landau theory. In order to deterine quantitative agreeent of () with exeriental data we treat variable araeters T, ξ, Δ C and ake a fitting in teerature range T < T o, because of the aroxiation used. After that we get following araeter values that give the best fit: T = 3.97K, 9.8A Δ C = J ol K (see the figure ), as well as the critical exonents β =. [7] and γ =.68 [8]. ξ =, ( ) β ) Figure. Secific heat above and below the transition teerature. Solid lines show theoretical data and circles and triangles show the exeriental data [4]. 5

7 SPMCS Journal of Physics: Conference Series 394 () doi:.88/ /394// The agreeent of theoretical forula () with exerient shows us that the use of eirical aroxiation rocedure of Haller for OP and siilar rocedure for the longitudinal suscetibility in the Landau-de Gennes odel gives a satisfactory descrition of heat caacity exerients in neatic hase. The critical exonent β of order araeter is in good agreeent with the exeriental data for MA [9]. The regular art of the heat caacity Δ C in neatic hase is a bit less than the corresonding value in the area higher than the transition oint, and it coincides with exerient. 3. Conclusions In this aer we have resented the analysis of the secific heat anoaly near the NI transition. Exerients show a colex behavior in which Gaussian and non-gaussian fluctuations cannot be neglected []. The absence of the divergence at the NI transition oint allows us to exect of using the Gaussian aroxiation or weakly interacted fluctuation liit. The existence of the cubic invariant in () is robably the ain reason for the failure of the ean-field descrition in the vicinity of the NI transition. Fro the analysis due to Landau-de Gennes odel we obtained for the critical exonentsγ =.68, β =., which are in qualitative agreeent with those obtained for MA fro different sources. The teerature deendence of the heat caacity in the neatic hase is connected with the change in the degree of neatic order and with the increase of the longitudinal and biaxial fluctuations. In the isotroic hase the fluctuation contribution is the only one. It is interesting to note that fro the renoralization grou study [] it follows that the critical anoalies can be connected either with the resence of the fixed oint, which described the transition near the isolated singular oint, or with the two fixed oints which are due to the secific character of the fluctuations in the neatic. The last of the are a saddle and a stable focus. In the neighbourhood of a first-order transition line of the hase diagra of the renoralization grou equations the transition is controlled by saddle fixed oint, in the vicinity of which γ =.58, β = References [] Anisiov M A 99 Critical henoena in liquids and liquid crystals (Gordon and reach Science Publishers) [] De Gennes P G and Prost J 993 The Physics of Liquid Crystals nd Edition (Oxford: Clarendon Press) [3] Lei Lin 98 J. de Phys [4] Anisiov M A et al 977 Sov.Phys.-JETP [5] Stinson T W, Litster J D 973 Phys. Rev. Lett [6] Haller I 975 Prog. Solid State Che. 3 [7] Kobinata S et al 98 Mol. Cryst. Liq. Cryst [8] Aver'yanov E M 99 Sov.Phys.-JETP [9] Sioes M and Sieao D S 6 Phys. Rev. E [] Korshenevskii A L, Shalaev N 979 Sov.Phys.-JETP