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The Elastic Constants of a Nematic

Hans Gruler

Gordon McKay Laboratory, Harvard University, Cambridge, Mass. 02138, U.S.A.

(Z. Naturforsch. 30 a, 230-234 [1975] ; received November 4, 1974)

The elastic constants of a nematic liquid and of a crystal were derived by comparing the Gibbs free energy with the elastic energy. The expressions for the elastic constants of the nematic and the solid crystal are isomorphic: k oc | D | /1. In the nematic | D | and I are the mean field energy and the molecular length. In the solid crystal, | D | and I correspond to the curvature of the potential and to the lattice constant, respectively. The measured nematic elastic constants show the predicted I dependence.

1. Introduction The mean field of the nematic phase was first derived by Maier and Saupe 3: The symmetry of a crystal determines the number D(0,P2) = -(A/Vn2)P2-P2±... (2) of non-vanishing elastic constants. In a nematic which is a uniaxial crystal, we find five 0 is the angle between the length axis of one mole- elastic constants but only three describe bulk pro- cule and the direction of the preferred axis (optical perties 2. Here we calculate the magnitude of these axis). P2 is the second Legendre polynomial and three elastic constants by determining the elastic P2 is one order parameter given by the distribution function f{0): energy density in two different ways. On the one hand the elastic energy density can be expressed by the elastic constants and the curvature of the optical p2 = Tf (@) Po sin e d e/tf (&) sin e d 0 . o o axis. On the other hand we determine the change of the Gibbs free energy of the deformed state. By Vn is the mole volume and A is a coefficient which comparing these two energy expressions the elastic determines the field of the second Legendre poly- constants are determined. nomial. The changes of the internal energy AU and the 2. Mean Field Theory AS are 4:

AU =lND = -h N(A/Vn2)P22, The can be described with the Gibbs free energy: TAS = ND + NkT In Z

ji/2 Gs = Gi + AIJ -T AS+ pV (1) with Z = const / e~D,kT sin 0 d 0. 0 Gi is the free energy with an isotropic distribution The free energy is now: of the direction of the . AU = U^ — U\ is the increase of the internal energy and AS = S$ — Si GN = Gi + i N(A/Vn2) P22-NkT\nZ. the change of the entropy between the nematic and The order parameter of the nematic phase is deter- the isotropic phase. T, p and V are the temperature mined by in °K, the pressure and the volume, respectively. dG/dP2 = 0 (AFi is the change of the volume at the nematic to isotropic phase transition temperature, which will be which leads to: neglected in the further discussion.) AU and AS are (A/Vn2) P2 = kT(dZ/Z dP2) . (3 a) normally very complicated expressions, but if the This equation can be solved and fixes P2 as a func- interaction energy of one with all others tion of T (see 5). The nematic-isotropic phase tran- can be described with a mean field, then the problem becomes much easier. sition is given by the following equation:

1 [A/Vn2) P2 = k 7Vi In Z . (3 b) Reprint requests to: Experimentelle Physik III der Uni- These two equations define A/Vn2 and /^(^Vi) at versität Ulm, D-7900 Ulm, Oberer Eselsberg, West-Ger- many. the phase transition as: H. Gruler • The Elastic Constants of a Nematic Liquid Crystal 231

2 A/Vr? = 4.54 • k T$i, P2 = 0.44 . (3 c) system *> . The elastic energy density of a nematic liquid crystal is then: If A is independent of the temperature then the order parameter P2 is for a reduced temperature ^ela = 2 kn [LX,X + LY,Y) 2 + 2 k22(Lxy — Lytx)2 2 5 (T/Tm)(Vn/Vn{Tm)) a universal curve . + ^33 (Lx>z +Lytz) 3 3. Elastic Constants — (k2o + k24) 2 (Li, i Ljt j — Lu Ljt j) i

The change of Gibbs free energy which is pro- kn, k22 and k33 are the splay, twist and bend duced by a small deformation, can be expressed elastic constants respectively (Figure 2). A compari- with a deformation angle a similar to the descrip- son of Eqs. (6) and (7) leads to the elastic con- tion of Nehring and Saupe 1 and Frank 2: stants :

/ /./ / / / &0 is the angular distribution of the undeformed xVM'//'// i'l',111 state. The deformation angle, a, at a position, T, \\ \ \ // // ll'l l l'l' ', can be expressed with the spatial derivative of the \\V\V\ V deformed optical axis at the origin. The orientation SPLAY TWIST BEND of the optical axis L(r) at r is (see Fig. 1): k11 k22 k33

Fig. 2. The main deformations in a nematic liquid crystal.

<5 G = f Gela dr . vn

The volume integrals over the forms which are con-

nected to (k22 + k2i) and can be transformed Fig. 1. Illustration of the spatial derivatives of L (0). into surface integrals and therefore these terms do not contribute to the Euler-Lagrange differential 1 L (r) = Z + (Lx, x X + Lx, y Y + Lx, 2 Z) X^ equations . Now we will solve for kn , k22 and k3S . For a pure bend deformation, we get the following + {Ly< x X + Ly. yY+ Lyt 2 Z) Y +... relation: The undisturbed optical axis points into the Z-direc- 2 2 tion. The deformation angle a is defined by i k33 (Lx! + Lyl) = (Lx% + Ly ) Z 1 (N/V„) ( - D) or a?=(Lx,xX + Lx,yY + Lx,zZ)2 k33 = Z2HN/Vn)(-D) . (8 a) + (Ly,xX + Ly

This value is far away from the measured values 6' ' This reduced elastic constant, Cn, is independent of the temperature if there are no abnormal changes ^11 * ^22 * 33 == • 1 • 3.5 • in long range order (A = const) and if there are no If we calculate the volume which one molecule structural changes in short range order (m = const occupies, we get ~ 7.2 Ä for the width, assuming and "/a = const). The curvature elastic constants kn that the box has the length of a molecule 17 Ä). of a nematic liquid crystal are therefore propor- The elastic constants now become: tional to the square of the order parameter and inversely proportional to the mole volume with the kn l &22 • = 1 I X I 5.5 • power 7/3. If we assume, as Saupe6 did, that m molecules If we are far enough away from the nematic to form a steric unit in order to reduce the steric isotropic as well as the nematic to smectic phase tran- hindrance, we find, for m = 2: sitions, only small changes in short range order ^11 * ^22 • ^33 = 1:1:1-4. structure may be expected; therefore Cn should be independent of temperature. Near phase transitions The calculated value of fc33 is: where rearrangements of short range order can Ä;33 = 2.7 • 1(T6 dyne occur, we expect a temperature dependence in Cn. for For 4,4'-di(methoxy)-azoxybenzene (PAA) the A = 13 • 1CT9 [erg cm6] <3>, Vn = 2.25 cm3® , elastic constants as well as the reduced elastic- P2 = 0.6, m = 2 and Z2~(17Ä)2. constants are shown in Fig. 3 as a function of

The measured value of k33 is approximately 1.2-10~6 dyne 6- 7. Another approach can be made if we introduce a unit cell which describes the short range order structure. Then we get for the mole volume

V n = (x2-z)L/m . (9) In this formula m describes the number of mole- cules per unit cell and x and z are appropriate aver- age distances in the x, y (= x) and z direction. If we assume that only the nearest neighbor interaction is important, then with Eqs. (8) and (9) we are able to find two sets of equations.

First we eliminate the mole volume Vn so that we obtain an equation which de Gennes 8 derived from dimensional analysis

kn = §(-D/z),_

k33= f (z/x)2(-D/z) . (10)

These equations can be very helpful for estimating the absolute value of the elastic constants from TEMPERATURE °C molecular properties, as we will show later. But the Fig. 3. Elastic and reduced elastic constants of PAA versus temperature dependence of the elastic constants can temperature (see Ref. 7). be better discussed if only (z/x) occur in the for- 7 mula because then the effect of thermal expansion temperature . As we see, the reduced elastic con- at a constant symmetry is eliminated. In this case stants are independent of the temperature far away we derive from Eqs. (8) and (9) a reduced elastic from the nematic to isotropic phase transition, as constant which first was introduced by Saupe 6. expected. Near the nematic to isotropic phase transition, 2 l 2 Cii = kn VV* P2~ = 3/2(L-m~ -7ir )• A C33 shows a large temperature dependence, whereas

i = 1, 2, 3 (11) Cn and Coo are almost temperature independent. = J'22 = z!x 733 = (x!z) 2 • This strong temperature dependence of C33 can be H. Gruler • The Elastic Constants of a Nematic Liquid Crystal 233

.5 1 1 / Cn^rO-O-N-N-ö-O-Cn Hzn.1 'ref'7° w 0) ML c 4r TX .H h9 ref 2' s .3 ID /H V - Q6 o A N=C-^C-N-O-0-C8 H,7 ref. 12 -- .2 / mh t• TSN H CH3 CH A CHrO-C-N-<ö- 2'9 C2HS ref. 11 ^SN ' TNI Z 07 .1 H

^ 1 1 () 20 30 10 JSN -tNL. Q&

• C7H,5OO>-N=N-O-C7H15 REF 7 TSN ^•10~8l°K/cm1

Fig. 4. Comparison between the measured splay elastic constant kn at Txi and the calculated kn as a function of the molecular length.

explained because structural changes in short range constants depend only on the ^-direction of the unit order are more strongly weighted in C33 than in cell which we assume to be proportional to the mole-

Cn and C22 [see Equation (11)]. The predicted cular length. On the other hand the abnormal tem- temperature dependence of C33 is in accordance perature dependence of kn and &22 near the nematic with experiment, but on the other hand Cn and to isotropic phase transition is small so that the ex-

C22 should increase with increasing temperature. trapolated value kn(Tyi) of the experimental data However, experiments show only a small tempera- can be compared with the theoretical value. ture dependence of Cn which is actually opposite Figure 4 shows the extrapolated elastic constant to be aforementioned description. kn (7Vi) for several different nematic molecules. Another explanation of the strong temperature The length of the molecules was determined from a dependence of C33 near the nematic to isotropic molecular model. We see that the elastic constant phase transition is the fluctuation of the order can be rather well described by the inverse of the parameter P2 . Such fluctuations give rise to some molecular length. But if we take the order parameter uncertainty in the preferred direction. Therefore we •P2(7yi) which the Maier-Saupe theory3 predicts can expect that those elastic constants which have a ^2(^Ni)=-44 then the predicted elastic constants connection to the preferred direction should show an are in general too large as we can see in Figure 4. abnormal temperature dependence near the nematic This is not surprising because the order parameter to isotropic phase transition. Equation (7) shows ^2(^x1) predicted by the Maier-Saupe theory is too that from the main elastic constants only the bend large compared with measured ones. The enlarged elastic constant kS3 is connected to the preferred Maier-Saupe theory (Humphries- James-Luckhurst 15 2-direction. Therefore only k33 should have an ab- theory) never predicts a universal order parameter normal temperature dependence as it was observed for all nematics as the Maier-Saupe theory 3. There- in experiment. fore we should have measurements of the order The pretransitional effects of the elastic constants parameter in order to compare theory and experi- near the smectic phase are not included in this cal- mental values of the elastic constants. But since not culation. Theory 9'10 and experiment were described enough order parameter measurements are available elsewhere 11 ~14. we make the best fit for P2(^NI) t0 elastic con- stant data, which then gives .38. This value is close The elastic constants can also be described by the to the value which Humphries et al.15 found for nematic mean field D and the dimensions of the several compounds. unit cell as we see from Equation (10). The average value of the nematic mean field at the nematic to In Fig. 4 there are still deviations of the elastic isotropic phase transition is given by the mean field constants from the straight line which we will dis- theory [see Eq. (3 c)]. The splay and twist elastic cuss in more detail. For the n = 1 and 2 homolog of 234 H. Gruler • The Elastic Constants of a Nematic Liquid Crystal 234

the alkyloxyazoxybenzenes (M) the degree of order If the displacement (u2 — ux) is small in com- of n = 2 homolog is about 10% higher than for the parison with the lattice constant a, the displacement n = 1 homolog 5. This means that the n = 2 homolog energy may be written in Hooke's law approxima- fits the theory much better as shown in Figure 4. tion: 16 For MBBA (•) Jen et al. found an extrapolated ß/2(u2-u1)* + ... value for P2{TNI) of about .32, so that this nematic ß is the currature of the elastic potential. compound also fits the theory quite well. McMil- The energy density for a homogeneous pure strain lan's mean field theory 17 for the smectic A phase is then: leads to an enlarged order parameter if the nematic temperature range becomes small enough. There- ß{elx +ely +e%) +... (13) fore heptyloxyazobenzene (O) with its relatively ' ii small nematic temperature range also fits theory where exx, eyy and ezz are strain components. The better than Fig. 4 illustrates. Figure 4 shows that energy density can also be written with phenomeno- the measured elastic constants versus molecular logical coefficients c: length fit quite well the Maier-Saupe theory if the u = h c {elx + e%y +(%) +... (14) degree of order is known or can be estimated. Now we have the possibility to predict quite well the A comparison between Eqs. (13) and (14) gives elastic properties for a nematic molecule where the the elastic constant, c: phase diagramm is known. c = a2(LjVn) ß. (15) However, we have to be careful if the molecules This expression for the elastic constant c is iso- are very long, because we based our calculation on morphic with the expression for the elastic constants the mean field approximation. For very long mole- of a nematic liquid crystal [Equation (8)]. Equa- cules the Onsager theory which includes the steric tion (15) can be changed hindrance can describe the nematic phase better than the mean field approximation 18. For example, c^ß/a (16) for PAA the mean field approximation is quite a if the mole volume is expressed by the lattice con- good approximation since the measured Grüneisen stant. Equation (16) is again isomorphic with Equa- constant is about 3.7 19. For pure van der Waals tion (11). Since D in Eqs. (8) and (11) and ß in interaction (mean field) we expect a Grüneisen con- Eqs. (15) and (16) have different dimensions, the stant of 2 and for hard core interaction (Onsager) dimensions of the elastic constants of a solid crystal infinity 19. and a liquid crystal are different.

4. Solid Crystal Acknowledgements

The elastic constants of a solid crystal can be The author would like to thank all members of derived with the lattice theory 20 which is similar to the Division, especially R. B. Meyer, for their kind hospitality during his stay at Harvard. This research the mean field approximation in a nematic liquid was sponsored by the National Science Foundation crystal. In this chapter, we will show the iso- and by the Division of Engineering and Applied morphism in the elastic constants between a solid Physics, Harvard University. and a nematic liquid crystal.

1 J. Nehring and A. Saupe, J. Chem. Phys. 54, 337 [1971]. 12 L. Cheung, R. B. Meyer, and H. Gruler, Phys. Rev. Lett. 2 F. C. Frank, Disc. Faraday Soc. 25, 19 [1958]. 31, 349 [1973]. 3 W. Maier and A. Saupe, Z. Naturforsch. 14 a, 882 [1959] ; 13 P. E. Cladis, Phys. Rev. Lett. 31, 1200 [1973]. 13a, 287 [I960]. 14 M. Delaye, R. Ribotta, and G. Durand, Phys. Rev. Lett. 4 See besides Ref. 3 L. D. Landau and E. M. Lifshitz, 31,443 [1973]. "", Addison-Wesley Publishing Comp. 15 R. L. Humphries, P. G. James, and G. R. Luckhurst, J. Reading, Menlo Park, London 1970. Chem. Soc., Faraday Trans. II. 68, 1031 [1972]. 5 A. Saupe, Angew. Chem. Internat. Edit. 7, 97 [1968]. 18 S. Jen, N. A. Clark, and P. S. Pershan, Phys. Rev. Lett. 6 A. Saupe, Z. Naturforsch. 15a, 810 and 815 [I960]. 31, 1552 [1973]. 7 H. Gruler, Z. Naturforsch. 28 a, 474 [1973]. 17 W. L. McMillan, Phys. Rev. A 4, 1238 [1971]. 8 P. G. de Gennes, "The Physics of Liquid ", Ox- 18 L. Onsager, Ann. N. Y. Acad. Sei. 51, 6271 [1949]. ford University Press, London 1974. 19 J. R. McColl, Phys. Lett. 38 A, 55 [1972]. 9 K. K. Kobayashi, Mol. Cryst. & Liquid Cryst. 13, 137 20 See, for example, C. Kittel, Introduction to Solid State [1971]. Physics, J. Wiley & Son, Inc., New York 1963, pp. 85. 10 P. G. de Gennes, Solid State Comm. 10, 753 [1972]. 21 I. Haller, J. Chem. Phys. 57, 1400 [1972]; H. Gruler, 11 L. Cheung and R. B. Meyer, Phys. Lett. 43 A, 261 T. J. Scheffer, and G. Meier, Z. Naturforsch. 27 a, 966 [1973]. [1972].