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Tricritical behavior at the isotropic-nematic transition
Volume 67A, number 2
PHYSICS LETTERS
24 July 1978
TRICRITICAL BEHAVIOR AT THE ISOTROPIC-NEMATIC TRANSITION * P.H. KEYES Universityof Massachusettsat Boston, Boston, MA 02125, USA Received 26 October j 977 Revised manuscript received 17 May 1978
It is argued on the basis of a reexamination of the Landau-de Gennes theory and with reference to experimental data that the isotropic-nematic transition exhibits pretransitional fluctuations characteristic of a tricritical point.
According to de Gennes [l] the order parameter for the nematic-isotropic transition is a symmetric, traceless tensor, usually taken to be the anisotropic part Qas of the magnetic susceptibility: 1
(1)
Qcyp=~p-Apx,,.
Expressed in diagonalized form Qap may be written as $(-Q+P)
Q,@= 0 10
0
0
+(-Q-P>
0 ,
0
Q :I
(2)
which allows for the possibility of a biaxial nematic if P # 0. The free energy is expanded in terms of the order parameter as F=Fu +~AQ~~-~BQ,,Q,,Q,,~~cQ~~Q::, (3) with B and C constant and A = A’(T - T*)‘Y. The exponent 7 = 1 in a mean field theory. The parameters xa and oa are the anisotropies in the magnetic and electric susceptibilities, respectively. In the presence of suitable external fields both critical and tricritical points appear in this model. For the purposes of this discussion we assume xa > 0 and oa < 0. Then a t&critical point [2,3] is found for H = 0, sE2 = &B3/C2, at a temperature Ttcp = T* * Work supported by the Research Corporation and through a University of Massachusetts Faculty Development Grant.
132
+ & B2fA’C. Both Q and P undergo second order changes at this point. For E = 0, xaH2 = $J B3/C2, the nematic-isotropic transition terminates at an ordinary critical point with Q = l/9, P = 0 and T = T* t $gB2/A’C. (These parameters are obtained “%y setting the first and second derivatives of F with respect to Q equal to zero. Previous calculations [4,5] of the location of this critical point have been done within the context of molecular field theory.) In the absence of applied fields, eq. (3) with the substitution of eq. (2) becomes
F = F0 t 9~~2 - $BQ3 t $CQ~ + +AP~ + $CP4 t ;CQ2P2 +$BQP2.
(4)
In this case a transition from the state Q = 0, P = 0 to Q = 2/9, P = 0, takes place at temperature T, = T* t Y&B2/A’C. The transition is first order, but it is weakly so, and large pretransitional anomalies are found. According to current notions [6] concerning phase transitions one might expect these anomalies to be critical-like or tricritical-like according to which type of second-order point is closer. Inspection of the foregoing results yields (Tcp - Tc)/(Ttcp - T,) = 16/31 and T, is seen to be only half again as close to Tcp as it is to Ttcp. (Actually the closest critical point lies along the line of critical points joining Tcp to Ttcp and is somewhat closer than T, .) In view of the well known inability of mean field & eory to confidently locate transition points, these calculations provide no compelling reason to regard the transition as either critical-like or tricritical-like.
Volume 67A, number 2
24 July 1978
PHYSICS LETTERS
The purpose of this letter is to put forward the hypothesis that the isotropic-nematic transition in the absence of applied fields, as it is normally studied, is effectively tricritical in nature. It has now been established [7] that the three-state Potts model for the isotropic-nematic transition, has tricritical exponents. Since the three-state Potts model and the de Gennes model are identical in the mean field approximation [8,9] , it would not be surprising ifthe latter model were also found to violate mean field theory. It should also be noted that the most recent evidence [lO,ll] seems to favor the position that the transition in the three-dimensional three-state Potts model, like that in the isotropic-nematic system, is weakly first order rather than continuous. It is essential for tricritical behavior to take place that there be two competing order parameters both showing critical fluctuations. This feature is already contained in the de Gennes theory, although this fact seems not to have been fully appreciated before. In eq. (4) it can be seen that both the Q2 and P2 terms have coefficients varying as T - T*. Thus the isotropic phase is expected to exhibit diverging fluctuations in the biaxial order parameter P, as well as in Q, even though the nematic phase has P = 0. Experimental evidence for the existence of these biaxial order parameter fluctuations has not been presented before. Unfortunately, light intensity measurements cannot provide this information, it turns out that the angular dependences and the depolarization ratios are the same, whether P = 0 or not, as long as Qap is traceless [ 121. The same is not true for the spectrum of the scattered light, however. In this case fluctuations in P will result in deviations of the Rayleigh line from a simple single lorentzian shape. These deviations, calculated from an extension of the de Gennes theory to include gradient terms in the free energy [ 13,141, are expected to be most pronounced close to T,., where the correlation length is largest. Effects of this sort have now been found in the isotropic phases of cholesteryl oleyl carbonate [ 151 and cholesteryl2-(2.ethoxyethoxy) ethyl carbonate [ 141. Electric birefringence experiments on materials with cu, < 0 could also be used to confirm the existence of biaxial fluctuations. According to the view presented here, the apparent success of the de Gennes mean field theory is actually a result of the fact that most of the critical exponents
measured so far happen to coincide with those expected for a tricritical point. The results 7 = 1 and v = l/2 are such examples [ 161. The finding of mean field exponents for a system under the influence of short range forces appears as a mystery [ 171 if the transition is critical or nearly so; near a tricritical point, however, mean field exponents are the anticipated result [18]. Most of the other tricriticalexponents do not happen to agree with the exponents calculated from the de Gennes theory. For example, the specific heat in the de Gennes model has exponents OL = 0, and 0~’= l/2 above and below T,, respectively. At a tricritical point both of the exponents are l/2, which is also the experimental finding [ 191. This deficiency in the de Gennes theory has been corrected in an ad hoc fashion by Imura and Okano [ 201, using the method originally proposed by Fixman [21] to explain a similar disparity between experiment and the classical theories of the binary liquid and liquid-gas critical points. Since the method in its original application is known to be faulty, its application to the liquid crystal problem must be viewed with caution. There is no such difficulty, however, if the tricritical hypothesis is correct. Within the nematic phase the order parameter Q is expected to follow a power law Q= Q, +A(T**
- T)fl,
(5)
with Q. and A constants, T** the effective second order point seen from below T,, and fl is l/2 in the de Genres theory but l/4 near a tricritical point. Poggi et al. [22] have shown that eq. (5) holds using the de Gennes parameters if the temperature range is limited to a degree or two. A least squares fit to their MBBA data, but with p fmed at the tricritical value of l/4, yields A = 0.49, Q0 = -0.01, T** - T, = O.l8’C, and a chi square only two percent higher than they obtained. Fig. 1 shows the data plotted under the assumptions that fl= l/4 and Qu = 0, which it nearly is. The T** found in this case agrees well with the values obtained from thermal expansion measurements [23]. If the temperature range is expanded it becomes quite clear that the value fl= l/4 is preferred over l/2. Indeed, Haller [24] has made the observation that thirteen different liquid crystals follow power laws over their entire nematic ranges with p’s of about 0.2 + 0.03. The tricritical hypothesis furnishes an explanation for this empirical law. 133
Volume 67A, number 2
PHYSICS LETTERS
24 July 1978
References
[II P.G. de Genres, Phys. Lett. 30A (1969) 454; Mol. Cryst. Liq. Cryst. 12 (1971) 193.
[Xl C. Fan and M.J. Stephen, Phys. Rev. Lett. 25 (1970) 500.
T-T,
(“Cl
Fig. 1.
Application of the de Gennes model to the nematic phase is subject to the criticism that the theory is intended primarily to describe the pretransitional phenomena in the isotropic phase. The exponent fl can in effect also be measured in the isotropic phase, however, through a scaling analysis of a field induced birefringence experiment. The appropriate scaled variables are h/Q6 and &IQ, where h E H2 and E 2 (T - T*)/T*. Such an experiment can also yield the gap exponent A defined in terms of a2Q/ah2 w (T - T*)-r-*. The values for 6 and A are, respectively, 3 and 2 in the de Gennes theory, but 5 and 5/4 in the tricritical case. Experiments to determine these exponents are now in progress. Finally, it should be noted that logarithmic modifications to mean field behavior may occur near a tricritical point [ 181. Such effects have been reported for the isotropic-cholesteric transition [ 15;25] . These results transcend simple mean field theory.
134
]31 R.G. Priest, Phys. Lett. 47A (1974) 475. ]41 J. Hanus, Phys. Rev. 178 (1969) 420. [51 P.J. Wojtowicz and P. Sheng, Phys. Lett. 48A (1974) 235. [61 E.K. Riedel, Phys. Rev. Lett. 28 (1972) 675. 171 T.W. Burkhardt, H.J.F. Knops and M. den Nijs, J. Phys. A9 (1976) L179, and references cited therein. [81 T.C. Lubensky and R.G. Priest, Phys. Lett. 48A (1974) 103; Phys. Rev. B13 (1976)4159. 191 J.P. Straley and M.E. Fisher, J. Phys. A6 (1973) 1310. ilO1 A. Aharony, K.A. Mtiller and W. Berlinger, Phys. Rev. L&t. 38 (1977) 33. 1111 B. Barbara, M.F. Rossignol and P. Bak, J. Phys. Cl1 (1978) L183. [121 M. Kerker, The scattering of light and other electromagnetic radiation (Academic Press, New York, 1969); sections 10.1 and 10.2 can be specialized to the case of a traceless tensor. [ 131 C.C. Yang, Ph.D. thesis, Harvard Univ. (1972). [ 141 P.H. Keyes and C.C. Yang, to be published. [ 151 P.H. Keyes and D.E. Ajgaonkar, Bull. Am. Phys. Sot. 23 (1976) 548, and to be published. [ 161 T.W. Stinson and J.D. Litster, Phys. Rev. Lett. 25 (1970) 503; J.D. Litster and T.W. Stinson, J. Appl. Phys. 41 (1970) 996; T.W. St&on, J.D. Litster and N.A. Clark, J. Physique Suppl. 33 (1972) Cl-169. [I71 See for instance: P.G. de Genres, The physics of liquid crystals (Clarendon, Oxford, 1974) pp. 52-53. 1181 E.K. Riedel and F.J. Wegner, Phys. Rev. Lett. 29 (1972) 349; Phys. Rev. B7 (1973) 248; B9 (1974) 294. [191 G. Koren, Phys. Rev. Al3 (1976) 1177; H. Gruler and F. Jones, J. Physique 36 (1975) Cl-53. [201 H. Imura and K. Okano, Chem. Phys. Lett. 17 (1972) 111 1211 M. Fixman, J. Chem. Phys. 36 (1962) 1957. [221 Y. Poggi, P. Atten and J.C. Filippini, Mol. Cryst. Liq. Cryst. 37 (1976) 1. ~31 D. Armitage and F.P. Price, Phys. Rev. Al5 (1977) 2496. v41 I. Haller, Prog. Solid State Chem. 10 (1975) 103; J.R. McCall kindly pointed out this reference. 1251 P.H. Keyes and D.B. Ajgaonkar, Phys. Lett. 64A (1977) 298.
PHYSICS LETTERS
24 July 1978
TRICRITICAL BEHAVIOR AT THE ISOTROPIC-NEMATIC TRANSITION * P.H. KEYES Universityof Massachusettsat Boston, Boston, MA 02125, USA Received 26 October j 977 Revised manuscript received 17 May 1978
It is argued on the basis of a reexamination of the Landau-de Gennes theory and with reference to experimental data that the isotropic-nematic transition exhibits pretransitional fluctuations characteristic of a tricritical point.
According to de Gennes [l] the order parameter for the nematic-isotropic transition is a symmetric, traceless tensor, usually taken to be the anisotropic part Qas of the magnetic susceptibility: 1
(1)
Qcyp=~p-Apx,,.
Expressed in diagonalized form Qap may be written as $(-Q+P)
Q,@= 0 10
0
0
+(-Q-P>
0 ,
0
Q :I
(2)
which allows for the possibility of a biaxial nematic if P # 0. The free energy is expanded in terms of the order parameter as F=Fu +~AQ~~-~BQ,,Q,,Q,,~~cQ~~Q::, (3) with B and C constant and A = A’(T - T*)‘Y. The exponent 7 = 1 in a mean field theory. The parameters xa and oa are the anisotropies in the magnetic and electric susceptibilities, respectively. In the presence of suitable external fields both critical and tricritical points appear in this model. For the purposes of this discussion we assume xa > 0 and oa < 0. Then a t&critical point [2,3] is found for H = 0, sE2 = &B3/C2, at a temperature Ttcp = T* * Work supported by the Research Corporation and through a University of Massachusetts Faculty Development Grant.
132
+ & B2fA’C. Both Q and P undergo second order changes at this point. For E = 0, xaH2 = $J B3/C2, the nematic-isotropic transition terminates at an ordinary critical point with Q = l/9, P = 0 and T = T* t $gB2/A’C. (These parameters are obtained “%y setting the first and second derivatives of F with respect to Q equal to zero. Previous calculations [4,5] of the location of this critical point have been done within the context of molecular field theory.) In the absence of applied fields, eq. (3) with the substitution of eq. (2) becomes
F = F0 t 9~~2 - $BQ3 t $CQ~ + +AP~ + $CP4 t ;CQ2P2 +$BQP2.
(4)
In this case a transition from the state Q = 0, P = 0 to Q = 2/9, P = 0, takes place at temperature T, = T* t Y&B2/A’C. The transition is first order, but it is weakly so, and large pretransitional anomalies are found. According to current notions [6] concerning phase transitions one might expect these anomalies to be critical-like or tricritical-like according to which type of second-order point is closer. Inspection of the foregoing results yields (Tcp - Tc)/(Ttcp - T,) = 16/31 and T, is seen to be only half again as close to Tcp as it is to Ttcp. (Actually the closest critical point lies along the line of critical points joining Tcp to Ttcp and is somewhat closer than T, .) In view of the well known inability of mean field & eory to confidently locate transition points, these calculations provide no compelling reason to regard the transition as either critical-like or tricritical-like.
Volume 67A, number 2
24 July 1978
PHYSICS LETTERS
The purpose of this letter is to put forward the hypothesis that the isotropic-nematic transition in the absence of applied fields, as it is normally studied, is effectively tricritical in nature. It has now been established [7] that the three-state Potts model for the isotropic-nematic transition, has tricritical exponents. Since the three-state Potts model and the de Gennes model are identical in the mean field approximation [8,9] , it would not be surprising ifthe latter model were also found to violate mean field theory. It should also be noted that the most recent evidence [lO,ll] seems to favor the position that the transition in the three-dimensional three-state Potts model, like that in the isotropic-nematic system, is weakly first order rather than continuous. It is essential for tricritical behavior to take place that there be two competing order parameters both showing critical fluctuations. This feature is already contained in the de Gennes theory, although this fact seems not to have been fully appreciated before. In eq. (4) it can be seen that both the Q2 and P2 terms have coefficients varying as T - T*. Thus the isotropic phase is expected to exhibit diverging fluctuations in the biaxial order parameter P, as well as in Q, even though the nematic phase has P = 0. Experimental evidence for the existence of these biaxial order parameter fluctuations has not been presented before. Unfortunately, light intensity measurements cannot provide this information, it turns out that the angular dependences and the depolarization ratios are the same, whether P = 0 or not, as long as Qap is traceless [ 121. The same is not true for the spectrum of the scattered light, however. In this case fluctuations in P will result in deviations of the Rayleigh line from a simple single lorentzian shape. These deviations, calculated from an extension of the de Gennes theory to include gradient terms in the free energy [ 13,141, are expected to be most pronounced close to T,., where the correlation length is largest. Effects of this sort have now been found in the isotropic phases of cholesteryl oleyl carbonate [ 151 and cholesteryl2-(2.ethoxyethoxy) ethyl carbonate [ 141. Electric birefringence experiments on materials with cu, < 0 could also be used to confirm the existence of biaxial fluctuations. According to the view presented here, the apparent success of the de Gennes mean field theory is actually a result of the fact that most of the critical exponents
measured so far happen to coincide with those expected for a tricritical point. The results 7 = 1 and v = l/2 are such examples [ 161. The finding of mean field exponents for a system under the influence of short range forces appears as a mystery [ 171 if the transition is critical or nearly so; near a tricritical point, however, mean field exponents are the anticipated result [18]. Most of the other tricriticalexponents do not happen to agree with the exponents calculated from the de Gennes theory. For example, the specific heat in the de Gennes model has exponents OL = 0, and 0~’= l/2 above and below T,, respectively. At a tricritical point both of the exponents are l/2, which is also the experimental finding [ 191. This deficiency in the de Gennes theory has been corrected in an ad hoc fashion by Imura and Okano [ 201, using the method originally proposed by Fixman [21] to explain a similar disparity between experiment and the classical theories of the binary liquid and liquid-gas critical points. Since the method in its original application is known to be faulty, its application to the liquid crystal problem must be viewed with caution. There is no such difficulty, however, if the tricritical hypothesis is correct. Within the nematic phase the order parameter Q is expected to follow a power law Q= Q, +A(T**
- T)fl,
(5)
with Q. and A constants, T** the effective second order point seen from below T,, and fl is l/2 in the de Genres theory but l/4 near a tricritical point. Poggi et al. [22] have shown that eq. (5) holds using the de Gennes parameters if the temperature range is limited to a degree or two. A least squares fit to their MBBA data, but with p fmed at the tricritical value of l/4, yields A = 0.49, Q0 = -0.01, T** - T, = O.l8’C, and a chi square only two percent higher than they obtained. Fig. 1 shows the data plotted under the assumptions that fl= l/4 and Qu = 0, which it nearly is. The T** found in this case agrees well with the values obtained from thermal expansion measurements [23]. If the temperature range is expanded it becomes quite clear that the value fl= l/4 is preferred over l/2. Indeed, Haller [24] has made the observation that thirteen different liquid crystals follow power laws over their entire nematic ranges with p’s of about 0.2 + 0.03. The tricritical hypothesis furnishes an explanation for this empirical law. 133
Volume 67A, number 2
PHYSICS LETTERS
24 July 1978
References
[II P.G. de Genres, Phys. Lett. 30A (1969) 454; Mol. Cryst. Liq. Cryst. 12 (1971) 193.
[Xl C. Fan and M.J. Stephen, Phys. Rev. Lett. 25 (1970) 500.
T-T,
(“Cl
Fig. 1.
Application of the de Gennes model to the nematic phase is subject to the criticism that the theory is intended primarily to describe the pretransitional phenomena in the isotropic phase. The exponent fl can in effect also be measured in the isotropic phase, however, through a scaling analysis of a field induced birefringence experiment. The appropriate scaled variables are h/Q6 and &IQ, where h E H2 and E 2 (T - T*)/T*. Such an experiment can also yield the gap exponent A defined in terms of a2Q/ah2 w (T - T*)-r-*. The values for 6 and A are, respectively, 3 and 2 in the de Gennes theory, but 5 and 5/4 in the tricritical case. Experiments to determine these exponents are now in progress. Finally, it should be noted that logarithmic modifications to mean field behavior may occur near a tricritical point [ 181. Such effects have been reported for the isotropic-cholesteric transition [ 15;25] . These results transcend simple mean field theory.
134
]31 R.G. Priest, Phys. Lett. 47A (1974) 475. ]41 J. Hanus, Phys. Rev. 178 (1969) 420. [51 P.J. Wojtowicz and P. Sheng, Phys. Lett. 48A (1974) 235. [61 E.K. Riedel, Phys. Rev. Lett. 28 (1972) 675. 171 T.W. Burkhardt, H.J.F. Knops and M. den Nijs, J. Phys. A9 (1976) L179, and references cited therein. [81 T.C. Lubensky and R.G. Priest, Phys. Lett. 48A (1974) 103; Phys. Rev. B13 (1976)4159. 191 J.P. Straley and M.E. Fisher, J. Phys. A6 (1973) 1310. ilO1 A. Aharony, K.A. Mtiller and W. Berlinger, Phys. Rev. L&t. 38 (1977) 33. 1111 B. Barbara, M.F. Rossignol and P. Bak, J. Phys. Cl1 (1978) L183. [121 M. Kerker, The scattering of light and other electromagnetic radiation (Academic Press, New York, 1969); sections 10.1 and 10.2 can be specialized to the case of a traceless tensor. [ 131 C.C. Yang, Ph.D. thesis, Harvard Univ. (1972). [ 141 P.H. Keyes and C.C. Yang, to be published. [ 151 P.H. Keyes and D.E. Ajgaonkar, Bull. Am. Phys. Sot. 23 (1976) 548, and to be published. [ 161 T.W. Stinson and J.D. Litster, Phys. Rev. Lett. 25 (1970) 503; J.D. Litster and T.W. Stinson, J. Appl. Phys. 41 (1970) 996; T.W. St&on, J.D. Litster and N.A. Clark, J. Physique Suppl. 33 (1972) Cl-169. [I71 See for instance: P.G. de Genres, The physics of liquid crystals (Clarendon, Oxford, 1974) pp. 52-53. 1181 E.K. Riedel and F.J. Wegner, Phys. Rev. Lett. 29 (1972) 349; Phys. Rev. B7 (1973) 248; B9 (1974) 294. [191 G. Koren, Phys. Rev. Al3 (1976) 1177; H. Gruler and F. Jones, J. Physique 36 (1975) Cl-53. [201 H. Imura and K. Okano, Chem. Phys. Lett. 17 (1972) 111 1211 M. Fixman, J. Chem. Phys. 36 (1962) 1957. [221 Y. Poggi, P. Atten and J.C. Filippini, Mol. Cryst. Liq. Cryst. 37 (1976) 1. ~31 D. Armitage and F.P. Price, Phys. Rev. Al5 (1977) 2496. v41 I. Haller, Prog. Solid State Chem. 10 (1975) 103; J.R. McCall kindly pointed out this reference. 1251 P.H. Keyes and D.B. Ajgaonkar, Phys. Lett. 64A (1977) 298.